n If n I ) 1 lilf iltJij injfliMij J li . . 11 u } \ n itmmmnmimmi ummiiminhmimir ' ^ nmmmmmm irinijjinu 11 ( uj nmuijwmmmnii mmmmmm 1{IL„,. ijBilWffipl MnijjiiiuifmuiMJwimiJ* '•%iiiii|iipiM" mmmm^Si mmnmmmmm 'xmmvmnammx (imimimMmBiim mmmmmmumm. .. nimriifijjijrumiJHHilj mwumw.munmm..., immmmimhuimmim . ujirr»}Um|iiutH|tivmjiiji- — •■ —mmmnm mmumm i q^o IN MEMORIAM FLORIAN CAJORI UNIVERSITY ALGEBRA BY C. A. VAN VELZER and CHAS. S. SLIGHTER, PROFESSORS m THE UNIVERSITY OF WISCONSIN. MADISON, WIS. TRACY, GIBBS & COMPANY. 1892. COPYRIGHT, C. A. VAN VELZER, CHAS. S, SLIGHTER. 1892. Tracy, Gibbs & Co., Printers and Stereotypers. rs 5 PREFACE. ' The present volume is a modified form of a treatise on •algebra which appeared, as a preliminary edition, about four years ago. The results of the authors' experience with the preliminary edition, together with the suggestions of several friends, have been used in preparing the pres- ent work. The authors have aimed to produce a work which should not be too difficult for the average student in college and universit}^ classes, and which should be ex- tensive enough to include all that is given of algebra even in the largest American institutions, and to be useful as a work of reference. Although not intended for absolute beginners in algebra, it was thought best to begin at the beginning and to make the book complete in itself. In preparing this work, the authors have freely con- sulted other texts. Those which have been found most useful, are the following: Serret's Algebre Superieure, Comberouse*s Algebre Supelrieure, Laurent's Traite d' Algebre, Chrystal's Algebra, C. Smith's Treatise on Algebra, Oliver, Wait and Jones' Algebra, and Klempt's Lehrbuch zur einfiihrung in die Moderne Algebra. Many thanks are due to Professor Plorian Cajori of Colorado College, who has not only offered valuable suggestions, but has written all the historical notes. C. A. VAN VELZKR. CHAS. S. SI.ICHTKR. Madison, Wis., January, i8pj. CONTENTS. PAGE. Chapter I. Introduction, 1 Negative numbers and quantities, - - - 9 Evaluation of expressions, - - - - 12 Chapter II. Addition, 15 Union of similar terms or addition of monomials, - 18 Addition of expressions, - - - _ 19 Chapter III. Subtraction, 22 Subtraction of expressions, - - - - 23 Insertion and removal of parentheses, - - 26 Chapter IV. Multiplication, - - - - - 28 Product of monomials, - - - - - 33 Product of a polynomial and a monomial, - 34 Product of a polynomial by a polynomial, - - 37 Special products, - - - _ _ 4Q Chapter V. Division, 48 Division of a monomial by a monomial, - - 51 Division of a polynomial by a monomial, - 51 Division of a polynomial by a polynomial, - - 53 The fundamental laws of algebra, - - - 59 Historical note, 61 Chapter VI. Mathematical Induction, - - - 63 Chaptor VII. Factors and Multiples, - - - 67 Expressions of the form rt-^ — ^c^, - - - 71 Expressions of the form x'^+ax-\-d, - - 71 Expressions of the form <^;r 2 _^^_;,;^^^ . . - 73 Expressions of the form «3 —<^3^ - - _ 75 Expressions of the form a^-\-d^, - - - 76 Expressions of the form x'^-\-a^x^-\-a^, - - 78 Expressions of the form %" — «", - - - 79 Expressions of the form :?c" + ««, - - - 83 Miscellaneous factors, - - - - - 87 . H. C. F. of expressions easily factored, - - 89 H. C. F. of expressions not easily factored, - - 91 Lowest common multiple, - - - - 97 L. C. M. of expressions not easily factored, - - 99 Chapter VIII. Fractions, 102 Addition of fractions, - - - - 104 Subtraction of fractions, - - - - 105 Multiplication of fractions, - - - U)7 Miscellaneous fractions, ... . uj CONTENTS. Chapter IX. Powers and Roots, - Square of a binomial, - - - - Cube of a binomial, - - - - - Roots of monomials, - - - - Square root of polynomials, - - - - Cube root of polynomials, . - - Chapter X. Simple Equations, - - - Literal equations, - - - - - Symbolic expression, - - - - - Problems, ------ General equation of first degree, - - - Generalized problems, - - - - Historical note, ------ Chapter XI. Simultaneous Equations, Elimination by subtraction, - - - - Elimination by comparison, - - - Elimination by addition and subtraction, Special expedients, - _ - - Simultaneous equations containing three unknown numbers, ------ Literal simultaneous equations, Problems, ------ General system with two unknown numbers, Chapter XII. Quadratic Equations, Pure quadratic equations, - - - Solution by factoring, - - - - - Affected quadratics, - - - - Literal quadratic equations, - - - - Solution by factoring, - - - - Problems leading to quadratic equations, Equations solved like quadratics. 120 121 123 128 132 141 142 145 150 151 154 156 157 159 160 163 168 169 173 177 179 180 184 185 186 192 Chapter XIII. Theory of Quadratic Equations and Expressions, - Discussion of the roots, - - - - 199 Historical note, ----- 204 Chapter XIV. Theory of Indices, - - - - Fractional exponents, - - - - 207 Negative exponents, ----- 214 Zero equations, ----- 221 Chapter XV. Surds, Reduction of surds, - - . - 226 Operations on surds, - - - - 230 Rationalization of expressions containing surds, 233 Functions of surds, _ - - - 238 Square root of binomial quadratic surds, - - 241 119 135 155 176 194 206 224 CONTENTS. Chapter XVI. Single Equations, - - - - 245 Rationalization of equations, - _ _ 256 Graphic representation of expressions and equations of the first degree, - - . - 259 Graphic representation of quadratic expressions and equations, ----- 265 Chapter XVII. Systems of Equations, - - - 268 Linear quadratic system, - - - - 275 Systems of two quadratics, - - - - 277 Special expedients, _ _ _ - 280 Graphic representation of systems of linear equa- tions, 283 Problems, ------ 286 Chapter XVIIL Theory of Limits, - - ^289 Theorems on limits. - - - - - 293 Indeterminate forms, - - - - - 298 Chapter XIX. Ratio, Proportion and Variation, - 304 Properties of ratios, - - . - 305 Incommensurable numbers - - - - 308 Compound ratios, - - - - - 311 Proportion, - - - - - -314 Variation, 320 Chapter XX. Progressions, 325 Arithmetical progressions, - - - 325 Geometrical progressions, - - - - 331 Infinite geometrical progressions, - - 336 Hnrmonical progressions, - - - - 338 Miscellaneous exercises, - - - - 340 Chapter XXI. Arrangements and Groups, - - 343 Chapter XXII. Binomial Theorem, - - - 367 Properties of the expansion, - - - - 370 Multinomial theorem. - - - - . 373 Historical note, 375 Chapter XXIII. Theory of Probabilities, - - 378 Simple probability, . - - - 379 Total probability, 383 Compound probability, - - - - 384 Mathematical expectation, - - - - 388 Successive trials, ----- 391 Miscellaneous exercises. - - - - 394 Chapter XXIV. Convergence and Divergence of Series, -------- 396 Chapter XXV. Undetermined Coefficients, - - 424 CONTENTS. Chapter XXVI. Summation of Series, - - - 438 Series reducible to the form of t^-^ — zt^-\-Uj^—tt^ ^u.^-ti^-^.,. , - - - - 438 Summatiou by undetermined coefficients, - - 440 Method of differences, - - - - 442 Recurring series, - - _ - - 445 Chapter XXVII. Binomial Theorem for Fractional and Negative Exponents, - - - - 453 Chapter XXVIII. Continued Fractions, - - 459 Chapter XXIX. Derivatives, - - - - 482 Chapter XXX. Incommensurable Exponents and Logarithms, 505 Laws of incommensurable indices, - - 509 Logarithms, ------ 511 Properties of logarithms, - - - - 513 Characteristic and mantissa, - - _ 517 Tables of logarithms, - - - - 522 Computation by logarithms, - - - - 536 Exponential and logarithmic series, - - 538 Historical note, ------ 544 Chapter XXXI. Complex Numbers, - - - 547 Modulus and amplitude, - - - - 559 Chapter XXXII. The Rational Integral Function, 566 Properties and constitution of derivatives oif{pc) 595 Binomial coefficients, ----- 599 Chapter XXXIII. Special Equations, - - - 602 Reciprocal equations, - - - - - 602 Binomial equations, - - - - 606 Cubic equations, ----- 615 Biquadratic equations, - - - - 620 Historical note, ------ 625 Chapter XXXIV. Separation of Roots, - - 627 Sturm's theorem, ----- 641 Theorems of Fourier and Budan, - - 649 Chapter XXXV. Numerical Equations, - - 659 Chapter XXXVI. Decomposition of Rational Fractions, ------- 669 Rapid method, 681 •Chapter XXXVII. Graphic Representation of Equations, - - 686 Equations of the form y=f{x), - - - 686 Graphs of equations of the form /(x,jj/)=0, - 691 Graphs of quadratic systems, - - - 692 Chapter XXXVIII. Determinants, - - - 695 Rationalization of any algebraic expession, - 731 I CHAPTER I. INTRODUCTION. 1. Number and Quantity. Anything which can be measured by a unit of the same kind is called a Quantity. Thus, 10 bushels is a quantity, the unit being a bushel, and this unit taken 10 times gives the quantity 10 bushels. Also, 10 cords is a quantity, the unit in this case being one cord, and this unit taken 10 times gives the quantity 10 cords. Also, the abstract number 10 is a quantity, the unit in this case being the abstract number 1, and this unit taken 10 times gives the quantity 10. Of course the unit itself is a quantity. The word quantity as above defined plainly includes number, but while a number is a quantity, a quantity is not always a number. Five miles would always b<^ called a quantity and never be called a number, but the number 5 may be called either a number or quantity indifferently. The word quantity is usually used as here explained, but some writers on Algebra never use the word quantity to include number. 2. Letters used for Numbers. In Algebra letters are used to represent or stand for numbers. Any letter may be used to represent any number provided the same letter represents the same number throughout the same discussion. The answer to a problem in Algebra is often something like 5 miles or 4 tons or 3 dollars or some other concrete quantity, but the reasoning is always conducted by nurn- bers, and so the letters used in Algebra always represent 1 — U. A. 2 UNIVERSITY ALGEBRA. numbers, and the result reached is the number of miles or tons or dollars or whatever it may be, and the name of the thing we are considering may be added at the end to the nufnber we have obtained by solving the problem. 3. The Sign of Addition is +, read ''plus.'' When this sign is placed between two numbers it signifies that the two numbers are to be added together. Thus a-\-b denotes the sum of the numbers represented by a and b. When two or more numbers are added together the result i'§ called the Sum. 4. The Sign of Subtraction is —, read ''minus.'' When this sign is placed between two numbers it signi- fies that the second of the two numbers is to be subtracted from the first. Thus a—b denotes the result obtained by subtracting the number represented b}^ b from the number represented by a. When one number is subtracted from another the result is called the Difference or Remainder. 5. The Sign of Multiplication is X , read "times" or ''into" or "multiplied by" When this sign is placed be- tween two numbers it signifies that the two numbers are to be multiplied together. Thus aY^b denotes the result obtained by multiplying together the numbers represented by a and b. When two or more numbers are multiplied together the result is called the Product. 6. In Algebra it is usual to omit the sign X except between numbers represented by Arabic numerals. Thus, instead of writing 7 X <^, we write simply lb (read ' 'seven b"), and instead of ^xaxb we write simply Zab (read "three ab"^. Sometimes a dot is used' instead of the sign X to indicate multiplication. Thus 2.3.7 means the INTRODUCTION. 3 same as 2 X 3 X 7. The dot should not be used when two numbers represented by Arabic numerals are multiplied together for fear of confusing it with a decimal point. 7. Factor. When two or more numbers are multiplied together to form a product, each of the numbers or the product of any number of them is called a Factor of the product. Thus, if 5, a and b are multiplied together, the product is 5^^, and the factors of the product are 5, a, b,ha, ab^ 5b, 8. Any factor of a product is called the Coefficient or Co-factor of the product of the remaining factors. Thus, in the product a^ be, a"^ is the coefficient or co-factor of be, a^b is the coefficient of c, ab is the coefficient of ac, etc. 9. Numerical and Literal Factors. When a prod- uct is made up partly of numbers represented by figures and partly of numbers represented by letters, we call the numbers represented by figures Numerical Factors and those represented by letters Literal Factors. 10. Numerical Coefficient. The product of all the numerical factors of any product is of course the coefficient of the product of all the literal factors, and the former is often called the Numerical Coefficient of the latter. 11. Power of a Number. When a product consists of the same number repeated any number of times as a factor, the product is called a Power of that number and is usually written in a simplified form. Thus : aa is written a'^ , read ''a square^' or '' second power of a*' ^ aaa is written a^ , read •*« eube'^ or ''third power of a-y aaaa is written a^, read ''a fourth'' or ' fourth power of a ;^^ and so on. 4 UNIVERSITY ALGEBRA. 12. Exponent or Index. The small figure written above and to the right of a number is called the Exponent or Index of the power; it shows how many times the number occurs as a factor in the power. According to this notation, a^ means that a is used once as a factor, but when a number is used only once as a factor it is cus- tomary to omit the exponent and write simply a instead of ^i. 13. The Sign of Division is -r-, read ''divided by,'' When this sign is placed between two numbers it signifies that the first number is to be divided by the second num- ber. Thus, a-T-b denotes the result obtained by dividing a by b. Usually, however, division is indicated by a fraction with the dividend above the line and the divisor below. Thus, a-T-b is written -r, and when written in this form it o is often read ' 'a over b. ' ' 14. It is stated in Art. 2 that letters are used to rep- resent numbers. Several letters may occur in the same problem, each letter standing for some number. Not only may -several letters occur in the same problem, but several letters may be combined by means of algebraic signs and this combination of letters also stands for some number. Thus, if ^ stands for 6, b for 10, and c for 5, then a-^bc-\-b'^ stands for 6 + 50+100, or 156. Anything, whether short and simple or long and com- plicated, which stands for some number is called an Expression. The number which an expression stands for is called the Value of the expression. 15. When an expression is broken in parts just before each of the signs + and — , each part thus formed is INTRODUCTION. 5 called a Term of the expression. Thus, in the expres- sion Sad-i-4c'^—2e—ld the terms are Sad, +4:C^, —2^, —15. We speak of the terms of an expression in a manner somewhat analogous to the way in which we speak of the syllables of a word. A syllable may nol convey an intelligible idea when taken by itself, but when joined to other syllables to make up a word, the whole word does convey a definite idea. So a term taken by itself may not, at this stage, express any idea, but the whole expression does convey an idea. See Art. 14. Indeed, each terra would, even now, convey a definite idea were it not for the sign + or — which always goes with each term after the first. 16. The terms of an expression which have the sign + or no sign at all are called Additive Terms, and those terms which have the sign — are called Subtractive Terms. Thus, in the expression n-\-2d—4:a-\-5a—7d, the terms n, -}-2d, and -\-5a are additive terms, and — 4« and —73 are subtractive terms. In comparing several additive terms they are spoken of as terms of the same sign, and several subtractive terms are also spoken of as terms of the same sign ; but when additive and subtractive terms are spoken of together, they are said to be terms of imlike or opposite signs. Notice that when we speak of the signs, without any further quali- fication, it is only the first two of the four fundamental signs of Algebra, +, — , X, and -h, that we have reference to. 17. Terms whose literal parts are identical and whose signs and numerical coefficients may or may not differ are called Similar Terms. Thus, in the expression 9a2<^— 3 What with respect to jj/? What with respect to x and j/? What with respect to a and x1 What with respect to a, X, and J^'? 20. When the degree of a monomial is spoken of with- out specifying the letters with respect to which the degree is taken, it is usually understood to mean the degree with respect to all the letters it contains, and is then equal to the number of literal prime factors, or what is the same thing, the sum of all the exponents of the letters in the expression. INTRODUCTION. 7 21. The Degree of a Polynomial with respect to any letter or letters it may cojitain is the degree of that one of its terms whose degree with respect to the specified letters is highest. Thus, a'^x^ -\-abc^+e'^x^y^ is of the second degree with respect to a, because the first term is of the second degree with respect to a and neither of the other terms is of so high degree with respect to a. The same expression is of the fourth degree with re- spect to X, because the third term is of the fourth degree with respect to x and neither of the other terms is of so high degree with respect to x. 3. What is the degree of ax'^y-\-bxy'^-\-x'^y'^ with re- spect to ^? What with respect to jk? What with respect to X and yl What with respect to a? What with respect to a and b'> 4. What is the degree of x^y+xy^+x^ with respect to jr? What with respect to y? What with respect to X and y? Ans. Five, for the degree with respect to x alone is 5, and the term that determines the degree has no y, so it leaves the degree 5. 5. What is the degree of x^ + ax^y+ dxy'^-j- a by ^ with respect to x? What with respect toy? What with respect to X and y? 6. What is the degree of a'^bx+d^xy^+cxy^ with re- spect to a? What with respect to jf ? What with respect toy} What with respect to a and x? What with respect to a, b, c, X, and jj/? 22. When the degree of a polynomial is spoken of with- out specifying the letters with respect to which the degree is taken, it is usually understood to mean the degree with respect to all the letters it contains, and is then equal to 8 UNIVERSITY ALGEBRA. the number of literal prime factors in that term which contains the greatest number of such prime factors. 23. The Sign of Equality is =, read ''equals'' or **w equal to.'' 24. The statement of equality which exists between two expressions is called an Equation, and the parts on either side of the sign = are called the Members of the equation. The expression on the left-hand side of the sign = is called the Left or First Member, and the ex- pression on the right-hand side of the sign = is called the Right or Second Member. 25. The Sign of Inequality is >, read ''greater than' ^ or <, read ''less than." Thus a^b signifies that a is greater than b, and a6 or 6<10. Now it is found convenient to extend the meaning of the words ' ' less than ' ' and ' ' greater than ' ' so that throughout the whole scale any number will be greater than any number to the left of it, and less than any number to the right of it. Thus we would say that ~5<-3 and -2<0. It should be carefully noticed that this is a technical use of the words ' ' greater than ' ' and ' ' less than ' ' and con- forms to the popular use of these words only when the numbers are positive. Of course it would be wrong to say that —2 is less than if we use '* less than " in the popular sense, because no number can be less than nothing at all, in the popular sense of "less than." But if we keep in mind that the words are used in a technical sense, there is no objection to such inequality as — 2<0. KVAI^UATION OF KXPRKSSIONS. 41. When the values of all the letters of an expression are given the value of the expression may be found by putting in place of the letters the given values and per- forming the indicated operations. For example, if ^=5, <^=4, and ;z^lO, then the value oi (a^ —Zb'^^n-\- an'^ is easily found as follows : a3=53 = 125. ^2=42_i6^ ^nd S^^^g^ 16=48. ... a3-3^2=,i25_48=77, .-. (^3 _3^2)^_77x 10=770. Also, ;z2 = i02 = l00, .-. a7^2 = 5x 100=500, ... («3_3^2)^+^^2_770-f-500=1270. INTRODUCTION. 1 3 KXAMPI.KS. Find the value of each of the following five expressions for the values of a and b given : 1. [- |-+2/^)-2a^whena=5, ^=3. \ a-\-o / ~aAl) 2. — ; when a=3, <5=2. a-\-b 3. (3a2-4^2)(^3^2_|.4^2>) ^j^en a=3, ^=2. 5- ftl|J+^^'-^') w^^^ ^=2' ^=^- Find the value of each of the following : 6. a^— 3a - + 2^ — 14 when <2=4. 7. {a''-—^\a^-^)^\i^na=^. 8. ^3 — 2^2-^ + 2 when ^=5. g. (a--2)(a — 1)(<2 + 1) when a=5. ^3_27 ^3+27 ^ 10. ^ — y- when <2=4. a — o a-\-€> 11. ^i — ;r when a=o. a — 2 a-f-2 12. <2'^ — ^2 — 5^ — 50 when <3^=6. 13. —-\ — r when a=l(J. a—1 a-j-1 14. If 10:i:=50, what is the value of jt? 15. If 10x—2o, what is the value of ^? 16. If o.r=50, what is the value of .;»:? 17. If 7-r=42, what is the value of .r? 18. If 7:r+2=44, what is the value of.;*:? 19. If 5:r— 10=40, what is the value of .r? 20. If 5.^+5=55, what is the value of ;r? 14 UNIVERSITY ALGEBRA. 42. A careful inspection of examples 6 to 20 above shows that, when there is only one letter in an expression, two cases may arise : first, the value of the letter may be given and the value of the expression required ; second, the value of the expression may be given and the value of the letter required. In the first case the value of the letter is kfiown or given, and in the second case the value of the letter is unknown or required. Thus we see that in Algebra there are two kinds of numbers, called respectively Known and Unknown, either of which may be represented by a letter ; and, as it is possible for both kinds of numbers to appear in the same discussion, it is customary to distinguish between them by representing the known numbers by the first and intermediate letters of the alphabet, and the unknown numbers by the last letters of the alphabet. To determine the values of letters when the values of the expressions which contain these letters are given, is one of the most important questions of Algebra. Some- times this can be done easily, sometimes it is quite diffi- cult, and sometimes it cannot be done at all. CHAPTER II. ADDITION. 43. Addition is the process of finding the result of taking two or more numbers together. The result is called the Sum, and the numbers added are called the Summands. This defines addition in Arithmetic, where numbers have no direc- tion, and, as we shall see, is sufficiently broad to include the case of negative numbers. 44. To indicate that the sum of several numbers is to be found we supply the sign + to numbers having no sign and then write the numbers down one after another with their signs unchanged. Thus, to indicate the sum of 8, —2, — 5, and 6, we write + 8-2-5 + 6. 45. When we add +3 and +8 we get, by the defini- tion, of addition, +3 + 8=+ll, because the result of taking together 3 in a certain direc- tion and 8 in the same direction is 11 in that direction. When we add —3 and +8 we get, by the definition of addition, —3 + 8= +5, because the result of taking together 3 in a certain direc- tion and 8 in the opposite direction is 5 in the latter direction. When we add +3 and —8 we get, by the definition of addition, +3—8=— 5, because the result of taking together 3 in a certain direc- tion and 8 in the opposite direction is 5 in the latter direction. 1 6 UNIVERSITY ALGEBRA. When we add —3 and —8 we get because the result of taking together 3 in a certain direc- tion and 8 in the same direction is 11 in that direction. In exactly the same way we would get, by the defini- tion, the sum of -i-2a and +7<2, or 2a and 7a in same direction, =-}-9a —'2a and +7<2, or 2a and 7a in opposite directions, = + 5a + 2« and —7a, or 2a and 7a in opposite directions, = —5<2 — 2a and —7a, or 2a and 7a in same direction, = — da In general, if a and d stand for any two numbers, we have as above, by the definition of addition, sum of -i-a and -\-d=-{-a-i-d sum of -\-a and —d=-\~a—d sum of — a and -j-d=—a-\-d sum of —a and —d=—a—d 46. Thus it follows from the definition of addition that the sum of two numbers of the same sign is the arith- metical sum of their absolute values with the common sign of the summands, and the sum of two numbers of opposite signs is the arithmetical difference of their abso- lute values with the sign of the summand having the greater absolute value. It is well to notice one important difference between addition in Arithmetic and addition in Algebra. In Arithmetic addition implies augmentation, but in Algebra this is not necessarily the case. 47. To add 6, —7, 2, and —4 we write + 6-7 + 2-4. To find a shorter form for this sum, we may put -f 6 and —7 together (as in Art. 45) giving —1 ; then we may combine this result, —1, with -f 2, giving +1 ; then this ADDITION. 17 result, +1, with —4, giving —-3 as the sum of 6, —7, 2, and —4. Also, to add ^, — ^, — r, and ^ we have -j-a—d—c+d. This result cannot be shortened. 48. Since addition is the process of finding the result of taking several numbers together, it follows that addi- tion may be performed in many different ways by taking the numbers zn different orders. That is, addition may be performed in any order. This is called the Commutative Law of Addition. It may be symbolized as follows : —a^h^c—d=h—d-^c—a—c-{-h—d—ai etc. 49. It is also apparent from the definition that addi- tion may be performed in many different ways by first adding the numbers into certain groups and then adding these groups together to get the required sum. Thus, the sum of 3, 5, 7 and —3 may be found by first adding 3 and 5, giving 8, then adding 7 and —3, giving 4, finally adding 8 and 4 and obtaining 12 as the sum of the original numbers. Symbolicly this may be written 3 + 5 + 7-3=(3 + 5) + (7-3). Since we might have used any numbers, as a, 3, — ^, —d, and e, and have grouped them in many different ways, we may write more generally a^rb—c—d\-c-^(a-\-h—c)-\-(^—d-\-e^ = (« + 3) + (-^~^+^) = (a-f ^) + (— ^-— ^ + 2+1^2-'12xy. 2. ^m—10ts—^n—Ats—2n + 2m—^ts—^m. 3. ba-\'W^c—1d—2a—'db'^c+2d—2a+2b^c+M. 4. —10m+ll—bx—12—4:m—^x-\-l + ^x—bm, 5. 9^-7^+3^— 8^+7^— 3^— 5^— 8^. ADDITION. 19 7. llxy-^-'lab—^^xy—^bab+ab+lO. 8. 25— 25;i:+25j/+13— 30j/+20;i;-8. 9. 7^2— 24-^2+2— 46r2+9>^2^5 + ^^ 10. lSx—6y + 8^—5x-\-9j/—llz—Sx—6j/-h^. 11. 4m—62n + 18x-^62m—Gx + 4:2?z + 10m-^lSn—14tx. 12. 10m + ll—5a'^ — 12 + 4m-6a'^-{-l + 18a'- — 10m. 13. ^— 2^^+18r^— 14<^^-21^^— 3<^^+5^^. ADDITION OF KXPRKSSIONS. 52. When two or more expressions are to be added, we may enclose each expression in a parenthesis, supply the sign + to each parenthesis, and then write the paren- theses one after another. lyCt us find the sum of the three expressions x-\-y, X — ^ and 2x-\-^y — 2. First, we enclose these in parentheses, write them one after another separated by plus signs, and get (-^4-jr) + {x-2) + (2x-^^y-2). Second, we remove parentheses and get x-\-y-\-x — ^4-2jr4-3j/— 2. • Third, we arrange these terms so that similar terms shall come together, and get x+x+2x+y+Sy—^—2. Fourth, we unite each group of similar terms into a single term and get 4xi-4y—z—2. and this is the simplest form possible for the sum of the three given expressions. 53. It is easy to see that we can find the sum of any number of expressions, whatever those expressions may be, in a manner similar to that just pursued. To do this 20 UNIVERSITY ALGEBRA. we enclose each expression in a parenthesis and write these parentheses one after another separated by plus signs; then we remove parentheses from the expression. Next we arrange the terms of the expression thus found, so that similar terms shall come together, and then unite each group of similar terms into a single term. Kxampi,e;s. Find the sum of the following expressions: 1. ^m—4cst—r'^, —llst+lm+^r'^ and lOst—lr'^-'m. 2. 3;t:2+^j/+3j/2, x'^—Zxy-^y'^ and 3jr2 + 3jj/2. 3. Sxy+2y, bxy—x, Sx—5y and 7xy—x—2y, 4. Sa—Ad—6cd+2e, 10<^+3^— 10^^ and 9^—20^+14^^. 5. 6^+7— 4a— 5^, 6/5+7^— 4— 5a. and 6a-\-7d—14:C—5. 54. Arrangement of \A^ork in Addition. It evidently comes to the same thing if, instead of writing the expres- sions one after another within parentheses, as in Art. 53, we write them one below another without parentheses, arranging the similar terms in the same vertical column. Then, when this is done, we may draw a line under the last expression, and the example is arranged in exactly the saine form as in Arithmetic. Now the similar terms in each column may be combined into a single term, and this term placed under the line as one term of the sum. When every column has been thus treated all the terms of the sum are found. Thus, suppose we are required to find the sum of ^ 2^2+ 3^2 _5^^^ 6^2-2^2 and W^ -^a'^ -^cd. By the present method we write 2^2+3^2^5^^ 6a2-2^2 -4a2 + 8^2_4^^ 4^2 +9^2 _9^^ ADDITION. 21 Now it is very evident that we have here exactly the same terms to combine that we had by the other method, after the parentheses had been removed, and the similar terms brought together. It is plain that the only differ- ence is in the arrangement of the work. l^XAMPIvKS. I. 2. 4^2+5^3 + 932 2ad- bc+bca 3. Add 2b+Sc—ba, 8^-33+4c, and Tb-lbc-la. 4. Add n+2r-\-Zs—4:t, r—As—bt—^n, s—^t—^n^&r. 5. Add x+Za+2b-c, 2y—Sb+2c+a, 2>z-Zc—2a—b. a-Sd-{-2c +2r 6. Add 13:i;8— 4;»;2 — 6;ir + 17, 22.^^+ 20.^2+ 3:r-- 10, 2^2_i7^8_2;»;-14, and 3^3_i2jtr2 + 12 + 5^1;. In arranging expressions which involve different powers of the same number it is usual to place all of the terms containing the highest power of the letter in the first column, all the terms con- taining the next highest power in the next column, and so on Thus, this example would generally be arranged thus : 13x3- 4x^-Qx+17 22x3+20x;2+3x-10 -17^3+ 2x^-2x-U 3x3-123c8+5jc+12 7. Add 24xy+15de'-12/g', \Zfg--Z2xy, V^xy-Zde, Me'-hxy'-2fg, and ^fg—'lxy. 8. Add 9|a»-7a2^+54^^2 + 111^3^ _7^2_^5^^+9^2^ and 7ia3-2^a2^_4^^2_i2^8-5a2+4a^-20/^2^ CHAPTER III. SUBTRACTION. 55. Subtraction is the process of undoing- Sidditioii; that is, subtraction is the process of finding from two given numbers, called the Minuend and Subtrahend, a third number, called the Remainder or Difference, such that the sum of the subtrahend and remainder shall equal the minuend. 56. Any operation, like subtraction, which is the un- doing of another operation is often called the Inverse of the other operation. 57. Notation. To denote that one expression is to be subtracted from another expression we enclose each in a parenthesis and write the subtrahend after the minuend with a minus sign between them. Thus, to indicate that 2:tr— 5 is to be subtracted from x'^—4:X+2 we write 58. Since subtraction is defined as the undoing of addition, it follows that the principles of subtraction must be based on those of addition. Thus: +12-(+5)=+ 7 because + 7 + (+5) =+12, and +12-(-5)= + 17 because + 17 + (—5) = + 12. Also, in general, a—[-\-h)-=a-h [1] because a—b-\-(^-\-b) = a, and a-{-h)--^a^h [2] because a + b-{-{^—b)=a. SUBTRACTION. 23 Since a and b may stand for any two numbers, we may deduce the following principle from [1] and [2] : To subtract a given number from any expression^ we annex the nuTnber with its sign changed to the expression from which it is to be subtracted, 59. Associative La^v of Subtraction. The associa- tive law of addition states that we may add a given ex- pression to another by adding the expression as a whole or by adding its terms in succession. Hence, since sub- traction is the inverse of addition, to subtract a given expression from another we have merely to subtract its separate terms in succession. Thus: =-a — b-\-c — d, 60. Removal of a Parenthesis preceded by the sign — . Comparing the two sides of the equality a—{b—c-\-d')^=a—b-\-c—d, we derive the principle of the insertion or removal of a parenthesis preceded by the minus sign : A parenthesis preceded by the minus sign may be inserted or removed, provided that the sign of every term, within the parenthesis be changed. SUBTRACTION OF KXPRKSSIONS. 61. I^et us find the difference between Sx^—4:xy+5y^ and 2x^—'2xy-'4y^. First, we enclose each of these expressions in a paren- thesis, and write the subtrahend after the minuend with a minus sign between them, and get (Sx'^ —4xy + 5y''')-'(2x^- —2xy—4y^). 24 UNIVERSITY ALGEBRA. Second, we remove each of these parentheses, taking care to change all signs within the second parenthesis, and get Third, we arrange these terms so that similar terms shall come together, and get Sx'^—2x^ — 4xj/+2xy+6y-i-4:j/^, Fourth, we unite each group of similar terms into a single term, and get x^-'2xj/+dy^, and this is the simplest form possible for the difference of the two given expressions. 62. It is easy to see that we can find the difference of any two expressions, whatever the expressions may be, in a manner similar to that just pursued. To do this we enclose each expression in a parenthesis, write the min- uend first and the subtrahend second, with a minus sign between them, and then remove parentheses. Next we arrange the terms of the expression thus found so that similar terms shall come together. Finally, we unite each group of similar terms into a single term. EXAMPLE. 1. From x^+y^ take x^-^y^. 2. From x+aSd^ take —x—Sa+b^. 3. From a—b+c—d take a+b'-c+d, 4. From 2;i8+3a»— r»— ^« take «»— a»+r«— 25«. 5. From a^+2ab+b^ take a'^—2ab+b^. 6. What must be added to r^+s^ + tl to produce 3? 7. What must be subtracted from abc^ to produce m'\-r^ 8. What must ^ab be subtracted from to produce —abl SUBTRACTION. 25 9. From ahc^ — lab'^c+Za'^bc take Zabc^ +1ab'^c+a'^bc. 10. From the sum of a^ + ^^ and —^ab subtract the sum of a^ — b'^ and 3^^. 11. From x^-{-ax'^-\-a'^^x+d^ subtract ^ax'^-'O^x^ and from this difference subtract ^ax'^—a'^x, 63. Arrangement of Work in Subtraction. In find- ing the difference of two expressions by the method already learned, we place each expression in a parenthesis, write the subtrahend after the minuend with a minus sign between them, and then remove the parentheses. But, evidently, it comes to the same thing if, instead of writing the subtrahend with all its signs changed after the minuend, we write the subtrahend with all its signs changed under the minuend, with similar terms of the minuend and subtrahend in the same vertical column, and then unite similar terms exactly as in addition. For example, if we wish to subtract 2^2__4^2__g irova 9(3^2_{_3^2_7^ ^g arrange the work thus: Minuend, 9^2^3^2__7 Subtrahend with signs changed, — 2^2_j_4^2_|_g Remainder, 1 a'^ ^1 b''- — \ The signs of the subtrahend need not actually be changed if the student will imagine them changed as he proceeds in the work. KXAMPi^KS. I. 2. From 15a— 73+3^— 7^— 8(? 7^— 2>/— z+A+a take 10^ + 7^—3^4- 4^+ 4g x^ y^b2—2 +n 3. From 4:x'^-\-2xy+^y'^' take x'^-'Xy-\-2y^. 4. From 1x^-—^x—l take bx'^—^x^-Z. 26 UNIVERSITY ALGEBRA. 5. From Sx^—2x'^ + Sx—4 take x^—ix'^-Sx+l. 6. From 6ay—5xy+2a^x^ take 4:xy—Say—a'^x'^, 7. From ia+^d—^'-9d+\ take |«— f^— f^+i^— «. 8. From 4:ady^—5axy+2a^d'^ take a2^2_^^^_3^^^2^ 9. From 2x+lla+10d—5c'-2S take 2^-10+5^—33. 10. From x^ + Sxy—y^ +j/2:—2j/^ take jr^ + 2;r^ + 5;r5r — 3y2_2^^ 11. From 6x'^ + 7xy—5j/^—12xyz—8yjs take 8;r|/— Ty^f 12. From 4r^+62;^2_26w— 23;^2 ^ake 9;e2_2r^+21;;^ + 2n'^—Srs, INSERTION AND R^MOVAI, OF PARBNTHE;s:^S. 64. Parenthesis within a Parenthesis. It may sometimes happen that an expression within a paren- thesis is itself an expression which contains a paren- thesis, so we have a parenthesis within a parenthesis. Indeed, we may have several parentheses one within another. These complicated expressions present no difficulty, for we can take the parentheses one at a time, and if we know how to remove one, we may do this and then remove another, and so on until all are removed. For example, if we wish to remove the parentheses from we begin by removing the inner parenthesis first and write the expression in the form a + (id—c+d+e), and now by removing the remaining parenthesis we write the result in the' final form a-\-d—c-\-d-\-e. SUBTRACTION. 2/ 65. Until some skill has been attained it is usually best for the beginner to remove the innermost paren- thesis first, and then the innermost parenthesis of all that remains, and so on until all, or as many as may be de- sired are removed. KXAMPLKS. Remove the parentheses from the following expressions: I. a^-ib-{c+dy]. 3. 5a3-(4^3_[-3(^2_j.^2)_4(^__2)]) + 2. 4. i_[i_(i«[i_(i_^)])]4.;^. 5. Enclose the last three terms of a-\- b—4:C—he'^ +Q>r^ —n^ — 16 within a parenthesis preceded by a + sign. 6. Enclose the third and fourth terms of a~\-d—4c -f5^2+6r^ — ;^* — 16 within a parenthesis preceded by a — sign, and the fifth, sixth, and seventh terms of the same expression in another parenthesis preceded by a — sign. 7. Fill out the blank parenthesis in the equation 8. Fill out the blank parenthesis in the equation CHAPTER IV. MUIvTlPIylCATlON. 66. Extended Definition. To Multiply one number by another, we do to the first what is done to unity to produce the second.* The number that is multiplied is called the Multiplicand, and the number multiplied by is called the Multiplier. In the present chapter we will understand the sign X to mean ''multiplied bf (not ''times'"^, so that ay.h means that a is to be multiplied by b, and axbxc means that a is to be multiplied by b and the result multiplied by c. The original meaning of muhiplication in Arithmetic is that of repeated addition, and, with this meaning in mind, we would define multiplication to be the taking of one number as many times as there are units in another. Thus, 3 multiplied by 5 means 3+3+34-3+3, and I multiplied by 5 means f+f+l+f+f. As soon, however, as the multiplier is a fraction, it is found that this meaning of multiplication does not apply; for while 3 can be repeated 5 times, yet 3 cannot be repeated \ a time, nor can | be repeated f of a time. Now, although the operation of multiplying | by f cannot be looked upon as repeated addition, yet this operation does occur in Arithmetic, and is called multiplication. It is plain, therefore, that the word is used with some other meaning than that originally given it, which new meaning we have stated above. 67. Illustrations. To multiply 3 by 5 we must do to 3 what is done to unity to make 5. But 6=1 + 1 + 1 + 1 + 1; whence 3x5=3+3+3+3+3, or 15. ♦Boset Algebra Elementaire, Charles Smith's Algebras. MULTIPLICATION. 29 To multiply f by 5 we must do to f what is done 10 unity to make 5. But 5=1 + 1 + 1 + 1 + 1; hence 2 x5=|+|+|+|+|, or -i/. To multiply 5 by f we must do to 5 what is done to unity to make f ; that is, we must divide 5 into 4 equal parts, giving f , and take one of these parts 3 times. Hence, 5 X f =| X 3=-^^. To multiply f by f we must do to f what is done to unity to make f ; that is, we must divide f into 5 equal parts, giving y^-, and take one of these parts 4 times. 2 4 2, 2x4 Hence, oXf=^s — p^^—^ — i-* 3 5 3x5 3x5 In the general case of the product of two fractions we would find, n s_nXs r.^-, 68. Law of Signs in Multiplication. The definition of multiplication given above is sufficiently broad to in- clude the case of multiplication by negative numbers. To multiply —6 by 3 we must do to ~6 what is done to unity to produce 3. But 3 is 1 + 1 + 1; hence, —6x3 is —6—6—6, or —18. To multiply 6 by — 3 we must do to 6 what is done to unity to produce —3. But -3 is -(1 + 1 + 1); hence, 6x —3 is —(6 + 6 + 6), or —18. To multiply —6 by —3 we must do to —6 what is done to unity to produce —3. But -3 is -(1 + 1 + 1); hence, — 6x — 3 is —(—6-6—6), or +18. [2] 30 UNIVERSITY ALGEBRA. Likewise it may be seen that the definition includes the case where the multiplier is a negative fraction. Hence, if a and b stand for any two numbers, positive or nega- tive, integral or fractional, we have (+a)X(+&) = +a& (1)1 (-a)X(+&) = -a& (2) (+a)X(-&):=-a& (3) (-a)X(-&)=+a& (4) These equations express the Law of Signs in mul- tiplication, which is often stated thus : Like signs give + and unlike give — . 69. Commutative Law of Multiplication. From the law of signs we observe that the sign of a product will be the same no matter in what order the factors are multiplied together. Thus, the product (4-<2) x ir-U) has the same sign as the product (— <^)x( + a). We can show a similar truth respecting the absolute value of the product. First, suppose both multiplicand and multiplier are positive whole numbers: we wish to prove axb=bxa. We may write down b rows of units with a units in each row, thus : 1 1 1 1 1 . . . « columns. 11111... 11111... 11111... b rows. Then, since there are a units in 1 row, in b rows there are axb units. But since there are b units in 1 column, in a columns there are bxa units. Of course the number of units is, in either case, the same, so axb=bxa, if a and b are positive whole numbers. iM ULTIPLICATION. 3 I Second, suppose multiplicand and multiplier are pes- 7Z S itive fractions, say — and -. In Art. 67 we found r t n s nxs y/_—— . r t rxt Similarly we would find s n sxn -x-=- t r txr But we have just shown that nXs=sXn^ and also that rXt=tXr, hence 71 s s n -X-=-X — r t t r Hence, for all positive values of a and h^ integral or fractional, a x b=^ bxa. But, from the law of signs, the sign of the product does not depend upon the order of the factors ; therefore, for all values of a and b, positive or negative, integral or fractional, axh=bxa, [3] That is, zn the product of any two numbers, we may take the factors in either order. This is called the Commu- tative Law of Multiplication. 70. Case of Three or More Factors. Suppose each of the units in the rectangular arrangement in the pre- vious article be replaced by a number whose value is represented by a. I^et there be b columns and c rows, thus: a a a a a , . , ^columns. a a a a a , , , a a a a a . , , a a a a a , , » c rows. Now the total number of units in the arrangement may be estimated in different ways. The number in each row 32 UNIVERSITY ALGEBRA. is axd, and hence the number in c rows is axdxc. The number in each column is aXc, and hence the number in d columns is aXcxb, Therefore, axbxc=axcxb, (1) Since axb=bxa, axbxc=bxaxc, (2) Putting bxaiox axb and cXa ior ax c in (1), bxaxc=cxaxb. (3) From (1) we see that a product remains the same if the second and third factors be interchanged, from (2) we see that a product remains the same if the first and second factors be interchanged, and from (3) we see that the same is true if the first and third be interchanged. Whence we write, ax:f>xc=axcxh=^bxaxc=hxcxa=cxaxh=cxl>xa [4] This is the commutative law for three factors. The law may be shown to hold for fractional and negative factors exactly as in the last article. 71. Associative Law of Multiplication. The num- ber of a's in the rectangle in the last article is be. Whence the total number of units is ax{bxc). By the last article the total number of units equals axbxc, ox (axb^Xc, whence we write, aXhXc={aXh)Xc=aX{hXc) [5] Thus, 3x7x5 is the same as 21 x 5 or 3 X 35. This fact, that multiplication may be performed by grouping the factors in diJfferent ways, is called the Associative Lavv of Multiplication. By means of [4] and [5] the commutative and associa- tive laws can easily be extended to the product of any Qumber of factors, negative or fractional. MULTIPLICATION. 33 72. Index Law of Multiplication. From the defi- nition of a power of a number we have a^ x a^ = aaXaaa = ^5_^2-t-3 Also a^ Xa^ = aaaXaaaaa=a^=a^^^, In general : a''=aaaa ... to « factors ar=aaaa ... to r factors a'*xa''= {aaa ... to ^ factors) X (aaa ... to r factors) =aaaa , , , to (n + r) factors by associative law, =<2""^'' by definition of an exponent. What we have now proved may be stated in the fol- lowing words : The product of two powers of the same number is equal to that number with an exponent equal to the sum of the exponents of the two factors, 73. Formulas. Statements like the above expressed in the symbolic language of Algebra are called Formulas. Thus : a« X a^=Oabxy, by commutative law. (2) Find the product of Zax"- by —2bx^. (^ax'^) X {-2bx^')=-(^ax'')(2bx^), by law of signs, = — 6^^Jt^^ by associative, commutative, and index laws. Thus the product of any two monomials can be obtained by means of the four laws already set forth. 34 UNIVERSITY ALGEBRA. KXAMPI^KS. Multiply 1. ^''x^ by hh^p^x'^, 5. 1^-^hx by -5^3^^ 2. -8/^2 by i^3^3^ 6, -33^^ by lOi^y . 3. 60;i:j/2 by .05^3jK. 7. (^+jk)'^ by 4(jr+jj/)^ 4. -20^2^ by -.3^2^. 8. 3(;r--5)2 by ^(^-5)*. PRODUCT OF A POI^YNOMIAI. AND A MONOMIAL. 75. Distributive Law of Multiplication. We wish to prove {a-{-U)n=^an-\-bn, where a, by and n stand for any numbers whatever. First, we let nhe a, positive whole number. Then (a+d')n=(a + d) + Ca-\-d)-i-(a-^d) + . . . to ;z terms, by definition of multiplication, =a+d+a+d+a + d+. . . by associative law of addition, '=a+a+. . to w terms + ^4-^+. . to n terms, by commutative law of addition, sszan+dn, (1) by definition of multiplication. Second, let ^ be a positive fraction, represented by -• Then, if division be defined as the process of undoing multiplication , (a-i- d)-i-r=a-^r-j- d-^r, (2) if r be a positive whole number; for if we multiply either side of the equation by r we shall obtain a-j-d. But (/i+^h=—j-xs, by definition of multiplication, = (a--/+^-T-0^, by (2), =a-r-/X^+<^-r-/X^, by (1), . =ax^+^x^ (3) by definition of multiplication. MULTIPLICATION. 3 5 Third, let —r represent any negative multiplier, inte- gral or fractional. Then, from (1) and (3), {a-^U)r—ar-\-br, (4) whence —{a + b) r= — {ar-\-bf), (5) —{cL-\-by-=—ar—br, (6) by removing parenthesis. Now, by the law of signs, the members of (6) may be written as follows : (« + ^)(-^)=«(-^) + ^(-r). (7) Therefore, for all values of n, {a+b)n=an-{-bn. [8] 76. More than Two Summands. In (a+b)n sup- pose a=x-{-y. The equation {a-\- b)n=an-[- bn becomes (x-{-y+b')n=^(x-\-y)n-{-b7i^xn-\-yn-{-bn, Likewise, ['OC-^y-{-z-\-w-{-. . .)n=xn-{-yn+zn+wn+. . . [9] That is, ike product of any polynomial by a monomial equals the sum of all the terms found by multiplying each term of the polynomial by the given monomial. The above is called the Distributive Law of Multi- plication. 77. The following examples illustrate the principles of multiplication we have thus far proved : (1) Find the product of {Zax-\-^.bx'^—^y^ by Zxy, (Sax+4bx'^—4y')xSxy==dax'^y-hl2bx^y—12xy^. (2) Find the product of 6x^—4x^+8x by ~^ax^, {^x^—Ax'^-\-Sx'){-'\ax'^)=—?>ax^-\-2ax^—4:ax^, 78. Arrangement of Work. The following illustra- tions will be sufl&cient to explain the usual arrangement of work when the product of a polynomial by a monomial is sought: 36 UNIVERSITY ALGEBRA. (1) Multiply a-\-b—c by 5. Multiplicand, a+ b— c multiplier, 5 product, ha-\-hb—hc (2) Multiply :r-3;r2 + 2;t:3 by -5^. Multiplicand, x -- Zx'^+ 2x^ multiplier, —hx product, —bx'^ -{-Ibx^ — lOx^ (3) Multiply ^ay'^ by la'^y'^—^ay. Since multiplication may be perforTned in a7iy order, we arrange this just as though ^ay'^ were the multiplier. KXAMPi^KS. I. 2. Multiply 2x—5y+B Multiply ba'^ — Aa-^Q by 11 by —4:ax Multiply 3. 2x—4y-}-2 by Sx. 4. a^d by 4:a+2ac^—Sd^ — l, 5. Sab-\-4a'^b—5cb-{-6 by —Aa^d\ 6. 6a4-4^2_|_2^^3_3^2 by -_i.^2^^^ Simplify each of the following expressions : 4(4^-73)=16«-283; _7(2^_5^) = _i4^-|-35^. Therefore, 4(4^-73)-7(2a-5<5)=:16«-28^-14«+35^. Combining similar terms, =2^+7^. 8. a(ia+b'-c)—b(a—b+c)-hc(a-Jtb), 9. 3«^(«— ^)— ^(r(2^— 3a)-f6a32 — ^2(3^_|.2^). 10. ab'^(iax—bx+eax^—5bx^)4:a^b. IX. i;r)/2(^3— ;ry2+9;i:2jj/— 7>/^)8^^ MULTIPLICATION. 37 PRODUCT OF A POI.YNOMIAI, BY A POI.YNOMIAI,. 79. Law of Product of two Polynomials. In the equation, (a+d+c+, . ,)n=an+dn+cn + , . . (1) put n=x+y; then we have, (a + d+c+. . ,)(xA-y)=aix+y) + d(ix+y) + cCx+y) + , . . by commutative law, ={x-{-y)a-\-(x+y)b+(x+y)c+, . , by distributive law, ■=xa-\-xb+xc+ . , . Also, by putting (x-}-y+z+. . .) for n in (1), we get (a + b+c+. . .)(x+y+2+, . ,')=xa-\-xb-{-xc+, . . +ya-j-j/d+yc+. . . -hs^a-^- ^d-{- 2c-{-. . ., etc. Stating this in words, fke product of a polynomial by a polynomial is the aggregate obtained by placing one poly- nomial as a factor in each term of the other. Or, in other words : The product of one polynomial by another is the sum. of all the terms found by multiplying each term of one polynomial by each term of the other polynomial, 80. We illustrate this by a few examples : (1) Multiply ;r— 4 by ^+9. Placing the second polynomial as a factor in each term of the first polynomial, we have ;r(;r+9)-4(;i:+9). Multiplying by x and —4, we obtain x'^-\-^x—\x—Z^. Uniting similar terms, we have the required product, (2) Multiply 3^2^ 2^~9 by 2a^-6^. Placing the second polynomial as a factor in each term of the first polynomial, we have 3ij2(2a^— 6^) + 2a(2«^— 6<^)--9(2^^-6^). 38 UNIVERSITY ALGEBRA. Multiplying by 3^^, la, and —9, we obtain Uniting similar terms, we have the required product, ^a^b-\Wb-?>^ab-\-hU. 81. Arrangement of Work. Let us go through the work of multiplying the polynomial 2<22 — 3^ — 3 by 3^2— 2<2+2. We first place the second polynomial as a factor in each term of the first polynomial, and obtain 2a2(3a2-2« + 2)-3^(3^2_2a+2)-3(3^2_2^_^2). (1) Then multiplying the expressions in the parentheses by 2^^^ — 3<3:, and —3, respectively, we obtain Now, uniting similar terms, we get the required product, ^a^—\Za^+a''-^, (3) This work may be arranged in a more convenient form : Multiplicand, 2^2— 3^ —3 (4) multiplier, 3^2_ 2a +2 (5) first partial product, 6a4_ 9a3_9^2 (6) second partial product, - 4a=^ + 6a2_|.e^ (7) third partial product. 4^2_e^_6 (8) product. 6a4_i3^34. ^2 _e (9) which is the usual arrangement of work in the multipli- cation of two polynomials. 82. The following examples will tend to show the advantage of arranging the work of multiplication in the way explained : x-\-y a — b x'^ +xy-\-y^ x-\-y a-\-b x—y x'^ -\- xy a^ — ab x^ -\- x^y -\- xy"^ xy-^y^ ab—b^ —x'^y—xy'^—y^ x'^-\-2xy-\-y'^ a^ —b'^ x^ — jK* MULTIPLICATION. 39 ' ' The student should observe that, with the view of readily bring- ing similar terms of the product into the same column, the terms of the multiplicand and the multiplier are arranged in a certain order. We fix on some letter which occurs in many of the terms and arrange the terms according to the potvers of that letter. Thus, taking the last example, we fix on the letter X', we put first in the multiplicand the term x^, which contains the highest power of x, namely the second power; next we put the term xy which contains the next power of X, namely the first; and last we put the term y^ which does not contain x at all. The multiplicand is then said to be arranged according to descending powers of x. We arrange the multiplier in the same way. "We might also have arranged both multiplicand and multiplier in reverse order, in which case they would be arranged according ta as cendiftg powers of x. It is of no consequence which order we adopt, but we should take the same order for the multiplicand and the mul- tiplier." — Todhunter's Algebra for Beginners. EXAMPI.KS. Multiply the following expressions : 1. x—lZ by x—14:. 3. ^x'^ + 2y'^ hy^x'^ — ^y'^ 2. x+1^ by .r— 20. 4. x'^—xy+y'^ by x+y. 5. 2.r2-3+5.;t:8 by Qx-S-^Ax^. Arranging according to the descending powers of x, we obtain 5x^-\.2x»-3 4x^-{-Qx -8 20a:«+8a;6 -12^8 +30^*4-12^3 __i8^ -40^8-16.y« +24 20^«+8^5_{.3o,c4_40;»:8_i6;»:8_lg;P4-24 6. ab^+3aH'-2aH^ by 2a^-ad-5d^. 7. x^'-5ax—2a^ by x^+2ax+Sa^. 8. 7x'^+y'^—Sxy by 2x^+y—x. 9. 2x^—4x'^—4x--l by 2x^—4^'^'-Ax-'l. 10. 5.r— 7.;i;2_^.r3 + l by l+2x'^-'4cX. 40 UNIVERSITY ALGEBRA. II. a^-\-b'^-\-c'^ — hc—ca—ab by a-\-b-\-c, 13. x''" -{-y"^ -^ z'^ -{■ xy—y2-\-xz by x—y-\-z. 14. ^^2-1^-1 by 1^2 ^1^-3. 15. 2^_|^4-5^ by \m\-\r-^p, SPBCIAI, PRODUCTS. 83. Product of Sum and Difference. By actual multiplication we find (a+&)(a-5)=:a2-&2. [10] Since a and b may stand for any numbers, we may say: The product of the sum and the difference of any two numbers equals the difference of their squares. 84. Square of a Binomial. By actual multiplication we find (<:e+&)2=as+2a&+&S [11] {a-b}^=a^-2ab+b^. [12] Since a and b may stand for any numbers, we may say: The square of the sum of two numbers is equal to the square of the first, plus twice the product of the twOy plus the square of the second. The square of the difference of two numbers is equal to the square of the first, minus twice the product of the two^ plus the square of the second. 85. Products of the Form {x-^a^{x-^b). By actual multiplication we find (ir-ha)(iZJ+&)=a5»+(a+&)ir+a&. [13] That is, the product of any two binomials in which the first terms are alike is equal to the square of the first term plus the first term with a coefficient equal to the sum of the second terms, plus the product of the second terms. MULTIPLICATION. 4 1 Of course due attention must be paid to the signs of the terms in writing out such products. Thus : KXAMPI^ES. Write down the square of the following binomials : I. ad+c. 5. A:x—^ab, 9- m^-r'-s\ 2. c—ab. 6. oab—Sac, 10. (al>^y + i2xy. 3. x-y. 7- i-i- II. (2xy+2x-'. 4. lax^b. 8. 100-1. 12. a+a. Write by inspection each of the following products : 13. (ix-\-a)(ix—d). 18. (x+yX^—y')' 23. (ix-\-2y)(x—2y) 14. (x+lX^-1). 19. (b''+cXb''-c). 24. (jr2+4)(^2_4) 15. (a + 25)(«-4). 20. (a-2)(a-3). 25. (a'^-l')(a'^-j-2) 16. (^-^)(a + 3). 21. (a + 2)(a + 3). 26. Ca + bc)(a-cd) 17. (a + 10)(a-5). 22. (^— 6)(«+5). 27. (a+^^)(^ + ^^') 28. (2^-^)(2a + 3^). 31. (a^^+^2)(^^2_2). 29. (2«— <^2)(2a~^*'^). 32. (mn + o^^mn—A), 30. (.;i:H-3j/2')(^— 2z^z;). 33. (;r2jj/+^jj/2)(^2^_;rK2)^ 86. Generalized Law of Distribution. I^t us con- sider the product (^1+^2+^3 + - . .+^(^1+^2+^3+. . .+^r) (1) where ^j, ^3, ^a . . ., ^1, ^2) <^3 • . .» stand for any num- bers whatever. We have supposed that there are n terms in the first parenthesis and r terms in the second paren- thesis. By the distributive law the product will consist of the sum of all the partial products found by multiplying 42 UNIVERSITY ALGEBRA. each term in one parenthesis by each term in the other parenthesis. Thus, the product may be written, '\-a^hr^-a^br-\-a^br-\-. . .-\-aJ)r (2) As we have written the result, there are n columns of terms with r terms in each column. Therefore, the total number of terms in the product is nr. Next consider the product («^4-«, + ^3+. . ^.)(^, + ^,+ ^3+. . ^.)(^, + ^, + ^3+. . 6^ (3) where there n terms in the first parenthesis, r in the second, ^ in the third, and so on. This product may be written («x^x+^A+- .+^A+^A- .)(^.+^.+^3+- •^^)' W where the first parenthesis is suppose to contain the nr terms written in (2) above. We now have a parenthesis containing nr terms to be multiplied by a parenthesis containing ^ terms. This will give nrxs or nrs terms, which may be written out just as in the case of (1) above : ^i Vi + • • + ^^ Vi' + ^i Vi + • • + ^;. Vi' + • • ) + aj)^c^ + . . + a„brC^ ab^Cs-\- . . + a,fi^Cs, + aj?^:^ + . . + a,fi^c,, + • • , + ab^c^^- . . + a,Jb^^ By a continuation of this process we could evidently write out the product of any number of parentheses. Whence we say: The product of k parentheses is the aggregate of Aix, the possible partial products which can be made by multiplying together k terms, of which one and only one must be taken from each parenthesis. MULTIPLICATION. 43 If the number of terms in the differe7it parentheses be n, r, s, t, . , , respectively ^ the total number of terms in the product will be nrst . . . The student will observe that the above is merely a general statement of the distributive law. The state- ment of the law in this form is of the greatest import- ance. The following examples will tend to make its application clear: (1) Write out {a + b')(ic+d)(ie+f). Here we write ace + bee + ade + bde -\- acf-\- bcf+ adf+ bdf for the product of the three parentheses is the aggregate of all the possible partial products which can be made by- multiplying together three terms, of which one and only one must be taken from each parenthesis. The number of terms in the result is 2 x 2 x 2 or 8. (2) Writeout (« + ^)(^+0(^+^)• This gives abc-\-b'^c-\-ac'^-\- bc'^-\- a'^b+b'^a + a'^c+ bca. Here the first and last terms are alike, so we write: 2abc^b''c-{-ac'^^bc'^+a^b-^b'^a-\-a'^c, (3) Write out {a + /^) (^ + b) (,a + b). The possible products of three factors each, which can be made from the two letters a and b are a^ , a'^b, ab'^ , and b^. Thus, while the above product may be written as 8 terms, it can evidently contain but 4 distinct terms. We can readily determine what terms are repeated. The term a^ can evidently occur but once, since a can be taken from each of the three parentheses in but one way. The same is true of b^ . The term a'^b, however, may be made in three, ways: first, by taking b from the third parenthesis and a from each of the other two ; second, by taking b from the second parenthesis and a from each of the other two ; third, by taking b from the first paren- 44 UNIVERSITY ALGEBRA. thesis and a from each of the other two. lyikewise, the term ab'^ occurs three times. So we write the above product, a^-^-Za'^b-^Zab'^-^b^, 87. Type Term. All terms that can be derived from one another by interchanges among the letters are said to be terms of the same Type. Thus, a'^ , b'^ , etc., are of the same type, also a^ , b^ , etc., also a'^b, a^c, b'^a, etc. The number of terms of the same type will depend on the number of letters we have. Thus, if we have the four letters a, b, c, d, there are four terms of the same type as <3^2, six terms of the same type as ab, twelve terms of the same type as a'^b^ etc. 88. ^ and U Notation. It is often convenient to indicate the sum of all the terms of the same type by an abbreviation. This is done by prefixing the letter ^, which means ''sum of all terms of same type as'' or ''sum of all such terms as, ' ' to one term of the type. Thus, if we have the letters a, b^ and c, the symbol '2a'^b means a'^b-\-a'^C'\-b'^a-\-b'^c-\-c^a-\-c^b. If we have four letters, a, b, c, and d, the symbol ^a'^b means the sum of more terms than those just written. In using this notation it is always necessary to know how many letters are to be made use of, but this is generally shown by the context. Thus: (a + by=^:Ea^ + Z:EaH. The product of several factors of the same type is often indicated in a similar way, namely, b}^ prefixing the letter iT, which means ' 'product of all factors of the same type as, " or ' 'product of all such factors as, ' ' to one factor of the type. Thus, if we have the letters. <3^, b, and c, na'^b=a'^bxa'^cy:b'^axb'^cxc^^axc'^b, n(a—b)={a — b)(^b'-c)(j:—a), etc. MULTIPLICATION. 45 89. Cyclical Order. There is often a convenience in arranging the letters of an expression in a peculiar order. Thus, in the expression bc-{-ca-\-ab, the second term is made from the first by changing b into c and c into a, the third term is made from the second by changing c into a and a into b. A similar change in the third term would give the first. In each of these cases one term is said to be made from the other by a Cyclical Change or by Advancing the Letters. The expression a(^b—cY + b(c—a)'^-\-c(a—bY has its letters in cyclical order. 90. Number of Letters. The adjectives Binary, Ternary, Quaternary, etc., are useful in describing expressions which are made from two, three, four, etc., different letters respectively. Thus: a^b is a binary term, ab'^c'^ is a ternary term, xyzw is a quaternary term, and so on. 91. Product of Form (^x+a){x^-b){x-\-c) . . . Sup- pose that the product of n binomials, in which the first terms are alike, is sought: say {x-\-a){x-[-b')(^x-\-c) . . . By the generalized distributive law we may proceed thus: All the ^'s must be multiplied together, which gives x'' as one part of the product. Also, a must be multiplied by the ^'s in all the other parentheses, which gives ax''~'^ ; b must be multiplied by the x's in all the other paren- theses, which gives bx*'~'^ ; by continuing this we get as as another part of the product, {a+b-]rc-\-. . .);r'*"~^. We must also multiply ahy b and this by the product [)f the ^'s in all the other parentheses, which gives abx''~'^ ; ike wise a\yY c and this by the x's in all the other paren- theses; etc.; by continuing this we get another part of le product, {ab-\-ac+, . .-^-bc+bd-^- , . ,^x''~'^. 46 UNIVERSITY ALGEBRA. In like manner we find another part of the product, (abc-\-abd-\- . . . + bcd-\-bce-\-, . .')x"~^^ and so on. Finally, all the numbers, a, b, c, etc., must be multi- plied together, so the complete product may be written, (x-\-a){x-\-b')(x-{-c) ... to ^ factors =x''-\-{a-^b+c+. . .)x''-^-\-{ab-^ac^, . .)x''-^ -\-{abc-\-abd-\-. . .)x''~^-\-. . .-{-abc . . . As we have written this, the number of terms is n + 1. If the parentheses be removed the number of terms will be 2 X 2 X 2 ... or 2". Using the ^ notation, we write : ['3C-ra)[X-{-h)[0C-\-C) . . . to n factors =x''+2a.x''-^-}-2(ib,x''-^+2abc.x*'-^+. . ,+abc,. . [14] 92. Square of a Polynomial. Consider the following important product : (a + b+c+. . ,Xa+b+c+, . .) The result will consist of nxn or n'^ terms, if n repre- sents the number of terms in each parenthesis. By our generalized distributive law we must take a from one factor with a from the other, b from one factor with b from the other, etc., which gives a^ -j- b"^ -\- c^ + . . . Also we must take a from the first factor with b from the second, and b from the first with a from the second, which gives two terms equal to ab. Likewise for the terms ac^ ad .,. be ,,.; so that we say: (a+b+c+, . ,y = a^ + b'^+c'^ + . . ,+2ab + 2ac+. . .+2bc+. . . or {a-^b+c-\-. . .)^=2a^-{-22ab. [16] The number of terms in the result is nxn or n^, of which n are like a^, and n^—n like ab. The number of distinct terms \s n-\-\{n'^ —n^ ox \n{7i-\-V). MULTIPLICATION. 47 KXAMPIvKS. Write out the products in the following ten examples : I. (^b—c){c—a)(a—b). 5. {b-\-c)(ic-^a){a-\-b)(^b-'C){c-a){a-b). 8. (;r4-j^+-s'— ^— ^— ^)^ g. (c+a + n-\-t—d+o—z + ty, 11. Prove (;t:+j/)* = 2(;»;2+^2)(^^^^)2_(^2_y)2^ 12. Write down all the quaternary products of the three letters a, b, c, and tell how many different types they fall into and how many products there are in each class. 13. Do the same for the binary and for the ternary products of the four letters a, b, c, d. 14. Prove that (^a-^b-\-c)(J)-\-c—d){c-^a---b^{a-{-b----c) = 2^^V2-:Sa4^ 15. How many distinct terms in the result of the last example ? 16. Prove :2'^^.:^^^-7I^^:S'^ = ^ + ^+^:. o^ 17. Prove ^(^-^)-^==3i7(^-^). 18. Find the value 01 :S'[3— (<:— a)]^ when «=1, ^=3, and ^=5. 19. Prove that {x—d){x—b~){x—c) . . . equals x^—^ax^'-^-V^abx''-'^ — , . . + (—1)"^^^. . . CHAPTER V. DIVISION. 93. Division is the process of undoing multiplication ; that is, division is the process of finding from two given numbers, called the Dividend and Divisor, a third num- ber, called the Quotient, such that Divisor X Quotient =I>ividend, [1] Thus, to divide 12 by 3, we must determine the factor which, when multiplied by the given factor 3, will produce 12. In an expression like a-^b-^c-^d we will understand that a is to be divided by b, and this result divided by c, and so on. Since division is the inverse of multiplication, either of the expressions axb-^b or a-T-bY.b is equivalent to a, 94. Law of Signs in Division. We may write the following equations : (+a&)^(+5)=+a (1)^ (+a&)^ (-&)=-« (2) (-a&)^(+&)=-a (3) (-a&)^(-6)=+a (4)^ from [2], Chapter IV. Since a and b stand for any num- bers whatever, we conclude that when the signs of the dividend and divisor are alike, the sign of the quotient is + , and when the signs of the dividend and divisor are unlike, the sign of the quotient is — . Whence the La^v of Signs in division: Like signs give + and unlike give — . [2] DIVISION. 49 95. Commutative Law of Division. We shall prove that a-r'b-i-c=a-^c-T-b. For a-^b-^cX (bXc) = a-T-b-T'CXcXb, by associative and commutative laws of multiplication, = a~bxb, =a. Also, a-i-c-^bx (bXc) = a-^c-7-bxbXc, by associative law, = a-^cxc, = a. Whence, a-irb-i-cx(bxc) = a-^c-i-bx(bxc). Therefore, since these products are equal and the mul- tipliers C^xc) and (bxc) are alike, the multiplicands must be equal also ; that is, a^h^c—a-^c-^h, [3] The same principle can easily be proved for more than two divisors, so that we may say: In an expression co7itaining several divisors the order of the divisors is indifferent. This is called the Commutative Law of Division. Again, we know axb-^cxc=axb. Also, a-^cxbxc=a-^cx{bxc), by associative law of multiplication, =a-^cx{cxb^, by commutative law of multiplication, =zaxb. Whence, axb-^cxc=a-T-cxbxc, Therefore, axh^c=a-^cxh. [4] That is, if a succession of nu^nbers be connected by the signs X and -r-, the order of the nu7nbers may be changed^ provided each move with its proper sign. I 50 UNIVERSITY ALGEBRA. 96. Associative Law of Division. We will first prove that a X {h-T-c)=a x b-^c. We know ay.{b-^c)Y.c=^ay.\{b-T-c)y.c\, by associative law of multiplication, = ax^; but axb-^cxc—axb] Therefore, a x {,b-^c) xc=ax b-r-cxc. Whence, aX{b^c)=aXb-^c. [5] Also, a-i-(b Xc)x(bx c)=a, but a-^b-r-cX(bXc)==a'^b-7-cXbXc, = a-7-bxb-7-cXCy by commutative law of division and multiplication, =a. Therefore, a-^{bxc)=a-^b-^c. [6] I^ikewise it may be proved that a-^{b-i-c)=a^bxc. [7] From [5], [6], and [7] we say: In a series of numbers connected by the signs X and -^ a parenthesis preceded by the sign -r- may be inserted or removed, provided the signs X or -7- of all numbers within the parenthesis be changed. 97. Index Law of Division. We know «^-T-a^=a2 =«^~*, because a^ xa^=a^; also a^-^a^=a^=a^~^, be- cause a^xa^=a^. In general, a«^a'-=a«-'', [8] if n is greater than r, because a**~*'xa*'=a**. Of course n and r must stand for positive whole numbers. That is, the quotient of any power of a number divided by a lesser power of the same number is equal to that number with an exponent equal to the exponent of the dividend minus the exponent of the divisor. This is called the Index Law of Division. DIVISION. 51 DIVISION OF A MONOMIAlv BY A MONOMIAL. 98. The four laws established above enable us to find the quotient of any monomial divided by another mo- nomial. Thus : (1) Divide Udchyic. Since the dividend equals the product of the divisor and quotient, it follows that the quotient of one monomial by aiiother monomial is found by removing from the divi- dend all the factors which occur in the divisor. By the laws, the factors 4 and <; may be removed from the divi- dend in any order, so that the result is Zb, (2) Divide X^abx^'y^ by Uxy'^ , Using the index law, etc., we obtain ^ax'^y'^. KXAMPIyE^S. Divide 1. V^ax^ by ^a, 6. Vla'^chy a'^c. 11. m^ny*^ by ny'^ , 2. 28^/3 by Ac. 7. 42ac'^ by ac^, 12. ^x'^y^ by ^xy^ . 3. 81^ V^ by 9^2. 8. lam'^ by m. 13. Sc^xy^ by 2c'^y^ 4. Ibam'^ by \ba. 9. ba^x^ by a^. 14. 48m'^n by m'^n, 5. 7Qdx^ by 19d, 10. Sm^n^ by n^. 15. 72;t:5j/ i^y 72x^, DIVISION OF A POLYNOMIAL BY A MONOMIAL. 99. Distributive L#a^v of Division. The equation {oo+y-j-z+. . .)-i-n=x-^n-^y-^n-\-z^n+. . . [9] is true. For if each side of this equation be multiplied by n the result will be x-^y-\-2-\-, . .:=x+y-j-2+, . . which is true. Therefore [9] is true. That is, the quotient of any polynomial by a monomial eqiials the sum of all the tej'ms found by dividing each term of the polynomial by the given monomial. This is called the Distributive Law of Division. 52 UNIVERSITY ALGEBRA. 100. From the distributive and other laws it follows that a polynomial is divided by a Tuonomial by removing from each term of the polynomial all the factors which occur hi the divisor. Thus: (9^3jr— 15^2^2 _|_i2^^3)^3^^^3^2_5^^_|.4^2 because (Sa'^—bax+Ax^) xSax=9a^x — 15a'^x'^ + 12ax^y and the quotient is found by removing the factors 3, a, and X from each term of the dividend. If we wish to divide 16a—9bx-i-Scx'^ by 4:X, the factors 4 and x do not occur in some of the terms and conse- quently cannot be removed from them, so we can merely indicate the division in such cases, thus : (Wa^9bx-{-Scx')-^4.x=---~ + ~ X 4: 4: 101. The Arrangement of the Work when a poly- nomial is divided by a monomial is as follows : (1) Divide 16^4 + 24^^—20^1:2 by 4jr2. Divisor, 4;t:2 \J^6x^24:X^ -20x'- dividend, 4:X^-\- 6x — 5 quotient. (2) Divide 5x'^—7x^y-^4x'^y'^ by dx'^. Divisor, 5x^ I 5x^ —7 x'^y -\- Ax'^y"^ dividend, x^—lxy-{-^y'^ quotient. EXAMPI^KS. Divide 1. 15«2__9^54.i8^9 by 3^2. 2. oa^x^ — Sba'^x'^y^ +20axy'^ by —5ax. 3. 12a'^x^y^—2Aax^y^ — 18x^-y-i-6xy by —6xy, 4. 2x^y'^—Sxy^+4x^y—y^hySxy^. 5. a^x^ySa'^nx'^y-i-San'^xy'^ — n^xy^ by anxy. DIVISION. 53 DIVISION OF A POLYNOMIAL BY A POLYNOMIAL. 102. Suppose we wish to divide x^ + 5x+6 by x-\-S, The dividend, x'^ + 6x+6, is the product of the divisor, x-\-S, and another factor (the quotient) which we wish to find. Now we can undo this multiplication if we can write x^-\-bx+6 so that (:r+3) will be a factor in each of its terms; for, by the previous exercise, an expression is divided by (^ + 3) if we remove (;r+3) as a factor from each term. But we may say: ;i:2 4-5;r+6=;t;2 + 3;t:+2;t:+6, and by using parentheses, =;r(:r+3) + 2(;i:-f 3). Then, by removing the factor (^+3) from each term, we obtain, (:r2+5^+6)-T-(;r+3)=.r+2, which is the required quotient. As another example, let it be proposed to divide x^ -{-14:X+A5 by x-{-9. Now this can be done if we can write x^-{-14:X+4:5 so that x-{-9 will be a factor in each of its terms; for an expression is divided by (x-j-d) if we remove (x+9) as a factor from each term. But we know x'^ + Ux -\-4:5=x'^ -i-9x+5x+45, and by using parentheses , = x(x + 9) + 5 (jt: + 9) . Then, by removing the factor (^+9) from each term, we obtain, (x'^ -j-Ux-\-45)^(x+9)=x-\-5, which is the required quotient. It is noticed in each of the above cases that our process consists in breaking the given dividend into parts, so that the divisor is a factor of each part, and then removing that factor from each part. 103. We may formally state the above process thus: One polynomial is divided by a second polynomial, if the first polynomial be written so that the second polynomial is a factor in each term of the first, and this factor removed. 54 UNIVERSITY ALGEBRA. If it is impossible to write a polynomial in this manner, then that polynomial is not exactly divisible by the pro- posed divisor, and the division can be merely indicated. Thus, (x^'i-6x+2)-i-(x-\-S) is worked as follows: x^ -}-6x-i-2=x^ -\-Sx-hSx-\-2. Using parentheses, =x(x-i-S') + (Sx-\-2'). Therefore, (x''+6x-\-2}-^(x+S)=x+^~ KXAMPI.KS. Divide I. jt;2+3^+2 byj*;+2. 2. x'^ + 9x+U by :r+7. 3. ^2_|_4^__45 by ;r--5. x^-{-4:X—4t5=x^—5x-j-9x—45. Using parentheses, =x{x—5)-\-9{x—5). Removing the factor (;i^— 5) from each term gives the required quotient: (x«+4;c-45)-H(jr— 5)=5C+9. 4. ^2^3^-10 by:r-2. 5. x'^-4:x-21 by ;i;--7. 6. x^—4:X—4:6 by ^+5. X*— 4x— 45=^^+5x— 9x— 45. Using parentheses, =x{x-\-5)—9{x-^5). Removing the factor {x-\-5) from each term gives the required quotient: (x^-4x-45)-^{x-{-5)=x-9. 7. ^2-_3^_io by ^+2. 8. x^—4:x—21 by ;^;+3. 9. x^—l^x+Ab by ^—5. 5c«-145c+45=xs-5x-9;c-f45. Using parentheses, =x(%— 5)— 9(x— 5). Removing the factor (;c— 5) from each term gives the required quotient: (x«-14x+45)-J-(x-5)=^-9. 10. ^2_7^+io by ^—2. 12. 3«2_^19^+20by a + 5. II. ^2-5^+4 bya-1. 13. Aa'^ + Ua-^^ hy Aa+2. 104. Arrangement of "Work. Since division is the inverse of multiplication, we exhibit in connection with each other the arrangement of work in the two operations. nvisioN. 55 MULTIPLICATION, OR THE DIRECT OPERATION. Multiplicand, [ multiplier, first partial product, second partial product, product, X + 4: X + 9 x^+ 4x 9:r+36 x^-hlSx+SQ DIVISION, OR THE INVERSE OPERATION. Divisor. Dividend. Quotient. x+4: ) .;i;2 + 13;f+36 ( jr-f 9 J first partial dividend, x'^+ 4x first remainder, 9x-\-S6 second partial dividend, 9.r-f-36 [second remainder, In the work in division the process is as follows : We \ first arrange dividend and divisor thus : x+4)x'^ + lSx-\-S6 ( f We next divide x^, the first term of the dividend, by x, [the first term of the divisor, which gives x as the quotient. fWe now multiply the whole divisor by x and put the f product, x^+4:Xy under the dividend. We then have x+A) x^-\-lSx-i-36 (^x x''-\- 4x [by subtraction, 9jt: + 36 [ We next divide 9x, the first term of this remainder, by x^ I the first term of the divisor, which gives 9 as the quotient. fWe now multiply the whole divisor by 9 and put the Iproduct, 9;»;+36, under the last remainder. We then have x+4);ir2 + 13.r+36 (:ir+9 x'^-\- 4x 9.^+36 9.;t:-f36 I by subtraction, whence the quotient is ;r+9. 56 UNIVERSITY ALGEBRA. The student will observe that the partial products which, when combined, constitute the final product in multiplication, occur in the work in division as ''partial dividends," which, when combined, equal the original dividend. So the process of division here used consists merely in breaking up the dividend into the component partial products, from each one of which one term of the quotient is obtained. Thus the above work is merely a convenient way of breaking :r2 + 13;t:+36 into the two expressions, ("partial dividends"), (:r2+4;»;) + (9;i:+36), and when this is written ^(;f +4)4-9(^4-4) we readily obtain x-\-% for the quotient, by Art. 102. 105. We give a few more examples where multiplica- tion and division are exhibited together, so that the student may more clearly understand this manner of un- doing multiplication. (1) 2a -11 (2) a 4-9 3^ + 4 a -5 6^2_33^ ^2-1-9^ 8^-44 -5a-45 3a-ll ) 6^2-25^-44 ( 3«+4 ^_|_9 ) ^g_|_4^_45 ( ^_5 6^2-33^ a^^^a 8^-44 -5^-45 8^-44 -5^-45 Some prefer to write both divisor and quotient to the right of the dividend. Thus : (3) Dividend, 155C«— 445c432 I 3jc— 4 Divisor. 15x2— 20x I 5x— 8 Quotient. -24x432 -24x432 This arrangement saves space, and the divisor is where it is readily multiplied by each term of the quotient. DIVISION. 57 106. It is very important in the division of a polynomial by a poly7iomial that both dividend and divisor be arranged according to the powers of a commoii letter. It makes no difference whether the arrangement be according to the descending or the ascending powers of a common letter, but both dividend and divisor should be arranged in the same order. Any letter may be selected for this purpose, but the letter which occurs the greatest number of times in the given dividend and divisor is naturally preferred. If some powers of the selected letter do not occur in the dividend, then it is well to leave a blank space in the work for every such term. Thus: (1) Dividea2— ^2bya + ^. (2) Divide a^ — b^ by a—b, a^ — b^ I a-\-b a^ — b^ I a — b a^-\-ab 1 a—b a^—a^b a^j^ab-\-b^ -ab-b^ a^b —ab—b^ a^b-ab^ ab^-b^ ab^-b^ KXAMPLKS. Divide 1. x'^+Sx—AO by ^-5. 8. a^+ia—AB by a— 5. 2. x'^-{-5x—6 byji;— 1. g. a'^—4:a—S2 by a— 8. 3. x'^-\-7x—S0 byj»;-3. lo. ^2^7^—78 by a— 6. 4. x'^-\-4x—5 byjt:— 1. ii. a^ — 121 by ^ + 11. 5. x'^ — 2x—6S by Ji;— 9. 12. x'^—ax—6a'^ by x—Sa 6. ^2_^7.r— 44 by ;r— 4. 13. x'^—da'^ by x—Sa. 7. x^—A by jf— 2. 14. ;i;2— 49>/2 ^y ;r+7j^^. 15. a''-i-6ab + 9b'^ by a + Sb. 16. a'^ — 17ab+72b'^ by a—db. 17. 2;»;2— 9^+10 by 2;t:— 5. 19. 6;r2+5;t;-21 by 3;t:+7 18. 3.r2 + 2;t:— 1 by 3jf~l. 20. 9j»;2_64 by 3jr+8. 58 UNIVERSITY ALGEBRA. * 21. a^-i-d^+c^—Sadc by a + d-hc Arrange according to the descending powers of one of the letters, say a. It is important to ^eep this arrangement throughout the vvork, and to give the powers of ^ preference over those of c. —a^b—a^c —Sabc —a^b —ab^— abc —a^c-\-ab^—2abc —a^c — abc- ab^— abc^ac^-\-b^ ab^ -\-b^^b^c — abc-\-ac^ —b^c f — abc —b^c—bc^ • -_ _____ ac^ -\-bc^-\-c^ Sometimes the use of parentheses simplifies such examples : a-\-b-\-c a^ — ^abc-Yb^-\-c^ a^-Ya^{b-\-c) \a^ — a{b-Yc)-\-{b^ —bc-^c^) -a^{l)-\-c) — ^abc-^b^-Yc^ —a^[b + c)-a[b-{-c)^ a{b^—bc-\-c^)-\-b^-{-c^ a{b^-bc-Yc^)^b^-\-c^ 22. 2^4— 6^3 + 3^2__3^_^l l3y ^2_3^_^i 23. 2w4-6;;^3 + 3;;^2— 3^+1 by m'^'-Zm-^l. 24. 4:j/4-18j/3+22ji/2— 7r+5 by 2jj/— 5. 25. 45.;*:* + 18;trS + 35j»;2+4;r-4 by ^x^+lx—l. 26. a'^b'^ — b^+a^—a'^b^ by a^ -^ a'^ b ^ b^ -\- ab''- , 27. x^+y^+Sxj/—l by x-i-y—1. 28. ^2—2^^4- ^2 —^24-2^^-^2 by a—b+c—d, 29. a^ + b^'-c^—2aH^ by a^ — ^2__^2^ 30. l+x^+x"^ by x^ + l—x, 31. «5_243 by «~3. 32. 1—6^5+5^6 by l_2;t;4-^2 33. ;t:^ — 2<2^;t:^+^^ by x^—2ax-\-a^, 34. 3.rS-3 by |^2 + i^+|. DIVISION. 59 THK FUNDAMENTAI, I^AWS OF AI.GKBRA. 107. The fundamental laws of Algebra are collected below for convenient reference. A. ADDITION AND SUBTRACTION. 108. Commutative ILaw. If a succession of numbers be connected by the signs + and — the order of the numbers may be changed^ provided each number moves with its proper sigii. 109. Associative Law. If a succession of numbers be co7tnected by the signs + and — , a parenthesis preceded by the sign + ^^JF be inserted or removed, without changing the value of the expression, and a parenthesis preceded by the sign — may be inserted or removed, provided the signs + a7id — of all the numbers within the parenthesis be changed. B. MULTIPLICATION AND DIVISION. 110. Commutative Law. If a succession of numbers be conyiected by the signs X aiid — the order of the numbers may be changed, provided each number moves with its proper sign. 111. Associative Law. If a succession of numbers be connected by the signs X and -r-, a parenthesis preceded by the sign X may be inserted or removed without changing the value of the expression, and a pare7ithesis preceded by the sign -r- Tnay be inserted or removed, provided the sig?is X and -r- of all the numbers within the parenthesis be cha7iged. 112. Distributive Law. The p7'odi(ct, or qicotie7it, of the sum of several numbers by a given 7iumber equals the sum of the respective products, or quotients, of each sum- mand by the give7i number. 6o UNIVERSITY ALGEBRA. 113. Index La^vs. The product of two powers of the same number is equal to that number with an exponent equal to the sum of the exponents of the two factors. The quotient of any power of a number divided by a lesser power of the same number is equal to that 7iumber with an ex- ponent equal to the exponent of the dividend minus the exponent of the divisor. « MISCEI.I.ANKOUS KXERCISKS. 1. Show directly from the meaning of the symbols employed that a-\-{b'-'C) — a-{-b—c, a—{b—c) = a—b-\-c. Simplify the following five expressions : 2. a-Z{b-1\_a-Zb'\-1a^. 3. a-(3« + ^-[4«-(3/^-^)] + 3^). 4. \la-Kh-a)\-\l\{b-\a)^%{a-\\b-\aY)\ 1c- a r2c—Sa ( a— 2c r^ 3a4-^~|\~l 6. 4(a-f[^-|^])(i[2a-^] + 2[^-.]). 7. From x{x-\-a—2b)(^x—2a-\-b) subtract (^x-\-a){x+b) (x-2a~2b). 8. {x—y-{-z){x-{-y—z) — (x-\-y-\-2)(^x—y—2)—Ay2=} 9. Divide ^a''b^-12a^b+U^ -^la^b'' -{-4.a'^-llab^ by 10. Divide (ac+bdy — {ad^bcy by (^-^)(^-^). 11. Dividea^— 0/2— 3<^2^^2_^^_^^2 4_^4 by ^2_^^_|_2^2 12. Divide x^{a-\-l) —xy(x—y){a-^ b)—y^ib—V) by x'^—xy-\-y'^. 13. Divide (l + a)2(l + ^2>)_(i_p^)2(i^^2) bya-^. 14. \2a''x''-2{Zb-4.cXb-c)y''-^abxv']-^[ax^2(b-c)y'] equals what? DIVISION. 6l Historical Note. The way in which Greek and Hindoo science reached the Occident during the middle ages forms an inter- esting study. Greek and Hindoo thought would have been in danger of being lost, or of reaching the new nations of Europe much later than it did, had it not been for the Arabs. As original investigators in mathematics the Arabs did not excel, but they were zealous in acquiring and recording in their own language those branches of Greek and Hindoo mathematics which they were able to understand. When the love of science began to grow in the Occident, they trans- mitted to the Europeans the intellectual treasures of antiquity in their possession. Among the many terms of Arabic origin is the word ' ' Algebra. ' ' Its earliest appearance is as the first word in the title of a work by Mohamed ben Musa Hovarezmi, of the 9th century, entitled Aldshebr Walmiikabala. "Aldshebr" has passed into "Algebra." The two title- words mean "restoration" and "reduction." Thus, ^jc^ — 2a;=5x+6 passes by "Aldshebr" into x;2=5xH-2%-j-6, and by "Walmukabala'' into 5c2=:7%+6. About the beginning of the 12th century, Arabic MSS. began in Europe to be translated into Latin. Thus, Abilard of Bath translated the works of Ben Musa Hovarezmi. This was done also (along with many other mathematical works) by Gerhard of Cremona. In this way Algebra, with its rules for solving linear and quadratic equations, was transplanted to Europe. It is interesting to observe to what great extent the progress of [ Arithmetic and Algebra has been dependent upon the apparently \ small matter of the kind of notation adopted. To peoples not familiar \ with the Arabic notation, arithmetical calculations with large hum- f bers were impossible or insufferably tedious. In Algebra, Diophantus and the Hindoos used symbols (different from ours), but to a much less extent than is done now. The general introduction of new sym- ; bols has usually been slow. Thus, the mode of designating powers I by indices wab suggested by Oresme in the 14th century, but remained lunnoticed ; it was brought forward again by Simon Stevin in Holland l(died 1620), but was not appreciated until the time of Wallis and Newton. The development of the notion of a general exponent (negative, fractional, incommensurable) first appears in a work of . John Wallis (published in 1665) in connection with the quadrature of 1 curves. The practice of denoting general or indefinite quantities by [ letters of the alphabet was introduced by Francois Vieta (died 1603), \ while Thomas Harriot (died 1621) first used for that purpose small 62 UNIVERSITY ALGEBRA. letters in place of the capital letters of Vieta. Harriot also suggested the use of the symbols > and < in the sense now current. The earliest book in which + and — are found is the arithmetic of John Widman of Leipsig, 1489, but they did not come into general use before the time of Vieta. Our sign for equality was invented in 1540 by Recorde, the author of the first English treatise on Algebra. To William Oughtred we owe the symbols X for multiplication and :: to designate proportion. Descartes in 1G37 denoted multiplication by a dot. The sign of division, -f-, was used by Pell in 1630; brackets were employed by Girard in 1G29 ; the sign v by Rudolf in 1536. The custom of using a letter to denote either a positive or negative number did not become familiar to mathematicians until the time of Descartes about 250 years ago. ' 'The establishment of the three great laws of Algebra was left for the present century. The chief contributors thereto were Peacock, De Morgan, D. F. Gregory, Hankel and others, working professedly at the philosophy of the first principles ; and Hamilton, Grassman, Peirce and their followers, who j:hrew a flood of light on the subject by conceiving Algebras whose laws differ from those of ordinary Algebra. To these should be added Argand, Cauchy, Gauss and others, who developed the theory of imaginaries in ordinary Algebra." To show the appearance of mathematical work before the intro- duction of the common symbols we give the following expression taken from Cardan's works (1545) : Bj v. cu. R 1087. 10 I »^ R z/. cu. Bs 108 m 10, which is an abbreviation for ' 'Radix universalis cubica radicis ex 108 plus 10, minus radice universali cubica radicis ex 108 minus 10." Or, in modern symbols, |^V108+10-l^Vl08-10 Here is a sentence from Vieta's work (1615) : Et omnibus per E cubum ductis et ex arte concinnatus, E cubi quad. + Z solido 2 in ^ cubum, acquabitur B plani cubo. This translated reads: Multiplying both members ("all") by -£* and uniting like terms, E^ E^-\r2-^=B^. CHAPTER VI. MATHKMATlCAIy INDUCTION. 114. In the following pages we shall occasionally [desire to use a method of demonstration known under ^ the name of Mathematical Induction. We shall there- i fore illustrate and explain the method here, so that it will not be necessary to stop on the method itself when occasion arises for its use. We shall first illustrate the method by an actual ex- I ample of its use, and then the description of it will be I intelligible. 115. Suppose the statement made that the sum of any I number of consecutive odd numbers is the square of the lnumber of numbers which are added. We readily see that 1 + 3= 4=2^, 1 + 3 + 5= 9=32, 1 + 3 + 5 + 7=16=42. In the first case there are two consecutive odd numbers ' added together, and their sum is 2 ^ . In the second case there are three consecutive odd numbers added together, J and their sum is 3^. In the third case there are four [ consecutive odd numbers added together, and their sum ;is 42. Now, although at present we do not know absolutely that the statement we started out with is true beyond the sum of /oar consecutive odd numbers, still from what we have observed we strongly suspect it is true in any case. ; "Now let us for a moment assume that the statement we \ are considering is true when 71 consecutive odd numbers 64 UNIVERSITY ALGEBRA. are added together; then, since the >^th odd number is easily seen to be 2n—l, we have 1 + 3 + 5 + . . . + (2;^— 1) = 7^2. Add 2n-\-l to each member of this equation and we obtain 1 + 3 + 5 + ;. . + (2;^-l) + (2^ + l) = ;^2+2;^ + l. But, as is readily seen by actual multiplication, .'. 1 + 3 + 5 + . . . + (2;^ + l) = (;^ + l)2. The left member of this equation is the sum of n-}-l consecutive odd numbers, and therefore if the statement we are considering is true when n consecutive odd num- bers are added together, it is also true when n-\-l con- secutive odd numbers are added together. But we know that the statement is true when four consecutive odd numbers are added together; hence, from what has just been shown, it is true when Jive con- secutive odd numbers are added together. Being true when ^ve numbers are added together, it is also true when SIX numbers are added together, and so on. Hence the statement is true universally. 116. The method just used may be divided into three parts: in the first part we ascertain by observation or trial that the statement we are considering is true in some simple case ; in the second part we prove that if the statement is true in any one case it is also true in the- next case ; in the third part we deduce that the statement is true for every case after the one ascertained by obser- vation or trial. 117. The student may have heard of the word induction as used in natural science, but the use of the word there is quite different from its use here. In natural science some law is observed to hold in a number of instances, and from this it is assumed to hold gener- ally. The more cases in which the law is observed to hold, the- MATHEMATICAL INDUCTION. 6$ stronger is the belief in the fact that it holds generally; but we cannot be absolutely certain that the law holds in any case except those that have been examined, and we can never arrive at the conclusion that it is a necessary truth. In fact, induction as used in natural science never amounts to absolute demonstration, but mathematical induction is just as rigid as any other process in mathematics. The important difference between induction as used in natural science and mathematical induction consists in the fact that in natural science the second part of the process as described in Art. 116 is always lacking, and it is this part that enables us to give rigid f demonstration to the fact we are considering. 118. When some statement is found by trial to be rue in several successive cases it is very natural for tudents to infer that the same statement is true univer- |sally. As this is one of the most common mistakes of I5 students, an illustration or two may well be given here. Consider the expression x^-}-x-\-17. If x= 0, then x'^-\-x+n= 17, a prime number. If x= 1, then x^ f ^-f-17= 19, a prime number. If x= 2, then x'^+x-\-Vl= 23, a prime number. If x= 3, then x'^-{-x+Vl= 29, a prime number. If x= 4, then x'^+x+Yl= 37, a prime number. If x= 5, then x'^+x+n= 47, a prime number. If x== 6, then x^+x+n= 59, a prime number. If x= 7, then x'^+x+17= 73, a prime number. If x= 8, then x'^+x+Vl= 89, a prime number. If x=^ 9, then x'^+x+n=101, a prime number. If x=10y then x^+x+n=127, a prime number. If the student tries values of x greater than 10 he will till find that the value of x'^+x+Vl turns out to be a prime number, and the conclusion is nearly irresistible that this expression must always represent a prime num- ber for any positive integral value of x. Just before this conclusion is reached the student should give x the 5— u. A. 66 UNIVERSITY ALGEBRA. value 16, whicli makes the expression x'^+x+n=2Sd, which is the square of 17, and therefore not a prime number. A similar illustration may be given with the expres- sion x'^-\-x-\-41, which will be found to represent a prime number for any positive integral value of x less than 40. Again, the expression 2jr^+29 is a prime number for any positive integral value of x less than 29. KXAMPI,:^S. Prove by mathematical induction : 1, 12+22 + 32 + . . .+n''=^\n(n + l)(2n + V), 2. 2 + 22+2^ + . . . + 2"=2(2'^-l). CHAPTER VII. FACTORS AND MUI^TIPI,:^. 119. A definition of factor has been given (see Art. 7), , and we have already learned that an expression may have \ several factors. For example, the different factors of lO^t:^ are 2, 5, 10, x, 2xy bx, lOx, x'^ , 2x^, 5x^. Of these factors, 2,5, and x may be called Prime, because they cannot be further factored. The expression 10;i;^ contains the prime factor x twice ^ so all the prime factors of IOjt^ are 2, h, x, x\ and as any expression equals the product of all its prime factors, we have, 10:^2 = 2x5^^. When an expression is written as the product of all its prime factors, it is said to be Resolved into its Prime Factors. Resolve the following eight expressions into their prime factors: I. Z^x'^y^, 3. 38a^V^ 5. ISSr^^^ 7. I^Zluv'^w^. 7,. 150a^<^2 4. 51;;^V^ 6. ^I^abn, 8. Ih^x^yz^, 9. What are the different prime factors of 15xyzu^v^ , and htc^v^. 16. 60^z;, ^hxz'^v, dOy^zuv, dxzu'^v, and 30.rz/^. 17. x^ — 1, x^ — 1, and x—1. 18. x^—x—6, x^+x—2, and x'^ — 4:X—12. 19. 4^<^(^2__3^^^2/^2) and 5^2(^2 4.^^_6^2)^ 20. ^— y, ^+JK, .^^— j»/^, and x^—y^. 21. xs— 4.;i:2 4-3;r, .r*+Jt;'^— 12.r2, .;r5 + 3x4— 4.;i:8. FACTORS AND MU-LTIPLES. 99 22. x^—7x-}-6, x'^—5x—6, and x^ — 1. 23. ;r2 +7^4-6, x'^-h6x—7, and x^— 6jt:— 7. 24. 5(^2-2^^), 10(^^ + 2^2)^ and lo^aH^-Ad^), I,. C. M. OF EXPRESSIONS NOT EASILY FACTORED. 163. When we wish to find the ly.C.M. of two ex- pressions not easily factored, we first find the H.C.F. of the two expressions by one of the methods alread}^ given. This H.C.F. is of course 07ie factor of each of the two given expressions, and the other factor is obtained by dividing each expression in turn by the H.C.F. Now represent the two expressions by X and V, their H.C.F. by /% and their L.C.M. by Af, and suppose X=Fu and V=^Fv. Now, since /^ is the H.C.F. of X and y, u and v contain no common factor. Therefore M=Fuv. This equation may be written in either of the forms : M=Fuy = Xy M=-Fv-p = Yy -. FuFv XY Therefore, the L.C.M. of two expressions is found by dividi7ig either of the expressions by their H.C.F. and multiplyiyig the quotieiit by the other expression, or the L.C.M. of two expressions is fou7id by dividing the product of the two expressions by their H. C.F. Again, from either of the last three equations, by mul- tiplying both members by F it is evident that MF=XY, i. e., the product of the H.C.F. and the L.C.M. of two expressions is equal to the product of the two expressio?is . lOO UNIVERSITY ALGEBRA. Let us find the L.C.M. of Qx^—Ux^y+2y^ and 9jt:3— 22x)/2— S^/S. ^Q. first find the H.C.F. of these two expressions as follows: 3 9x;3— 225cy*— 8;/3 ) ISx^— SSx^jj/ -j. 6>/3 ( 2 18x3 — 44xy2— 16y3 —11;/ ) — 33x^;t/+44x>/2+22;/3 3^2 _ ^xy — 2y^ Sx^—4xy—2y^ ) 9x^ —22xy^—8y^ ( Sx-j-iy dx^—12x'^y— Qxy^ 12x^y—lQxyZ—Sy^ 12x^y—lQxy^—S y^ From this work we see that 3^^ — 4^^;^ — 2y^ is the H.C.F. of the two given expressions, and if we divide each expression in turn by this H.C.F. we may obtain the other factors as follows: dx^—4:xy—2y^ ) Qx^—llx^y +2y^ ( 2x—y 6^3 — Sx^y — 4^xy^ — 3X2;/+I^j^2_|_2y3 — Sx-^y-\-4: x y^^2y^ The second expression has already been divided by the H.C.F. and the quotient found to be 3j^^4~4^- Hence we have 6x^—llx2y-{-2y^ = {dx2—4xy—2y'^){2x—y), and 9x^—22xyZ—Sy^ = {3x^—4:Xy—2y^){3x+4y). Now we have the two given expressions factored, and from these factors we can readily write down tne L.C.M. of the two given expressions. Plainly, this L CM. is (3x2— 4;cy— 2;i/2)(2%— ji/)(3x+4>/). KXAMPLKS. Find the ly.C.M. of the following expressions: 1. x^-\-%x'^-\-\^x^-Vl and x^ +1x'^ ^^x—Xh, 2. x^-V^x'^^X^x+Vl and jt^ + S.^^— 4;r-12. 3. x^'\r"lx'^\-nx—\h and x^ -\-Zx'^—4.x—Vl. 4. bx'' + llx-j-2 and 15.;r4 +48.^3+9.^2. 5. 4:X^-10x^+4x+2 and Sx^-2x^-Sx+2. ' 6. 6x'^-\-llxj/ + 4:j/'^ and 4:x'^ — 8xj/'—5y^. 7. x^+x'^—Bx+S and x^ — Sx'^-j-Sx-l. i(^y ^ FACTORS AND MULTIPLES. lOI 8. x^+xy'^-\-2y^ and x^ -\-x'^y-{-4y^. 9. x^ + 2x^ +x^ + Sx'^ ■i-lQx+^ and x^—4:X^+x'^—4c. 10. ;i;5_4^3 4.^2_4 ^nd x^-^r'^x'^ — ^x—ll. 164. If we wish the ly.C.M. of more than two ex- pressions, we first find the ly.C.M. of any two of them, and then the L.C.M. of this result and the third expres- sion, and so on until all the expressions are used. The result is the ly.C.M. of all the given expressions. Exampi,e;s. Find the ly.C.M. of the following expressions: 1. 2x'^-\-2x—l, Sx^—Ax-i-l, and 2;»;3-3;i:+l. 2. ^3+2:1:24-9, x3_8^+3^ and x'^-Sx+l. 3. x^—2x'-2, x^—Ax^+S, and x^-'Sx'^+2, 4. dx^+2x-i-l, Sx^—8x'^ + l, and x^Sx+l. 5. x^-Sx+2, x^-ex'^ + llx—Q, and x'^-dx+Q. CHAPTER VIII. , FRACTIONS. 165. We have already used the fractional form 7 as another way of writing a-r-b.so that 7 is an expression of division. We have already learned that in any case of division the quotient multiplied by the divisor equals the dividend, or, in the language of fractions, precisely the same thing may be written, Quotient X Denomznator= Numerator. 166. From this equation it is plain to see that if the denominator remains unchanged, multiplying the numer- ator by any number multiplies the quotient by the same number, and dividing the numerator by any number divides the quotient by the same number, or, as it is more often stated, tnultiplying the numerator by any num- ber multiplies the fraction by that number, and dividing the nuTnerator by any number divides the fraction by thai number. 167. Again, from the same equation, Quotient X Denominator^ Nurnerator^ it is also plain that if the numerator remains unchanged, multiplying the denominator by any number divides the quotient by that number, and dividing the denominator by any number multiplies the quotient by that number, or, as it is more often stated, multiplying the denominato? by any number divides the fraction by that number, and dividing the denominator by any niimber multiplies the fraction by that number. FRACTIONS. 103 168. Once more, from the same equation, Quotient X De?iominator^= Niimerator, it is plain that if the quotient remains unchanged, multi- plying the denominator by any number multiplies the numerator by the same number, or, stated in another way, viultiplying both 7iumerator and denominator by the same 7i2imber does not alter the value of the fraction. It is also evident from the same equation that dividing both numerator and denominator by the same number leaves the quotient unchanged, or, stated in the usual form, dividing both numerator a?id denominator by the same number does not alter the value of the fraction. 169. When the numerator and denominator of a frac- tion have no common factor the fraction is said to be in its Lowest Terms. Therefore, to reduce a fraction to its lowest terms we divide both numerator a7id denomi7iator by their H.C.F., for by so doing we obtain a fraction in which the numerator and denominator contain no com- mon factor. EXAMPLKS. Reduce the following fractions to their lowest terms : (3.r-2y)^-(2.y+2j/)^ aM;^^M- 11^ + 6 I. 2. ;t:2-3jr-70 ' 30^=^- 19^2 _^1 {a + by-{c-Vd y ^^-10^2+26^-8 {a-^cy-{b^dY ^' a3~9^2_^ 23^-12* 3;j;3_e^2_|_^_2 8^^— lO^2__i0^_3 I 4. — z^r-n-v^ — ^o x^-lx-\-^ ' 6^4 -22^^ + 31^2 _23^_7 5- x^-x-Q ' "• 3^4 + 14^3-9^+2' x^-i-a^ 1 + 2^-3^2 6. -^r-^ —7;' 12. x^+2ax+a^ ' 1-3^-2^2+4^3 I04 UNIVERSITY ALGEBRA ADDITION OF FRACTIONS. 170. If two or more fractions have the same denom- inator, the fractions may be added by adding the numer- ators and placing the sum over the common denominator; but if the denominators are not the same we must mul- tiply the numerator and denominator of each fraction by such a number as will make all the denominators the same, and then add the numerators and place the sum over the common denominator. 171. The process of changing the numerators and denominators of fractions so that each fraction shall pre- serve the same value it had before while the denomina- tors are all made alike is called Reducing to a Common Denominator. For example, suppose we wish to add -^ and — . We iw mn must reduce to a common denominator, which of course must be a common multiple oi rn^ and mn. Any common multiple will do, but the lowest common multiple is preferable. The lowest common multiple is plainly m'^n\ hence we must multiply the numerator and denominator of the first fraction by n, and the numerator and denominator of the second fraction by m. We then have abn 4mx m'^n m'^n Plainly, then, the sum of the two fractions is adn -^Amx fn'^n Hence we may write the equation, ad 4:X _abn -i-4mx ni^ mn Tn'^n FRACTIONS. lOS If any number of fractions are to be added together we pursue a similar method. Therefore, to add several frac- tions together, i^educe all Hie fractions to a conimoji deiioTti- inator, preferably the lowest commoji denominator, add the resulti?tg numerators and place their sum, over the common denominator. KXAMPLKS. Add the following fractions : a-\-b J a-{-b 2. 3. 4 5. 6. x'^ -j-y'^ x^-y^' 1 and x{x—b^ , and x'^—y'^ x—y 2 , 3 a-^b-\-c a + b a-\-c a + 1 2a + S and 3^ + 4 ^2+5a-f6' (^ + 2)2' ""^ (a + Sy 2a+6 , 2^ + 7 a and ^2+7^ + 10 b . 1 ^3_^3> ^2+^^_^^2' and a — b X ^* x+a' X ^ x'^-\-c'^ — ,-7, and -^- 1 x+b x^-\-ax-}-ac 8. x^+a'^ x-\-a x^-i-a'^x'^ + a^' x^-hax+a ;, and x—a x^—ax+a^ a'^-bc b''- bc ac -ac - c^ and — -ab ab 10. ^2— 3^+2' ^2— a— 2 , and SUBTRACTION OF FRACTIONS. 172. If one fraction is to be subtracted from another ^and the fractions have the same denominator, we may [subtract the second numerator from the first and place I the remainder over the common denominator; but if the I06 UNIVERSITY ALGEBRA. denominators are different, we must first reduce the frac- tions to a common denominator and then perform the subtraction. KXAMPI^BS. ^ 2a-\-l ^ , Sa-i-2 1. From — o — take — ^— • 2. From ^^ take ^^. a a 3. From ^±^ take ^^. x—y x-\-y ^^-f-1/^ x~\-\ 4. From -::^r^^ take ^^ 5. From — — -^ take 0. From ^ . ^ — r^ take Jtri/ x"^ V ~\~ xv^ 7- From ^^ take -^,^^- 8. From ^!d^Z! take ^^%. 9. From £!±^Z! take £!z:^2^Z!. 10. From 7 ^7 r; take {x—a^ia—b) (x—b)(a—b) II. From » , . , ,. — ^ — 7 take x'^+ (a-\-b)x-\-ab x'^ -^(a-^c)x-\-ac In the following examples perform the additions and subtractions indicated and express the result as a single fraction in its lowest terms : 1.1 1 12. x+l^x+2 x+Z x+1 x+2 x^Z ^3- /^ I o^/<^_LQ^"^/ (^+2)(;c + 3)^(^+l)(^+3) (;^+l)(^+2) fa b c\ fa 3^ hc\ ^5- I2 + 3-4J+I4-T-6/ ^ a-^b , b—a iab 16. 7+ FRACTIONS. ^ 107 2 3 17- a—b a-\-b a'^ — b'^ 2^+1 3jt;+2 4;r4-3 (:r-l)(jr-2) (jt:-2)(jt:-3) (;r-3)(;t:-4) a <^=^* b b We may divide a fraction by multiplying the denominator; ac , a<: hence, S^'^^M ^ ^. r a c ac , Therefore, t^^— t:?* ■^ ' b d bd Hence, to multiply two fractions together, multiply the numerators together for a new numerator^ and the denom- inators together for a new denominator. I08 UNIVERSITY ALGEBRA. 174. Suppose we wish the product of three fractions, - . , ace as for instance, t x i x t* oaf We may multiply the first two fractions together, as ac just explained, giving — , and we can then multiply this result by the third fraction by the method just explained, . . ace ^. ^ ace ace givmg^. Therefore, ^x^x^=^. As this process may be extended to any number of fractions, we have : The product of any mimher of fractions is found by multiplying all the numerators together for a new numerator and all the denominators together for a 7iew denominator. The result should be reduced to its lowest terms if it is not in its lowest terms already. 175. Instead of actually performing the multiplications it will frequently be best to indicate them by using paren- theses, for sometimes in the result the numerator and denominator will contain a common factor, which can be struck out and thus save the trouble of multiplying by these factors. For example, if we wish the product of a+x b ^ c — X — T-, —, — , and — -— , b c-j-x a-\-x we may write the product thus : b{a-\-x){c—x^ b{c+ x^{a-\- x) Here the numerator and denominator contain the com- mon factor b{a-\-x^, which being rejected from both numerator and denominator leaves the result simply c — X c+x FRACTIONS. 109 KXAMPLKS. Find the product of the following fractions : 4 5 6 7 8 9 10 1 + ^2 a + ^2' ^3 — ^l 2ab ' a(a + 2d) /,4 and and and 1-a 1 ^2 a^— 2^;»;H-;«;2 ^^2 ^34-^3' and 4:a{a + b) and 2«2;,;2_|.^4 a2+ji;2 and (3^3;i:+ajr3 a^— ^^ «3 + 3a2^4-3a^2_j.33 a—b a'^ + b'"' and ^2_9jr-f 20 ^2__i3^_|.42 x^ — 5x a 'a+b ^2 and =• ;i:— 7 2~_y2 ^-1 ^^, and (^-•^>' ;r2— 2;t:+l ^^2-9 ' and x^ -{-y^ :^;2-4jr+3 DIVISION OF FRACTIONS. (t C 176. We are to divide - by -. o a We may write the quotient in the form of a fraction, where the numerator is itself the fraction -7 and the c ^ denominator is the fraction ^. a Hence, —= quotient. 1 no UNIVERSITY ALGEBRA. Let US multiply both numerator and denominator of this fraction by bd. We know this will not change the value of the quotient. a . , abd . Xbd=—r-=ad. o c , , bed , -■^bd-=-^-=^bc, d d Hence, quotient=-r— . T;herefore, a c ad b ' d be This result may be obtained by 7nultiplytng the fraction ^by the fraction — , which last is the divisor inverted. b e Hence, to divide one fraction by another, invert the terms of the divisor and multiply. EXAMPLES. I. Divide f!±2^J^^ by ^-^^ +-^^+ — -• ,. Divide £!±^^^£! by _£-^4_. x-\-y — z x'' — lxy+y''—z^ 3. Divide ^^ by ^^-^ 4. Divide -^^ by ^-3- 5. Divide ^;±^ by ^±^. 6. Divide o , o n — a2 — ^y — TTT-' a^-i-za-^-l — b^ ab+1 ^. . -, 1 , 1—a 7. Divide z ^ by yj-. — :-^- 1—a^ (l-]-ay FRACTIONS. Ill 8. Divide 2 — A2 ^y r^xh ' a^ — b^ a-\-o ^. ., a^ — b^-\-a—b. a—b 9- Divide ^^-^^^^ by -^. ,0. Divide ^^^ by :^-±^. {x—y}^ — 2^ x—y-\-2 II. Divide T — 77 by ^2-^2 12. Divide 1 — by x—l l-\-ax ' MISCKI^IvANKOUS FRACTIONS. 177. We may take an integral expression (/. e., a monomial or polynomial in which there is no fraction involved) along with one or more fractions, giving us a form partly integral and partly fractional. Such expres- sions are sometimes called Mixed Expressions or mixed numbers. Mixed expressions may always be reduced to the form of fractions b}^ writing the integral part in the form of a fraction with a denominator 1, and then per- forming the indicated operations. For example, suppose we wish to express x^—y^ I in the form of a fraction. We write the expression thus : x'^ +y'^ xy 1 x'^—y'^ Reducing to a common denominator and adding, in the x^—y^+xy usual way, we get ^ — ^ -^• This process is called Reducing Mixed Expressions to Fractions. 112 UNIVERSITY ALGEBRA. KXAMPIvKS. Reduce the following mixed expressions to fractions : 3 11 1 2. x-{-a 4. x+y-\ 1 — 6. ax+by a X y ' ax— by , r , b c a , T , , 1 7. a + b+c -r 9. a + b+c-{ r^-r' a b c a+b-^c S. x'^+x+l+~'^' 10, x'^—ax+a^+ ^^ x+1 ' x—a 178. When the given mixed expression has only one fraction, as in the example x^-{-y^+ » o > we reduced x^ — y^ to a common denominator by multiplying numerator and x^ ~\- v^ denominator of — —^ by jr^ —y'^ , the denominator of the fractional part of the given expression. This amounts to multiplying the integral part by the given denominator and adding the numerator, exactly as in Arithmetic. Since the fraction was produced from the mixed ex- pression by multiplying the denominator by the integral part and adding the numerator, it follows that to reverse this process and go back from the fraction to the mixed expression we would divide the numerator by the denom- inator as far as possible and write the quotient for the inte- gral part and the rernainder over the deyiominator for the fractional part, which again is exactly as in Arithmetic. KXAMPI.BS. Change the following fractions to mixed expressions: x^ — x'^y-\-xy'^—y^ x^-\- Zax'^ -\- Sa'^x-\-a^-{-a'^ x-\-y ' x-\-a x'--\-ax+a'^ x^+a'^x'^-^a^-^x+a x'^+a ' x'^ -\- ax -i- a^ FRACTIONS. 113 5. 6. 10 r:2 — 1/2 7. ■JV +-2'^ ji; 2— _y2 8. x+y x-\-y 9. 10. 4a 179. By combinations or repeated applications of the preceding processes we are able to deal with more com- plicated cases than have yet been given. For example, let us take the fraction x-\-a x—a X — a x-\-a The numerator x-^a x—a x — a x-{-a x-\-a x—a_{x-\-a)^—{x — a)* x—a x + a x^—a^ _ [x^ + 2ax-^a^)- {x^ -2ax-\-a^) ~" x^—a* 4ax The denominator = x-\-a x—a x — a x-\-a _ {x-{-a)^-\-{x—a)^ x^—a» _ {x^ + 2ax + a^)-\-{x^-2ax+a^) x^—a^ x^-a^ Therefore the original fraction _ Ux ^ 2(x^ .a^) x^—a^ iax x'^—a^ A^ax 2ax ~2(x^-\-a2)~x» + a^ This final result is much simpler than the fraction we started with, so this kind of work may be called Simplifying Complex Fractions or Expressions In- volving Fractions. 8— U. A. tI4 UNIVERSITY ALGEBRA. KXAMPIy o~~o' ^^ would by the previous process combine each of these expressions into a single fraction as follows: 2^ a 1 6^2 + 4^ + 3 and 2'*'3^4~ 12 a l_3a-2 2 3~ 6 6^2 + 4^ + 3^ 3^-2 [a* a l\[a 11 6^2 + 4^ + 3, Therefore. [_+-+-| [---J =_^^-X 72" FRACTIONS. 117 But we may also multiply these expressions together in the same way as integral expressions are multiplied, as follows: 1 2'^3"^4 a 1 2 3 a^ a 1 ~"6"~9" "12 a^ a a 1 T"^ 8~9~ "12 The terms of this result may be combined, after reducing to a common denominator, 72. Combining terms, a^ a a^_\ _ 18g3 + 9^-8^— 6 _ 18^^ +^-6 T'^8~9~12"" 72 ~ 72 which is the same as was reached by the other process. As another illustration, let us divide -^ ttft-^-^ by o— s- o do 4 o J The work may be arranged as follows: a \\a^ 13^2 1/^_^_1 3~2;"6" 36" "^iVT 3~2 "T a^ 1 --TT +7 9 "^6 z 1 5 + 4 The terms of this quotient may be combined, giving 3^2_2^— 3 6 which is the same result as would be obtained by reducing both divi- dend and divisor to single fractions and proceeding by the method already given for dividing one fraction by another. Il8 UNIVERSITY ALGEBRA. KXAMPI,E)S. By the method used in these two illustrations work examples 1 to 9. 1. Multiply ;t: 2 H — by ;t: 2. Multiply ^+1+2 by 1+^- X^ X 1 X X 3. Multiply 1— r— I by -r- ^ '^ a b ab a 4. Divide ■g—y2- + i2 + l-3^ by 2-3- /I 1\2 11 5. Divide(-+^)-lby-+^-l. ^ ^. .^ x^ 1 ^ x^ 1 6. Divide jg-^ by -J---. ;i;2 4 1 1 7. Divide ;»; 3 — o-+-^"~o ^byjr+ — 27 3 8. Divide 8x^-j — 3 by 2x-i-— and multiply the result by^+i. 9. Multiply ;ir^ ^ by ;rH — and divide the result by X . X x-^y y-\-2 z+x ^ ' • (^-^)(^-j^) (^-ji/)(^— ^) {y—2){y—x)~''^ Simplify each of the following : 12. 13 x^^^-l x^- ~1 x^- X X— -1 x^-1 * X- -X V -1 x'' x^ ■1 1_1 14. a+ CHAPTER IX. POWKRS AND ROOTS. 181. The process of raising any number or expression to any power is called Involution. 182. In Art. 121 we learned that the product of any number of factors, each of which is a power of the same number, is equal to that number raised to a power whose i?idex is the sum of the indices of the factors. In Art. 122 we learned that the rth power of the nth power of a num- is equal to the rnth power of that number. In Art. 123 we learned that the n th power of the product of any num- ber of numbers is equal to the product of the n th powers of those numbers. Also that the n th power of the quotient of two numbers equals the quotient of the nth powers oj those numbers. Thus we may raise to any desired power, First, a power of a number; Second, the product of several numbers ; Third, the quotient of two numbers. These three cases include every monomial that can be proposed. Hence any monomial can be raised to any required power by the methods already given. 183. As any even power of a monomial having a — sign is the product of an even number of subtractive terms, it follows that any even power of a monomial having a — sign is a monomial having a + sign or no sign. Thus, (-2)2 = +4, (-3)4= +81, (-2^^2)6=, 64^6^1 ^ etc. I20 UNIVERSITY ALGEBRA. 184. Again, as any odd power of a monomial having a — sign is the product of an odd number of subtractive terms, it follows that any odd power of a monomial having a — sign is a monomial having a — sign. Thus, e;xampi,e)s. Write down the values of the following expressions : /2£2£\8 /5£y\3 / 4^2^ \^ ^* KhmrV ^' \ 2uv )' ^* V u'^vzaV' ^' \2^j)' ^' \ 2uvy' ' [j^y \y)' ^°- V^jrJ Kbyz)' ^^' \2x''y) ' \2xyV' ^^ \2xz) *• VW' ^^- \xyV ' V rj/V* SQUARE OF A BINOMIAI,. 185. In Art. 84 we found that (^a + dy = a^+2ad+5\ or, stated in words, the square of the sum of any two num- bers is equal to the square of the first, plus twice the product of the two, plus the square of the second, 186. In Art. 84 we also found that or, stated in words, the square of the difference between any two numbers is equal to the square of the first, minus twice the product of the two, plus the square of the second. The two statements in italics are so important and are used so often that they should be thoroughly familiar to the student. POWERS AND ROOTS. 12 1 KXAMPLKS. According to the two statements in italics, write down the square of each of the following binomials : 1. 40-3. 5. 10x+:y. 9. (Sad^y-\-2aH. 2. x^—3y^, 6. 2x^+^2^ 10. (Sx^y-i-CBxy, X a ^ 2xyz , 2rst m ^^o / 1 \ ^ CUBK OF A BINOMIAI,. 187. In Art. 84 we found that Multiplying each member of this equation by « + 3, we get (a + ^)^ = ^3 + 3«2^+3a^2 + ^3^ The student should actually go through the work of multiplying the second member of the jQrst equation by a-\-b to see that this result is correct. As a and b may stand for any numbers whatever, we may say that The cube of the sum of a7iy two numbers is equal to the cube of the first, plus three times the square of the first mul- tiplied by the seco7id, plus three times the first mtdtiplied by the square of the secojid, plus the cube of the second, 188. In Art. 84 we found that ia—by^a'^—2ab^-b'^. Multiplying each member of this equation by a—b, we get {a-bY^a^-Za''b-\-Zab''-b^, As a and b may stand for any numbers whatever, we may say that The cube of the difference of two numbers is equal to the cube of the firsts minus three times the square of the first 122 UNIVERSITY ALGEBRA. multiplied by the second, plus three times the first multiplied by the square of the second, minus the cube of the second. The two statements in italics are so important and so frequently- used that they should be thoroughly familiar to the student, KXAMPI^KS. By means of the two statements in italics, write down the cube of each of the following binomials : I. Zx-\-^' 7. uv^ — ^. 13. — h-- 3 uv^ ^ a a 4-1+4 ^0.4-1 .^^'^' -'^' a b ' 4:X^ ' 1x In 5. -^ II. — : + l. 17. 2^2 ab ac rst a ^ a b Zx 20/^3 ^2 22 6. -+- 12. -^ — 18. — b a y X 2 a 189. In Art. 92 we learned that the square of any polynomial may be written down directly by writing first the sum of the squares of each of the tenets of the given polynomial, and to . this adding twice the product of each term by each term that follows it in the given polynomial. In applying this method it must be remembered that the letters we have used may stand for negative as well as positive numbers, and in those terms which are made up of twice the product of each term by each term that follows it, the rule of signs must be observed. By the method just explained the square ot a—x — z is ^2 +;i;2 -i-2'2 — 2a;j; — 2a2+2x2, the last term having the sign + because the two terms multiplied together to produce it have like signs. POWERS AND ROOTS. 1 23 KXAMPI^KS. Write down the squares of the following polynomials : 1. x^-Sx'^ + Sx-l. 9. a^ + C^dy + iScy-x. 2. x^ +Zx'^y+2>xy'^+y^, 10. xy+ab—c'^+2'^. 3. a-\-b-\-c--d—e. 11. abc—xyz+x'^+y'^. 4. 2« — 2b— c — d—e. 12. mrs'^—uv+st—w. 5. x+y+z—a—b—c. 13. (a^)^— ;»;2^+(2;t:)2+j/. 6. u+2v-\-Sw—4:X, 14. a + 2:r+^;r+2a;ir2. 8. w + 2r— ^^+/. 16. a^;i;2— e^z^2_|.(;^^)2^ ROOTS OF MONOMIAI^. 190. The process of finding a number or expression when a power of that number or expression is given is called Evolution. The number or expression found by evolution is called a Root of the number or expression given. 191. As there are square, cube, fourth, fifth, etc., powers, so there are square, cube, fourth, fifth, etc., roots. The square root of a given expression means that expression which squared will produce the given ex- pression. The Tzth root of a given expression means that expression which raised to the ^th power will pro- duce the given expression. For example, 2^ = 8, there- fore the cube root of 8 is 2; 2^ = 16, therefore the fourth root of 16 is 2, etc. 192. A root is indicated by the sign V , called a Radical Sign. A horizontal line usually extends from the upper end of the radical sign over the expression of which the root is to be extracted. See Art. 27. 124 UNIVERSITY ALGEBRA. To indicate what root is to be extracted, a small figure called the Index of the root is placed in the angle of the radical, except in the case of the square root, in which the index is not used. Thus, the square root of 16 isjndicated by 1/16, the cube root of 8 is^ indicated by 1^8, the ^th root of a is indicated by i/ a, A letter may be used as the index of a root. Thus, ly a means the ;^th root of a, that is, a number which raised to the ;^th power will produce a. 193. We must notice one important distinction be- tween raising to a power and extracting a root. If we have given an expression to be raised to a given power, we obtain only one result ; but if we have an expression given to extract a given root, we may sometimes obtain more than one result. For example, 5^ = 25, hence_we say l/25=5; but also (—5)2 = 25, hence we say l/'25=— -5. It appears thus that there are two numbers, +5 and —5, either of which is a square root of 25. The two results are often written together by means of the double sign ±. Thus, l/25=±5. 194. If any number be raised to any even power, that number is used an even number of times as a factor, and therefore the result must be positive; but this same result can be obtained by raising to the same power as before the original number with its sign changed. Therefore, any even root of a positive number is either pos- itive or negative, 195. If a number be raised to an odd power, that number is used an odd number of times as a factor, and POWERS AND ROOTS. 12$ therefore the result is a number of the same sign as the one given. Therefore, any odd root of a number has the sam.e sign as the number itself. 196. If any number be raised to an even power, the result \^ positive. Therefore, there is no positive or nega- tive number which raised to an even power will give a negative result. Therefore, we cannot find an even root of a negative number. An even root of a negative number is called an Impossible or Imaginary Number. 197. To find any root of any expression we naturally look to see how the corresponding power was obtained, and then go through the work backward if possible, thus returning to the expression from which we started in the case of involution. We have found that {a**y=a**''; that is, a" is an expres- sion which raised to the rth power gives a*"") therefore, Hence, to extract the n th root of a power of an expres- sion^ we divide the exponent of the given power by the in- dex of the root; but in order to perform the division, the exponent of the power must be a multiple of the index of the root. We cannot extract the square root of a^ because 5, the exponent of the power, is not a multiple of 2, the index of the root. 198. Root of a Product. To find the nih root of the product of two factors, we have a-b''^{aby\ therefore, 1^'^^= i/(aby-=ab. In this result the first factor a may be found by taking the 71 th root of a"", the first factor of the given expression; 126 UNIVERSITY ALGEBRA. and the second factor b of the result may be found by taking the nth. root of b"*, the second factor of the given expression. Therefore, the n\h root of the product of two factors is equal to the product of the nth. roots of those factors. 199. Of course the same argument may be used with more than two factors, and hence, evidently, the nth root of the product of several factors is equal to the product of the n th roots of those factors. 200. Root of a Quotient. To find the wth root of the quotient of two expressions, we have a"" (a^ ¥\b) In this result the numerator, a, is found by taking the n\h. root of the given numerator, and the denominator, ^, is found by taking the n th root of the given denominator. Therefore, the nth root of the quotient of two expressions is equal to the quotient of the n th roots of those expressions, EXAMPLES. Find the square root of each of the following twelve expressions : „ . 9^2 49^V ^' ^* 2. ^a^x^, 5. -2~2- 8. -v-g. II. -- --. a^y^ y^z^ b^ d^ ** 49a2;t2 ^ AiX^y^ b^ POWERS AND ROOTS. 127 Find the cube root of each of the following nine expressions : 13. SaH^c^. 16. — 64a9;t:i^ 19. -^Qia^ 5^ -r-Sa^ , 27 '* 27:%:^ /^ • ^3^3^3 '^3^6 ?^6;t:l5 27 x^ ^^^^ r^^g ^5- -"8^* ^^- -8;^ 64' ^^' "^^ * ~^' 22. Find the fourth root of a^d'^^c'^. 23. Find the square root of l^a^x^y'^^, and then the square root of this result. 24. Find the fourth root of IQa^x'^y'^'^, 25. Find the cube root of ^^^^^^^24^ ^^^ Xh.Qn the fourth root of this result. 26. Find the sixth root a^^b'^'^c'^^y and then the square root of this result. 27. Find the twelfth root oi a^^b^'^c'^'^. 28. Find the square root of a^^b^^c^^j then the cube root of the result, and then the square root of this second result. 29. Find the fifth root of—S2a^x^^y^\ 30. Find the seventh root of 12Sa'^ b'^ c^ ^ . l^a^x'^^y^ 31. Find the square root of ^ ^ * 201. Before leaving the subject of roots of monomials, it is well to notice that what we have learned may be used to find the roots of arithmetical numbers when the numbers given have exact roots. We resolve the number into its prime factors, and ex- press it as the product of various powers of these prime factors, then divide each exponent by the index of the 128 UNIVERSITY ALGEBRA. required root. When the resulting factors are multiplied together the required root is found. Suppose we wish to find the square root ol 53361. 3 3 7 7 11 53861=. 1/63861 =S EXi square root 63861 17787 5929 847 121 Hence, therefore, I. Find the 11 32x72xll^• .x7xll=231 i.MPLES. of 5184. 2. Find the square root of 43264. 3. Find the cube root of 85184. 4. Find the cube root ot 32768. SQUARE ROOT OF POIvYNOMIAI^S. 202. To find out how to extract the square root of a polynomial we must see how. the polynomial was pro- duced by squaring. We know that (x+y) '^=x'^ + 2xy+y'^ ; therefore we know that the square root of x'^ + 2xy+y'^ is x+y. Our problem, then, is this : Given the expres- sion x'^+2xy-\-y'^, to find from it the expression x+y. The first term x of the root is the square root of the first term x'^ o^ the given expression. I^et us set down the term x already found, and subtract its square from the given expression. There remains of the given ex- pression 2;t:j/-j-j/2 or (2x-\-y)y. From this we see that the second term y of the root will be the quotient when the remainder just found is POWERS AND ROOTS. 1 29 divided by 2x-\-y. This divisor 2x+j/ consists of two terms, the first of which is twice the portion of the root already found, and the second is the new term y itself. The work may be arranged as follows: x'^-i-2xy+j/^ ( x+j/ x^ 2x+y 2xy-\-y^ 2xy-\-y'^ After the first term x of the root has been found, its double 2x is used as a trial divisor by which to divide the remainder '^xy+y'^. We see that the first term 2xy of this remainder, when divided by the trial divisor 2:r, gives y, from which we judge that y is the next term in the root. When the y is thus found, it is added to the trial divisor 2;r, giving the complete divisor 2x+y, and this is multiplied by r, giving the expression 2xy+y'^. 203. Of course we may obtain by this process the dif- ference of two numbers for our square root as well as the sum, as in the example just given. This will be plain by u working out another example. To find the square root of Aa^ — 12ad+ d^ , Arrange the work thus : 4^2 iaSd i -12a5-\-9d^ I -12ad + 9d'^ Here the first term of the remainder — -12«^, when divided by the trial divisor Aa, gives the quotient —Sd. Hence we judge that --3^ is the next term of the root, and upon trial this proves to be right. L 9 — U A. I30 UNIVERSITY ALGEBRA. KXAMPI^KS. Find the square root of each of the following : 1. a^-{-4ab + 4:d\ 5. x^ + 2x^+x'^. 9. a'^ — 2aH^ + d^. 2. 4a'^—4ad+d\ 6. x^ — 2x^+x^, 10. a^ — 2ad^+d\ S. 4:a^—Sad+Ad\ 7. x^—2x^+x^, 11. a^ + 2a^d'^ + d^, 4. 4^2— 16«+16. 8. x'^ + 2x'^-\-l. 12. 9x^ — 18x'^ + d, 13. a^—2a^x^+x^, 16. Ax^ —4:nx^j/-i-n^y'^ , 14. a'^d^ + 2adcd+c'^d^, 17. a2^4_2^^;^;3 + ^2^2^ 15. a4^*-6a2^2_^9. 18. a^d^-2a''d^c^+c^\ 204, Thus far, the polynomials of which we have ex- tracted the square root have been in every case those of three terms. The above process, however, can be ex- tended so as to find the square root of any polynomial which is a perfect square, no matter how many terms the polynomial contains. For example, to find the square root ofa'^+d^+c'^-i-2ad+2ac-^2dc. First arrange the expression according to powers of some letter, say a, and write a"^ -\-2a3 + 2ac+ d^ + 2dc-i-c^ . The first term ^^ of this polynomial is produced by squaring a. Therefore the first term of the root is a, and the whole root is <3^-f something, and this something is what we wish to find. Proceeding as before with the first term of the root, a part of the process may be arranged thus : a^+2ad+2ac-i-d'^ + 2dc+c^ ( a a^ 2ab+2ac+b'^+2bc+c'^ Now twice a used as a trial divisor would suggest b for the next term of the root. Call the next term b and pro- ceed as before, and the work will stand thus : POWERS AND ROOTS. I31 ^2 2a + d 2ab-^2ac-^d'^+2dc+c^ 2ab +<^2 2ac +2bc+c'^ There is sfzll a remainder, so we have not yet found the entire root; but the root is ^-y 3r+9_ 5y+16 _^^5 284--^ _ 2-4-7 7. 4- 5 +br-U. 3. h—y_ \\—Zy 5y— 6 By— 6 4j/ "7 ~ 23 * -7 9 ir 2j/-6 3>/_,y-4 3.^ ^.\-ly 11. ^(^-2)-K^~3)-f K^-4)=4. 12. 3(^+3)2+6(^+5)2=8(^+8)2. 13. K^-3)-K^-8) + i(^-5)=0. ^' 3 4j/ 12 I^ITKRAI, EQUATIONS. 229. If the known numbers in an equation are repre- sented by letters instead of by figures, the equation is spoken of as a Lriteral Equation. Of course the same principles hold in the solution of such equations as in the solution of numerical equations. It must be remembered that the first and intermediate letters of the alphabet stand for numbers supposed to be known or given. 142 UNIVERSITY ALGEBRA. EXAMPI^KS. Solve the following literal equations: ax—b bx-Yc I. =^abc. c a Multiplying both sides by ac, a{ax -b)-c {bx-\- c)=sa*dc*. Removing the parentheses, a^x~ad—l>cx—c^=:a*dc^. Transposing known numbers to right side, Uniting with a parenthesis, {a^—dc)x=a^dc^ + ad-\-c^. Dividmg both sides by {a^ — oc), x= — r— ^ — a^—bc 2. {b—V)y=b—y. 4. ad—dy=-my—am. 3. {y. — b^x^a—x. 5. p{x—V)+x=^q—p. 6. {b+V)X'\-ab=b{a^-x^-\-a. 7. {a-\-b^x+{a—b^x—a'^, 8. {a + bx'){b^ax)=^ab{x'^—V). 9. {x-Ya\x+b)^{x'--a){X'--b)-Y{a+by. 10. ax{x-\-a^ + bx(jK-{-b^ = (jx-\-b)(^x-\-a){x-\-b). 11. -rx^, — x=^b'^+x'^, b a Zb(^x-a) {x—b''') ^ (^4:a+cx)b 5a 15b 6a nx r — X , n(r—x) ^3- -y-^f"^-^--' 14. 15. 2ax--2b ax— a ax 2 b+c _a ^ , a(a-l)^b{b-r)+c XX X SYMBOI.IC EXPRESSIONS. 230. In solving a problem in Algebra we must not only select some letter to stand for the unknown number, but we are required to find expressions containing the SIMPLE EQUATIONS. 143 unknown letter which wall symbolize all other numbers occurring in the problem. Such may be called Symbolic Expressions. Thus, if the sum of two numbers is 100, and if x stands for one of them, then the expression 100— x stands for the other number. If x is the price of one horse, then lOx stands for the cost of 10 horses ; if 5 yards of cloth cost X dollars, then the cost of one yard is represented by X the expression ^, etc. Drill in the formation of symbolic expressions is of help in the solution of problems. 1. What two numbers differ from 100 by 7? What two numbers differ from 100 by ^ ? What two numbers differ from nhy x"^, 2. A train traveled at the rate of 20 miles per hour for 3 hours; how far did it go ? A train traveled at the rate of r miles per hour for 3 hours; how far did it go ? A train traveled at the rate of r miles per hour for t hours; how far did it go ? 3. The rate of a train is r. How far will it go in time /? 4. A man can do a piece of work in 10 days. How much of it can he do in one day ? A man can do a piece of work in n days. How much of it can he do in one day ? 5. One pipe will fill a cistern in a hours, and another pipe will fill it in b hours. What part does each pipe carry in one hour? What part do both pipes together carry in one hour ? 6. The digit 5 stands in tens' place; what number is expressed ? The digit, represented by x, stands in tens* place; what number is expressed ? A digit represented by x stands in hundreds' place; what number is expressed? 144 , UNIVERSITY ALGEBRA. 7. A can do a piece of work in 9 days and B can do it in 12 days. What part can each do in one day ? What part can both, working together, accomplish in one day ? A can do a piece of work in a days and B can do it in b days. What part can both, working together, accom- plish in one day ? 8. The three digits of a number beginning at the right are represented by x, x+2, and xS; what is the number expressed by them ? 9. Write three consecutive even numbers of which 6 is the first. Write three consecutive even numbers of which 2n is the first. 10. Write three consecutive odd numbers of which 5 is the first. Write three consecutive odd numbers of which 2n + l is the first. 11. Write three consecutive numbers of which n is the greatest. Write three consecutive even numbers of which 2n is the greatest. Write three consecutive odd numbers of which 2n+l is the greatest. 12. Show that the squares of any two consecutive numbers differ by an odd number. 13. Show that if n stands for a whole number, then nCn + Y) is divisible by 2. 14. Show that if n stands for a whole number, then n(Sn—l) is divisible by 2. 231. It is well to note that if n stands for any whole number, then 27^ is the symbolic expressions for any even number, since it is exactly divisible by 2, and 2n-\-l is the symbolic expression for any odd number, since when divided by 2 the remainder is 1. SIMPLE EQUATIONS. 145 PROBI^KMS. 232. Directions for Solving Problems. The stu- dent will find that the following is the usual course pur- sued in solving problems by Algebra : I. Represent one of the unknown numbers, preferably the one whose value is asked for, by x. II. Make symbolic expressio?is to represent each of the other unknown numbers mentioned in the problem. III. Find, from the problem, two of these symbolic ex- pressions that are equal to each other. IV. Solve the equation thus formed. Solve each of the following problems : 1. The sum of two numbers is 33 and their difference is 7. Find the numbers. Let 5C— one of the numbers; then 33 — ^=the other number. And since the difference of the two numbers is 7; therefore, 5C-(33-;r)=7. Removing parenthesis, ^—33 + ^=7. Transposing and uniting terms, 2x=40; whence, ^=20. Therefore, one number is 20 and the other 13. 2. In a yard there are chickens and rabbits, and alto- gether they have 14 heads and 38 feet. How many rabbits and how many chickens in the yard ? 3. A man was engaged for 80 days under the agree- ment that he was to receive $4 for each day he worked, but was to forfeit $1.50 for each day he was idle. He received $276 in all. How many days was he idle? 4. A number consists of two digits, the sum of the digits being 10. If the digits be reversed the new num- ber is 18 larger than the original number. What is the number? 10— u. A. 146 UNIVERSITY ALGEBRA. Let ^=the digit in tens' place. Since the sum of the digits is 10; therefore, 10— ;i;=the digit in units' place. Hence the number expressed by these digits is 10;c+10-x. But the number expressed when the digits are reversed is 10(10-^) + ^. Since this is 18 larger than the original number; therefore, l0(10-^) + ;c-(10^+10-^)=:18. Removing parentheses, 100 — 10;c + x — lOx — 10+x=18. Transposing terms, — 10x-|-;r— 10^ + x=18 — 100+10. Uniting terms, —lSx= — T2. Therefore, x=4, one digit. Therefore, 10—^=6, the other digit. Hence the number is 46. 5. A number consists of two digits of which the sum is 12. If we subtract 18 from the number we obtain the number with digits reversed. What is the number? 6. A man is 32 years old and his son 11. In how many years will the father be 4 times as old as his son ? 7. A man is / years old and his son s. In how many years will the father be n times as old as his son? 8. A mistress promised her servant $150 a year and a new dress. The servant left at the end of 10 months and received $120 and the dress. What was the value of the dress ? 9. A train leaves a station and travels at the rate of 25 miles an hour. Two hours later another train follows it, traveling at a rate of 35 miles per hour. How long before the second train will overtake the first ? 10. The body A travels one yard a minute and is pur- suing B, which is 10 yards ahead of it. If A moves 12 times as fast as By how many minutes before A and B are together? SIMPLE EQUATIONS. 147 11. At what time between 2 and 3 o* clock are the hands of a clock together ? The minute hand moves over one minute space in one minute. At 2 o'clock the minute hand is pursuing the hour hand, which is 10 spaces ahead of it. The minute hand moves 12 times as fast as the hour hand. 12. Chicago and Madison are 130 miles apart. At 6 p. M. a train leaves Chicago for Madison and runs at the rate of 30 miles an hour. A 7 p. m. a train leaves Madison for Chicago and runs at the rate of 20 miles an hour. When and where will the two trains meet ? 13. A can do a piece of work in 50 days, B can do the same work in 60 days, and C can do the same work in 75 days. How many days will it take them to do the work together? Let X represent the number of days it takes the three men to do the work together. Then - is the part of the work they all do in one day. Since in one day A alone does ^ of the work, B alone does ^^ of the work, and C alone does ^ of the work, therefore, 1 1 ' 1_1 14. A bath can be filled by the cold water pipe in 10 minutes and by the hot water pipe in 15 minutes. It can be emptied by the waste pipe in 8 minutes. A man hav- ing left all these pipes running, returns after a time and finds the bath half full. How long was the man absent ? 15. Suppose that in the preceding problem it had been stated that the man returned after an absence of 16 minutes. In what condition would the bath be found ? 16. Two barrels, holding 48 gallons each, are to be filled with varnish worth 70 cents a gallon. The varnish is to be taken from two sorts, one worth 60 cents per gallon and the other worth 90 cents per gallon. How much of each sort is required ? 148 UNIVERSITY ALGEBRA. 17. The difference of the squares of two consecutive numbers is 19. What are the numbers? 18. A person had $1000, part of which he loaned at 7 percent, and the rest at 6 per cent.; the total interest received was $63. How much was loaned at 7 per cent.? Let 5C=the number of dollars loaned at 1%. Then 1000— ^=the number of dollars loaned at Q%. Therefore, — — =:the interest received at 1%. and L-p-r =the interest received at 6%. Since the whole interest received was $63, therefore, 7%_^6(1000-ic)___ 100+ 100^-^^' whence x=:300. 19. A man invested $1100 at 5 per cent, interest, and 5 years later he invested $1000 at 7 per cent, interest. How many years from this last date until the principals and interests of the two investments will equal each other? 20. A man invested a dollars at n per cent., and d years later he invested b dollars at r per cent. How long until the two sums, principal and interest, will be equal ? 21. The sum of $1325 is borrowed, to be paid back in two equal annual payments, allowing 8 per cent, simple interest. Find the annual payments. 22. A man can row 5 miles an hour down stream and 3 miles an hour up stream. If he starts down stream at 1 o'clock how far can he go so that he may rest ashore an hour and get home at 6 o'clock ? 23. A man walks from A to B and back in a certain time at the rate of 3|- miles an hour. When he walks 3 miles an hour to B and 4 miles an hour back it takes him 5 minutes longer. Find the distance from A to B. SIMPLE EQUATIONb. I49 24. A train leaves ^4 at 11 a. m. for B, and travels at the rate of 25 miles an hour. Another train leaves C at noon, and runs through A to B at the rate of 35 miles an hour, arriving at B 24 minutes later than the first train. The distance from C to A being 21 miles, find the dis- tance from A to B. 25. A passenger train, 431 feet long, going 41 miles an hour, overtakes a freight train on a parallel track. The freight train is going 28 miles an hour and is 731 feet long. How long does the passenger train take in pass- ing the other? 26. Susie was born in eighteen hundred and seventy x. Agnes was born the next year and Mabel was born two years later than Agnes. When Susie was x years old the sum of the ages of the three was 20 years. Find the year of birth of each. 27. The metal of a solid sphere, radius r, is made into a hollow sphere, internal radius r; required its thickness. 28. A cylinder of hickory encloses a cylinder of iron, the whole containing 30 cubic inches and weighing 52 ounces. If the hickory weighs .5 ounce per cubic inch and the iron 4.2 ounces per cubic inch, how many cubic inches of iron are there in the cylinder? 29. A mass of tin and lead weighing 180 lbs. loses 21 lbs. when weighed in water, and it is known that 37 lbs. of tin lose 5 lbs. and 23 lbs. of lead lose 2 lbs. in water. What are the weights of tin and lead in the mass ? 30. At two stations, A and B, the prices of coal are $4.50 and $5.00 per ton, respectively. If the distance from A to B is 160 miles and the freight rate is f cents per ton per mile, find the distance from A of sl station at which it is immaterial whether coal be bought at A or B. 150 UNIVERSITY ALGEBRA. GKNKRAI. EQUATION OF FIRST DKGRKE. 233. General Equation. It is evident that a;r+^=0 represents any equation of the first degree containing one unknown number ; for it provides for any coefficient of x and for any term, or Absolute Term as it is called, not containing the unknown number. The root of this equa- tion is and it is seen that the equation may be put in the form, ^~\ J"^ * That is, any equation of the first degree may be put in the form x—theroot=0. If it is not plain that every simple equation containing but one unknown number may be put in the form ax-\-b=.0, the following will tend to make it clear. Take the equation Ifx + l ■y-3]_[.r+l x-l 41 2 6 J~[ 8 "^ 12 . ^+1 ^—3 x+1 x—\ Multiplymg by 4, -^ —z=:—^Jr—^ Multiplying by 6, 3x + 3-(^-3)=3x + 3+2^-2. Transposing, 3;ir-x-3%— 2x + 3 + 3-3 + 2=0. Uniting terms, -3:?c4-5=0. This is in the form ax-\-b=Q. The root is |, and the equation may be written x— 1=0. 234. Discussion of the Equation. Since the root of the equation ax-^-b^^O is , it follows that the root is negative if a and b have like signs, hnt positive if a and b have unlike signs. The three following cases are im- portant : I. Suppose b=0 and a^O. In this case the equation becomes ax^^Q and the root takes the form . It is . a plain that in this case the value of x is 0. The symbol =^ stands for "not equal to,'' and-^t! for **not less than.*^ SIMPLE EQUATIONS. 151 II. Suppose a=0 a7id d=0. In this case the equa- tion becomes Ox+0=0. This equation can be satisfied by any value of x whatever, since times any number is itself 0. In this case the root takes the form %, which, because of the fact just mentioned, is often called the Indeterminate Form. III. Suppose a=0 and b^O. In this case the equa- tion becomes Ox-{-b=0 and the root takes the form — -tt. If in the equation ax+b=0 we put ^=3-^, we get x= — 1^0b ; if we put ^=tto o"' we get x= — 1000^ if we put ^=To"i7nr' we get jr=— 10000<^. Thus we see that in the equation ax-\-b=0, as we take a numerically smaller and smaller, the value of x becomes numerically larger and larger. Evidently, then, so long as a is not zero, there is a value of x which will satisfy the equation, no matter how near zero we take a, but for ^=0 there is no finite value of x which will satisfy the equation. That is, if ^=?^0 the equation Ojtr+^=0 is an absurdity for finite values of x, GKNKRAI^IZKD PROBLKMS. I. Two couriers, A and By are traveling the same road, the former at the rate of a miles per hour, the latter at the rate of b miles per hour. At noon they are d miles apart. When are they together? Solution of the Problem. Let x equal the number of hours that elapse before the couriers are together. Therefore, «5C= number of miles A goes until they meet, and <^x=number of miles B goes until they meet. Since the first must go d miles more than the second, therefore, whence ^= 7. d ^~ That is, they meet in 7 hours. 152 UNIVERSITY ALGEBRA. Discussion of the Problem. First, suppose that the couriers are traveling the road in the same direction. Then a and b are of the same sign. d I. \i ayb, that is, if A travels faster than B, then r, or the value a — b of X, is positive ; so that they will meet at some time after noon. It will be but a short time after noon, (1) if ^ is small, or (2) if a is much greater than b. On the other hand, it will be a long interval after noon, if (1) d is large, or if (2) b is nearly equal to a. II. \i a minutes. Discuss the problem for : bc-\-ac — ab ^ X 1 1 1 J XT 1 1 1 I, -<-+t; and II, -=-+T- cab cab 4. A barrel holding q gallons is to be filled with var- nish worth n cents per gallon. The varnish is to be taken from two sorts, one worth a cents per gallon and the other worth b cents per gallon. How much of each sort is required ? , . gin — b) ,, , q[a — n) The result is 7-^ gallons at a cents and — 7-^ gallons at b a—b a—b ^ cents per gallon. Discuss this result for: I, ayw^b; II, n^a^b; III, b'>n>a: IV, a>byn\ V, a—b—n\ VI, a=n>b. 5. Let s^ and ^2 ^^ ^^^ specific gravities of two sub- stances of which a compound is to be made whose specific gravity is 6" and its absolute weight w. How much of each substance is required ? tJAt O ( O ,__ X \ 12) S ( S S \ The result is ^^ ^ — -^ of the first and -^ ^ of the second substance. Discuss the problem for: I, S=Sj^:^s^; 11, s^^—s^z^S; III, s^=s^ = S; IV, s^>S>S2; V. s^>S2>S. 154 UNIVERSITY ALGEBRA. 6. At two stations, A and B, the prices of coal are $^ per ton and %b per ton, respectively. If the distance be- tween A and B is d miles and the freight rate for coal is %r per ton per mile, find the distance from A of a station on the line at which it is immaterial to a dealer whether he buys coal at A or at^. The result is ^ miles from A. Discuss the problem for : I, d-\-rd>a; II, d-\-rd (1) and jj/+23=3.r (2) Finding the value oi x in terms of y from (1), we get ;j;=3r-19. (3) Substituting ^y—Vd for x in (2), we get jj/+23=9y-57 (4) Transposing, j^—9ji/= —57—23. (5) Uniting terms and dividing both sides by —1, 8r=80; (6) whence y=lO, (7) SIMULTANEOUS EQUATIONS. I 5/ Substituting this value for y in (3), we get :r=30— 19=11; whence x=\l and j^=10. To verify, we substitute these values in the original equations and get 22=60-38 and 10+2^=33. (2) Find ;r and J/ if 1x+^y=im (1) and 3:r-j/=20 (2) Finding the value of y in terms of x from (2), we get jK=3^-20 (3) Substituting 3:r— 20 for j in (1), we find 7-r+ 9^-60= 100. (4) Transposing and uniting terms, 16:r=160; - (5) whence x=10. (6) Substituting this value for x in (3), we have j^=30-20=10- w^hence ;i:=10 and y=10. 241. It is easy to see that the method of elimination used in these examples may be applied in the case of any set of two simultaneous equations. Hence we may say: From either equatiori express one of the unknown num- bers in terms of the other, and substitute this value in the other equation. BI.IMINATION BY COMPARISON. 242. We give a few examples of elimination of an un- known number by the method of comparison. (1) Find ;»; and j>/ if Zx=lZ—y (1) and 2;r=j/+32 (2) From (1), ^=^^- (3) 158 UNIVERSITY ALGEBRA. From (2), ^=^^^- (4) Therefore. ^=-^- © Clearing of fractions, 2(73-j^)=3(^+32). (6) That is, 146-2:k=3j/+96. (7) Hence, by transposing and uniting terms, 6^=50; (8) whence j/=10. (9) 7Q 10 Then, from (3), ^=:i£^=21. (10) Hence, ;i;=21 and _7=10. (2) Find ^ and jj/ if 9jj/+8;r=41 (1) and ll;r-7j/=37 (2) From (1), ^=^^T^- (^) From (2), ^=^^^^- (4) Whence — g — = j^ (5) Clearing of fractions, 7(41-8;»;)=9(ll;t;~37). (6) That is, 287-56;t:=99.;t;-333. (7) Hence, by transposing and uniting terms, 155;i;=620. (8) Therefore, x=4. n^^ r /ox 41-32 , Then, from (3), j^= — ^ — =1. Hence, x=4: and jk=1. 243. The above is sufficient to show us how to elim- inate an unknown number by the method of comparison. Express the same unknown number in terms of the other from each equation^ and equate the expressions thus fou7id. SIMULTANEOUS EQUATIONS. I59 ELIMINATION BY ADDITION AND SUBTRACTION. 244. The elimination in the following examples is done by the method of addition and subtraction. (1) Find ;tr and j>/ if x+y=b1^ (1) and ;»;~jj/=333 (2) Addilig the left members and the right members of these two equations, we obtain ^+^,= 579 (3) x—y=^Z^ (4) 2;tr=912 (5) whence x=Ab^ (6) Subtracting the members of (4) from the corresponding members of (3), we get 2j/=246; whence j/=123. Therefore, ;r=456 and j^=123. (2) Find ;t: and j^ if 15;r:-8r=30 (1) and Zx^2y=lb (2) Multiplying both members of the second equation by 5, we have for the two equations Ibx- 8jj/=3,0 (3) \bx-\-10y=lb (4) By addition, — 18>/=— 45 (5) whence J^=2^. (6) Hence, from (1), 15;tr~20=30. (7) Therefore, ^=3^ (8) Or the value of x may be obtained in another way. Multiplying both members of the second of the given equations by 4, we have for the two equations, lbx-Sy=ZO (9) 12.r-f8^=6Q (10) By addition, 27^=90 (11) whence ^^fy or 3-|-. l60 UNIVERSITY ALGEBRA. (3) Find ;»; and J/ if ll;i;4-12j/=100 (1) and dx-i- 8y= 80 (2) Multiplying both members of the first equation by 9, and both members of the second by 11, we obtain 99j^ + 108r=900 (3) 99^+ 88j/=88Q (4) By subtracting, 20>/= 20 • (5) whence J^=l. (6) Substituting this value for y in either of the original equations, we find x=S, 245. The above examples show us how to eliminate an unknown number by the method of addition and sub- traction. Multiply both members of the equations by such numbers as will make the 7iumerical coefficients of one of the unknown numbers the same in the resulting equations; then by addi- tion or subtraction we can form an equation co?itaining only the remaining 7ium.ber. SPKCIAI. EXPKDIKNTS. 246. The student will find that elimination by addi- tion and subtraction is in most cases the shortest method. Occasionally, however, an example will be found which is more readily solved by one of the other methods. Some- times, too, special expedients will still further abbreviate the processes. We give a few examples of this. (1) Solve 23^4 39j/=193, 39;i;+23j/=241. By adding the members of the two equations, we get 62x-{-62y=4S4. (1) By subtracting the members of the second equation from those of the first, we have SIMULTANEOUS EQUATIONS. l6l 16;ir-16>/=48. (2) From (1) we get x-hy=7. (3) From (2) we get x—j/=S, (4) Whence, by addition and then by subtraction, x—6 and j/=2. If this example be done in the ordinary way, large numbers will be encountered. (2) Solve ^--^=1, -^+-^=6. To solve these we must first clear each of fractions, giving 9x— 8)/= 12, and 14.r+5j/=36, which can now be solved in any of the usual ways. ,^^ ^ , 9 4, 18 , 20 ,^ (3) Solve =1, — h— =16. ^ ^ X y X y If these be cleared of fractions, the resulting equations will involve the product xy^ and we would have equa- tions of a kind not yet considered. But by considering — and — as the unknown numbers, we may solve by the x y methods already used. For example, by multiplying both members of the first equation by 2, we get ---=2. (1) X y ^ ^ But T+V=16- (2) X y 28 By subtraction, — =14, (3) 2 whence -=1 or j>/=2. y Therefore, from (1), x=Z, We could have eliminated x more readily if we had first divided both members of the second given equation by 2. 11— U. A. l62 UNIVERSITY ALGEBRA. (4) Solve i^+l;=2, 1-^=1. Multiplying the first equation through by 12, we have ^+^=24. (1) X y Muhiplying the second through by 4, we have ^-^=4. (2) y X Subtracting (2) from (1), |=20; (3) whence x=^\ and y=^. EXAMPIyKS. Solve by any method : I. 3^+8y=19, 2. 2;i:-7j/=8, 3. 19jr-21j/=100 ^x—y=l. 4y—9x=19. 21x—19y=U0 4.10.+|=210, 8. 1+1=3, -.f+|=3i X 15 4 . X y 1 10^-^=290. ---=4. 3-g-2 X 12 1 1 11 5. f +7,-261. 9. -+--10, .3.j+--g5. ^■H-«. -M-F «-5^4=^' 2 1_ £,Z_2 ?_^±=?? x~j)/~ 2"*" 3"" 6' 4;r''"5j|/ 10* 7. x+|=4, IX. 1+31^=139. 15. ^^=^x-j, 24=1. I + -.-31. 1^^=^- SIMULTANEOUS EQUATIONS. 1 63 16.-—+-- o, 17. ___+d ^. x+j^ x-y_^^ o x-2y_x y 18. 7^-13^=6;tr-10jc-8=0. X9. ^f'-^^=2.-4. ' 2,-4-^-£=^=3.. 20. ~3~ 12""" "60"' SIMULTANEOUS EQUATIONS CONTAINING THRKB UNKNOWN NUMBERS. 247. If we have two equations containing three un- known numbers, such as 2x+?>y—b2= 9, we can eliminate one of these unknown numbers by the methods already explained, giving one equation con- taining two unknown numbers. Thus, in this particular case, by multiplying the members of the second equation by 2 and subtracting, we find 7j/-19^=-17. Since an indefinite number of values will satisfy one equation containing two unknown numbers, it follows from this that an indefinite number of sets of values will satisfy two equations containing three unknown numbers. Suppose, however, that we have three equations con- taining three unknown numbers, as 2x+?yy-b2= 9, (1) x--2y+l2=l^, (2) Zx- y-22= 8. (3) 164 UNIVERSITY ALGEBRA. Multiplying both members of (2) by 2 and subtracting from (1), we get 7y— 19-^= — 17. (4) Multiplying both members of (2) by 3 and subtracting from (3), we get 5y— 23^=-31. (5) Now we can eliminate y from (4) and (5) by multiplying both members of (4) by 5, and both members of (5) by 7, giving 35j/— 95<^=— 85, (6) 35>/-161^=-217; ' (7) whence, by subtraction, 66<3'=132, (8) w^hence ^=2. Substituting 2 for z in (4), 7^-38= -17, (9) whence, j^=3, and substituting j/=3 and -3'= 2 in (1), we find ^=5. Therefore, ^=5, jv=3, and ^=2. Here we notice that we have been able to find the values of three unknown numbers from three equations. 248. It is evident that we may proceed in a similar way in any case of three linear equations containing three unknown numbers. That is, to solve three simul- taneous equations containing three unknown numbers: ^ I. Obtain from two of the equations an equation which contains only two of the unknown numbers^ by any method of elimination, II. From the third given equation and either of the former two obtain another equation which contains the same two unknown numbers, III. From the two equations containing two unknown numbers thus found find the values of these unknown num- bers. IV. By substituting these values in one of the given equations the value of the remaining unknown number may be found. SIMULTANEOUS EQUATIONS. 165 249. We further illustrate this subject by working a few examples. It should be observed that while it makes no difference which one of the unknown numbers is elim- inated first, yet the work is often lessened by selecting for this purpose that one of the unknown numbers whose numerical coefficients have the smallest I^.C.M. (1) Solve 4x- by+ ^= 6, (1) 1x-ny+2z= 9, (2) x+ j>/+3^=12. (3) The unknown number z has the smallest numerical coefficients, and it is easier to eliminate it than any of the other unknown numbers. Multiplying both mem- bers of (1) by 2, we have 8jir-10j/+2^=12. (4) Subtracting (2) from this, gives x-\-y=^, ^ (5) Now multiplying both members of (1) by 3, we get 12;i:-15>/+3^=18. (6) Subtracting (3) from this, we find ll^-16j/=6. (7) We have now to find the values of x and y from (5) and (7). Multiplying both members of (5) by 11, we get ll;t:-fllj/=33. (8) Subtracting (7) from this, we find 27j/=27; (9) whence jr=l. From (5), jr=2, and from (3), 2 + 1 + 3^=12; whence 2'= 3. Therefore, x=2, y—1, and z=Z, l66 UNIVERSITY ALGEBRA. (2) Solve ^+:k=5, (1) jK+^=7, (2) ^+£'=6. (3) This is quickly solved by special expedient. Thus, adding the members of the three equations, we get 2x+2y-^2z=18, or x+y-i-2=d. (4) From (Ij, x+y=5; therefore, from (4), ^=4. From (2), y-\-z—l\ therefore, from (4), x=2. From (3), x+z=6; therefore, from (4), j/=3. (3) Solve 1+^+1=4. (1) i+-l^-^=4. (3) X y z ^ ^ Here we should consider -, — , and — as the unknown X y z numbers. Subtracting (1) from twice (2), we get ^-f^=4. (4) Subtracting (3) from three times (2), we get y z ^ We are now to find — and ~ from (4) and (5). Subtract- y z ^ ^ ^ 20 ing (4) from (5), T""^' whence <3'=5. From (4), jj/=4. From (1), ;tr=3. SIMULTANEOUS EQUATIONS. 167 EXAMPLES. Solve the following simultaneous equations: 1. ^+j/=37, 3. 2:r+j/=5, 5. ^t+jK— ^=17, y-\-z=22, by— 2=10. y-\-2—x=1, 2. jj/+-^=14, 4. 5xSy==—l, 6. 5x-\-7y—2z=lS ^4-:i;=18, 9j/-2^=12, 8;i:+3r+-^=17 x+y=24:. Sx+S2=n, x-4y-i-10z=2S II. ;»;— ^+-s'=5, 7. x-{-y+z=SO, 8x-\-4y+2z=50, 27x+dy+Sz=64:. 8. 3.r+2jK+^=23, 5:r+2j/+4<^=46, 10^ +5y 4-4.3'= 75. 9. 4:X-2y+52=lS, 2x+4:y-S2=22, 6x+7y—2=6S. 10. .;r+J^+4^=23, 3;i;-j/+22'=ll, x+4y—z=4:. Sx+4y-5z=lS, 12. ;r+2j/+3^=4, 2;r+3j/+4^=6, 3;»;+4j/+5^=8. 13. ;r+2j/+3^=32, jj/+2^+3;i:=40, 2+2x+^y=4:0. 14. ^+3j/+3^=13, j/+3^+3;r=15, ^+3jr+3jF=17. 2^3 15. ^+1=12, 5 6~ ' 12+7-^- 16. x+y 2 X-\-2 3 = 1, = 1. 17- -+-=1, X y y^ z 2 ;»;^0 3 i9.f+|+J=62. 3^4^5 '' 4 + 5+6-^^- 1,23, 20. -+ =1, X y z ^+i+^=24, .a; ^ 2" l-«+-^=14. X y 2 l68 UNIVERSITY ALGEBRA. LITKRAI. SIMUWANBOUS EQUATIONS. Solve the following: I. x-{-y=2a and x—y=23. By addition, 2x=2a + 2if; whence x=a-^d. By subtraction, 2y=2a--2d; whence y=a—b. 2. jr+j/=3«— 2<^, 4. ax+by=^c, 6. "Ix—Zy^^hb—a, x—y=2a—Bd. nx+ry=s. Sx'-2y=a-^5d, 1,1 n Jl =^. — — =0. ax— •€}=—— -' X y X y -^ bd 8. ax-\-by=2ab, 9. a(^a-\- x)^b{b-^y), 2bx-^2ay=Zb'^—a'^. ax+2by=d. 10. (^-<^):r+(^ + %=2(a2-^2)^ ax^by=a'^-^b'^. 11. .r4-:K=2a, 13. ;ir+jK+^=^, 15. ax-^by—cz=2ab, x+z=2b, x—y+2=b, by-]-c2'-ax=2bc, y+2=2c. x+y—2=c. c2+ax—by=2ac, 12. ax-hy=r, 14. x—nx+y=0, 16. j»r+J^+<3'=a, ^— JK=^, x—ry+2=0, ny=rx, bx-^2=t x-\-2=t, p2=qy, 17. x-\-ay^a'^2-\-a^=^^^ x-i-by-{-b'^2-^b^=^0, x+cy+c^2-}-c^=0, o ^ , ^ ^ 1 1 a b c ^ 18. -+ =;z. 19. -=a 20. -+_- f-=3, X y 2 "^ X y X y^ 2 X y 2 ' y 2 X y '2" ' —- + -4--=/ 1=^—1. ?^__i_f_o ;^ 7 -3' * 2 X X y ^"" * SIMULTANEOUS EQUATIONS. I69 PROBLEMS. 1. The sum of two numbers is 70, and their difference is 24. Find the numbers. Problems similar to this have been worked with the use of one un- known number. We will now work it using two unknown numbers. Let t 5C=the first number, and let _>/=the second number. Then, because the sum of the numbers is 70, and the difference of the numbers is 24; therefore, ;r+^=70, and 5C— >'=24. Solving these, we find x=47 and 7=33. 2. Find a fraction such that if we add 1 to the numer- ator, it becomes equal to ^y but if we add 2 to the denom- inator, it becomes equal to ^. Let 5C=the numerator of the required fraction, and let 7= the denominator of the required fraction. Since the fraction with 1 added to the numerator equals i, therefore ^=-. (1) Since the fraction with 2 added to the denominator equals J, there- From (1) and (2) we get, by clearing of fractions, 2^+2=jjr, (3) 3x=y + 2. (4) Eliminating y, we get x=4; whence, from (3), jj^=10. The fraction is therefore ^. 3. Find a fraction such that when 1 is added to both numerator and denominator it equals ^, but when 3 is subtracted from numerator and denominator it equals ^. 4. A number is formed of two digits of which the dif- ference is 3. If the digits are reversed, a number is ob- tained which is -f- of the original number. What is the original number? Let X represent the digit in tens' place, and y represent the digit in units' place. I/O UNIVERSITY ALGEBRA. 5. Find a fraction such that when 11 is taken from both numerator and denominator it equals -J-, but when 12 is taken from both numerator and denominator it equals f . 6. Two masons, A and B, are building a wall, which they could finish, working together, in 12 days. A works 8 days and B 2 days, when the wall is \ done. How long would it have taken each to have built the wall ? Let x=the number of days it would take A, and jj/=the number of days it would take B, In one day A builds - of the wall, and B -. But together they build jij of the wall in one day. Therefore, o o In 3 days A builds - of the wall, and in 2 days B builds - of the wall. "^ X y 3 2 1 Therefore, from the problem, — \--—-z' (2) ^ X y ^ ^ ' If we solve the simultaneous equations (1) and (2), we shall find the values of x and y. 7. A and B can together do a certain work in 80 days; at the end of 18 days, however, B is called off and A finishes it alone in 20 days more. Find the time in which each could do it alone. 8. A cistern holding 4500 gallons is filled by two pipes. If the first pipe be opened 8 minutes and the second pipe 1 minute, 400 gallons will run into the cistern ; but if the first pipe be opened 1 minute and the second 7 minutes, 600 gallons will run in. How much water does each pipe carr}^ in one minute? How long will it take bpth pipes to fill the cistern if they are opened together? 9. A and B can do a piece ot work in 12 days ; B and C can do it in 20 days ; A and C in 15 days. How long will it take each to do the work alone? SIMULTANEOUS EQUATIONS. I7I 10. A cistern can be filled by two pipes. If both pipes be opened for 15 minutes they will fill ^ ot the cistern ; but if the first pipe be opened for 12 minutes and the second for 20 minutes, \ of the cistern will be filled. How long will it take each of the pipes when opened alone to fill the cistern? 11. A man receives $2160 yearly interest on his cap- ital. If he had loaned the same capital at \ per cent, higher interest he would receive $240 more interest each year. Find the amount of his capital and the rate per cent. 12. A cistern is filled with three pipes. The first and second will fill it in 72 minutes, the second and third in 120 minutes, and the first and third in 90 minutes. How long will it take each of the pipes to fill it ? 13. Three cities, A, B, and C, are at the comers of a triangle. From A through ^ to C is 82 miles: from B through C to A is 97 miles ; from C through A to B is 89 miles. How far are the cities A, B, and C apart? 14. A certain number consists of three digits, whose sum is 15. If the first two digits be reversed the number becomes 180 larger, but if the last two digits be reversed the number becomes but 18 larger. What is the number? 15. The sum of three numbers is 70. The second divided by the first gives 2 for the quotient and 1 for the remainder, but the third divided by the second gives 3 for the quotient and 3 for the remainder. What are the numbers? 16. A man rows 30 miles and back in 12 hours. He finds he can row 5 miles with the stream in the same time as 3 against it. Find the time he was rowing up and down, respectively. \^2 UNIVERSITY ALGEBRA. 17. In round numbers, it takes 72 English and 51 Ger- man yards together to make 100 meters. Also 48 English and 84 German yards make 100 meters. How many inches (English) in the meter? How many inches (English) in the German yard ? 18. A man has two sums, one of $10000 and another of $15000, at interest, and receives therefrom $1200 yearly. If the first sum had been loaned at the rate that the sec- ond bore, and if the second sum had been loaned at the rate that the first bore, he would receive $25 less per year. At what rates were the sums loaned ? 19. A bicyclist starts from A to B and back, and at the same time another bicyclist starts from B to A and back. They meet at 1 o'clock 15 miles from A, and again at 3 o'clock 9 miles from A. What is the distance from A to B, and what are the rates of the bicyclists ? 20. Two bodies move upon the circumferance of a circle which is 100 feet in length, and meet every 20 seconds when their directions are the same and every 4 seconds when their directions are opposite. How many feet does each body move per second ? 21. A railway train, after traveling an hour, is de- tained 15 minutes, after which it proceeds at f of its former rate and arrives 24 minutes late. Had the deten- tion taken place 5 miles further on, the train would have been but 21 minutes late. Find the original rate of the train and the distance traveled. 22. The specific gravity of lead is 11.36, that of gutta percha is .966, and that of sea water is 1.03. It is re- quired to form of gutta percha and lead a mass of 5 pounds which shall have the same weight as an equal volume of sea water. SIMULTANEOUS EQUATIONS. I73 23. There are two stations, A and B, 9.1 miles apart, established for determining the velocity of sound. At the same instant a cannon is fired from each station. At A the report of the cannon at B is heard 42 seconds after its flash is seen, and at B the report of the cannon at A is heard 44 seconds after its flash is seen. Determine in feet per second the velocity of sound and the velocity of the wind at the time of the experiment. GKNKRAI. SYSTEM WITH TWO UNKNOWN NUMBERS. 250. It is evident that ax+by^c represents any equa- tion of the first degree containing two unknown numbers, for it provides for any coefficient of x^ any coefficient of J/, and for any term not containing an unknown num- ber. Hence, a^x+b^y^Cy^, (1) a^x-^b^y^c^y (2) may be taken to represent any system of two equations of the first degree containing two unknown numbers. The symbols a' .a' , a!" , etc., read **a prime,*'* "a second t'' **a third,** etc.; or a^, a^, a^, read "a sub 1," **a sub 2," **a sub 3,** axe often used for numbers which have a common relation to some other num- ber or symbol. Thus, a^ stands for the coefficient of * in the Jlrst equation, and ^z j stands for the coeflQcient of x in the second equation, in this case the suffixes showing from what equation the symbol is taken. Of course such symbols stand for entirely different numbers, just as though different letters of the alphabet were used. Multiplying (1) through by a 2 and (2) through hy a^, we have a^a^^^x+a^b^y^a^c-^, (3) a^a^x-^-a^b^y^a^c^, , (4) Subtracting (3) from (4), (a^b.^—a^b^)y=-a^c^—ac^c^\ (5) whence l/^gg^g W 1/4 UNIVERSITY ALGEBRA. Again, multiplying (1) by b^ and (2) by b^, we have a^b<2,x-^b^b^y=b^c^, (6) a^b^x^-b^b^y^b^c^, (7) Subtracting, {a^b^—a^b^x—b^c^ — b^c^y (8) Equations [1] and [2] may be used as formulas for solving any system of simple equations containing two unknown numbers. 251. Since the denominators in the right members of [1] and [2] are the same, we may write X __ y 1 ^2^1 ~" ^1^2 ^1^2 ""^2^1 a-^b^—a^bi 252. Discussion of the Systems. From [1] and [2] we observe the three following cases : I. Suppose a -^c 2— a 2C 1=0 and aib2^ci2bi^0; that is^ suppose — ^=^=7^=-^. In this case the value oi y is 0. ^2 ^2 ^2 What will result if ^2^i~"^i^2=0 and a-^b^—a^b-^^^O} II. Suppose ^1^2 — ^2^1=0 ^^^ <3^i^2""<^2^i=0; that is, suppose ~=~=T^* I^ ^^is case the values of x and y ^2 ^2 ^2 a c b each take the indeterminate form 7^. Now if — =-^=-^ U ^2 ^2 ^2 say=r, then aiX+b-i^y=^c-i can be made from a2:^^+^2j^ =^2 by multiplying both members by a certain number r. Thus, one equation is merely a repetition of the other equation, and since one of these equations is satisfied by an indefinite number of values of x and j/, it follows that an indefinite number of values of x and y satisfy the system a-^^x+b^y^c^ and <22jr+<^2j^'=^2- SIMULTANEOUS EQUATIONS. 1 75 When the values of x and y take the form % the equations a'^x-\-b^y=c^ and a^x-^-b^^y^c^ are said to be Dependent and the system is satisfied by an iridefinite number of values of X a?id y. The student may try to solve the system 3:x;+3>'=6, 8^+8y=16. III. Suppose a -^c 2^(1 2C 1=^0 and a -^b 2 — a^bi^O) that is, suppose —=7^=7^—. In this case the denominators in ^2 ^2 ^2 the values of x and y become 0. It is plain that by dividing the members of each equation of the system by the coefficient of x the system reduces to x+ky=l, x+ky=m^ where l^m. It is now evident that one equation of this system contradicts the other. A B When the values of x and y take the form -^ and -^ the equations a^x+b-^y^^c^ and a2X-\-b2y=C2 are said to be Incompatible, and the system is satisfied by no finite values of x and y. The student may try to solve the system 2x + 5y-=9, 4;c + 10/ -15; i. e.t the system x-k-\y=%, x+fjK=^. CHAPTER XII. QUADRATIC EQUATIONS. 253. In Art. 21 was given a definition of the degree of a polynomial with respect to any letter or letters. The Degree of an Equation is its degree with respect to the unknown numbers; i, e, , it is the degree of that one of its terms whose degree with respect to the unknown number is highest. Remember that the last letters of the alphabet are used to stand for unknown numbers. 254. A Quadratic Equation is an equation of the second degree. In this chapter we deal only with equa- tions of one unknown number. Quadratic equations are divided into two classes: Pure or Incomplete, and Affected or Complete. A pure or incomplete quadratic equation is one which contains the second but not the first power of the un- known number, as 3jf2 = 12 and — = — =2. o An affected or complete quadratic equation is one which contains both the second and first powers of the unknown x^ X number, as Zx'^-\'^=Z^ and —+-=3. 255. A Root of an equation is any number which substituted for the unknown number will satisfy the equation, i. ^., will cause the equation to be true. For example, 2 is the root of the equation x^+x=6y for if 2 be written in place of x we get 2^+2=6, or 4-|-2=6, which is true. Again, —3 is also a root of x^+x=6y QUADRATIC EQUATIONS. 1/7 for if — 3 be written in place of ;i;we get (—3)2 — 3=6, or 9—3=6, which is true. Although we deal in this chapter only with equations of the second degree, still this definition of root will hold good for an equation of any degree whatever, but it must be understood that the word can be used only with ref- erence to an equation of one unknown number. The student must not confuse the root of an equation with the root of an expression. See Art. 190. 256. The Solution of an equation is the process by which the roots are found. PURK QUADRATIC EQUATIONS. 257. If -^^=4 we know that x must be some number which raised to the second power will give 4. Now there are two such numbers, 2 and —2; therefore x=^ or —2. If JT^ be placed equal to some other number, there will be two values of x of opposite signs but other- wise just alike. Solve the equation 2 (^r^- 3)— 22=4. Performing indicated operation, 2x2 -6-22^:4. Transposing, 2;*:2 =4+6+22. Uniting terms, 2x2=32. Dividing by 2, 5C«=16. Hence, ^=4 or —4. 258. To solve a pure quadratic equation, collect all the terfns containing the unknown number in the first member of the equation and all the known terms hi the second mem- ber; unite each group of terms into a single term; divide by the coefiiciejit of the square of the unknown number, and extract the square root of each side of the resulting equation. 178 UNIVERSITY ALGEBRA. 259. In solving a pure quadratic equation we usually write both values of the unknown number at once. For example, in the equation x^=4j after extracting the square root of each side, we write :r=±2, using the double sign to show that 2 or --2 will satisfy the given equation x^ = 4:. The student may think that we should write the double sign on do^/z sides of the equation instead of one side only, thus: ±^=±2. But this we know means x=2 or x=—2 or —x=2 or —x= — 2. The third of these equations is really the same as the second, and the fourth is really the same as the first, so that we really get no more values by writing ±x—±2 than we do by writing x=# ±2. 260. Whenever we extract the square root of each side of the equation we should write the double sign ± on one side of the equation obtained. 261. When in treating a pure quadratic equation by the method described in Art. 258 we arrive at an equa- tion in which the right-hand member is not a perfect square, we cannot find the square root exactly, but can do so approximately. In such cases we merely indicate the square root; for example, if we have the equation x^ = S we write ;r=±l/3. KXAMPi^BS. Solve the following equations : 1. x^+S=4. 5. (:r2 + l) + (^2 + 2)-f (;r2 + 3) = 306. 2. x^-hS=7. 6. S(ix^'-l)+4(x'^-2)=5x'^ + S9. 3. 3:^2-4=204-^^ 7. (;r2-4)-f (jr2-f 2)=^2+4^ 4. 5x''-7=2dS + 2x\ 8. 2(;r2 + i)^3(^2 + 3)_5^^ 9. (x''-l)-(x^-2)-(x^-S')=SiSx\ QUADRATIC EQUATIONS. 1 79 "•^H3+5 = ^- ^3.-y s 10=0. 12. 2;»;H =Sx 14. -^ — -+-=-. XX X x^—1 4 2 15. 2 (:v2_i)_3 ^^2 4.1)^10=0. 16. 9;t;2_ie==o. 18. (2^+l)2=4;f+2. 17. 12;t2_75=o. 19. 8^2_i99_(;^+l>)2_2;i;. SOI^UTION BY FACTORING. 262. The equation ;r^ = 4 may be written in the form ^2—4=0. Now, as the first member of this equation is the difierence of two squares, it may be factored, and hence the equation may be written (;r— 2)(^+2)=0. 263. The product of two or more factors is equal to zero whenever any one of the factors is zero. Therefore the equation (x—2)(ix-{-2)=0 may be satisfied in either of two ways: first, when jt— 2=0, z. e., when x=2; and second, when x-\-2=0, z. e., when x==—2. As another example take the equation 5;r2 -9=2:^2 + 18. Transposing all the terms to the first member, we get 5^2__2;r2-9-18=0, or 8;»;2__27=0. Dividing by 3, x'^ —9=0. Factoring, (ji;— 3)(^-f3) = 0. This equation can be satisfied in either of two ways : first, when x—S=0, z. e., when x=S; and second, when .^+3=0, z. e.y when x=—S. » 264. By the method illustrated in these two examples we get the same roots as would be obtained by the former method. Thus we have another method of solving a pure l80 UNIVERSITY ALGEBRA. quadratic equation, viz. : Collect into one term all the un- known numbers^ and into another term all the known num- bers; write these two terms on the same side of the equation, making the other side zero; divide both members by the coefficient of the square of the unknown number (remem- bering that when zero is divided by any nu7nber the quotient is still zero); factor the resulting first member; put each factor separately equal to zero, and solve the resulting simple equations, KXAMPIvBS. Solve the following equations by the method just explained : 1. .^2-100=0. 6. (^+2)2=4(:r+5). 2. 4^2__ioo=0. 7. (;r+2)(:r+3)=5^+42. 3. (2;i;+l)2=4;t:+82. 8. j^2_^.r+l=;t:+101. 4. 5;i;2=80. 9. x2-2:r--3=33-2;r. 5. ;i;2 + i==26. 10. (;c+^+^)2 = 2(« + ^);i;+2(«2+^2)^ AFFBCTKD QUADRATICS. 265. If we have given the equation ^2 = 25 we solve it by one of the preceding methods, and find x=±:5. Similarly, if we have the equation (^+1)^ = 25 we find x+l=±B. If we take the upper sign we get ^+1=5, or^=4, and if we take the lower sign we get x-{-l=—5, or x=^—6. Thus we find the roots of the equation (;i:+l)2=25, or what is the same, x^-{-2x+l=25, or x^-\-2x=-24:/ 266. Therefore, to solve x^+2x=24y we first add 1 to each member to make the first member a perfect square, and get x'^-\-2x+l = 2b\ then we take the square root of each member, and get :r4-l=±5, whence x=4: or —6. QUADRATIC EQUATIONS. l8l Similarly, to solve x'^-\-6x=7, we add to each member a number that will make the first member a perfect square. Plainly, 9 is the number; hence x'^ + 6x+9=16. Next we take the square root of each member, and get ;r+3==b4, whence x=l or— 7. 267. Similarly, to solve any affected quadratic equa- tion, we first reduce the equation to a form in which the terms containing x^ and x are in the first member, and the term not containing x is in the second member; sec- ond, if the coefiicient of x^ is not unity, we divide each member of the equation by that coefficient, so that the coefficient of ^^ shall be unity; third, we add to each member of the equation a number that will make the first member a perfect square, and then take the square root of each member and solve the resulting simple equations. 268. Adding to a given expression a number that will make the sum a perfect square is called Complet- ing the Square. 269. When the coefficient of jt^ is unity what number is it that we must add to each member of an equation to make the first member a perfect square? To answer this let us see how a perfect square is produced. We know that (ix+ay^x'^-\-2ax+a^, and (x — a)^ = x'^ — 2ax+a^. Notice here that whatever number is represented by a, the third term is the square of one-half the coefficient of x; hence the number to be added to each member of the given equation is the square of one-half the coeffi- cient of X. 1 82 UNIVERSITY ALGEBRA. 270. Hence, to solve any affected quadratic equation: I. Reduce to a form in which both x^ and x are in the first member and all terms not contai7iing x are in the sec- ond member, II. If the coefficie7it of x'^ is not already unity, divide each member of the equation by that coefficient^ thus making the coefficient of x'^ unity. III. Complete the square by adding to each member the square of one- half the coefficient of x. IV. Extract the square root of each member of the equa- tion and solve the resulting simple equations. KXAMPivKS. Solve the following equations : 1. .a:2+4x=5. 7. ^x'^ —^x-=%, 13. x'^ — Vdx-r- —9. 2. ^24.6;r=16. 8. ;r2-7.r=-6. 14. 1x''-\hx=m. 3. 2;i;2— 20;r=48. 9. x'^—ax=^a'^. 15. x'^-\-^x= —15. 4. ;tr2+3;c=18. ID. ^2_2a^=3«2. 16. 3;i:2_|.i2;r=36. 5. ;r2+5.r=36. \i,x''-x=^X 17. 2.r2 + 10x=100 6. 3;r2 + 6.a:=9. 12. ^2 +^^^^2+^^ 18. ^2_5^^_4 19. 3;r2 — 12a.a;=63^^ 20. ^x'^ — Vlax=\^a'^, 271. The above method will enable us to solve any affected quadratic equation that may be given, but fre- quently it will oblige us to use fractions, and unless the terms of the fractions are small numbers it will be easier to complete the square by another method, which we will now consider. 272. We know that {ax-{'by=^a'^x'^-\-1abx-\rb'^, and lax—by = a'^x'^—2abx-^b'^, so that each of these two second members is a perfect QUADRATIC EQUATIONS. I 83 square. We therefore seek to reduce the given equation so that the first member shall be in the form of one of these two second members. 273. Notice two things: first, that the coefiicient of ;i:2 is a perfect square, and second, that the third term equals the square of the quotient obtained by dividing the second term by twice the square root of the first term. Therefore, to reduce the first member of any given quadratic equation to either of the forms a'^x'^+2adx-i-d^ or a'^x^ — 2adx+d'^, I. Reduce the equation to a form in which the terms con- taining x'^ and X are in the first member^ and all terms not containing x are in the second member. II. Multiply each member of the equation by a number that will make the coefficieiit of x'^ a perfect square. III. Add to each member the square of the quotient obtained by dividing the second term by twice the square root of the first term. The rest of the process of solution is like that already given, viz.: take the square root of each member and solve the resulting si77iple equations. Let us solve by this method the equation 3;c^ + 7^ + 11=2^+33. Transposing %x and 11, we get 3^2 + 5;*:=: 22. Multiply each member by 3 or 12 or 27 or 3 times any square number and the coefficient of 5C* will be a perfect square. Taking the first of these multipliers, we get 9^2^15x=66. Adding to each member {^Y, or (f )2, we get Taking the square root of each member, we get 3x+f=±Y-. Hence, 3x=:6 or —11 and x = 2 or — V** I 84 UNIVERSITY ALGEBRA. If we had multiplied by 12 instead of 3 we would have obtained 36:jc2 + 60;^H- 25=264+25=3289. Hence, 6?cH-5=±17. Hence, 6;c=12 or -22, and x=2 or — V- kxampi.es. Solve tlie following equations: 1. Sx^+4x=7. 5. 2x''-S5=Sx. g. 2;i;2 + i0:r=300. 2. Sx^+6x=24:. 6. 3.^2— 60=5.r. lo. 3;t:2 — 10^^=200. 3. 4.x'^—5x=26. 7. 3.^2—24=6^^. ii. 4x'^—7x-i-^=0. /^, 5x^-7 x==24, S.2x^-Sx=104:. 12. |:r2-|:i;=-if 13. 9x^ + ex-AS=0. 17. 2.^2 -22;^;=- 60. 14. 18;i;2_3^_6e:=0. 18. 3jr2 + 7;i;-370=0. 15. ^x^-Sx+{i=0. 19. 5^2_l^__7__0. Ic-flO 5 hours less than in the first instance, we have 600 600__ X :r+10~ From this equation we have 5^2 -}-50x=600 (^+10) - 600^, or 5xZ-\-o0x=m00, or ^24-10^=1200. Completing the square, we have %2 + 10x+25=1225. Extracting the square root, we have x+5=±35. Hence x==30 or -40. The first of these results (30) agrees with the conclusion reached above; but here another question arises, what is to be done with the result — 40 ? So far as the algebraic work goes —40 is as good a result as 30, but a train traveling —40 miles an hour is something void of meaning, so this result is rejected and 30 is retained as the true result. 277. It will often happen, as in the example just worked, that the solution of the equation formed as above described leads to a result which does not apply- to the problem we are solving. The reason of this is that the algebraic statement of the problem (by means of the equation formed as above described) is more general than the statement in words. It will, however, usually be quite easy to select the result which belongs to the problem we are solving, and then we reject the other result as inapplicable. 1 88 UNIVERSITY ALGEBRA. PROBLEMS. I. A cistern can be filled by two pipes in 33-|- minutes. If the smaller pipe takes 15 minutes more than the larger one to fill the cistern, in what time will it be filled by each pipe singly? Let us see if the smaller pipe will fill the cistern in 45 minutes. If so, the larger pipe will fill the cistern in 30 minutes. If larger pipe will fill the cistern in 30 minutes it will fill ^ of the cistern in Icminute, and if smaller pipe will fill the cistern in 45 minutes it will fill :^ of the cistern in 1 minute. Therefore together they will fill s^-faV=tV of cistern in 1 minute. But together they fill 1 3 oqi^^Tofj of cistern in 1 minute, and as ^ is not equal to y§o we con- clude that 45 minutes is noi the time in which the smaller pipe will fill the cistern. Let ^= number of minutes required to fill the cistern by the smaller pipe; then ^ — 15=number of minutes required to fill the cistern by the larger pipe. If larger pipe will fill the cistern in % — 15 minutes, it will fill r-= of cistern in 1 minute; and if smaller pipe will x-16 -^ ^^ fill the cistern in x minutes, it will fill - of the cistern in 1 minute. ^11 2x 15 Therefore the two pipes together will fill — -j — or — -— of I x — 15 X x{x — \^) cistern in 1 minute. Together they fill ^— - or j-g^ of the cistern in 1 minute. Therefore, we have the equation 3 2x-15 100"^(5C-15) Clearing of fractions, 3^^ -45:>c=200x - 1500. Transposing 200;c, 3^^ _ 2455c = - 1500. Multiplying by 12, 36x2-2940;*:= - 18000. Adding (245)^ to each member to complete the square, 36x2 -29405C+60025=42025. Extracting the square root of each member, 6x- 245 =±205. Hence. 6x=450 or 40; therefore, x=75 or 6f . From this it appears that the smaller pipe will fill the cistern in either 75 or 6| minutes; but evidently 6| is not admissible, for it takes the smaller pipe 15 minutes more to fill the cistern than it takes the larger pipe; but it takes the larger pipe some time to fill the cistern. QUADRATIC EQUATIONS. 1 89 So it is plain that it must take the smaller pipe more than 15 minutes to fill the cistern. We therefore reject the result 6| as being inad- missible; the other result, 75, is admissible, however, and satisfies the requirements of the problem. Hence it takes the smaller pipe 75 minutes, and therefore the larger one 60 minutes to fill the cistern alone. 2. A merchant selling some damaged goods for $72, finds that his loss per cent, is \ of the number of dollars the goods cost. Find the cost of the goods. Let ;*:= number of dollars the goods cost; then 5= the loss per cent. Therefore, ^^=the entire loss. By the statement in the problem, we have 5c2+21600=300x, ^8-300x=-2160a ;*r2-300x+22500=r900, 3C-150=±30, 5C=:180orl20. Each of these answers fulfills all the requirements of the problem, and each is admissible. 3. One of two numbers is f of the other one and the sum of their squares is 208. Find the two numbers. 4. Divide the number 60 into two such parts that the quotient of the greater divided by the less may equal one more than twice the less. 5. A merchant bought a quantity of cloth for $120; if he had bought 6 yards more for the same sum, the price per yard would have been $1 less. How many yards did he buy, and what was the price per yard ? 6. A merchant sold two pieces of cloth which together contained 40 yards, and received for each piece twice as many cents per yard as there were yards in the piece. For the smaller piece he received ^ as much as for the larger one. How many yards were there in each piece ? I go UNIVERSITY ALGEBRA. 7. A merchant sold some goods for $39, and in so doing gained as much per cent, as the goods cost him. What was the cost of the goods ? .8. Find a number such that 3 more than twice the number multipHed by 3 less than twice the number may- give a product of 112. 9. A man traveled 105 miles, and then found if he had gone 2 miles less per hour he would have been 6 hours longer on his journey. How many miles did he travel per hour? 10. A man bought two farms for $2800 each; the larger contained 10 acres more than the smaller, but he paid $5 more per acre for the smaller than for the larger. How many acres were there in each farm ? 11. The length of a rectangle is 10 feet more than the breadth, and the area is 600 square feet. Find the length and breadth of the rectangle. 12. A flower bed 9 feet long and 6 feet wide has a path around it whose area is equal to the area of the bed itself. What is the width of the path? 13. A number consists of two digits. The digit in unit's place being the square of the digit in ten's place, and if 54 be added to the number the digits are reversed in order. What is the number? 14. Find a number such that if it be added to 94 and again subtracted from 94 the product of the sum and dif- ference thus obtained shall be 8512. 15. Find a number such that if its third part be multi- plied by its fourth part and to the product 5 times the number be added the sum exceeds 200 b}^ as much as the number required is less than 280. QUADRATIC EQUATIONS. I91 16. A man bought a horse and sold it again for $119, by which means he gained as much per cent, as the horse cost him dollars. How many dollars did the horse cost him? 17. The combined area of two squares is 962 square feet, and a side of one square is 18 feet longer than a side of the other. What is the size of each square ? 18. A square field contains a number of square rods equal to 260 more than 32 times its perimeter. How many rods in one side of the square? 19. The sum of the ages of a father and son is 85 years, and ^ of the product of their ages in years exceeds five times the father's age by 200 years. What is the age of each ? 20. What is the price of oranges when 10 more for $1.20 lowers the price one cent each ? 21. A certain number is the product of three consecu- tive whole numbers, and if it is divided by each one oi these three factors in turn the sum of the three quotients thus obtained is 767. What is the number? 22. The sum of the squares of three consecutive odd numbers is 83. What are the numbers? 23. The sum of the squares of four consecutive even numbers is 120. What are the numbers? 24. Divide the number 18 into two such parts that their product shall exceed 30 times their difierence by 20 25. In a bag which contains 60 coins of silver and gold each silver coin is worth as many cents as there are gold coins, and each gold coin is worth as many dollars as there are silver coins, and the whole is worth 1505. How many gold and how many silver coins in the bag? 192 UNIVERSITY ALGEBRA. 26. A man bought a number of horses for $10000 ; each cost four times as many dollars as there were horses. How many horses did he buy? 27. A room whose length exceeds its breadth by 8 feet is covered with matting 4 feet wide, and the number of yards in length of the matting exceeds f the number of feet in breadth of the room by 20. Find the length and breadth of the room. 28. There are two numbers whose difference is 7, and half their product plus 30 is equal to the square of the smaller number. What are the numbers ? 29. A and B start together on a journey of 36 miles. A travels one mile per hour faster than B and arrives three hours before him. Find the rate of each. 30. Two workmen, A and B, are engaged to work at different wages. A works a certain number of days and receives $27, and B, who. works one day less than A, receives $34. If A had worked two days more and B two days less, they would have received equal amounts. Find the number of days each, worked. EQUATIONS SOIvVKD I,IKK QUADRATICS. 278. Some equations which are not quadratics may be solved by the methods explained in this chapter. We have had such equations as ^2-— 13^+36=0, and have seen that such equations are easily solved. Now it is plain that we can use some other symbol in place of x to designate an unknown number. Thus we might have an equation in which y'^ stands in place of x, and of course y^ in place of x'^ , and the equation would be y-13j/2+36=0, QUADRATIC EQUATIONS. 1 93 from which, by solving in the usual way, regarding y'^ temporarily as the unknown number, we obtain jj/2=4 or 9. Hence j/=±2 or ±3. In a similar manner we could treat equations in which more complex expressions stand in place oi x'^ and x in the equations before used, but whatever expression stands in place of x the square of that expression must stand in place of x*^, else the equation cannot be solved by the methods of this chapter. KXAMPI,:^S. Solve the following equations : 1. :r4 -29^2 + 100=0. 3.^4_i7y + ie=0. 2. jt:^ -35:^3 +216=0. 4. J/+ 81/^+15=0. 5. (^+3)«-28(^+3)3+27=0. 10. (;r2-5;i;+6)2-14(;ir2-5;i;+6)-24=0. 13— U. A. CHAPTER XIII. THEORY OF QUADRATIC EQUATIONS AND EXPRESSIONS. 279. Equations and Expressions. When all the terms of a quadratic equation are transposed to the left member, the right member is zero, and the equation takes the form ax^ + dx+c=0; or, if the equation be divided through by the coefficient of x^y it takes the form x^ +px+q=0. The left member of either of these equa- tions is a Quadratic Expression. Since a root of an equation has been defined as any expression which substituted for the unknown number satisfies the equation, therefore it is evident from either of the forms ax^ + dx -j- c=0 or x'^+px-\-q=^0 that a root of a quadratic equation may also be defined as any ex- pression which substituted for the unknown number in a quadratic expression causes that expression to vanish; that iSy to equal zero. Thus, the equation jr^— 3;ir=10, whose roots are 5 and —2, when placed in the form of a quadratic expression equal to zero, becomes ^2--3;r— 10=0. It is now seen that the roots are such numbers that, when substituted for X, cause the expression to vanish. For the expres- sion is x'^—Zx—\^, and putting 5 for x it becomes 25—15—10, which is zero. Putting —2 for ;i: the ex- pression becomes 4+6—10, which is also zero. If any other number than a root is put for x the ex- pression will not vanish ; thus when ^=—4, the expression becomes 16 + 12 — 10 or 18, ;tr=— 3, the expression becomes 9+ 9 — 10 or 8, i»= — 2, the expression becomes 4+ 6—10 or o, THEORY OF QUADRATICS. 195 x= — l, the expression becomes 1+ 3— 10 or -— 6, x= 0, the expression becomes 0+ 0—10 or —10, x= 1, the expression becomes 1— 3—10 or —12, x= 2, the expression becomes 4— 6—10 or —12, x= 3, the expression becomes 9— 9— 10 or —10, x= 4, the expression becomes 16—12—10 or — 6, x== 5, the expression becomes 25 — 15—10 or o, x=s 6, the expression becomes 36—18—10 or 8. 280. Factors of a Quadratic Expression. We have already factored some quadratic expressions. Let us now take the general quadratic expression x^+px+q. Adding and subtracting ^ we obtain or (^2+^^+^^)_(^_^) or writing this as the difference of two squares, we obtain (-l)-(^^^)■ Factoring this expression, we obtain From this it is evident that a quadratic expression can be resolved into the product of two linear factors; i. e. , two factors each of which is of the first degree with respect to the unknown number. 281. If we should solve the equation ^^+/jr+^=0 p [p^ we would find its roots to be —■^±:^~ — q. Repre- senting these roots by r^ and rg respectively, we have 196 UNIVERSITY ALGEBRA. »=2-V4 ^- Comparing these with the factors of the quadratic ex- pression x'^-\-px-\-q obtained in Art. 280 we see that the above factors are {pc—r^{x—r^. Therefore , if the roots of a quadratic equation are r-^ and r^ , the equation may be written in the form (x — r{){x — r2)=0. It follows at once from this statement that if all the terms of a quadratic equation are transferred to one m^ember that member is exactly divisible by x minus a root. Since with the meanings given to r^ and r,^. ^^ roots of x'^ +px+q=0 are the same as those of (;t:— ri)(^— r2)=0 it follows that the form (x—r-^)(x—r2')=0 may be used interchangeably with x'^ +px+q=0 to represent any quad- ratic equation, 282. Number of Roots. Representing, as before, the roots of x^+px-{-q=0 by r^ and ^3, the values given in Art. 275 show that there are two roots to any quad- ratic equation, but for certain values of p and q these two values are exactly the same. This is the case when ^ — ^=0, for then the second term of each root reduces to zero and each root of the equation reduces to — ^. In this case there is really only one value of x that will sat- isfy the equation. Instead, however, of saying that there is only one root of the equation we say that there are two roots, but that they are equal to each other. Of course this is only another way of saying that there is but one value of X, but further along it will be apparent that there is a great advantage in speaking of two equal roots rather than one root. THEORY OF QUADRATICS. 1 97 For certain values of p and q the number under the radical is negative and the roots are imaginary. (See Art. 196.) In this case there is 710 rea/^yalne of .;«; which will satisfy the equation. Instead, however, of saying that there is no real root we say that there are two roots, but they are imaginary. Of course this is only another way of saying that there is no real value of .;*:, but further along it will be apparent that there is great advantage in this form of expression. With the understanding just explained about equal and imaginary roots, we may say that every quadratic equatio7i with one unknown num- ber has two roots a7id only two, 283. Sum of Roots. Representing, as before, the roots of x'^-\-px-\-q=^ by r^ and r^ we have found ^--l+V?-^' p \p'' By addition we find r^-}-r2 = —p. Therefore whe7i a quadratic equation is in the form x^-\-fix+q=0 the coefficie7it of x with its sign cha7iged is equal to the sum of the roots, 284. Product of Roots. As in the previous article P , fp^ we have ^^"""2 \^ — ^* P IP' By multiplication, recognizing the product of a sum and difference, we find * By way of distinction, a number that is not imaginary is called real. 198 UNIVERSITY ALGEBRA. — (-l)-(Vl^-)=?-(^-)- Therefore when a quadratic equation is in the form x^'j-px-{-q=0 the term not containing x is equal lo the product of the two roots. In any equation the term not containing the unknown number is called the Absolute Term. Therefore when a quadratic equation is in the form x'^-\-px-j-q=0 the abso- lute term, is equal to the product of the two roots, 285. Upon the results reached in the two preceding articles we may found a new method of solving the quad- ratic equation x'^+px-k-q^^. We have ^1+^2 = —p\ (1) r^r.,^q, (2) Squaring (1) we obtain r2+2rir2 + rl=/2. (3) From (2), 4^1^2=4^. (4) Subtracting, rl-l?^ r^-\-r\=p'' -4.q. (5) Taking square root, rj —r^ = Vp'^ —iq. (6) ri + r, = -p. (7) Adding, 2r^ = -p+l/p''-4q, (8) Hence, r,»-|+Jv^^34^=-|+^ZEl£ Subtracting (6) from (7), 2r2 = —p—Vp'^—ig. Hence, r,= -|-iv//^^=-|-y^-y. THEORY OF QUADRATICS. I99 DISCUSSION OF "run roots. 286. Real and Imaginary Roots. The roots of x^+px+g=0 are ^^=-I+Vt^^> and ^2=~|-y^ — g- From the distinction between real and imaginary ex- pressions it follows that these two roots are real when the expression under the radical sign is positive, and imaginary when the expression under the radical sign is negative. Therefore the two roots are real when ~ — g is positive; that is, when ^>g; and the two roots are imaginary when ^ — q is negative ; that is, when ^<^. If the quadratic equation were given in a different form we could still find when the roots would be real and when imaginary, for we simply have to solve the equation and notice what expression appears under the radical sign, and we conclude that the roots will be real when this expression is positive, and imaginary when this expres- sion is negative. BXAMPI.KS. Solve the following equations and determine when the roots are real and when imaginary: I. x^-2ax-h2d=0. 6. ax'^—4:bx—4:=^0. 2. ax^-2bx+Zc^0. 7. x^-iax+b^O. 3. a a 8. x^—4x-{-a=^0. 4. 2x'^+Sax=^55. 9 a'^x^-}-4bx—4:C=0. 5. ax'^+Ux+^ab^O. 10. 2ax'^ + Sbx—4:abc==0, 200 UNIVERSITY ALGEBRA. 287. Equal Roots. In order that the above values of r^ and rg may be equal to each other we must have If this value of g be substituted for g in the original equation, that equation becomes x^+px+^=0, or ^i=n Therefore when a quadratic equation has ('+l)=» equal roots and all its terms are in the left member, that member is a perfect square. In the preceding article we found that the roots of x^ -\-px+g=0 are real when ^ — q is positive, and imag- inary when -t — g is negative, and in this article we have found that the roots are equal to each other when I — ^ ^^ ^^^^* ^^ ^^^^ appears that the case of equal roots is just on the dividing line between real and imag- inary roots. Kxampi,e;s. Determine when the roots of each of the following equations are equal to each other. 1. x^—Aax—5d=0. 4. x'^—2x+a=0. 2. x^-ax+2d=0. 5. x^+--l=0, a b 3. x'^-\-2ax-\-ab^=Q, 6. jt^— -^— ^=0. THEORY OF QUADRATICS. , 20I 288. Roots Numerically Equal but of Opposite Signs. In order that the above values of r-^ and ^3 may be numerically equal but of opposite signs we must have -IW^--IW1^- • • 2 2 289. Roots Real and Numerically Equal but of Opposite Signs. In the preceding article we found that the roots are numerically equal but of opposite signs when p=0. If, in addition, the roots are to be real, the expression under the radical sign must be positive ; but when p=0 the expression under the radical sign reduces to —q, and for —q to be positive it is necessary for q to be negative. Hence ifi order that the roots of x'^ -\-px-j-q=0 may be real and ntimerically equal but of opposite signs it is necessary that p=0 and that q be negative, KXAMPLKS Find when the roots of the following equations are real and numerically equal but of opposite signs. 1. 2x'^+4:ax—U=0, 4. ^x'^-^ax=a'^—b'^, 2. ax'^—2abx+bc=0, 5. x'^—2{a—b)x-\-b'^—2ab=0 3. x''-+^ax—4:=0. 6. bcx'^ + (^b'^-^c'^)x-'bc=^0, 290. One Positive and One Negative Root. When an expression consists of two terms, that term which has the greater numerical value, that is, the value differing most from zero, is called the Dominant Term. Now it is plain that when the dominant term is positive the whole expression is positive, and when the dominant term is negative the whole expression is negative. 202 UNIVERSITY ALGEBRA. Applying this principle to the case of the values of r^ and ^2 found above it is plain that if we wish one root to be positive and one negative the part containing the radical sign must be the dominant term. 4 ^-q>-^^,.:i^-g>i^;.:-g>0. .'. ^ is negative. Therefore the equation x^+px+g=0 has one positive ayid one negative root when q is negative, 291. Both Roots Negative. In order that the values of r^ and ^3 found above may both be negative it is plain that —~ must be the dominant term and must be nega- P tive. Evidently, for — ^ to be negative / must be pos- itive, and for — ^ to be the dominant term we must have — ^ numerically>'%/=^^^ — q, ... ^>^-q; ..0>-g; ,'. —q is negative; /. q is positive. But q must be less than ^ in order that the roots may be real. Therefore hath roots of the equation x^-{-px+q=0 are negative when both p and q are positive and ^^q. 292. Both Roots Positive. In order that the values of r^ and f 2 found above may both be positive it is plain P that — ^ must be the dominant term and must be posi- P tive. Evidently, for —^ to be positive p must be neg- tive, and for — ~ to be the dominant term we must have THEORY OF QUADRATICS. 203 .*. —q is negative; .'. ^ is positive. But q must be less than ~ in order that the roots may be real. Therefore both roots of the equation x^+px+q=0 are positive when p ts negative y q is positive ^ and ^>^. BXAMPLKS. Solve the following six equations and determine when one root will be positive and one negative. 1. ^2 + 6^^_4^=0. 4. 2:r2+6^^— 5^=0. 2. ^2+1^+23=0. 5. x'^—x-U^^, 3. ;i;2— |^;r— 5<^=0. 6. (;r+^)2— ^2^0. Find when both roots of the following four equations are negative. 7. ;*:2 4-10^jr— 10<^=0. 9. x'^—hax-^a'^ +b=^, 8. ;»;2_3^^_2^==0. 10. (;r— «)2 + (^+^)+^2^0 Find when both roots of the following four equations are positive. 11. (^— a)2 + 3^— *=0. 13. x'^-\-%{x--d)^ah-^^. 12. x'^-\-a{x—U)=Q, 14. (;r+3a)2 + (;»:+3)+^=0 15. Show that the roots of the equation x'^-^-a'^x-^-b'^^Q are both negative when a'^'>U. 16. Show that the roots of the equation x'^—a'^x-\-b'^=^^ are both positive when ^2^2^. 17. Show that the equation x'^—ax—b'^^==^^ has one pos- itive and one negative root. 18. Show that the equation x*^ +ax—b^=0 cannot have imaginary roots. 204 UNIVERSITY ALGEBRA. 19. If ri and rg are the roots of x^-i-px+g=0 find the value of rl +r| in terms of p and g, ^1 + ^2 = -/. (1) ^i^s==^- (3) From (1), r} + 2r^r2 + r|=/2. and from (2), ^^1^2='^^ • Therefore, rl + rl=_p^ -2c^. > 20. Find the value of 1 — in terms of p and q. 21. Find the value of --\ — '^- in terms of p and c 22. Prove that (r^ — ^2 ) ^ =/ ^ — 4^ . 23. Form an equation whose roots are the squares of the roots of the equation x'^ -\-px-{-q=^. 24. Find the value of a such that the roots of the equa- tion x^—'4:X-^a=0 will differ by 2. Historical Note. The origin of the solution of quadratic equations has not been definitely traced to any one man or any one race. It is a curious fact that the geometric solution of quadratic, as also of cubic, equations was invented before the analytic method. The former was known to the Greeks. (See Euclid, book VI., props. 27-29; data, 84, 85. Negative values of roots were ignored by them. Constructions of the positive roots of equations were made by the intersection with one another of straight lines, conic sections, or higher curves. Among the Arabs and the people of the Occident down to the time of Cardan, the geometric methods of the Greeks preponderated. The algebraic solution of quadratic equations was known to Dio- phantus, an Alexandrian Greek of the fourth century, A. D., who wrote a treatise on arithmetic. He rejected — roots and failed to recognize two roots in quadratic equations even when both were -f-. To the Greeks the idea of multiple-valued solutions was entirely foreign. The first to observe that a quadratic has two roots and to recognize the existence of absolutely negative quantities were the Hindoos. The earliest complete method of solving these equations, together with their application to practical problems, is found in a THEORY OF QUADRATICS. 205 Brahmagupta, work of a Hindoo astronomer of the seventh century A. D. There is considerable resemblance between the writings of Diophantus and of Hindoo mathematicians. We have reason to believe that during the time of commercial intercourse between Rome and India, by way of Alexandria, Diophantus got the first glimpses of algebraical knowledge from India, while the Hindoos afterwards drew upon the writings of the gifted Alexandrian. The Hindoos were no mean mathematicians. While the Greeks excelled in geometry they surpassed in arithmetic and algebra. To the Brahmins we owe that great invention the "Arabic Notation." The Hindoos solved prob- lems in indeterminate analysis which taxed to the utmost the powers of Euler and Lagrange in rediscovering methods of solving them. CHAPTER XIV. THEORY OF INDICES. 293. The exponents whicli we have considered here- tofore are defined by the following equation : a**=aaaa , , . to n factors. In other words, a" is an abbreviated way of writing the product of n factors each equal to a, 294. It has been proved that positive integral ex- ponents follow the five laws expressed by the following formulas : a''a'•=a"+^ Art. 73. A ar-^ar^ar-\ ifn>r, Art. 97. B {ary^ar^ Art. 122. C (abcy=a''b''(f\ Art. 123. D Art. 123. E \b) ^r We shall find it convenient to refer to these formulas ^s A, B, C, D, and E, respectively, and we shall speak of them collectively as the Laws of Exponents or Indices. 295. It is plain that in order that oT may have a meaning by the definition already given the exponent n must be a positive whole number, but in the general investigations of Algebra it is very desirable that the formulas used shall not be thus restricted. We shall therefore endeavor to extend the meaning of a" so that fractional and negative exponents may be used. THEORY OF INDICES. 207 FR ACTION AI, EXPONENTS. 1 H 296. Since expressions like a'' and a^ have no meaning by the definition previously given of a number affected with an exponent, we are at liberty to define such expres- sions in the manner that is most convenient. Now if we can discover a definition that will make law A hold for fractional as well as integral exponents, that is the defi- nition we shall adopt. If fractional exponents can be defined in accordance with law A, then, r being any positive whole number, we ought to have 1 i 1^ ^ r ^ M-i+^-... tor terns a'-a'^a''. . . to r factors=«'^'' ~ =^a\ 1 that is, w ought to mean one of the r equal factors which multiplied together produce a ; that is, dr ought to mean the rth root of a. Further, n and r being any positive whole numbers, we ought to have a^ara^. . . to r factors =a^^^*** '' ""ss^"; n that is, ar ought to mean one of the r equal factors which n multiplied together produce a'*; that is, a^ ought to mean the rth root of a". Thus we see that 71 and r being positive whole num- n bers it is possible to give d^ a definition that will make law A hold for firactional as well as integral exponents. Therefore we adopt the following definition : Any posi- tive fractional exponent indicates a root of a power of a number; the numerator indicates the power and the denom- inator the root. This definition is purposely chosen to make law A hold for fractional as well as for integral exponents, and we shall presently see that by this same definition the other laws of indices will also hold. 208 UNIVERSITY ALGEBRA. 297. Root of Power Equal to Power of Root. I,et J=x. (1) Raising both members to rth power, we get a=x^, (2) because the rth root of the rth power of a equals a. Raising both members of (2) to the nth power, we get a-=x*'''=(x''y. (3) Extracting the rth root of each member, we get (a")'=^^ (4) because the rth root of the rth power of x"" equals x". Raising both members of (1) to the nth power, we get (^ary=.x^. (5) Hence, from (4) and (5), (a**y = (a^y; that is, the rth root of the nth power of a number equals the nth power of the rth root of that number. 298. Comparing this with Art. 296 we see that we have n found two meanings for a^\ first, the rth root of the n\\\. power of a ; second, the ^th power of the rth root of a. This may be written in the form of an equation as follows: a-=l/a^=(^^)«, [1] or, writing exactly the same equation, but using frac- tional exponents instead of radical signs, n 11 a^= (««)'-= (a'')^ [2] Kxampl:^. Write each of the following sixteen expressions, using fractional exponents in place of radical signs : I. l/a. 5. V~a^. g. Vx^, 13. i^a—b. 2. V^^, 6. (i/a)«. 10. {flcy. 14. {i/~^by, 3. ^~a^. 7. Va^. II. 1^^. 15. Va'^-b'', 4. 1^^. 8, (T/a)5. 12. {V^y, 16. i\a-\-by. 33- ^^. 37. n^- 34. li 38. ^i 3. 35. ;;^4. 39- <^*. 36. xT. 40. /i^. THEORY OF INDICES. 209 Find the numerical value of each, of the following six- teen expressions : 17. 4i 21. 625i 25. 8lt 29. 2561 18. 27^. 22. 64^. 26. 125i 30. 64^ 19. 9i 23. 216^ 27. 32I 31. 512^ 20. lei 24. 16* 28. 81^. 32. 1287-. Write each of the following expressions zn two ways^ using radical signs instead of fractional exponents : 127 n 41. r^. 45. a^. 42. X^, 46. ^2«. 1 w+1 43. JV'. 47. -^ '' • - o ^+-^ 44. <2J»'. 48. <2 ^ . 299. Value the Same whether Exponent is in its Lowest Terms or not. Let 71, r, and t be any positive whole numbers. We are to prove that a^=a^i. We know that because the rt th power of the rt th root of a number equals the number itself. Taking the rth root of each side, Raising both sides to the n th power, Each side is now a power of a root. By the meaning of fractional indices (Art. 297) we write this a'^-a'^f. [3] Therefore, a number with a fractional exponent has the same value whether the exponent is expressed in its lowest ter^ns or not. 14— u. A. 2IO UNIVERSITY ALGEBRA. 300. We have defined fractional exponents in such a way that law A holds for fractional as well as for integral exponents, and we now proceed to prove that with the same definition all the other laws hold for fractional as well as integral exponents. 301. Fractional Exponents follow Law B, Let Hy r, s, and / be any positive whole numbers, so that - and - are any positive fractions. We are to prove - - 71 S d^^d^=ar~, if — >- r t n s nt sr We know ar-^a^—an^atr^ by Art. 299. = {a^^Y^ic^'Y, by equation [2]. by law B for integral indices, since nt and sr are whole 71 S numbers and nf>sr if ->-.- nt—sr ■=^a rt ^ by definition, Art. 296. But this last fractional exponent is what we would get s ti if we should subtract - from — . Therefore Therefore, fractioTial expoTients follow law B. 302. Fractional Exponents follow Law C. Let 71, r, s, and t be any positive whole numbers. n ns Cask I. To prove (ary=a^ . n n n n We know Qi^y=a^a^a^ . . . to ^ factors, by definition of an integral index. ^^?+^^.. .to. terms ^ by kw ^. ns =ar ^ by adding fractional indices. Therefore, (aO^=a''. \p\ THEORY OF INDICES. 2X1 Cask II. To prove (a^y=ari, ft 1 ni 1 We know (ary=a(riy, by Art. 299. = ([«^«0V by meaning of a fractional index. by law C for integral indices. by taking the /th root of the /th power. n =^ari^ meaning of fractional index. «1 n Therefore, [ay^w^'. [6] n s ns Casb III. To prove (aO'=^''^ We know («^)^= [(^^)^^ by the meaning of a fractional index. = [a^y, by Case II. =a^^, by Case I. « £ ns Therefore, (an'=a''^ [7] Therefore, fractional exponents follow law C. 303. Fractional Exponents follow Law D, n n n n To prove {abcf—a^b^cr. We know {abcy=^\{abcYfy by meaning of a fractional index. by law D for integral indices. by law C, (Art. 302, Case I.) by law D for integral indices. n n n T=za^b^(^, taking rth root of rth power Therefore, {ahcY=arlr&. [8] Therefore, fractional exponents follow law D, 212 UNIVERSITY ALGEBRA. 304. Fractional Exponents follow Lavr £. S)-[©7 by the meaning of a fractional index. Yn J t>y law^ for integral indices. = ft-? law C, (Art. 302, Case I.) To prove We know \br) =[©7 law E for integral indices* = — taking r\h root of rth power. br Therefore, (|f =^ ^ M Therefore, fractional expone7its follow law E, Perform the*indicated operations in each of the follow- ing examples by means of the laws of exponents, now proved to hold for negative and fractional exponents. I. d^y. a'^. tfSx J=^i+f (by A)=J'^'^=J^. 11 ^ A. JL 4. x^xx^, 6. x^»xa^», 3 2 ^ L. 5. dJx a^, n. a''X ^2«. 8. a'^-^a'^. 11. d>aH^-^Aa'^b^, 13. ^a^-^^a^. ^1 n 1 n_ 12. 9^5-^(2^. 14. ab^-T-a'^b^^ 2. x^X-r^. 3. ;f5x;ri^. 2 1 9. h^-^m, % 8 ID. mJ-i-mT^. THEORY OF INDICES. 21 3 15. C^*)"^. {J)iz=Jo (by C)=a^^ (by Art. 299). i6. C^^)^. 18. (at)F 20. [(^r^^^ 17. (ki)i. 19. (a^^)^, 21. (jr6^)^ 22. (a^x'^j/^)'^, («3j»;iyl)f =(c3)l(;c2)5(;i)t (by Z>)=«5;^f>'i% (by C). 23- (aH^^i. 25. (36a4j^2^3)i 27. (S2xiyi)i, 24. (ad^)i. 26. (^"2-:rt>'^^*. \^4/ (^4)6 <^8 28. (ia^^3^)i 30. il)* 3. cy 34. <> 31- ^^^\i Uv 33- i-^Y 35. 36. (a^+a^-^l)(ai+a — a^). We arrange the work thus: «3+«3_^^ — a^ — a — a^ a^ + 2a^ + a^ -a^ 37. (x+2y^-^Sy^)(x—2jfi+Syi). 38. (.;r^+;K^)(^^-_y'^. 39. (a^-3^"^^-2-+4a-i-<^- 3 2 12 1 40. (a''—2a''+Sa")(2a''—a"'). 39. (a^-3^"^^-2-+4a-i-<^-^^<^^)(^^— 2a^^i). 214 UNIVERSITY ALGEBRA. 41. (2ab^-Zaib^y, 3 11 3 11 42. (x^^xy'^+x^y—y^)'^(jx^---'y^. We arrange the work thus: x^^y^ ) x^-xy^+o^y-y^ ( x-\-y x^—xy^ • x^y-y^ xiy -yi It is just as important to keep dividend and divisor arranged according to the powers of some letter in case the exponents are fractional as in the case they are integral. Fractional and integral exponents must take the order of their respective magnitudes. 44. (^-l)-i-(^i-l). 45. (.ai-'2aJx^+x^)^(ai—2aixi-hx). 46. (a^-d^-A+2dici)'^(ai+di-c^. 2 1 47. Find the square root of x^+2x'^+l. 48. JPactor x—2x'^y^-{-j/. NKGATIVK :^XPONBNTS. 305. If the product of two numbers is unity, either of the numbers is called the Reciprocal of the other num- ber. Thus ^ is the reciprocal of 2, f is the reciprocal of f , etc. In other words, the reciprocal of a number is 1 divided by that number. 306. Since expressions like a~" have no meaning by the definition previously given of a number afiected with an exponent, we are at liberty to define such expressions in the manner that is most convenient. Now if we can discover a definition that will make law A hold for neg- ative as well as positive exponents, that is the definition we shall adopt. THEORY OF INDICES. 21 5 If negative exponents can be defined in accordance with law A then, n being any positive number, integral or fractional, and hence —n any negative number, inte- gral or fractional, we ought to have Let r be taken greater than n ; then we know r Hence we ought to have a''a~''=— , or dividing by a'', <2~'*=— • 1 ^ That is, ^ " i^z^p-A/? to mean — • Thus we see that it is possible to give a~'^ a definition that will make law A hold for negative as well as positive indices. Therefore we adopt the following definition : . Any nutnber with a negative exponent is equal to the reciprocal of that number with a numerically equal but pos- itive exponent. 307. It follows necessarily from Art. 306 that ^nh—r^s n"h~^r^d~* — = = = = etc "^ ^ d' b'd' c-'b^d' a-^'bU-'d' a-^c-^' ^^^' That is, any factor 7nay be transferred from one term of a fraction to the other ter^n provided the sign of the exponent of that factor be changed. Thus. Ir^x b'^x 4"^«-i^2 3i^2^2 Y , etc. BXAMPI.KS. Find the numerical value of each of the following : 1. 2-1. 4. 10-^ 7. 2-^. 10. 1024~t 2. 4-2. 5. 1-1. 8. 16-i II. 512-i 3. (-2)-^ 6. 2-2. g. Sri 12. 625-i 2l6 UNIVERSITY ALGEBRA. 1 5 5-2 16-1 2 , l-» ■ 32-5- 7-1 14.3=,. i6.g3j. 18.-2^. 20.—,. Write each of the following expressions without using negative exponents : 21. x-'^, 25. ha-^ . 29. {x+yy^, 33. 2a^;i;-2j^~^. 22. x'^y-'^. 26. 3a-2ri 30. (--;r)-3. 34. (~a2)-3. 1_ 2^-2 ^4 ^ -\jj\ 24. —1-^- 28. _., _^ ' 32. ^ — 36. ^ _., . „ — -> x^ X ^y ^ 5^~t<5 ^^ ^ ^~ Write each of the following expressions in one li7ie: \ ^7- ^* ^^* ?^* "^^^ 47=275* 43. 5^.7^- „ 4 2x-^y^ Ax-^y-^ ab^ \ 45. / ^ 4 -^- 46. -3+^+-+--^- (^_^)-3^-2 X X X X 308. We have defined negative exponents in such a way that law A holds for negative as well as for positive exponents, and we now proceed to prove that with this same definition all the other laws hold for negative as well as for positive exponents. 309. Negative Exponents follow Law B. I^et n and r stand for two positive numbers, integral or fractional; then —n and —r stand for two negative numbers. THEORY OF INDICES. 21/ Case I. To prove a^'^ar^^ar-'^-^^. We know a''-^a~^=a'' x -n;,, by properties of fractions. by meaning of negative index. = a^-^-''), bylaws. Therefore, a-^a-=a^-i-r)^ [10] Of course —{ — r) may be written +r, and n — [ — r) may be written « + r. The form n—{—r) is kept merely to show the subtraction of the negative index. Cask II. To prove a~^-^ar^a~ We know <2~'*-f-a''=a~'*x— , by properties of fractions. «a~^Xa~'', by meaning of negative index. = «~"~^, by law ^. Therefore, a-^^a^^a-*"-^. [11] Cask III. To prove a~''-r-a~''=^"'*~(~''). _ _ _ 1 We know a ^'■^a ''=a ''x-^-^, properties of fractions. =^-''x^~(~''), by meaning of negative index. = «-«-(-''), by law^. Therefore, a-«-^a--=a-«-(-^). [12] Therefore, negative exponents follow law B, 310. It should be noticed in the above demonstration of law B that no restriction whatever is placed upon the relative magnitudes of the exponents n and r. Conse- quently, law B is proved for all kirids of exponents^ whether n is numerically greater thafi r or not. Thus: a'^-^a^ = a-'^) a'^-^a^'^ = a-^] a^-r-ai==a-^) a~'^-~a'^ = a~^\ etc. 2l8 UNIVERSITY ALGEBRA. 311. Negative Exponents follow Law C. I>t n and r stand for any two positive numbers, in- tegral or fractional. Cask I. To prove (««)-''= ^-'"•. We know (^O'^^^T"^' t>y meaning of negative index. ="^?> t>y law C for positive indices. =«""'', by meaning of negative index. Therefore, {a-"Y=a-*'''. [13] Cash II. To prove {0-^=0-*^^ We know (^~'*)''=(-^), by meaning of negative index. ="i^j by law ^ for positive indices. =a'"'*'', by meaning of negative index. . Therefore, (a~«)''=a-«^ [14} Casb III. To prove (ar'^y^a^'^ 1 We know (a"") ''= ^ _„^^ y by meaning of negative index = -z;^» by Case II. a =«'*'', by meaning 01 negative index. Therefore, («-«)-''=««'•. [15] Therefore, negative exponents follow law C, 312. Negative Exponents follow Law D, Let n be any positive number, integral or fractional. To prove (adcy^a-^d-'^c'^**. We know {abcy*"^ . . ^^ > by meaning of negative index. g= ^.^ ^ » by law Z^ for positive indices. =—; t;; ~' by properties of fractions. THEORY OF INDICES. 219 by meaning of negative index. Therefore, {abc)-''=a-''h-"cr*'. [16] Therefore, negative expone?its vllow law D, 313. Negative Exponents follow Law E. Let n be any positive number, integral or fractional. _ (ay*" a"" To prove y =^.. We know (7) —~rT^' ^7 meaning of negative index. \b) =~' t)y law^ for positive indices. a"" X by reducing fraction. by Art. 307. [17] Therefore, Therefore, negative exponents follow law E, EXAMPL:es. Perform the indicated operations in each of the follow- ing examples by means of the laws of exponents : _2. \_ 1. ^8x^-5. 4. 8(2-4 X 3^2. 7. ^ 3X7;^ 3. 2. ^-i^xr-io. 5. 2/-iXzA. 8. ^ax-^'K\bx'^ 3. ^r'^-^^-^. 6. x^-^x~^^. 9. a~^b~*'-^ab~'', 10. (-7^-^^-2)(-4a2^-i)(-u:2^2^-i). 12 3 2 2 3 3 2 11. (2a^r^)(a"^3^— 1^3^24- ^T^-¥). 12. 7^-1^-2^-3-^8^-2^-^^-^. 14. 18^~^^^^-s-^6a'2-^i^-«. 13. ^^x^y-'^ z^ -^1 x-^y~^ 2-^^ . 15. 6xiy~^z^-^2x~^j/iz~^, 220 UNIVERSITY ALGEBRA. i6. («-3)2. 21. (c-^)^. 26. (x~id^)~^, 17. (^-2)-^ 22. (a^^)-*. 27. (a-5|^-10)-|. 18. (^^)-2. 23. (^-43V6)-3. 28. (— 1^3-)-4^ 19. (;^^)-3. 24. (j»;ij/^~^^ 29. (— ^-4)-3. 20. (r"4)~^- 25. (8x^y-^yi. 30. (— a^)-!. 31. (— 8-s^VV^- 32. (— l^^jj-^)i 33 34. 35 42. («2^-i + 3a3ji;-2) (4^-1— 5;t:-i+6a.r-2) 4«x-i— 5fl:2.x— 2+ Qa^x-^ 43. (2x~^—^x-^4:x^) {^x~^—2x~^+Zx~^). 4 2 2 4 2 2 44. {x~^—2x~^y^+y^) (x~^—y'^). 45. (3jr"t~fx"^+4)x2.;i;~i 47. (.r~^+J^-2) (^r"^— j/-2) 46. (.r~t4.;r~'^+l)(;r"i._l). 48. (:r^j/+jj/t) (^i— jz-i). 49. (2a^— 3ajtri)(3^~'^+2:r~^(4a^;r"^4-9a~^.;»;'i"). _3. JL _i 3. _^i i 50. {X ^—X~^J''2+X 2j/_j/2)-^(jj; 2— j^2). _3 1 5C ^y-y^ X~iy-y^ 51. (:r-'^+2jr-2— 3;t:-i)-T-.(.;i:-2+3.:r-i). THEORY OF INDICES. 221 52. (x-^-y-^)^(x-i-y-^). 53- (:r4+^-^ + 4[:r2 4-:r-2] + 6)-f-(;t:2+:r-2 + 2). Arrange the terms of dividend thus : x'^+4x--{-Q-{-4:X-^-\-x-'^. 54. (x~i—x-'^—4:X~'i-hQx-^—2x~^)-T'(x~i—4cX~^-\- 2). _ 1 1_ 55. Simplify ^^ ^ 56. Simplify ^ ' ^ 57. Simplify [{ahiy^xCa-h-^iy^^. 58. Simplify [«32(^^^3)i(^2^3)i]i. 59. Simplify (2^a-i-S^d)(S^a-2id')-6i(a'^-d^) + 2iad. ZERO KXPONKNTS. 314. Since an expression of the form a^ has no mean- ing according to the original definition of a number affected with an exponent, and since such an expression has not been considered in the treatment of fractional or negative exponents, therefore we must consider this form of expression if we wish our formulas to be perfectly general. 315. We shall endeavor to discover a meaning for a^ which will make law A hold when one or both of the exponents are zero. If zero exponents can be defined in accordance with law A, we ought to have Therefore a^ ought to mean unity, whatever number is represented by a. Thus we see that it is possible to give to <2^ a definition that will make law A hold for zero as well as for positive or negative, integral or fractional ex- ponents. Therefore we adopt the following definition : Any expression of the form a^ (a being any number not zero) is equal to unity. 222 UNIVERSITY ALGEBRA. 316. We have defined zero exponents in such a way that law A holds for zero as well as positive or negative, * integral or fractional, exponents, and we now proceed to prove that with this same definition all the other laws hold for zero exponents. 317. Zero Exponents follow Law B, Case) I. To prove a^^a''=a^-''\ i. e, or*'. Since a^ = \, ,', a^-^ar=—=^a-^. Cask II. To prove a''~a^= =^^. Since any number affected with a zero exponent equals unity, .*. {ary=-\ and aO = l; .-. {a:y=^a\ Therefore, zero exponents follow law C, 319. Zero Exponents follow Law D. Since any number affected with a zero exponent equals unity, .\ {abcy^l. Also «o = i^ 30^1^ ^o=,i^ because each member equals unity. Therefore, zero exponents follow law D, THEORY OF INDICES. 223 320. Zero Exponents follow Law E. Since any number affected with a zero exponent equals unity, .-. g)=l. Also a^^\ and b^ = \. .'. [-^)=-^> because each member equals unity. Therefore, zero expo7ients follow law E, EXAMPLKS. 1. Multiply ax^-\-a^x^ by ax^—\. 2. Multiply X, (^"2-+:r~'2-), {x^—x'^'), (^2_^0)^ 3. Divide x''^ ■\- a^ x'^ —^x^ by ji:"^'— 2. 4. Find the product of {x^''-\-xy-\-y'^*')'^, (^"^y")'^, 111 5. Multiply together (a'^+ad+d'^y, (a—dy, {a— by ia'^+ab-^-b'^y. ? 2 6. Simplify -3—^3 X --4 r /y 2 .v-2 7. Simplify f + 1 8. What must be the relation between :r andjj/in order that x-\-y^ may be the reciprocal of x— j/"^? 1311 111 JL 9 . Divide a—x+ 4:a^x'^-'4,a^x'^ by aJ + zd^x'^—x 2 . 10. Simplify [(^t4)-^(a-V*)"2-]-2 4. ^ ... «+^ ^fa+b\-'^ 11. Form an equation whose roots are -—7 and! \ 12. If ^1 and r2 are the two roots of ax'^-\-bx-\-c=0^ find the value of (b-\-aj ^y^ + {b+ar^y^ in terms of a, b^ and <:. CHAPTER XV. SURDS. 321. The student has learned that there are two nota- tions in use for indicating the root of an expression, one notation using the ordinary radical signs, and the other using fractional exponents. While it is unnecessary to have two ways of writing the same thing, yet, because each notation has special advantages in particular cases, the two methods are retained. Of course the same laws (namely, A, B, C, D, and £ of the last chapter) govern the operations with roots, whatever form of notation be used. 322. Rational and Irrational. An expression is Rational with respect to any number or numbers when the numbers named are not involved in any manner by the extraction of a root. Thus, is rational with respect to x, but irrational with respect to c and d; the term Irrational being used in just the opposite sense to rational. 323. A Surd is the indicated root of a commensurable number (integral or fractional) if that root cannot Jpe ex- actly taken; as l/f or 1^'3. Expressions like l/4, 1^8, etc. , are said_to be in the Form of a Surd. Expressions like V^a, ^^ ab, etc., are often called surds, although, of course, they are such only when the letters stand for commensurable numbers whose roots cannot be exactly taken. SURDS. 225 It should be noticed here that we make a distinction between the terms incommensurable irrational expression and surd, a distinction w hich is not always made. According to the definition given above y 2 + V2, V V3, V TT, are not surds, but they are irrational and in- commensurable. This limited meaning of the word surd is convenient and is growing in use. 324. Orders of Surds. Surds may be conveniently classified by their indices as Quadratic, Cubic, Quar- tic, Quintic, . . . n-tic, etc., as the case may be. 325. The operations with surds depend upon prin- ciples established in the last chapter. For convenience of reference we restate below those principles which are used in the present chapter. 326. T/ze rth root of the product of several numbers is equal to the product of the r ih roots of the several numbers. That is, i/~^'. = i/ai/'hi/~c [1] 1 ILL because (abcY^a^b^c'', by equation [8], Chapter XIV. 327. The r th root of the quotient of two numbers is equal to the quotient of their rth roots, because (■i)'^='T' equation [9], Chap. XIV. \bJ J^r 328. The rtth root of a number equals the rth root of the tth root of the nujnber. That is, Va='^~V^ [3] i L L because a'''=(a^)'-, by equation [7], Chapter XIV. 16- u. A. 226 UNIVERSITY ALGEBRA. 329. The rtth root of the ntth power of a number equals the rth root of the nth power of that number. That is, i/^=i7^ [4] because a^t=ar^ by equation [3], Chapter XIV. 330. The n th power of the r th root of a number equals the r th root of the n th power of that number. That is, (i/ar=i/^ [5] This is equation [1], Chapter XIV. REDUCTION OF SURDS. 331. If any factor of the number under the radical sign is an exact power of the indicated root, the root of that fac- tor may be extracted a?id written as the coefficient of the surd, while the other factors are left under the radical sign. (1) Thus, y8=l/4x2^ = T/4y2 by [1]. = 2l/2 (2) Also, 1^81 = 1^27 x3_ = 1^27^3 by[l]. = 3f/3 (3) Also, V\^ax'^ = i^%x^y.'lax = t^8x~^t2^ by [1]. = 2jt:#'2^ r4) Also, f'a''+''b=iya"xa''b ^i^^^i/Vb by [1]. ^aiy^b SURDS. 227 332. It is sometimes convenient to have a surd in a form without a coefficient. The coefficient can always be introduced under the radical sign by reversing the process of Art. 331. (1) Thus, 2v^6=V^22i/6 = 1/22x6 by[l]. = 1/24 (2) Also, 50t/50= 1/502 1/50 = 1/502x50 by [1]. = y '125000 (3) Also, 41/5=1/ 4^1/5 = #'43_x3 by [1]. = #320 333. As the same process may evidently be applied in any case, we say: Any coefficient of a surd may be intro- duced as a factor under the radical sign, provided that the coefficient be first raised to a power equal to the index of the surd. 334. The expression under the radical sign of any surd can always be made a whole number. (1) Thus, -^1=1^^1 x|=l/i| =r2Vxi8 = 1/31^1/18 by[l]. =il/l8 (2) Also, 1^7 = 1^1x1=1/11 = l/^xl£ = 1^/^ 1/14 by [1]. =il/l4 228 UNIVERSITY ALGEBRA. (3) Also. V^Vf4S X ab''-'' by[l] 335. As the same process may evidently be applied in any case, we may say: The expression under the radical sign in any surd can be made ifitegral if both numerator afid denominator be multiplied by such a nu7nber as will render the denominator a perfect power cf the indicated roof.^ and if then the required root be taken of the denominator thus found. 336. We may change the index of some surds in the following manner: (1) Thus, 1^4=l^j/4 by [3]. = V2 since V^4=2. (2) Also, VM0= l/ r 1000 by [3]. = t/10 since #"1000= 10. (3) Also, 1^256^2^8 ==-^1/256^2^8 ^y [3]. = fl6ca^, since V236c'^a^ = 16ca^ 337. Since [3] is true in all cases, we know that the index of a surd can be lowered if the expression under the radical sign is a perfect power corresponding to some factor of the original radical index. SURDS. 229 338. A surd is in its Simplest Form when (1) no factor of the expression under the radical sign is a per- fect power of the required root, (2) the expression under the radical sign is integral, (3) the index of the surd is the lowest possible. . 339. Methods of making the different reductions re- quired by this definition have already been explained. We give a few examples. (1) Simplify ^g-^-^. by Art. 337. =l/4ai, by Art. 335. (2) Simplify 1^^. ^^400 = |/^ by Art. 337. =il/60 by Art. 335. =|1/15 by Art. 331. (3) Simplify fl^fH- mH=|i/f by Art. 337. =5l/| by Art. 331. = 1/10 by Art. 335. 340. In any piece of work it is usually expected that all the surds will finally be left in their simplest form. BXAMPI.KS. Reduce each of the following surds to its simplest form : a" ■■VI- '■# =VI -i_ 230 UNIVERSITY ALGEBRA. OPERATIONS ON SURDS. 341. Surds which differ only in their coefficients are said to be Similar. Thus 6V''2 and 15l/2 are similar surds; also fl^f and fl^-f-; also 5^ad^ and nV ab'^ . 342. The addition and subtraction of surds involves no principle not already used in the addition and sub- traction of other expressions, as the following examples show: _ _ _ (1) Combine the terms of 10l/7~8l/7+5l/7. 101/7-31/7+51/7=121/7, by the usual process of addition of terms. (2) Combine the terms of 7 1/2 - 1/I8 + 2^/8. Putting each surd in its simplest form, we have 7v/2-l/i8 + 2^8=7l/2-3l/2+4l/2 =8t/2 by the usual process of addition of terms. (3) Combine the terms of 5l/4+2l/32-l/l08. Putting each surd in its simplest form, we have 6i/4 + 2l/32-l/l08=5l/4+4]/4-3l/4 =61/4 (4) Combine the terms of fl/f—fl/f. Putting each surd in its simplest form, we have =il/6 _ . _ (5) Combine terms of 11f^ aH—^a'^i/ ^^b + ha^a^b. Putting each surd in its simplest form, we have 22l/^-8«2i/64^+5al/^ = 15«2|K^ SURDS. 231 343. We observe the advantage of reducing each of the surds in any given expression to its simplest form, for then it can be told whether or not some of the surds are similar to each other, and consequently whether or not they can be combined ; for 07ily similar surds can be combi7ied iyito a single surd. 344. The product of any number of surds of the same iiidex can always be expressed as a single surd by means of equation [1]. (1) Find the product of t/2 x 1^5 X Vn , l/2xl/5 ysVn= = l/2x5x7 = 1/70 (2) Find the product of V^x i/Ts. l/2> c v'i8= = V2xl8 = 6 :XT^9. (3) Find the product of #^54 by [1]. by [1]. f/54x 1^9=1^54x9 by [1]. = 3l/2 The result should always appear in its simplest form, 345. The quotient of two surds of the same index may be expressed as a single surd by means of equation [2]. (1) Find the quotient of i/28-t-i/7. V^28 H- 1/7 = 1/-^ by [2]. = 1/4=2 (2) Find the quotient of 1^81-^ l'/6. =3l^^J by Art. 331. =11^4 by Art. 335. The result should always appear in its simplest form. 232 UNIVERSITY ALGEBRA. 346. If the product or quotient of surds of different indices is sought, the surds may first be reduced to a common index by Art. 337 or equation [4]. (i) Find the product of Vhy, ^1. l/5x ^4=15/125x1^16 =1^125x16=15/2000 (2) Find the quotient of 1^9^1/3. ^9^i/3=i5^81h-i5/27 =r||=l5/3 __ (3) Find the product of V ab^ x V a''-b. i/^x -¥'^b= T^^TH X '^'^1^2 (4) Find the quotient of l5/|x2'^'^. =iTl6xl6=^ri6 347. Any power of a surd may be expressed as a single surd by means of the principle of Art. 330. Thus : (1) Square #"2. (^2)^ = ^25 by [5]. = ^4 (2) Cube 3l/2. iZViy=Z\V^^ by lawZ» for indices. =27l/2» by [5]. =54l/2 by Art. 331. The result should always be left in its simplest form. SURDS. 233 348. Any root of a surd may be expressed as a single surd by means of the principle of Art. 328. Thus : (1) Find the square root of 1^4. 1/^/4=1^4=1^1/4 by [3]. = ]^'2 (2) Find the cube root of |t/3. ^|v'3=2l^il/3 by Art. 331. =21^1/^ by Art. 333. =21^1/^ =2v'r^ by [3]. =2l/i=|l/3 by Art. 335. (3) Find the ^th root of aVb. ^ai/b=^Va'-b by Art. 333. = Varb by [3]. 349. This last process is a general one, but if for any particular values of a, b, r, and t this result should not happen to be in its simplest form, it should be so reduced. RATIONAI.IZATION OF EXPRESSIONS CONTAINING SURDS. 350. To Rationalize an expression is to perform an operation upon it that will free the expression of surds. Thus the binomial quadradc surd 3 + l/2 is rationalized when multiplied by 3— 1/2, for the product is 9—2 or 7, which is rational. 351. Any multiplier which when applied to an irra- tional expression will free the expression of surds is called a Rationalizing Factor. Thus 3 — 1/2 is a rationalizing factor for the binomial surd 8+1/2. 234 UNIVERSITY ALGEBRA. 352. It is often convenient to perform an operation upon both terms of a fraction, so as to render either the numerator or the denominator rational. It is sufficient for present purposes to show how this may be done when all the surds are of the second order. The following are examples. 2 (1") Rationalize the denominator of 7=. ^ ^ 2 + 1/2 Multiplying numerator and denominator by 2— 1/2, we get _ _ 2 _ 2(2-l/2) _ 4-21/2 2 + t/2"(2 + V'2)(2-i/2)~4~ (1/2)2 3 (2) Rationalize the denominator of —7= 7=. ^ ^ 1/5-1/2 Multiplying numerator and denominator by l/5-fV^2, we get , _ _ 3 _ 3(1/5 + 1/2) l/5_ 1/2 "" (1/5 - 1/2") (1/5 + 1/2 ) =5(4±i^=^5+l/2. 353. This work is based on the very evident principle that any binomial quadratic surd is made rational by mul- tiplying the surd by itself with the sign of one of its terms changed, for the product is then the difference of two squares. 354. Considerable labor is often saved in computing the value of an irreducible fraction if we first rationalize the denominator. Thus, to compute the value of 3 1/5- V^2 SURDS. 235 to five decimal places, the two square roots must be taken to at least five decimal places, and the quotient of divided by the differ e7ice of these roots must be found. This division by a five-place number will be avoided if we firs^t rationalize the denominator; for the result is Vb-\-v1, the value of which is found without the ' ' long division ' ' of the former method. 355. Rationalization of Trinomial Quadratic Surds. Any trinomial_ quadratic surd may be repre- sented by V a-\-Vb-^V c. Multiplying by V a-\-V b—V c we obtain ( l/^-f V~b-\- Vc^ (Va + V~b- Vc) = (Va-\- Vb) ^ - ( j/^) 2 = a-{-b—c-\-2Vab, (1) which is rational as far as c is concerned. Now multi- plying this last expression by a + b—c—2Vab (2) we obtain (a + b—c+2\/ab)(a-{-b—c—2Vab) = C^-j-b—cy—4:ab, (3) which is rational with respect to a, b, and c. The ration- alizing factor for the original trinomial quadratic surd is thus seen to be (l/^-f V'^-V'c) {a-\-b-c-2Vab) (4) The second parenthesis in (4) is composed of the factors (l/^- V"b+ Vc) (Va- Vb- Vc) Hence the rationalizing factor ot V a-\-V b-\-Vc may be written (V ^+ V~b— Vc) {Va— Vb+ V~c) (Va— Vb—Vc) . Observe that the terms of each of the component tri- nomial factors of this expression are those of the given surd, except the signs are those exhibited in the scheme + + -, + - +, + . 236 UNIVERSITY ALGEBRA. Now it is evident that there is no other arrangement of signs, keeping the first sign unchanged, than the arrangements written in this scheme, except the arrange- ment + + + , which is the arrangement of signs in the given trinomial. Therefore; The rationalizing factor for any trinomial quadratic surd is the product of all the different trinomials which can be made from the original by keeping the first term unchanged and giving the sig7is + and — to all the re- maining terms in every possible order except the order occurring in the given trinomial. As an example, find the rationalizing factor for The above method shows it to be (V^5-t/7-t/3 )(l/54- 1/7 + 1/3 )(l/5 + 1/7-1/3 ), and multiplying the original trinomial by this, the ra- tionalized result is found to be —59. 356. Rationalizing Factor for any Quadratic Surd. The above problem is capable of generalization, but its proof need not be given here. The generalized statement is as follows : The rationalizing factor for ajiy polynomial quadratic surd is the product of all the different polynomials which can be made from the original by keeping the first terfn unchanged and giving the signs + and — to all the re- maining terms in every possible order except the order occurring in the given polynomial, BXAMPLKS. Rationalize the denominator of each of the following : . 1-1/24-1/3 ^* 1 + V2-V^3* SURDS. 237 30 2 + 1/^-1/2 * 2-1/3 + 1/5 _ _* 2-1/6+T/2 3- Show that j/^l-Xlvr '-^'^'^'^''~'^- 357. Rationalization of any Binomial Surd. By reducing the fractional exponents to a common denomi- nator any binomial surd may be put in the form <2«=h^«. We shall show how to rationalize each of these forms. s t I. To rationalize the form a^ — b^. For convenience let a»=x and d^=y; whence ««— ^«=;r— jj/. Now multiplying x—y by x^'-^ + x^'-y+x^-y + . . . 4-y'~^ (1) we obtain, by Art. 137, (x—y) (x''-^+x'*-y+x'*-y + , . .+jj/"-^)=;t:''— jj/". But ;i;«— j/"=(a^"— (^-0«=^'— ^'» which is rational. Therefore (1) is the rationalizing factor for a»—d». s_ I s II. To rationalize the form a^ + b^. As before, let ««=;»; t £ i and b^=y] whence a'^-\-bn=^x+y. Multiplying x+y by x^^-i-.x^-'^y^^n-z^z__^ . .±.y"-^ (2) the product, by Arts. 138, 141, is (x+y) (^x''-^—x"-y+x''-y — . . .ijv'*"^) =x"—y" if n is even, ^x^'+y" if n is odd. But x**—y''= (any— (^«)"= a^— b'; :r''+j/«== (a"y-{-(b-y=a'+b'. Each of these results are rational ; therefore (2) is the £ -£ rationalizing factor for a»+b^. The following examples illustrate the processes ex- plained above. 233 UNIVERSITY ALGEBRA. 2. i (1) Rationalize d^ — r'^. With a common denominator for the exponents, this oecomes ^ ^e_;/6- whence n—Q, j-=4, /=3; then x^d^, y=r^. ± 3. ISL l_e 3 12. 6. 8 9 iL !_?. 1_5.' 4 3. (2) Rationalize 6 + 31/^5. With a common denominator for the fractional exponents, this 4. 1 4 becomes 6^ + (3*X5)^; whence ??=r4, ^=4, /=!; then :>c=6^ and y=(34X5)4. Since {x-\-y){x^ -x^y + xy^^ -y^)=x't-\-y*, [64 + (3^X5)4] [63-62x3X5^4-6X3^X5^-33x5^] [64]4 + [(34x5)4J4=:64-34x5=891. FUNCTIONS OF SURDS. 358. Function of a Number. A Function of a nnmber is a name applied to anj^ mathematical expres- sion in which the number appears. Thus, 2ax, x—y'^x'^, -^ — ^^ va—x'^, are all functions of x. In the same manner we speak of functions of several numbers. The second expression abovej may be called a function of x and y. Obviously, a funcl:ion of a number might be otherwise defined as any expression which depends upon the number for its value. 359. Rational Integral Function. A Rational Function of a number is one in which the number is not involved in a radical or affected with an irreducible fractional exponent. An Integral Function of a num- ber is one in which the number does not appear in the denominator of a fraction or is not affected with a nega- tive exponent. SURDS. 239 A function may be both rational and integral, in which case it is called a Rational Integral Function. If n is a positive whole number and a^, a^, a^^ . . . ci-n stand for any numbers whatever, then is a general expression representing any rational integral fmictioii of X of the n th degree. Such expressions as function of x, function of a, func- tion of x+h, etc., are abbreviated into F(x), F(a)^ F{x-\-li), or /(-^), f{ct),f(^x+h'), or a similar expression. It must be kept well in mind that F, /, etc., are not coefficients. 360. Function of a Quadratic Surd. Let V^a be any quadratic _surd and /(l/<3^) any rational integral function of y_a, say _ _ a^(y~ay + a^{V~ay-'+. . .+a^_y a + a„. (1) Now all even powers of V ^ in this _expression will be rational, and any_odd power, say (V ay"^^ , will reduce to the form a''\/ a. Therefore (1) will reduce to a form containing but two different kinds of terms, and may be put in the form A + BV a, (2) where A and B are rational with respect to a. Whence we say, every rational integral function of va may be expressed as the sum of a rational term and a rational multiple of V a. _ It follows frotn the above reasoning that f(—l/a) dif- fers from f^y a ) only in the sign of the irrational part, so that if _ _ fiVa^^^A-^BVa, (3) then /(-T/^)=^~^l/a; (4) whence /(T/a)/(-l/^) = ^2_^2^^ which is rational with respect to a. 240 UNIVERSITY ALGEBRA. 361. Function of any Surd. I^et a'p be any surd 1 1 and f{a^') any rational integral function of a^, say «o(^V + «i(^V"' + . . .+««-,«^ + ««. (1) 1 1 Now any power of a^ greater than /—I, say {a^y"^', s_ will reduce to the form a'^ap, where s is less than p. Whence (1) may be put in the form aJ-^^aJ-^+ . . . +^«-i«^V^n, (2) where A^, A^, A^, . . . are rational with respect to a. In the chapter on determinants it will be shown that a rational- izing factor can be found for (2), and thereby prove that a rational- izing factor can be found for any irrational expression whatever. 362. If ^^ ^, A, B are commensurable and V a, V b are incommensurable^ then v a cannot equal A+BVb. Suppose that l/a=A+BV^b, if possible. Then we must have, by squaring, _ a=A^-\-2AB\/b-\-BH, -A'^-BH or T/^=^ 2AB But by hypothesis the right side of this equality is commensurable. Thus we have an incommensurable number equal to a commensurable, which is absurd. Whence Va cannot equal A + BV^b. Since A+Bj/b may represent any rational integral function of l/b (Art. 360), we may say: One quadratic surd cannot be expressed as a rational integral furiction of another, 363. If CL, b, py q are commensurable and vb and V~q incommensurable y and if a+V b=p+vq, then a=p a?id V'j^l/q. If a does not equal py suppose a=p + d. Substitute this value for a in the given equation, and we have SURDS. 241 or d+Vb=Vq. From the last article this equality cannot be true. Therefore a cannot differ from p] and if a=p, V b must equal V q. squar:^ and square root of binomial quadratic SURDS. 364. Square of Binomial Surd. Let V a-\-V~b rep- resent any binomial quadratic surd. We then have The square_ of any binomial quadratic surd is of the form A + VB. See also Art. 360. 365. Square Root of the Form a+j/^ Since the square of Va + }/b takes the form A + l^B, it follows that the_square root of a+l/)8 can be be expressed in the form V x+Vy for some values of a and /?. We proceed to find for what values of a and P this is true. Suppose ±:^a+Vp=^\/x+\/y, (1) where the positive roots are to be taken in the right member. Then squaring, a+V'^=x+2Vxy+y (2) By Art. 363 we have x+y=a; (3) 21/^=1/^; (4) or from the last, 4:xy=p. (5) From (3), y^a—x. (6) Substituting in (5), ix'^ — Aax^—p. (7) Whence, solving, x=^(a±:\/a'^—fi), (8) and from (3), jv==i(a=f:l/a2— ^). (9) 16— U. A. 242 UNIVERSITY ALGEBRA. Therefore, substituting these values in (1), we have v'a-fl/iS=d=(l/i-(a+l/^:^^) + V'l-(a-T/a2_^)) [6] Similarly we would find Va-V~P=±{y\(a+}/^J^fi)^y'^Ca^V^^)) [7] Thus we see that the square root of an expression of the form a+V /? can be expressed as the sum of two simple surds if o-'^ — P is a perfect square, and can be expressed as the sum of two complex surds (real or imaginary) for all values of a and (3. KXAMPLKS. Express the square root of each of the following by means of two simple surds : 1. 47-121/15. _ _ Here a=:47 and T//?=12l/l5; whence iS=2209. Hence /a- t/)S= d=:(3l/3-2l/5 ). 2. 87-121/42. 4. 57-121/15. 3. 17-31/21. 5. 75 + 121/21. 6. Prove that |(l/3 + 1)2-2 (l/2- 1)2 = 1/59-24 1/6 7. Express in the simplest possible form the value of Vx+\+Vx—\ when 4:X=V a-\--^' V a MISCELI.ANKOUS KXKRCISKS. 1/12 1. Find the value of ,— ,— ,— (1 + 1/2) (1/6-1/3) 2. Simplify l/l2+-jV^75 + 6V^S. 3. Simplify 1 + 1/8+ 1/2- l/27-l/l2+l/75-T^^19 + 6l/g. 4. Simplify 1^2r+7l^x 1^^^21-71/2. 5. Find the value of x''-^x-\-l if ;r=3-l/3. SURDS. 243 6. Find the value of x^-\- 2j^^-{- 22^+ 6xy^ when 7. Find the value, when x=vS, of the expression 2:r— 1 2.;tr+l (x-iy (x+iy 8. Find the value of (35l/l0+77v^+63v/3 + 28l/l5) (i/T04-i/2-t/3) 9. Prove that the greater of the two fractions 1/3 + 1/2 1/3- ^2 V 2 + 1^2 + 1/3 t/2 - 1/2-1/3 exceeds the less by 2 — 1/2. 10. Multiply together V2x-{- V2{2x—1) —-^ and -1 V2;i: -4=:+i/2(2^-l)-l/2^. V 2x 11. If /> and ^ are whole numbers, find the factor which * 1 1 will rationalize aP-\-b^. 12. Find the rationalizing factor for 32"+ 2^, and show that the rationalized result is 23. l/x+a \/x—a\f ^^x^—a^ 13. Simplify {~--=--y==\l ^V X — a V x-\-ay^ -\-ay^y (x-\-aY — ax^ 14. Form an equation whose roots are -^ and ;= 3 + 1/5 3-1/5 15. Extract the square root of Vlc+Jx-- ^ \ X / — 1 16. Find the value of ; =: when 2jr= V a + -— = ,/- /I Va V X '^"V^-^ 244 UNIVERSITY ALGEBRA. 17. If 2:^=x'^+j/'^, find the simplest form of V{x+y-\-z) {x-^y—z) {2-\-x—y) (y-^z—x). 18. Prove 2 + 1/3 is the reciprocal of 2 — l/S; and find what must be the_relation between the two terms x and \/y so that x-{- Vy shall be the reciprocal of x—Vy, ig. Prove that if />=3 and ^=5, e^-'+pq-"^ -Vqp-^ +g^-^ ^ 3g^ + 5 CHAPTER XVI. SINGI.K EQUATIONS. 366. Roots and Factors. Every equation contain- ing one unknown number can be put in the form function of x=0 by transposing all the terms to the left side of the equa- tion. If it is an equation of the first degree it will always reduce to the form (Art. 233) ax-\-b=0 or x—r^=^0, (1) in which r^ stands for any number whatever, positive or negative, integral or fractional, commensurable or incom- mensurable. It is evident that this equation has the root ^1 and no other. An equation of the first degree may be defined as an equation which can be placed in the form of a rational integral linear function of x equal to zero. We have seen (Art. 281) that every quadratic equa- tion can be placed in the form ax'^ + bx+c=^0 or (ix—r^)(^x—r,^)=0, (2) which has the two roots r^ and ^2 and no others. Thus every quadratic equation can be placed in the form of the product of two rational integral linear functions of x equal to zero. It will be proved in a subsequent chapter that every cubic equation can be put in the form ax^-\-bx'^+cx+d=-0 or {x—r^){x—r^)(x—r^)=^0, (3) and that it has three roots, r^, r^, r^, and no others. That is, every cubic equation can be placed in the form of the product of three rational integral functions of x equal to zero. 246 UNIVERSITY ALGEBRA. Like truthvS will be shown in a subsequent chapter to hold for equations of the fourth and higher degrees. 367. We are led to inquire what operations may be performed upon the members of an equation without modifying the number of values of the unknown num- ber. Now, by the principles of algebra, a7i equation re- mains true if we unite the same number to both sides by addition or subtraction ; or if we multiply or divide both members by the same number; or if like powers or roots of both members be taken. But these operations may affect the 7iu?nber of values of the unknown numbers. Thus the roots of the equation Z{x-h^=x{x-h)-\'x''-1^ (1) are — 1 and 5 ; for either of these when substituted for x will satisfy the equation. But dividing the equation through by x—h, the resulting equation is Z=x+x-\-b, (2) Now this equation is not satisfied for x=^h, the only root being —1. Hence, although equation (2) must be true if (1) is true, yet the equations are not equivalent since their solutions are not identical. One root has disap- peared in the transformation. Just how this change occurs will be best seen after we place (1) in the form {x—r^{x—r^=^^. Since the roots of (1) are — 1 and 5, by the principle of Art. 281 the equation is equivalent to (jr-5)(:i:+l)=0. (3) Now if we divide this through by x—h we remove the factor in the left member which is zero for x^h, and consequently the equation will be no longer satisfied for x—h. If we divide through by jr+1 the equation will be no longer satisfied for x= —1. SINGLE EQUATIONS.' 247 Also consider the equation x''~6x+S=0, (4) which is satisfied for x=2 or x=4. Multiplying both members by :r— 3. we obtain Cx-SX^^-^^+8)=0. (5) But this equation is satisfied for either x=S or ;r=2 or x=4:. Hence, although multiplying both members of (4) by x—S has not altered the fact of the equality of the members of the equation, yet a value of x extraneous to the original equation has been introduced. Again, the equation 2x-l=;r + 5 (6) is satisfied only by the value x=6. Now by 'squaring both sides of the equation we obtain 4:x'^-4x-\-l=x''+10x-\-25, (7) which is satisfied for either x=6 or x=—^. Here, obviously, an extraneous solution has been introduced by the operation of squaring both members. In a like manner notice the effect of taking a root of both members of an equation. Thus suppose 4x'' = (x-6y. (8) This is satisfied for either x=2 or —6. Taking the square root of each member, we obtain 2x=x—6, (9) which is satisfied only by x=—6. We have lost one of the roots of the equation by this transformation. Equa- tion (8) is really not equivalent to (9), but to the two " equations f 2x= + (-^—6) \ \2x=-(x-6')) ^ ] We have given examples enough to show that certain operations upon an equation may modify the number of values of the unknown number. Thus we see that dur- ing a series of transformations which an equation must sometimes undergo it is possible that the values of the un- 248 UNIVERSITY ALGEBRA. known number that satisfy the original equation may all be lost and that any number of new values of the un- known number may be introduced. Thus none of the final results may satisfy the original equation, and con- sequently they may have no relation at all to the prob- lem in hand. It is now proposed to formulate certain propositions which will enable us to tell the exact place in the process of any solution where roots may be lost or new ones may enter. We shall then be able to perform operations on the members of an equation if we will note at the time their effect on the solution and finally make allowance for it in the result. This fact must be em- phasized*: the test for any solution of an equation is that it satisfy the original equation. '*No matter how elaborate or ingenious the process by which the solution has been obtained, if it do not stand this test it is no solution; and, on the other hand, no matter how simply obtained, provided it do stand this test, it is a solution." — Crystal, 368. Legitimate and Questionable Transforma- tions. If one equation is derived from another by ah operation which has no effect one way or another on the solution, it is spoken of as a Legitimate Transforma- tion or derivation ; if the operation does have an effect upon the final result, it is called a Questionable Trans- formation or derivation, meaning thereby that the opera- tion requires examination. 369. Equivalence of Equations. Two equations Such that any solution of the first is a solution of the second, and also that any solution of the second is a solution of the first, are said to be Equivalent. Thus the equations x'^ — ^x—Z, 2.;r(;i;-4)4-6=0 SINGLE EQUATIONS. 249 are equivalent, since each is satisfied by the values x= 3 and x= 1 and by no other values of x. One equation is said to be equivalent to several others when any solution of the first is a solution of one of the latter equations, and any solution of any of the latter equations is a solution of the first. Thus X'^=4:X — B is equivalent to the two equations \x-S=^0] Note that the solution to any equation consists merely of the simple linear equations (of the form xz=a) which together are equivalent to the original equation. Thus x=l, x=2, x=S, which is the solution of x^ — Qx^-{-llx — Q=0, are equivalent to the latter equation. A solu- tion to an equation is another equation or a set of equations. 370. The transformation of an equation by the addition or subtraction from both members of either a known number or a function of the unknown number is a legitimate deri- vation. An equation containing one unknown number, as it commonly appears with an expression on each side of the sign =, may be generalized in thought by the ex- pression a function of x^= another function of x, or using L to represent the left-hand side of the equation, whatever it may be, and J? to represent the expression on the right-hand side, we can represent the equation very conveniently by L=^R. (1) Now suppose that T, which may be either a known num- ber or a function of the unknown number, be added to both members of the equation, making L-\-T=R+T. (2) 250 UNIVERSITY ALGEBRA. Now it is plain that (2) cannot be satisfied unless L=Ry and that it is satisfied if L^*R. Therefore, by Art. 368, the derivation is legitimate. Thus: x^=ix-S by adding — 4x to each side is equivalent to X^-4:X=4:X~3-4:X, and this latter, by adding 4 to each side, is equivalent to ;»r2-4x + 4— -3+4. 371. Transposition of terms from one memhe> to the other^ changing the signs at the same time^ is legitimate. Thus, a L=R, to pass to L—R=r^O is merely subtracting R from both members. 372. Multiplying both members of an equation by the same expression is legitimate if the expression is a known number, not zero, but questionable if the expression is a function of the u?iknown nufnber. Represent the equation by L=R. (1) Multiplying both members by T, we obtain LT=RT. (2) Now this may be written, iL-R')T==0. (3) If Z is a known number not zero this can be satisfied only by the supposition that L—R\ that is, the equation is equivalent to (1). If 7" is a function of the unknown number, then (3) may be satisfied by any value of the unknown number that will make 7^=0, as well as for the values that make L=R. Whence (3), on this supposition, is not equiva- lent to (1), but to the two equations, { ^=0 } bINGLE EQUATIONS. 251 As an example, suppose L is x^, R is 4^—3, and T is 2. Then X^=4:X-^ (1) is equivalent to 2x'^=^x—Q, (2) But if T is ^—4, we get {x-'i)x^ = {^x-^){x-4:), (3) which is not equivalent to (1), since it is satisfied by ^=1, ^=3, x—A, and (1) is satisfied only by x^l, x—^. Also, if T is x^ — 4:, the new equation will be satisfied by x^—'Z, x:=z\, x=2, x=3. 373. If any equatio7i involves fractions with only knozvn numbers in the denominators^ it is legitimate to clear of fractions. The multiplier in this case is a known number. 374. If any equation involves fractions, some or all of whose denominators are furictions of the unknown 7iumber, and if these fractioyis are all in their lowest terms and 710 two denominators have a common factor, it is legitimate to integralize the equatio7i by multiplyi7ig by the product of the deno7ni7iators. To illustrate the reasoning, take the equation ARC ^+|- + -J-+Z?=0, (1) ^1 ^2 ^3 in which the fractions are supposed to be in their lowest terms and X^, X^, X^ represent different integral func- tions of the unknown number, and in which A, B, C, and ^ are either known numbers or functions of the un- known number. Multiplying by the the product of the denominators, we obtain AX^X^^BX^X^ 4- CX^X^ +DX^X^X^=^. (2) Now, since X^, ^2, X^, A, B, C, and D have no com- mon factor, no common factor has been introduced into all the terms of the equation by multiplying by ^1X2^3, and consequently no additional solutions can appear. 252 UNIVERSITY ALGEBRA. As an example under the above theorem take: (1) Solve IT^+fef=2. (1) These fractions are in their lowest terms and their denominators are prime to each other. Multiplying through by the product of the denominators, we obtain (3^-l)(7-5c)+(ll-2^)(4;t:-5)=2(ll-2x)(3:r-l). (2) Now we can see that although (1) has been multiplied through both by (11 — 2^) and (3:^—1), yet neither of these has been introduced as a factor through the equation. Hence there is no additional solution introduced. The roots of (2) will in fact be found to be 4 or —10, which values also satisfy (1). But an extraneous solution may be introduced if the denominators are not prime to each other, or if some of the fractions are not in their lowest terms. Thus (2) Solve _-=__+__. (1) These fractions have two denominators alike, and consequently not prime to each other. Multiplying through by the common denom- inator, x^—9, we obtain Sx{x-\-S)=Q{x-S)-\-9{x+d), (2) or reducing, x*-2xz=d, (3) whose roots are 3 and —1. Now if we put the original equation (1) in the form Sx-9_ 6 x-d "^ + 3 that is, 3=^ (4) it is seen that it is satisfied only for x=—l. Hence a solution was introduced in clearing (1) of fractions. It is easy to see that (1) is really equivalent to (4), and hence that in clearing (1) of fractions by multiplying by ^'^ — 9 we multiplied by ^ — 3 when it was not" neces- sary; this is where the solution ^=3 was introduced. (3) Solve "^'"^ -+4^+7=15. (1) Clearing of fractions, we get a:»-5^+6-f4x;2 + 7;»r-12x-21=15^-45. (2) or x*—5x=—Q; whence x=2 or 3. Putting (1) in the form ^-2+4;c+7=:15 {?>) it is seen that the value ^=3 was introduced in clearing (1) of frac- tions, since ^=3 does not satisfy (3). Equation (1) is not a case of Art. 374, since the fraction is not in its lowest terms. SINGLE EQUATIONS. ^53 375. Every equation can be integralized legitimately. If the several fractions in the equation are not in their lowest terms, they can be so reduced. Then these frac- tions can all be transposed to one side of the equation, their least common denominator found, and then added together. This will now give but one fraction in the equation, and when this is reduced to its lowest terms we shall have an equation of the form which by Art. 374 will take on no additional solutions N when multiplied through by D, since ^ is in its lowest terms by supposition. 376. The raising of both members of an equation to the same poiver is equivalent to multiplying through by a func- tion of the unknown number, and hence is a questionable derivation. Take the equation L=R (1) and raise both members to the n\,\i power, obtaining L^=R\ (2) Now (1) is equivalent to L-R^O, (3) and (2) is equivalent to Z'*--i?"=0. (4) But (4) can be derived from (3) by multiplying both members by 'L"-^+L''-'R+L*'-^R^ + , . ,+L''R''-^ + LR''-''+R'''^ whence (2) is equivalent to the two equations L=R, Thus, squaring jc— 2=3^ — 8 will introduce the solution of Z + ^=0 or, in this case, of 4x— 10=0; that is, x=f . 2 54 UNIVERSITY ALGEBRA. 377. Drnding both me^nbers of an equation by the same expression is legitimate if the expression is a knoivn num- ber^ but qjiestionable if it is a function of the unknown number. Suppose both members of the equation to be divisible by T and write the equation LT=RT. - (1) Now if 7" is a known number this equation, by Art. 372, is equivalent to L—R, (2) whence division by T would be legitimate. But if 7" is a function of the unknown number, then (1) is equivalent to the two equations { L=^R\ \ T=Q. I Division by T would give us but one of these, and con- sequently solutions would be lost. Hence the division by a function ot the unknown number is a questionable derivation. Solve the equation {x — ^)x'^=z[4:X—^)[x—^), Dividing by ^ — 5, we get .r2=4x — 3. Transposing and completing square, whence x^i or 3. Restoring the value lost by division by ^—5, x=zl or 3 or 5. 378. The extraction of the same root of both members of an equation is equivalent to dividing by a functio7i of the unknown nu?nber^ and hence is a questionable derivation. For we may pass from L^^R"* (1) to L^R (2) by dividing both members of (1) by Lr-'^+Lr-^R'\'Lr-''R^+, . .+l^R''-^+lr''-^+R''-'' Hence, by Art. 377, root extraction is a questionable derivation. SINGLE EQUATIONS. 255 Solve (2;r-l)2 = (jt- + 3)2. (1) Taking the square root of each side, 2x-l=x-j-3- (2) whence x=4. The solution lost is x= — |, which would have been avoided by writ- ing (2) as follows: 2^—1= ±(^+3). EXAMPLES. Solve each of the following equations : Dividing by x~5, we get S{x - l){x - 2)=(x+ 2)(^ + 3). (1) Expanding, 3x^—dx-\-G=:x^-\-5x + Q. (2) Transposing and uniting, 2;i?2_ 14^^—0. (3) Dividing by 2x, ^-7=0, (4) or ^=7. Restoring values lost by division, x=5 or or 7. 2. ax(cx—Sd') = da(Sd—cx). 3. a^x(c^x—a^d) = (a^d—c'^x)d'^c^, 4. a^—x'^^(a—x')(d—c — x'). 5. \-ax^-j--\-dx. a 6. x'^-^x=^n'^+n. Transposing, x"^ -n^^i^n — x. Dividing by ;c-«, x + n= — l; whence xz= — n~l. Restoring solution lost, x=?i or —?t- 1. 7. (a-xy—ia-dy^id-xy. Dividing by ^ - x, {a-x)^ + {a—x){a-d)-\-{a-3)^ = {3—x)^. Transposing, {a—x){a — d)-{- {a—dy^ ={d — x)^ —{a- x)^. Dividing by a —b, a— x-\-a — b=z — [b — x-\-a—x), or 3^=3« or x^^a. Restoring value of x lost, x=ia cr b, 8. (^-4)3 + (;r_5)3 = 31([x-4]2-[;t;-6]2). Divide through by 2^'— 9. 256 UNIVERSITY ALGEBRA. 9. (ia'-xy = (a—2xy—x'K 10. {x-ay={a-by--{x-by, Qx+b l + 8.r_l— ^ ^--x "• Sx-lb 15"""" 3 ""^~5~* If some denominators are monomials and the other denominators are not, it is good practice to clear of monomial denominators first. Thus, multiplying by 15, 15(6^+5) 8a;-15 --1— 8x=:5-5^+9-3x, 15(6£+5) or -^ T^— =15, 8^—15 whence 6:r4-5=8;c— 15, or ^=10. 9_^ 4 {x-V)Z "• 2 ^x-2 2 10-x \Z-\-x 7x+26 11+ Ax 13. .;i:+21 21 3^-4_ 7 14. —r- = X''-\-ZX — :-• x-\-l x+1 X"^ 3/ 1 1\ 23 ^5. — )U-i 3; ] 1 6\;ir-l 3/ 10(;t:-l) T, 1 , 2(^+1) 3^^+l . '• 3;ir— 1"^ x-1 3;t;2-4;c+l * ;tr2 — 1 ;»;—l 19. ;f— ^ AT— «__ 2(^— ^) 20. B(ab-x[a+d']) (2a + d')d'^x ^dx a^d^ ' a+b "^ a{a+by^ a {a+ty' RATIONALIZATION OF EQUATIONS. 379. If the unknown number in an equation appears under a radical sign, the equation must first be rational- SINGLE EQUATIONS. 257 ized before the value of the unknown number can be found. This is illustrated by the following examples. (1) Solve 3V4r-8=Vl3jc-3. Squaring both sides, 9{4:X—S) = 13x — d. Transposing and uniting, 23x=Qd, whence x=z3. (2) Solve vJ+9=5Vjt7-3. _ Squaring both sides, x-\-9=:2Dx—30^x-\-d. Transposing rational terms to one side and irrational to the other, 2ix=30S/x. Dividing by 6 and squaring, lQx^=25x. Solving this quadratic, "^=16 ^^ 0* 380. The most expeditious method for rationalizing any given equation depends upon the peculiar form of the equation, and can only be determined by the student after a little experience with this class of equations. (1) Given V9+I+^=ll. (1) Transposing everything but the irrational term to the right-hand side of the equation, we obtain ^fdTx=n-x, (2) Squaring both sides, 9 + ^=121-22^+^*, > (3) and solving this quadratic, we find x=*7 or 16. From (2) to (3) is a questionable derivation; for squaring both mem- bers of an equation, L—R, we have found (Art. 376) to be equivalent to multiplying through by Z + i?, and that the resulting equation is equivalent to the two equations \ Z+.?=0 \ . Therefore (3) is equivalent to the two equations I V9+^=ll-^ ) .. |V9 + ;f+ll-x=0 j ^' f +V9 + ^+:>r=:ll ) ,-, Hence, if we understand equation (1) to read the positive square root ^(9 + x)+5C=ll, 17— U. A. 258 UNIVERSITY ALGEBRA. then a new solution has been introduced between (2) and (3). But if we understand equation (1) to read either square root ^(9+x)4-^=ll. then it is equivalent to both the equations in (5), and no solution has been introduced. This is because the introduced equation ±Z-|-i?=0 is identical with the original equation ±L^=.R. In other cases the student will always find that rationalization 7nay or may not be considered a questionable derivation according as we con- sider the radical signs to call for a particular root or any root of the expressions involved. (2) Solve • V^+Vx+6=3. (1) Squaring each side of the equation, we obtain 5C+2Vx2 + 6x+x + 6=9. (2) Transposing all but the surd to the right-hand side, this becomes 2V:^^Tai=3-2x. (5) Squaring, we obtain 4x2H-24jr=9-12:>c4-4^^, (4) or %—\. What are the questionable steps? What is their effect ? The above solution is really equivalent to the following: V^+V;r 4-6=3. Transposing the 3 to the left member, v'^J-f-v:^^— 3=0. Multiplying through by rationalizing factor of left member (Art. 355), (\^+VA:-f-6-3)(\^+V.;r-i-6-|-3)(V^-V.r-H6— 3)(V^-V.^+6+3)=0; which reduces to 4^—1=0, (5) or x=\. The introduced equations are \^+V:r 4-6 + 3=0; V^-Vx+6— 3=0; V^-V7+6+3=0. Here, then, is an apparent paradox: three solutions seem to have been introduced, yet but one is found. This can be explained in the following manner. If we regard the radical signs as calling for either one of the two roots of the expression underneath, then the introduced equations are all identical with the original equation, and hence do not give rise to different solutions. If we restrict ourselves to using in each case that square root which has the sign given before the radical sign, then none of the introduced equations have any solution whatever, and hence no solution is introduced in this case. SINGLE EQUATIONS. 259 KXAMPIvKS. Solve each of the following equations: I. |/;c -4-4=4. 8. Vx—\/x—5=V3, 2. l/2jr-f6=4. 9. Vx—7=V''x—14:-\-l. 3. Vl0xTT^=5. 10. l/x—7=---']/x-^l—2. 4. •2;r+7 = V5:r— 2. II. x=7'-Vx'^—7. 5. 14 + ^4^-40=10. 12. l/^+20-l/:r-l-3=0. 6. Vl6x+9=4V4x-S. 13. T/;i;+34-T/'3;r-2=7. 7. V^^T~x=i+Vx. 14. T/2;r+l-hl/;i;-3=2l/^. 20;t: /-7^ ^ 18 15. - ^ — l/lOy— 9= , +9. ^ x—1 iZ-r+l i/^4.|/^__3 3 16. ,- = ^ ^ . 17. l/;i;-l -^—3 ' * ]/^__l/^^;_3 x—^ GRAPHIC RKPRESENTATION OF EXPRESSIONS AND EQUA- TIONS OF THE FIRST DEGREE. 381. Consider the expression 2;t:+3. If we assign different values to x we will obtain different values for 2;r+3. Thus, suppose ^=-4, -3, -2, -1, 0, +1, +2, +3; ... 2;i;+3=-5, -3, -1, +1, +3, +5, +7, +9. From this we observe that as x changes uniformly from —4 to +3 the expression 2x+S changes uniformly from —5 to +9. These changes may be pictured to our eye in a simple way. On the straight line of indefinite length, XX\ select any point, 0, and lay off equal spaces or units to the right and left of this point : y, -4 -3 -2 -1 +1 +2 +3 y ^ I I I I I I I I -^ Fig. 1.— Graphic representation of Algebraic Scale. 26o UNIVERSITY ALGEBRA. We may consider distances measured to the right of positive and distances measured to the left of negative. In this way there is pictured to our eye the values which are assigned to x above. To picture the values of ^x-\-Z we may draw at a perpencicular line 3 units long above XX' \ similarly one 5 units long at 1, one 7 units long at 2, etc. We may indicate the negative values of 2;t:+3 by drawing the lines below XX' \ thus at —2 we draw a line 1 unit long down. 2.— Graphic representa- tion of *lx-\-Z. 382. If in addition to the values assigned to x in the last ST'-^jr-r^p^j" article, intermediate fractional values be assigned and the cor- responding values of 2;i:-f3 be computed and placed in the fig- ure by drawing perpendiculars, it will be found that the ends of all the perpendiculars will be in the straight line P^ P^, Also if the same be done for values greater than 3 and less than —4 the ends of these perpendiculars will lie on P^ P^ produced. For this reason the line P^ P^ is called the Graphic Representation of 2;^;+3, or, briefly, the Graph or Plot of 2;t:+3. In Fig. 2 we are to understand that the line /^^ P^ is unlimited in length. BXAMPLKS. Represent graphically the following expressions: 1. 2;»;-3. 3- Sx+4:. 5. -x-2. 7. 4;»;+5. 2. x-1. 4. -'2x+h 6. -^x+S. 8. 5x--2. SINGLE EQUATIONS. 261 Fig. 3. — Graphic representa- tion of y=2x-\-Z. 333. To plot the equation y=2x+Z we draw the values of x as above, and the lines drawn perpendicular to XX' in Fig. 2 are the values of y. The graph then shows how y changes as x changes. We then have the Graph of an Equation instead of the graph of an expression. For this purpose the method of representation is elaborated, as the following def- initions explain. 384. The Axes. Any point in a plane may be located by a system of latitude and longitude; that is, by giving its distance from two fixed lines of indefinite length. These lines are called the Axes and are dis- tinguished as the :r-axis and the j/-axis. In Fig. 3, XX' is the Jtr-axis and YY' is the j^-axis. They are usu- ally, though not necessarily, taken at right angles. The point of intersection, O, is called the Origin. 385. Co-ordinates. The distances of a point from the two axes are called the Co-ordinates of the point The co-ordinate parallel to the ;j;-axis is called the Abscissa; the co-ordinate parallel to the j/-axis is called the Ordi- nate. These displace the words ''longitude" and ''lat- itude." Abscissas are always represented by an x and ordinates by a y. 386. Signs. Abscissas when measured to the right are reckoned positive, and when measured to the left are negative. Ordinates when measured up are positive^ and when measured down are negative. 262 UNIVERSITY ALGEBRA. -2 -8 -1 -5 — 2 + 1 + 1 4-2 +4 + 3 + 7 +4 + 10 387. Consider the equation 3;*:— 2=^. By assigning values to x and finding the corresponding values of j/, we may locate as many of the points of the graph as we please. The values of x and y are best given in the tabular form in the margin. Each pair of values on the same line lo- cates one point, as when :r is — 2, jj/ is —8. The graph is given in Fig. 4. 388. If we wish to plot the I ^ equation ax+b=y we may fio. 4.- Graph oi 3^-2=^. proceed as follows: For the value x=0 it is plain y='b\ therefore in the figure (Fig. 4) we draw OB=b, By transposing the equation, we obtain ^—=a. (1) X ^ Now if we let P be any point on the graph, then PD=^y and OD=x, Also PC-=y^CD=y—OB=y—b, since OB or b is negative in Fig. 4. Therefore (1) becomes PC CB^""' Thus we have shown that no matter where the point P be taken on the graph, the ratio PC : CB equals a fixed number a. Therefore, by plane geometry, the graph is a straight line. This proves that every equation of the first degree in tivo variables represents a straight line. 389. Graphic Representation of the Root. In the graph of a;r+^=^ what represents the root oi ax-\-b==^^l SINGLE EQUATIONS. 263 In the equation, when j/=0, x= ; in the graph, when 'i/=0, x=OA, Therefore OA^ , or the distance from the origin to the point of intersection of the x-axis and the graph of ax+b=y represents the root ofax+d=0. KXAMPivES. Form the graph of each of the following equations and point out the values of d, a, and the root of a;r+^=0. 1. x-^\=y. 3. —x-\-\=-y. 5. Ox-i-4:=y. 7. —^x+6=y 2. 2;tr— 3=jK. 4. ^=y. 6. 0;»;+0=j/. 8. —2x—A=y 390. Graphic discussion of ax-\-d=y. In Art. 234 we discussed the special forms of the equation ax +3=0. We may now represent graphically the conclusions there deduced. ' I. Suppose b=0 and a^O. Then in Fig. 4 OB [==b'] PC becomes zero and 777, [_=cl\ is not zero. Hence the graph takes the form of Fig. 5 or Fig. 6 depending upon whether a is positive or negative respectively. OA, or the root, is zero. Jf b=0 and a=^0 the graph passes through the origin. 264 UNIVERSITY ALGEBRA. II. Suppose a=0 and ^=0. Here OB becomes zero, and y is zero for all values of x. Hence the graph coin- cides with the ;»;-axis. OA, or the root, may be taken to aiiy poifit we please on the x-axzs, since the graph meets the X-axis at all points. This represents graphically the indeterminate value of the root. If a^=0 and b=0 the graph coincides with the x-axzs. T ^ Ml ^ Fig. 7. PC In this case 77^ is zero; III. Suppose a=0 and d=t=0. that is, the graph is parallel to the .r-axis. It does not coincide with the .r-axis, since d or OB^O. The graph takes the form of Figs. 7 or 8 depending upon whether d is positive or negative. r T ^ X ^0 r Fig. 9. Fig. to. 391. In Fig. 9 is represented the graph of ax+d=y when a is not zero, but very small. In Fig. 10 is repre- sented the graph ofax-j-d^y when a is very large. In Fig. 9 is ^ represented to be positive or negative ? SINGLE EQUATIONS. 265 -3 -2 -1 1 2 3 4 5 7 3 -2 -3 -3 -2 3 7 GRAPHIC RKPRKSKNTATION OF QUADRATIC KXPRKSSIONS AND EQUATIONS. 392. If we wish to draw the graph of any quadratic expression of the form ax"^ -\-bx-\-c, or any quadratic equa- tion of the form ax'^-\-bx-Yc=y, we must first obtain a table of values as indicated in the margin. This set of values came from the equation \x'^—^x—2=y. If the points determined by each set of values of x and y in this table be plotted they will be found to lie along the curved line represented in Fig. 11. If this table of values be extended by assigning other values to x and finding the corresponding values of _y, or if any number of fractional val- ues of X be interpolated and the intermediate values of y be obtained, in either case we would find that all the points located by the pairs of values would lie along the curve drawn in Fig. 11. For this reason the curve is said to be the graphic representation of the equation ^x'^—^x—2=y, 393. The curve represented in the figure is called the Parabola. Its properties need not be investigated here. It is suffi- cient to notice that the two roots of the quadratic equa- tion ax'^-{-bx+c=0 are OA and OB (in Fig. 11), and that in the case of example 7 below these two roots are equal to each other, and in example 8 the two roots are imaginary. \ Y \ \ \ Q< . J X a\ Fig. 11. 266 UNIVERSITY ALGEBRA. KXAMPLKS. Form a table of values and draw the graph for each of the following equations, pointing out the graphical rep- resentation of the roots of x^-i-ax+d=0 in each case. 1. x'^--2xS=y. 4. x'^=j/. 7. x'^—6x+9=j^. 2. x^—6x—7=jy, 5. x'^—S^j. 8. x'^—ex+lS=y. 3. x^+Sx+2=y. 6. x^-i-4:x=y. 9. jt^+S^j;— 3=j/. MISCELIyANEOUS EQUATIONS. Solve each of the following equations : 1. ;t;t+f=i^:ri 2. l/x^ —2 Vx + X=0, Divide through by A 3. x+5VW^x=4S. Subtracting 37 from each side of t he equ ation, we obtain ^-37 + 5 V37-x :zr6, which may be written —{S'7—x)-^5^ dl-x =6, or (37-x)-5V37-;r=-6. Putting y for V37 — ^, this becomes jk2-5j)/=— 6. Solving, J— 3 or 2. That is, V37-^=3or2, whence 37— ;«r=9 or 4, and 5C=28 or 33. The same example may be treated by the method of Art 379. /^. 2\/x^ — 6x+2—x^+Sx=Sx—78, 6. 4:X^—4x+20V2x^—5x+6=6x+6e. 7. x-^-'2x-^ = S. II. llQ.;^-4 + l=:2U-^ 8. x»—5x"+4:=0. 12. l/x-h4x~i=5. 9. 10;i;^'+.;i:«+24=0. 13. 8j»;*— 8;i;-*'=63. 10. a;trF:r+37^=^. 14. C-^— ^) — 7 ^,=2. SINGLE EQUATIONS. 267 15. Sx'^-Ax+15}/Sx^-4:X-6^42. 16. 2x^—Sx^-i-xi=0. Result: ;r=0 or 1 or 8. 6^-7 5(.r-l) ^ 1 ^^' 9;i;+6 12;i:+8 12* X hx'^ — \hx—% ^x—^ ___ ' 2"^ 10(;r~3) 5~~ "~ ' In example 20 rationalize the denominator of the fraction. CHAPTER XVII. SYSTKMS OF EQUATIONS. 394. The treatment of systems of equations of the first degree has already been explained. See chapter XI. The present chapter gives an account of legitimate and questionable transformations of systems of equations, and methods of solution of some systems in which all the equations are not of the first degree. See Arts. 239, 252. 395. It is assumed throughout this chapter that the equations of all systems considered are independent and compatible and that the number of equations of each sys- tem just equals the number of unknown numbers. 396. Equivalence of Systems. One system of equations is said to be Equivalent to another system ot equations when any set of values that satisfies the first system satisfies the second system also, and any set of values that satisfies the second system satisfies the first system also. Thus the system One system of equations is said to be equivalent to several systems of equations when any set of values that satisfies the first system satisfies one of the latter sys- tems, and any set of values that satisfies any one of the latter systems satisfies the first system. Thus the system x'^+y=l'i \ is equivalent to Jjr— 3=0) . J ^+2=0 X +J/= 5 j the two systems [_>/-2 = j ^^^ | j/— 7=0 SYSTEMS OF EQUATIONS. 269 397. Of course if but one equation of a system be operated upon in any manner during the solution, care must be taken that the tranformation be with a due regard to the theorems in Arts. 370-378. Obviously no operation which it is questionable to perform on an equation standing alone can be legitimately performed upon one belonging to a system. But in addition to the reductions which single equations may undergo, equa- tions of a system permit of certain transformations peculiar to themselves, and it remains to investigate the possible effect of these on the solution of the system. The following theorems are designed to point out the effect on the result of the ordinary steps in the process of elimination. 398. If from the system of equations we derive the system {a) ib) L„=R„ where all but the second equation remain unchanged, the derivatio7i is legitimate if T is a kjiown number not zero^ but questio7iable if T is a function of the unkjiown nurn- bers, it being indifferent whether S is a known number or a functio7i of the unknown ones. Write system {a) so that it shall read L,-R,=(i (1) L^-R^=^ (2) L„-R„=-Q W 270 UNIVERSITY ALGEBRA. and system (b) so that it shall appear SiL,-R,)-^TiL,^R,-)^0 (4) W L„-R„=Q Firsts suppose T a known number. Then any set of values that will satisfy {c) must make L^—R^, L^—R^ . . . and L„—R„ each zero. But, from the form of (3) and (4), any set that makes these zero must satisfy (d) also. Hence any solution of (c) is a solution of (^). It is seen from (3) that any set of values that satisfies (d) must make L^—R^ zero. Equation (4) will then become r(z.2-A)=o. (5) Now, since Z* is a known number not zero, this cannot be satisfied unless L^ — R^ is zero. Hence any set of values in order to satisfy {d) must make L-^ — R^ and L^—R^ and also . . . L„—R^ each zero. But any set of values that makes these zero will satisfy (r). Therefore any solution of {d) is a solution of (c). Now we have shown, first, that any set of values that will satisfy (c) will satisfy (d), and second, that any set of values that will satisfy {d') will satisfy {c). Hence, by Art. 396, the two systems are equivalent. Second, suppose T a function of some of the unknown numbers. In this case equation (5) may be satisfied by any set of values that will satisfy the equation 7^=0 without requiring that L^—R^, be zero. Consequently (d) can be satisfied without equation (2) being satisfied ; that is, without system (c) being satisfied. Therefore (d) is not equivalent to (r), but to the two systems L2—R2=0 L„-R„=0 and (L^'-R^=0 SYSTEMS OF EQUATIONS. 2/1 The derivation discussed in the above theorem is the one so fre- quently used in elimination. Thus take the system 2^+7=17 (1) ) 5;»:-10y= 5 (2) j" Multiplying (1) through by 5 and (2) through by 2 and obtaining a new equation by subtracting the former from the latter, the system becomes 2:>c+;/ = 17 (3) ) 25y=75 (4) ] We have eliminated x from the second equation, and consequently y is readily found to equal 3. From (3) x is then found to equal 7. The theorem also shows that it is legitimate to transform •^ ■^^= y \ into \ -^^-^ ^y x^ + (y=:xy f ^^^° ] 6-3x=0 by multiplying the first equation through by x and subtracting the resulting equation from the second. 399. /i^ ^s legitimate to derive from the system i\-k } w ike system L^=R^ \ .,. if T is a known number not zero. Rewrite {a) and (U) so that they shall read ^2-^2=0 (2)1 W and Zi^^i=0 (^)lr^^ It is evident that any set of values which will satisfy (<:) will satisfy (^); for whatever makes L^—R^ and Lc^—R^ each zero will satisfy (^). It is seen from (3) that any set of values that satisfies (^) must make Lx—R\ zero. Equation (4) will then become r(Z2~i?2)=o. (5) Now, since T' is a known number not zero, this cannot be satisfied unless L^—R^ is zero. Hence any set of values that satisfies {d) must make L^ — R^ and L^—R,^ €ach zero; that is, must be a solution of (r). 2/2 UNIVERSITY ALGEBRA. Now, since any solution of (c) is a solution of (^ ' and any solution of (d) is a solution of (^), the two systems are equivalent. As an example, consider the system ^ +7 --= 5 (1) ) ;^2+;/8=13 (2) S Multiplying (2) through by -1 and adding to this the square of the members of (1), we obtain the equivalent system x-\-y=b (3) \ x^-^2xy+y^-x-^-y^ = l2 (4) j" that is, x+y=5 (5) ) xy=Q (6) f Squaring (5) and subtracting 4 times (6) from it, we obtain the equiv- alent system x-\-y=5 (7) ) x'^-2xy+yz=zl (8) f which is equivalent to the /w^ systems ^+^=? i and \ ^^+^=5 , that is, to the solutions x=3 ) , j x=2 y=z2 \ ^^ }y=^ Note that the solution of any system of equations consists merely of the simple systems of linear equations (of the form x=a, y=^, etc.) which together are equivalent to the original system. Thus, [jr=r3, y—2] and [x=2, y=S], which is the solution of [x+/=5, x'^ -{-y^ = l3], are equivalent to the latter system. A solution to a sys- te?n of equations is another system or several systems. The above theorem evidently holds if the cube or any other power of the members of (1) be used instead of the square. 400. If from a system containing two unknown nu7nbers we derive the system L^=^R^ (3) I ^a\ L,L,=R,R, (4) I W the derivation is questionable if L^ and R^ both i7wolve un- known numbers y but legitimate if either is a known number not zero. First, suppose that L^ and R^ each involve unknown numbers. Any value of the unknown numbers which SYSTEMS OF EQUATIONS. 2/3 will satisfy the equation Z^=0 must satisfy equation (4), since the relation L^—R^ must hold if system (J?) is to be satisfied. Also any value of the unknown num- bers which will satisfy the equation R^=0 must satisfy (4), since the relation L^=R^ must hold if the system is to be satisfied. Moreover, au}^ value of the unknown numbers which will satisfy L^—R-i must satisfy (4), since the relation L-^=Ri must hold. Therefore, from these considerations, it is evident that the system (J)) is not equivalent to system (a), but to the three systems Second, suppose that either L^or R^\S2, known num- ber not zero. One of them, say R^, is the known num- ber. Therefore L^ cannot be zero, since the relation L^=Ri must hold. Hence the introduced systems (^j) and (^2) ^^^ absurdities, since they require that L-^ and R^ be zero. Consequently the derivation is legitimate, since the equations of the introduced systems are in- compatible. As an illustration of the theorem, consider the system x-4=G-y I 2x+y=13 \ This is satisfied by x=3 and y=7. Now form the system x-4=:C)-y I which is satisfied by either of the sets of values, x=3, y=l, or ^=4, ;j/=6. The additional solution may be obtained from either of the systems x-A^G—y) x—4^=G-y\ x-4=0 i" G-;/=0 i* As another example, consider the system x-{-2y=l S which is satisfied by ^=5, ;'= 1. From this we may obtain the system :tr2 -4/2=21 S 18— U. A. 274 UNIVERSITY ALGEBRA. From the first equation of the system, we obtain x=3 + 2y. Substituting this value for x in the second equation, it becomes 9+12>/+4>/2-4)/2r=21; whence J=l- Therefore, from the first equation of the system, x=5. In this case we see that no solution has been introduced. In fact, the introduced systems are incompatible, viz. : A7-2>/=3 ) x-2y-^ ) 5C-2y:=0f 7=0 f 40 It If from the system we derive the sy stern L^=^R^ ^2=^2 } w } (^) the derivation is questionable if both L^ and R^ involve unk?iown numbers, but legitimate if either is a k?iown number not zero. First, suppose that L^ and R^ both involve unknown numbers. Then, by Art. 400, if we pass from {U) to {a) we gain solutions. Hence to pass from {a) to {U) we lose those solutions. Second, suppose that either Z^ or 7?^ is a known num- ber. Then, by Art. 400, if we pass from {U) to {a) no solutions are gained. Hence none are lost if we pass from {a) to Qf). According to the above theorem it is legitimate to divide one equa- tion by another, member by member, if one member is a known number not zero. (1) Take the system x-\-y=i^ \ and derive the system jr+r=5 ) ,., x~;/=3 \ (^) The only set of values which will satisfy {a) is j:=4, y=\. This set satisfies {b), and no solution is lost. The system {a) is equivalent to the system {b) and to two other systems (see Art. 400), but the other two systems are incompatible. (^) SYSTEMS OF EQUATIONS. 275 (2) As an example of the case in which solutions may be lost, con- sider the system X =3—y I / \ x^ = C)-y2 \ ^^> which is satisfied by either of the two sets x=0, y=3 and ^=3, y=0. If we divide the second equation by the first, member by member, we pass to the system x=3—y which is satisfied only by the values x=zS, y=0. (3) Consider the system x —2y = S x^-4y^=21 x^-2y»=Sd-g By Art. 398 this is equivalent to X -2y = 3 x^-4:y^=2\ 2y^=12-z By the above theorem this is equivalent to x-2y= 3 x + 2y= 7 2y^=l2-z By Art. 398 this is equivalent to ^~^ ^ which, by Art. 399, ^ ^~^ y=i [Whi.... uy ^IL. ^VV,^ J 2y2 = \2-z) IS equivalent to ^^ ^ 10 UNKAR-QUADRATIC SYSTEM. 402. We now take up the solution of those systems involving two unknown numbers which consist of one linear and one quadratic equation. It is convenient to call this a Linear- Quadratic System. We proceed by first working the following example : x''-2y'' = l (2)1 W From equation (1) the value of x in terms of y is easily seen to be x==5—y. (3) Substituting this value of x in equation (2), we obtain (5-J/)2-2j/2 = l, (4) or 25-10y-\-y^-2jy^ = l. (5) Uniting and transposing terms, ^2 + i0j=24, (6) 2/6 UNIVERSITY ALGEBRA. whence, solving this quadratic, jk=2 or —12, and from equation (1), x=S or 17. Consequently there are two sets of values which satisfy (a), namely: y=2] j/=-12| 403. General System. The method used may be applied to the solution of any linear-quadratic system containing two unknown numbers. In fact, take the general case x-\-ay=b 1 ,^ x'^-\-cy'^+dxy+ex+fy=g ] where a, b, c^ d, e^ f, and g are supposed to stand for any real numbers whatever. The value of x in terms of y from the first equation of the system is x=b—ay (1) Substituting this for x in the second equation of the system and combining those terms in the second equa- tion which contain y^ and those which contain y and transposing all the known terms to the right-hand side, we obtain the system x=b-ay (2)1 {a'^^-c-ad')y'^-\'{bd-1ab-ae-{-f)y^g—b'^'-be (3) j in which (3) is a quadratic having y as the only un- known number; whence it can be solved. The values which may be found from this can be substituted in equation (2) and the values of x will be determined. EXAMPLES. Solve each of the following systems : 1. .;t:2— 3^2^52, 3. ;r2-jj/2 = — 9, 5. ;t:+j^=6, 2;tr— 3jj/=8. x—y=—\, xy=5. 2. 3jry-4)/2 = 20, 4. x''+y^ = 2, 6. 2y'^+xy=14:, Sx—iy^lO. x-i-y=2. 2y+x=7. 7. .A;2+3rj/-j>/2=3, 8. 5jr2— 3.ry— 2j/2 = 12, Sx-y=h 2x-y=B. SYSTEMS OF EQUATIONS. 2// SYSTEMS OF TWO QUADRATICS. 404. If we have a system of two quadratic equations containing two unknown numbers and attempt to elimi- nate one of the unknown numbers, it will be found in general that the resulting equation is of \h Clearing of fractions, 4+2v'^=9-9v (9) or 2v^+9v=5 Solving this, z;=-|- or — 5 Substituting each of these values in turn for v in (5) or (6), x=±2ordzll/S And since j^=z^jr, y=^l or qpfl/S Whence we have the four solutions _ ;i:=2| x=-21 x=l}/S I ^=-11/3 I y=li J/=-lj j,/=-|i/3i j|/=|l/3 I 406. General Case. The general system of two equations of the' above class may be represented by a^x'^+d^xy+c^y'^=d^ W 1 Ta^ a2X^-{-d2xy-}-C2y^=d2 (2) j ^ ^ Multiplying (1) by ^2 ^^^ (2) by d^ and subtracting, SYSTEMS OF EQUATIONS. 279 Dividing through hy y'^, we obtain an equation of the ^(£)Vi.(£)+c.o. From this equation two values of x-r-y can be obtained, say r-^ and r^. Then we have x==7^y x=^r^y Either of these being a linear equation, by combining (1) with each in turn we shall have two linear-quadratic sys- tems and shall obtain in general four solutions to the original system. 407. Case II. To show that any system of two sym- metrical quadratics can be solved we may take the general case of this system, as follows : x'^-^a^x^-b^xy+a^y+y'^^c^ (1) | x'^^-a^x-^b^xy^a^y-\-y'^=c^ (2) | W Substitute u-\-w for x and u—w for j/, u and w being two new unknown numbers. Then {a) becomes (2-f^2>2+2a22^ + (2-^2>'=^2 (4)3 ^^^ Multiplying f3) by 1—h^ and (4) by l—b^ and subtracting we obtain an equation of the form Au'^-^Bu^C, (5) from which two values of u may be obtained. Substi- tuting in (3) or (4), two values of w may be obtained. Finally x and y may be determined from the equations x=^u+w The above work shows that Case II can always be solved, but we do not pretend that the method used is always the most economical one to employ. The inge- nuity of the student will often suggest for particular examples special expedients which are preferable to the general method. 28o UNIVERSITY ALGEBRA. KXA.MPLKS. Solve each of the following systems of equations.. x^--2xy=8: x^—5j/^=: — l. 2, Axy=4Q, 5. ^2_3^_^_^2+29=0, 3. ^2+y,= 25, 6. Aix+jrj-^Sxj^O, xj/—jy^ = —4:, x^+x+y+j/^=20, SPECIAL EXPEDIENTS. 408. In solving systems of equations above the first degree special expedients are more to be sought for than general methods. The following solutions will tend to point out some of the more common artifices made use of. (1) Solve x2+>'2rr20, (1) x-]-y=(j. (2) Squaring (2), x^+2xy+y^=d6. (3) Subtracting (1), 2xy=lQ. (4) Subtracting (4) from (1), x^-2xy+y'i=4:; whence x—y=±2. But from (2), x+y=Q. Therefore ^=4 ) ^„ , j x=2 (2)cSolve x+y=Q, (1) xy=5. (2) Squaring (1), x^ + 2xy ^y^ =36. (3) Subtracting 4 times (2) from (8), x^—2xy+y^z=zl6; whence x —y= ± 4. But from (1), x-\-y=Q. Therefore ^^f [ and j ;=1 (3) Solve x^-\-y^=Zi, (1) xy=\b. (2) Adding 2 times (2) to (1), x^i-2xy-^y»=Q4:. (3) SYSTEMS OF EQUATIONS. 28 1 Subtracting 2 times (2) from (1), x2-2xy+y^=4. (4) Wnence, from (3) and (4), x-{-y=±S, x-y—±2. Therefore ^~5 ) x=^ ) 5C= -5 ) ar=-3 ) y=^S y=^S y=-^S y=-^o\ (4) Solve ^8+jj/8=72, • (1) x+y=:Q. (2) Cubing (2). 5c8 + 3;r«;/+3xr2+;/8=21C. (3) Subtracting (1) and dividing by 3, xy{x+y)=4S>'^ (4) whence, since x-\-y=(S, xy=8. (5) From (2) and (5) as in example (2) we find ^=4 ) x=.2 ) y=2 f y=i f (5) Solve x^+y^-12=x+y, (1) xy + 8=Z{x+y). (2) Adding 2 times (2) to (1), {x+y)^—5{x+y)=-4:, (3) or completing square, {x+y)^ —5{x-\-y)-\-^£-=^; (4) whence x+y=4: or 1. (5) Therefore (2) becomes xy=0 or — 6. (6) Solving (5) and (6) as in example (3), we find ^=0) x=4 I x=S I x=—2l y=4:\ y=os y=-2S y=B S (6) Solve x*+y*=2'72, (1) x-^y=Q. (2) Raising (2) to the fourth power, x'^-{-4x3y-{-Gx^y»-{-4:Xy^-\-y'i = l2d(j, (3) Subtracting (1) from (3), ^x^y-\-Gx^y^-\-4xy» = 1024:, (4) Factoring, xy{2x^-^dxy-\-2y^)=512. (5) Squaring (2) and multiplying by 2xy, xy{2x^-{-4xy + 2y^)=72xy. (6) Taking (5) from (6), {xy)^='72xy-512. (7) Solving this, xy=Q4: or 8. (8) Taking (8) with (2) we have two systems like example (2). x-\-y=Ql x-\-y=G ) xy=8 ) xy=64^ f Solving these, we find x=4: I x=2 I x=3-fV-55 j_ x=3-\^^ ) jK=2f jj/=4f ^=3-V-55) jK=3+V355f 282 UNIVERSITY ALGEBRA. KXAMPI^KS. Solve each of the following systems : I. ;r2+j/2=25, :r>/=12. 3. 2;r2+3j/2 = e2, 2^2_3^2=,38. ;irj/=20. 5. ;i:2 + 3xj/-2y = 26, 6. ;i;2-jv2 = 27, 7. jr»— j|/3=63, ^-jV=3. 8. Jt2+;rj/+j/2=:9i^ 9. ;r2+j/2+^+^_i8, 10. x^+y'^+x—y=78y xy=^24:. 11. 42(^2^^/2)^35^^^ lbxy=2b20. 12. x^— jj/^ = 19(^— -j^), jr^— JJ/2=5. 13. -^^+y=97, ^2^y_13, 14. j»;3_f.^3 = i52, 15- xy-\-x=20, xy—y==12, 16. 16^2 ==9y, 2xy—bx-h6y=S3. 17. .;i;S+jk3==i33^ 18. (^x^+y'^X^-^y') = 272, x'^+y'^+x-\-y=^A2. 19. ^2+^JI/=K-^+j/), 20. xy — ^2_^2__ — 21^ x^+y^+xy=22B. 21. xy-hxy'^ =^12, x-hxy^=:lS. 22. x'^-\-Sxy=o4:, xy-i-4y'^ = 115, 23. .;r2+xj/+2y2=2, 2^2_-._^^^^2=2. 24. x^+bxy+y'^=51, x'^—xy+y^=21. 25. ^2+y = 34^ 2x^-'Sxyi-2y^ = 23. 26. x2+j/2=;r>/4-7, ;r2— jj/2 = jt^ — I 27. x'^-\-xy=4:, x'^-\-2y'^—xy=8. 28. 2xy+y^ = 16, 2x^—xy=12. SYSTEMS OF EQUATIONS. 283 30. ^j+^=10, 11 9 32. ^^^^4-1/^= 12, ;i:^=1225. ^^ X j/""36' a I? 35. ^-2y^= 1. 36. ^+jj/=^— l/;i;+^, 37. (^+j/)(^y"j/*)=13, (;t:-j/)(jr*+j|/i) = 25. 38. j/+^ = 3xjj/3', 39. x{x+y-\-z)=a^, 2-\-x=2xy2, y(x-{-y-\-z)=^b'^, x-\-y—'^xy2. z{x-\-y+2)—c'^, GRAPHIC REPRKS:^NTATlON OF SYSTEMS OF I^INBAR EQUATIONS. 409. If we wish to repre- sent graphically the system of equations Zx—y=^2 x~2y=—6 we maj^ obtain a table of val- ues from each equation, then draw the graph of each equa- tion on the same axis of ref- erence, a shown in Fig. 12. Fig. 12 — Graph of the system 410. Geometrical Meaning of Solution of a Sys- tem. Any set of values of x and y that satisfies an equation containing x and y locates some point on the 284 UNIVERSITY ALGEBRA. graph of that equation. Consequently any set of values of X and y that satisfies both equations of a system of two equations contairiing x a?id y must locate some point com- mon to the graphs of the two equations. In other words, the coordinates of a point of intersection of two graphs is a solution of the equations of the graphs considered as simultaneous equations. Thus in Fig. 12 above the coor- dinates of P, the point of intersection of the lines, are 2 and 4, and ;tr=2, j>/=4 is the solution to the system Zx-y=1, x-2y=6. 411. Graphic Discussion of a System. In Art. 252 we discussed the special forms of the system a^x-i-d^y=c^ (1) | . . We may now represent graphically the conclusions there deduced. For this purpose it is convenient to have (a) in one of the following forms : (^) ^, .c. «2 ^2 W The equations in (^) are in the form of ax+b=y; there- fore, by Art. 388, the coefficients of x, ——^ and — ,-, determine the direction of the lines with reference to OX (the ratio PC\CB in Fig. 4), and -^ and ^^ determine the points of intersection of the lines with the j^-axis (the distance OB in Fig. 4). By analogy we may note that in the equations in {c) the coefficients of jk, and b ^1 — -, determine the direction of the lines with reference <^2 c Co to O K and — and — determine the points of intersec- tion of the lines with the .^-axis. SYSTEMS OF EQUATIONS. 285 I. Suppose aiC2—ci2Ci=0 and a^b<2^—a<2bx'=l^^\ that is ^ suppose -^=— ^ and —4^~, Whence from (c) tlie lines cross the ;r-axis at the same point. Since — ^--, the a^ ^2 directions of the lines are not the same. Therefore we have two different lines intersecting on the :r-axis. If a^c^—ciiC-^^^^ and a^b^—ct^b-^^O, the graph is two distinct lines intersecting on the x- axis. Ifb^c^ — ^1^2^=^^ and a-^b^ — d^^i^^i the graph is two distinct lines inter- seeling on the y-axis. II. Suppose a^c<2, — <3J2^i=0 and a^b2 — ^2^i=0; that is suppose — ^=— ^ and -^=-^. Then from (c) we see that a-^ a^ a^ a^ the lines have a common point on the ;r-axis and also have the same direction. Therefore the graph is, two coincident straight lines. As these lines intersect at ^ U X SL Fig. 13. r Fig. 14. every point, we see the appropriateness of calling the the solution indeterminate. Since the graphs of the sep- arate equations are coincident, it shows that the given system is really equivalent to but one equation contain- ing two unknown numbers. The graph is represented hi Fig. 14. If <52^i —^1^2—0 ^"^^ a ^b 2,— a 2b 1=^0 ^ the graph takes the form in Fig. 13. 286 UNIVERSITY ALGEBRA. If a ^c 2— (12^1=0 and a ^5 2— a 2^ 1=0, the graph consists of two coincident straight lines. If ^2^i"~^i^2=0 and ^ii^2"~^2^i~^> ^^^ ^^^^ ^^ true, III. Suppose a^C2—a2C-^^0 and ai^2~'^2^i=0; that is, suppose — and -1=-^. Then from (f) we observe that the lines intersect the jc-axis at different points and that the directions of the lines are the same; whence the graph consists of two parallel lines, as in Fig. 15. We notice that the lines of the system do not intersect, and consequently there is no finite solution to the system. The equations are said to be incompatible, but the lines are said to be parallel If a^C2—a2C^^^ and a^b2—a2b^=^0y the graph consists of two parallel lines. KXAMPLKS. Draw the graph of each of the following systems and point out the graphical representation of the solution to the system : 4. %x^-Zy=Z, 12jt:+9j/=3. 5. 5;r— 4j/=6, 8:i;=7r. 6. hx—y^^Xhy PROBI.KMS. 1. x-hy=6y x-y=2. 2. 7x-i-4j/=l, 9^ -1-4;/= 3. 3. 2;t:+jK=ll, 7. Sx+Sy=6, Sx+Sy=16. 8. x+^y=l 4:x+10y=15. g. Sx'-2y=0, 2xSy=d, I. What two numbers are those whose product is 24 and whose sum added to the sum of their squares is 62 ? I SYSTEMS OF EQUATIONS. 28/ 2.. A number consists of two digits whose sum is 15. If 31 be added to the product of the digits, the digits will he in the reverse order. What is the number? 3. The sum of the squares of two numbers is 410. If we diminish the greater by 4 and increase the lesser by 4 the sum of the squares of the two results is 394. What are the two numbers? 4. Find four consecutive integers such that the product of the first two shall be a number that has the other two for digits. 5. The hypothenuse of a right-angled triangle is 10 feet and its area is 24 square feet. Find the sides of the triangle. 6. Find the sides of a right-angled triangle ; given its hypothenuse equal to h and its area equal to a. 7. The perimeter of a rectangle is 16 feet and its ^rea is 15 square feet. Find the dimensions of the rectangle. 8. Find the dimensions of a rectangle ; given that its perimeter is p feet and its area a square feet. g. Show that if an isosceles triangle, the sum of whose sides is 2^^, be inscribed in a circle of radius a, one of the two equal sides of the triangle must equal l/3«2±al/9^2_2^^ 10. Find the side of an equilateral triangle, knowing that a side exceeds the altitude by d feet. 11. A and B attempt the same quadratic equation. A after reducing has only a mistake in the absolute term and finds for roots +8 and -f 2 ; B after reducing has only a mistake in the coefficient oi x and finds for roots —9 and —1. Find the roots of the correct equation. 288 UNIVERSITY ALGEBRA. 12. A man bought some horses for $1250. If he had bought 3 more and paid $25 less for each horse, they would have cost him $1300. How many horses did he buy, and at what price ? 13. If a carriage wheel 14|- feet in circumference take one second more to revolve, the rate of the carriage per hour will be 2|- miles less. How fast is the carriage traveling ? 14. A sets off from London to York, and B at the same time from York to London, and each travels uniformly. A reaches York 16 hours and B reaches London 36 hours after they have met on the road. Find the time in which each has performed the journey. 15. A man arrives at the railway station nearest his home \\ hours before the time at which he has ordered his carriage to meet him. He sets out at once to walk at the rate of 4 miles per hour, and meeting his carriage when it had traveled 8 miles, reaches home one hour earlier than he had originally expected. How far is his home from the station, and at what rate was his carriage driven ? I CHAPTER XVIII. THEORY OF I.IMITS. 412. Thus far in the sudy of Algebra the numbers we have used have always preserved the same value through- out the same problem. Letters have been used to repre- sent numbers. A letter may have stood for one number in one problem and another number in another problem, but any letter has always preserved the same value in the same problem. There are, however, many cases in Alge- bra in which it is desirable to consider expressions which change in value in the same problem, and thus we are led to consider two kinds of number defined in the next article. 413. Constants and Variables. When an expression preserves its value unchanged in the same discussion,, it is called a Constant ; but when under the conditions of the problem an expression may assume an indefinite num- ber of values, it is called a Variable. Constants are usually represented by the first or inter- mediate letters of the alphabet and variables by the last letters. The notation by which we distinguish between constants and var- iables is the same as that by which we distinguish between known and unknown numbers, but it must not be thought that any analogy is intended to be pointed out by this fact. When we are discussing a problem in which both constants and variables appear we usually do not care whether the constants are known or unknown. 414. Continuous and Discontinuous Variables. When a variable in passing from one value to another 19— u. A. 290 UNIVERSITY ALGEBRA. passes through all intermediate values, it is called a Continuous variable ; when it does not pass through all intermediate values, it is called a Discontinuous var- iable. 415. Limit of a Variable. When a variable changes in value by approaching nearer and nearer some constant which it can never equal, yet from which it can be made to differ by an amount as small as we please, this con- stant is called the Limit of the variable. 416. Illustrations. If a point move along a line ^^, starting at A and moving in such a way that the first second the point moves one-half the distance from A to B, the second second one-half the remaining distance, the third second one-half the distance which still re- mains, and so on ; then the distance from A to the mov- ing point is a variable whose limit is the distance AB, ^ ^ : ^ ^ For, no matter how long the point has been moving, there is still some distance remaining between it and the point B, so that the distance from A to the moving point can never equal AB\ but as the moving point can be brought as near as we please to B, its distance from A can be made to differ from the distance AB hy stn amount as small as we please. Thus we see that the distance from A to the moving point fulfills all the requirements of the definition of a variable, and the distance AB all the requirements of the definition of a limit. The student must note that it is not the pozn^ B that is the limit of the moving point, although the moving point approtaches the point B ; but it is the dista^ice AB that is the limit of the distance from A to the moving point. \ THEORY OF LIMITS. 29I If we call the distance the point moves the first second 1 (then of course the whole distance AB would be 2), the distance traversed the second second would be \, that traversed the third second would be ^, and so on, and the entire distance from A to the moving point in n seconds would be the sum of n terms of the series 1 i 1 1. _i -^J 2> T' 8' i^» • • • Now it is sure that the more terms of this series that are taken the less does the sum differ from 2; but the sum can never equal 2. Hence we say that the limit of the sum of the series l+i+i4-i+YV+- • • as the num- ber of terms is indefinitely increased is 2. Again, consider any regular polygon inscribed in a circle. Join the vertices with the middle points of the arcs subtending the sides, thus forming another regular inscribed polygon of double the number of sides. From this polygon form another of double its number of sides, and so on Now the polygon is always within the circle, and hence the area of the polygon can never equal the area of the circle; but as the process of doubling the number of sides is continued the less does the area of the polygon differ from the area of the circle. Hence we say that the limit of the area of the polygon is the area of the circle. Again, as a straight line is the shortest distance be- tween two points, any side of the inscribed polygon is less than the subtended arc ; hence the sum of all the sides, or the perimeter, of the polygon is less than the sum of all the subtended arcs, or the circumference, of the circle ; or in other words, the perimeter of the poly- gon can never equal the circumference of the circle. But as the process of doubling the number of sides is con- tinued the perimeter of the polygon differs less and less 292 UNIVERSITY ALGEBRA. from the ciecumference of the circle ; hence the circum- ference of the circle is the limit of the perimeter of the inscribed polygon. 417. The student should not infer from what has been said that all variables have limits. In fact the truth is quite the contrary, for most variables do not have limits. Thus in the illustration of the moving point given above the variable does not have a limit if we suppose the point to move at a uniform rate. For, if the velocity is uni- form, it is a mere question of time until the moving point passes B or, in fact, any other point to the right of B, however remote. Much more would this be true if the point moved with increasing instead of uniform velocity. Again, consider the fraction . If x be supposed to change in value, the value of the fraction changes and is itself a variable. Now suppose x to decrease in value. It is plain that the value of the fraction increases without limit as x decreases. In other words, the value of the fraction can be made as large as we please by taking x small enough. Hence as x decreases the value of the fraction increases, 418. It follows immediately from the definition of a variable that the difference between a variable, and its limit is a variable whose limit is zero. For if ;i: be a variable whose limit is a, then x may be made to differ from a by an amount as small as we please; hence a-^x may be made as small as we please. Yet as x can never equal a^ a—x can never equal zero ; hence a—x is a variable whose limit is zero. THEORY OF LIMITS. 293 419. Infinitesimal and Infinite. A variable which approaches zero as a limit is called an Infinitesimal. The difference between a variable and its limit is there- fore an infinitesimal. When a variable increases without limit in such a manner that it may become and remain larger than any assigned number, however large, the variable is said to be Infinite and is called Infinity and is often represented by the symbol 00. It is to be noticed that the words infinite and infinites- imal do not refer to definite magnitudes at all, but each refers to something which is essentially variable. These two words are introduced in order that we may have names to designate two particular kinds of variables which play an important part in mathematics. A num- ber which is neither infinite nor infinitesimal is called Finite. THEOREMS ON I^IMITS. 420. Theorem I. If two variables are continually equal and each approaches a limit, the limits are equal. Let X and y be the variables and let limit x=-a and limit y=^b. We are to prove that a^=^b. \i a and b are not equal, suppose a greater than b and let a—b^d. Let a--x=u and b—y=v', then a=x+u and'^=j/-f z/, and a-^b=^d becomes by substitution {x-]ru) — {y+v)=d, or (x—y)-\-(u—v)=^d. Since limit ^=«, .*. limit u=0] and since limit y=b, .'. limit v=0. Thus we see that u and v are each var- iables which may be made as small as we please, and hence the difference u—v may be made as small as we please, and hence may be made less than d. Therefore x—y equals some number; /. e, , x and y differ, which is 294 UNIVERSITY ALGEBRA. contrary to the hypothesis. Hence a cannot be greater than b, and in the same way it may be shown that b can- not be greater than a. Therefore a=b. 421. Theorem II, The limit of the sunt of a constant and a variable equals the sum of the constant and the limit of the variable. Represent the variable by x and its limit by a. There- fore limit x=^a. Let c be any constant. We are to prove that limit (c+x')=c-i-a. lyct a—x=u; (1) .-. x=a~u; (2) .'. c+x=c+a—u, (3) Now, since x may be taken so near a as to differ from it by an amount as small as we please, therefore by equa- tion (1) u may be made as small as we please, therefore c+a—u may be made to differ from c+a by an amount as small as we please. Therefore limit (^c+a—u')=c+a. From (3), limit (c+x)=c+a. 422. Theorem III. The limit of the variable sum of a limited number of variables equals the sum of their sep- arate limits. In this theorem we speak of the variable sum of a lim- ited number of variables because it is possible for two or more variables to be so related that when added together the sum is constant, and as it is only variables which have limits, it is necessary that the sum of the variables considered should itself be variable. Again, we speak of a limited number of variables because, as will be seen from a single illustration given at the end of the proof, the theorem is not necessarily true for an unlimited num- ber of variables. THEORY OF LIMITS. 295 Let the variables be x, y, z, etc., and let limit ;»;=«, limit j/=/^, limit z=c, etc.; we are to prove limit {x+y-\-2-^. . .) = <2 + ^H-^+. . . Let a — x^=^ic .'. x=a — u; d—y=^v .'. y=d — v; C — Z=W .'. 2=C — W'y etc. etc. Then (x+y+2+, . .) = (a + ^+^+. . .) — (u-^v+w-\- . . .) Suppose u to be numerically the greatest of the numbers u, V, w, etc., and suppose that there are n of these num- bers. Now, since x may be taken so near a as to differ from it by an amount as small as we please, it follows that u may be made as small as we please. Therefore X may be taken so that u will be smaller than any fixed number that may be named. Then letting k stand for some fixed number, no matter how small, x may be taken k so that[]u therefore limit { — )= limit —^- But by Art. X x^ \x) x^ 426, limit f 4)=-9 or -• Hence limit ("-)=-• \x^) a^ a \xj a This theorem is not proved when a is zero, for we do not yet know what is meant by -J-. This will be taken up presently. 428. Theorem IX. The limit of a power of a variable equals that power of the limit of the variable. With the same notation as before we are to prove that limit x^'^^a'', n being either positive or negative, inte- gral or fractional. 298 UNIVERSITY ALGEBRA. First. When ;^ is a positive integer. By Art. 425, limit {xxx . . .') = aaa . . . or limit x''=a''. i> Second. When ;2 is a positive fraction, say — • I^t ^^=jv; (1) then x-=y\ (2) hence by Art. 420, limit ;r=limit^. (3) If we represent limit jv by b, then by first case of this article. limit JV*'=^^• (4) hence from (3) and (4), a=b''. (5) From (1), .^l=y; (6) p hence by Art 420, limit ;i:^= limit y^=b^, (7) Bat from (5), b^=a^) (8) hence limit x^—ai, (9) Third. When ;^ is a negative number, either integral or fractional, say n=^—s. Since ^"^=-3., therefore limit :i:~^= limit — =r — -^ — ."=—,= ^~'\ x^ limit xf a" ' .*. limit x~^=a~\ INDKTKRMINATE FORMS. 429. Certain expressions occur in Algebra which for particular values of one or more of the letters assume an indeterminate form. For example, an expression may assume the form ^ and this result may be placed equal to any number whatever and the result satisfies the test of division, viz.: the quotient multiplied by the divisor equals the dividend; e.g., ^=6 because 6x0=0. Of course any other number would answer as well as 6. We shall presently consider some of these indeterminate forms. THEORY OF LIMITS. 299 430, From tlie properties of fractions we know that if the denominator is a fixed finite number and the numer- ator approaches zero the value of the fraction itself ap- proaches zero, and if the numerator increases without limit the value of the fraction itself increases without limit. In other words, if the denominator of a fraction is a fixed finite number the value of the fraction will be- come infinitesimal when the numerator becomes infinites- imal, and infinite when the numerator becomes infinite. Again, if the numerator is a fixed finite number and the denominator approaches zero the value of the fraction itself increases without limit, and if the denominator in- creases without limit the value of the fraction approaches zero. In other words, if the numerator of a fraction is a fixed finite number the value of the fraction will become infinite when the denominator becomes infinitesimal, and infinitesimal when the denominator becomes infinite. 431. The expression ■§■ considered by itself is abso- lutely void of meaning, and no meaning can be assigned to it until we know how the expression originated. It must be remembered, however, that an expression pre- sents itself in this form because of certain values being given to one or more of the letters in the expression. For example, the fraction — ^-; 77- when jr=2 assumes ^ ^ x^-]-x—b ^ the form ■^. Therefore in this case we are dealing with \' x'^ — Qx-\-S Is the value of the fraction — ^— 77- when ;r=2. By Uhe value of this fraction when x=2 we mean the /imzf which this fraction approaches as x approaches 2, and to find this limit is the problem before us. 300 UNIVERSITY ALGEBRA. x^ 6;t;+8 432. I^et us now find the limit of — tt-, tt as x x^-\-x—o approaches 2. To express this limit we use the notation limit — K-, ^. This is read the limit of — ^— j,- ^Q x^+x — o x^-\-x—Q as X approaches 2, the symbol ^ standing for approaches. ^ , ;r2-6^+8 (^— 2) (jr--4) x—A We know _^^_^=^-_^-^=— -. Therefore by theorem I, limit — 2~i ^=limit — -5- r >£--^__^ But plainly, ™ x+^~ b Therefore ^^'^ ^2+^__e --5' KXAMPI.KS. By a method similar to that illustrated above find the limit in each of the following examples : , limit r^^±^:^i ^ limit r c^+^)^-^n ^ limit r_^f_i - limit r^!+i"| . limit r^i=j^"| 6 Prove li^it r^^—l=^^''"^ 433. Sometimes we have to deal with the product of two expressions, one of which is infinite and the other infinitesimal : such an expression, for example, as Ix^—xi T ) when x approaches zero. Here the first factor approaches zero and the second factor in- creases without limit as x approaches zero. When x—^ this expression is absolutely void of nllaning until a THEORY OF LIMITS. 3OI meaning is assigned to it. By the value of this expres- sion when x=0 we mean ihe limit which this expression approaches as x approaches zero. To find this limit is the problem before us. 434. Let us now find the limit of {-^^ ""-^jf— tt ) as X approaches zero. We have by succesive reductions Therefore by theorem I, :;ToC(''-')(5TT-j)]-:j?i(>-')- KXAMPI^KS. In a manner similar to that just pursued find the limits in each of the following examples : - -'; [(^-,4i)(^-^)] 302 UNIVERSITY ALGEBRA 435. Sometimes we have to deal with the quotient of two expressions, each of which is infinite: Such an ex- pression for example, as f-^r j-^-f ) when x approaches zero. Here both dividend and divisor in- creases without limit as x approaches zero. When;r=0 this expression is void of meaning until a meaning is assigned to it. By the value of this expression when x=0 we mean the limit which this expression approaches as X approaches zero. To find this limit is the problem before us. 436. Let us now find the limit of (-^^^-:^(~^\ as X approaches zero. By successive reductions we have /_1 1\ , rx''+^ \ x-(x-l) X U— 1 x) \ X ) xlx—l) x^+S 1__ X ^ 1 '^x(x-'l) x^ + S^Cx-l^^x-' + S) • Therefore by theorem I, limit rf J__ 1 Wf^!±?^l=limit 1 But plainly _J™^q (x- 1X^2 + 3) = -3 Hence H-['fe-i)-r-^)]=-J- Exampi.es. In a manner similar to that just pursued find the limits in each of the following examples : 3 . o\ fx+1 THEORY OF LIMITS. 303 ■x+1 2 \ 12 , limit rf£±i_^_w-^i-l ^ limit rf.J__ lUf-ll-^l *• ^ 5 4 LV;tr-4 x) \x^ -WJ -. limit J 1 1-5- CHAPTER XIX. RATIO, PROPORTION AND VARIATION. 437. The relative magnitude of two numbers or quan- tities, measured by the number of times the first contains the second, is called the Ratio of the two numbers or quantities. Thus 12 contains 3 just four times; hence the ratio of 12 to 3 is 4, or ^. And similarly if a and b stand for a7iy two numbers the ratio of ^ to ^ is -7. 438. We may speak of the ratio of two quantities of the same kind as well as the ratio of two numbers. Thus, 12 feet contains 3 feet just 4 times, hence the ratio of 12 feet to 3 feet is 4 or ^-^-. A good opportunity is here afforded to call attention to the double use of the word division. When a stick 12 feet long is cut into four equal pieces each piece is a stick 3 feet long. In this case we say that 12 feet is divided by 4. This process is called division — the division of separation. A stick 3 feet long may be used as a measure with which to measure a stick 12 feet long, and the latter contains the former just four times. In this case we may say that 12 feet is meas- ured by 3 feet. This process is called ?neasurement ; it is also called division — the division of measurement. In the division of separation the divisor is always a number, and the quotient a quantity of the same kind as the dividend. In the division of measurement the measure is always 'a quantity of the same kind as the one measured and the ratio is always a number. The ratio of any two quantites of the same kind may be looked upon as the numbgr of units in the first divided by the number of units in the second. Plainly, quantities which are not of the same kind cannot have any ratio, for one cannot possibly be measured by the other, nor can one be contained in the other. For example, ten miles cannot be measured by two quarts, nor can two quarts be contained any number of times in ten miles. RATIO, PROPORTION AND VARIATION. 305 439. The ratio of a to d is denoted in either of two ways: yirs^, by writing the a before the d with a colon between them, thus, a:d; second, by a fraction in which a is the numerator and d is the denominator, thus, 7- a Whichever way the ratio is written, it is read ' 'the ratio oi a to d,'^ or simply ''a to d.^^ 440. In either way of writing the ratio of a to d, a is called the Antecedent or First Term, and d is called the Consequent or Second Term. PROPBRTIKS OP RATIOS. 441. Since a ratio is the quotient obtained by dividing the number of units in the antecedent by the number of units in the consequent, it follows that the properties of ratios may be obtained immediately from the properties of fractions. 442. Since a fraction may be multiplied either by multiplying the numerator or dividing the denominator, it follows that a ratio may be multiplied either by multi" plyiyig the antecedent or by dividing the consequent, 443. Since a fraction may be divided either by divid- ing the numerator or multiplying the denominator, it follows that a ratio may be divided either by dividing the antecedent or by multiplying the consequent, 444. Since a fraction remains unchanged in value when both numerator and denominator are multiplied or divided by the same number, it follows that a ratio remains unchanged in value when both antecedent and consequent are multiplied or divided by the same number, 20 -U. A. 306 UNIVERSITY ALGEBRA. 445. If the numerator of a fraction is greater than the denominator, the fraction is greater than 1 ; therefore, if the antecedent of a ratio is greater than the consequent y the ratio is greater than 1. , 446. If the numerator of a fraction is less than the denominator, the fraction is less than 1 ; therefore, if the antecedent of a ratio is less than the co7isequenty the ratio is less than 1. 447. If the numerator and denominator of a fraction are equal to each other, the fraction is equal to 1 ; there- fore, if the antecedent and consequent of a ratio are equal to each other y the ratio is equal to 1, 448. Theorem. A ratio which is greater than 1 is de- creased by increasing both antecedent and consequent by the same amount. Let 7 be a ratio which is greater than 1 ; then a^b, Now form a new ratio by increasing the antecedent and consequent by the same amount, x. The new ratio is a-^-x b+x If we multiply antecedent and consequent of the original ratio by ^+^, we get a^^ab-j-ax b^b^ + bx' ^^^ If we multiply antecedent and consequent of the new ratio by b, we get a+x ^ ab+bx b-^-x^b^+bx ^^ Now, as «>^, it is plain that ax^bx^ and hence ab-j-ax^ab+bx. Therefore, ab-\-ax ab-\-bx b'''^bx^b^-\-bx RATIO, PROPORTION AND VARIATION. 307 Therefore, from (1) and (2), a a-^x which is what was to be proved. Since a>dy it is plain that a+x>d+x. Therefore, b+x^ ' Therefore, as each ratio 7 and -j-. — is greater than 1, and o o-f-x as 7>T-^ — ' it follows that 7-; — zs nearer the value 1 b-\-x o+x a than 7 is. 449. Theorem. A ratio which is less than 1 is in- creased by increasing both antecedent and consequent by ike same amount, a Let 7 be a ratio which is less than 1 ; then a'>b. Now form a new ratio by increasing the antecedent and conse- quent by the same amount, x. The new ratio is a-\-x b+x If we multiply antecedent and consequent of the original ratio by ^+;r, we get a_ab'^ax b^b'^+bx W If we multiply antecedent and consequent of the new ratio by b, we yet a-\-x ^ ab-j- bx b+x'^b'^+bx ^"^^ Now, as a it follows that -7— — is nearer the value 1 than - is. b+x b-\-x b This last statement, together with the last statement in the previous article, shows that any ratio (except the ratio 1) is made more nearly the value 1 by increasing both antecedent and consequent by the same amount. INCOMMENSURABLE NUMBERS. 450. Two numbers are Commensurable wi^h each other when there exists some third number which is con- tained an INTEGRAI, number of times in each of the two numbers. For example, 14 and 6 are commensurable with each other, because there is a number, viz.: 2, which is contained 7 times in 14 and 3 times in 6. Again, f and f are. commensurable with each other, for there is a third number, viz. : -^, which is contained 14 times in f and 15 times in f . The idea may be expressed otherwise by saying that two numbers are commensurable with each other when their ratio equals the ratio oj two whole numbers. For example, |:f=14:15. 451. Two numbers are Incommensurable with each other when there is no number which is contaified in each Wk RATIO, PROPORTION AND VARIATION. 309 of the given numbers an inTKGRAI, number of times ^ or what is the same thing, two numbers are incommensurable with each other when their ratio cannot be expressed as the ratio of two WHOI^K numbers, 452. A Commensurable Number is one which is commensurable with unity; and an Incommensurable Number is one which is incom- mensurable with unity. Thus, 1/2, 1/5, 1/6 are examples of incommensurable numbers. 453. From the definition of an incommensurable num- ber it follows that the ratio of an incommensurable number to unity cannot be expressed as a fraction with an integral numerator and denominator. But the ratio of any number to unity is that number itself. Therefore, an incommensurable number cannot be expressed by a fraction with an integral numerator and denominator. Since any decimal which terminates can be expressed as a fraction with a whole number for numerator and denom- inator, therefore an incommensurable number cannot be expressed as a decimal which terminates, 454. But while we cannot find either a decimal or a common fraction with an integral numerator and denom- inator which will exactly express any given incommen- surable number, still we can find either a decimal or a common fraction which will differ from the given incom- mensurable number by an amount as small as we please. Thus, 1/2 is an incommensurable number, but, if we wish, we may write a number which will differ from V ^ by less than one ten thousandth ; the number is 1.4142. We may write a number which will differ from l/2 by 3IO UNIVERSITY ALGEBRA. less than one millionth; the number is 1.414213. A still closer approximation to l/2 is 1.41421356, but this is not exactly equal to 1/2. 455. It has been stated that an incommensurable number cannot be expressed as a decimal which termi- nates. But the value of an incommensurable number may be expressed decimally to a greater and greater degree ot accuracy by carrying it out to a greater and greater number of decimal places. We may then say that an incommensurable num^ber is a never ending decim>al. 456. Repeating decimal represents a Commen- surable Number. Consider, for example, the repeating decimal .126126, ....*, and let the value of this be represented by a. Hence, a=. 126126 Multiplying this equation by 1000, we have 1000^=126.126126 .... Subtracting 126 from each member, we have 1000a- 126=. 126126 .... Therefore, 1000a-126=«. Hence, 999^=126. Hence, «=Tll=rnr- In a similar manner it may be shown that any repeat- ing decimal is expressible as a common fraction with an integral numerator and denominator. A repeating decimal therefore cannot be an incommensurable number. Whence we conclude that an incommensurable number is a never ending decimal which does not repeat. Some incommensurable numbers have been computed to a great many decimal places. This is especially the ♦When dots of continuation are used in this way it is understood that the number represented is the exact number of which the portion ah eady written is only an approximation, and that closer and closer approximations are given by taking more and more figures in the decimal. RATIO, PROPORTION AND VARIATION 311 case with the incommensurable number which represents the circumference of a circle whose diameter is unity. This number, carried out to thirty decimal places, is as follows : 3.14159265358979328462643383280. 457. It has been shown that an incommensurable number cannot be expressed as a fraction with an integral numerator and denominator, that it cannot be expressed as a decimal 'which terminates, and that it cannot be expressed by a repeating decimal. The student may infer from all this that an incommensurable number is not an exact number at all, but such is not the case, as may easily be shown in the case of the incommensurable num- ber l/2, for we can draw a geometric representation of l/2. Take each of the two sides, CA and CB, of a right angled triangle equal to 1. Then AB, the hypotenuse, will equal V {ly + iXy^V^. Thus 1^2 is the exact distance from A to B, which is a perfectly A^ definite distance. Thus the idea that incommensurable numbers are indefinite or inexact must be avoided. This notion has arisen (j) because \h^ fractions ^e often use in place of incommensurable numbers, such as 1.4142+ for l/2, are merely approximations Q (I) iB to the true values. COMPOUND RATIOS. a c 458. If from two given ratios, ~ and -^ or a\b and c\ dy we form another ratio by multiplying the antecedents together for a new antecedent and the consequents to- 312 UNIVERSITY ALGEBRA. CiC getter for a new consequent, we get -j- or ac\bd, which is said to be Compounded of the given ratios a : b and c : d, 459. If ^>1 it follows that xy^y, or, expressed in words, if the multiplyer is greater than 1 the product is greater than the multiplicand. Therefore, if ^>l,^J>f Also, if J>1. |"^>i. n ^ a c ac , ^ But in each of these cases t 3 o^ — / is the ratio com- a ca a ^ c pounded of the ratios 7 and — Therefore, if a ratio be compounded of two ratios each of which is greater than 1 the result is greater than eithef of the given ratios, 460. In a similar way it may be shown that if a ratio be compounded of two ratios each of which is less than 1 the result is less than either of the given ratios, 461. If x a or in the common form of proportion, a — b'.b=c — d:d. This last proportion, a—b:b=c—d:d, is said to be derived from the proportion a :b=c:d by Division. 470. If a'.b=c:d, then by composition, a-\-b c+d and by division, ~~b~^~d~' ^^-^ Divide (1) by (2) member by member and we get a-\-b^c+d a — b c — d or written in the ordinary form of proportion, a-\-b\a—b^c-\rd\ c—d. This last proportion, a-\-b:a — b=c+d:c—d, is said to be derived from the proportion a:b=c',dhy Composition and Division. RATIO, PROPORTION AND VARIATION. 317 471 The products of corresponding terms of two or more sets of proportional nuTubers are proportional. Let a : d=c : d and e :f=k \k and n : r= s : t Writing each of tliese proportions as an equation in fractions, we have a__c ir~d. e^h n__s r~^~t and from these equations by multiplication we obtain a e n ^c h s 'bf~r~dkl aen__chs ^^ 'bfr'^m or writing this in the ordinary notation of proportion, we have aen : dfr=cks : dkty which is what was to be proved. 472. I^ike powers or like, roots of proportional numbers are proportional, Let a : b=c : d. Writing this proportion as an equation in fractions, we a c obtam I^T ^^ and raising each member to the n th power, we have a"" 2'=3, find the value of ;r whenjj/=3 and -3'= 6. 7. If ;t: varies inversely as the square oi y, and jr=l when jK=10, find the value of :r when jr=5. 8. If ;t: is directly proportional to y and inversely pro- portional to 2, and ;»;=20 when y=Q and -3*= 4, find the value of ^ when j/=3 and -3'= 10. g. li X varies asj^, prove that x^ varies asjj/^. 10. If X is inversely proportional to y and j/ is inversely proportional to 2, pro\^e that x is proportional to z. 11. If ;»; is proportional to 2 and j|/ is also proportional to 2, prove that xy is proportional to 2'^, also that x'^-^y^ is proportional to 2'^, 12. If Sx+7y is proportional to 3;r+13j/ and ;r=5 whenjK=3, find the ratio oi x to y and thus show that x varies as y. 13. The number of feet a body falls is proportional to the square of the number of seconds occupied in falling. Knowing that a body falls 16 feet the first second, find liow many feet it will fall in 5 seconds. 324 UNIVERSITY ALGEBRA. 14. With the same supposition as in the last example, find the height of a tower from which a stone dropped from the summit, reaches the ground in 3f seconds. 15. The w^eight of a metal ball is proportional to the cube of the radius, and a ball whose radius is 2 inches weighs 10 pounds, what is the weight of a ball whose radius is 5 inches? 16. If a heavier weight draw up a lighter one by means of a cord passed over a fixed wheel, the number of feet passed over by each weight in any given time varies directly as the difference of the weights, and inversely as the sum of the weights. If 10 pounds draw up 6 pounds 16 feet in 2 seconds, how high will 14 pounds draw 10 pounds in 2 seconds ? CHAPTER XX PROGRESSIONS. 485. An Arithmetical Progression is a series of terms such that each term differs from the immediately preceding term by a fixed number, called the Common Difference. The following are examples of arithmetical progressions : (1) 2, 4, 6, 8, 10. (3) 2^, 8f, 5, 6i 7^. (2) 31, 26, 21, 16. (4) (^-y), x, (x+j). (5) a, (a+d-), (« + 2^), Ca + Sd), 486. The first and last terms of any given progression are called the Extremes, and the other terms are called the Means. 487. The Arithmetical Mean of two numbers a and d is found as follows: Let A stand for required mean. Then, by definition A—a=d—A, whence A=^(a-j-h). [1] Thus, the arithmetical mean of 7 and 11 is 9, for 7, 9, 11 is an arithmetical progression having the common differ- ence of 2. By the arithmetical mean or average af several 7itimbers is meant something entirely disassociated from arithmetical progression. The term means the result found by adding several numbers and dividing by the number of them. Thus, the arithmetical mean of 1.06, 1.21, 1.93, is 1.40. 488. The n th Term. It is usual to represent the first term of an arithmetical progression by a, the common 326 UNIVERSITY ALGEBRA. difference by d, and the n th term by /. With this nota- tion we may represent any arithmetical progression by No. of term: 1 2 3 4 5 . . . Progression: a, (a-\-d), {a-{-2d), (a + 3^), (a-j-4d) We notice that the coefficient of d in the 2d term is 1, in the 3d term is 2, in the 4th term is 3, and, by the nature of the progression, the coefficient of d in any term is 1 less than the number of that term. Therefore, the n th term in this progression will be a+(n—l)d, or, representing the n th term by /, we have the formula l=a-\- {n—])d. [2] 489. Evidently the sum of an arithmetical progression is not changed if the order of the terms be reversed; thus, 3-f 5+7 + 9-f 11 may be written 11+9+7 + 5+3 2i+3i+4+4f may be written 4f +4+3^+2^ in which case the first term becomes the last term, the last term becomes the first term, and the common difference cha7iges signs. 490. In case the common difference is positive the progression may be called an Increasing Progression, and in case the common difference is negative the pro- gression may be called a Decreasing Progression. 491 The Sum of n Terms. If ^ stands for the sum of n terms of an arithmetical progression, we may write the two following equalities, the progressions being alike except written in reverse order : ^=a+(a+^) + (a + 2^) + (a + 3^) + . . .+a + (in'-l)d (1) ^=/+(/_^) + (/_2^) + (/-3^) + . . .+l-(n-l)d (2) PROGRESSIONS. 32/ Adding (1) and (2) together term for term, noticing that the terms containing the common difference nulify one another, we have 2^=(^+/) + («4-/) + («+/) + (^+/) + . . . + (^+/). Since the number of terms in the original progression was called n, we write the last equation whence the formula for s, 8=i.n(a.+i), [3] 492. Formula [2] enables us to obtain the value of I when a, n, and d are given, or the value of a when /, n, and d are given, or the value of a when /, n, and a are given, or the value of n when /, a, and ^are given. Thus: (1) Find the 20th term of 3 + 8+ 13 + . . . Here «=3, ^=5, n=20, therefore /rr:3 + 19X5--=98. (2) Find the number of terms in the progression 5 + 7+9-1-. . .+37. Here a==5, d=2, 1=37, whence 37=5 + (;^- 1)2. Solving for n, n=Vi. (3) Find the common difference in a progression of 11 terms in which the extremes are | and 30|. Here a=\, l=30\ and n=ll, whence 301=1 +(11- IK Solving for cl, d=3. (4) Insert 3 arithmetical means between 5 and 21. Here n=5, a=5, and /=21, whence 21=5 + {5-lK Solving for ^, ^=4. Therefore, the means are 9, 13, and 17. 493. Formula [3] enables us to find any one of the numbers s, n, a and /, when the value of the other three are given. Thus : 328 UNIVERSITY ALGEBRA. (1) Find the sum of 10 terms of the progression in which 5 is the first term and - 58 the last term. Here «=5, n=10, /= — 58, whence j=iXlO(5-58.) That is, s=—2Gd. (2) Find the number of terms in an arithmetical progression in which the first term is 4, the last term 22, and the sum 91. Here a=4, 1=22, and j=91, whence dl=^n{4 + 22.) Solving for «, n='7. 494. The two formulas ' U=a-\-(n-l)d (I) \s=\7i{a + l^ (2) contain five different letters, hence if any two of them stand for unknown numbers, and the values of the rest are given, the value of the two unknown numbers can be obtained by the solution of a system of two equations. Thus : (1) Find the sum of an arithmetical progression in which the last term is 149, the common difference 7, and the number of terms 22. Here /=149. d=l, and ?2=22, whence j 149=« + (22-l)7 (1) \ j=iX22(« + 149.) (2) From (1), a=2. Substituting in (2) .r=1661. (2) Find the first term of an arithmetical progression of 21 terms, whose sum is 1197 and common difference is 4. Here «=21, J=1197 and d=4:, whence j /=^+(21-l)4 (1) 1 1197=iX21{a+/). (2) From (1), l=a + m. From (2), /=114-«. (3) Whence. /^97 and ^=17. (4) PROGRESSION. 329 (3) Find the number of terms in a progression whose sum is 1095, the first term is 38 and the difference is 5. Here ^=1095, a=3S and d=5, whence j /=38+(;z-l)5 (1) \ 1095=J;/(38+/) (2) From (1), /=:334-5;^. (3) From (2), 21dO=SSn+nl. (4) Substituting the value of / from (3) in (4) we get 2190=71;/+5;/3. (5) Solving this quadratic equation, we find n=15 or —29.2. The second result is inadmissable, since the number of terms can not be either negative or fractional. KXAMPI^KS AND PROBI.BMS. Solve each of the following : 1. Given a=7, ^=4, 7^=13; find /and s. 2. Given a=^, d=6, n=SO; find /and s, 3. Given a=9, /=162, n==52; find s and ^. 4. Given a=57, /=30, ;z=19; find ^and d. 5. Given /=242, d=21, n=12; find a and s. 6. Given /=16, d=—S, n=5S; find a and s. 7. Given a=17, /=350, <3f=9; find n and s. 8. Given ^=—28, 1=28, d=7; find n and s. 9. Given a=l^, /=54, ^=999; find ^^ and (sT. 10. Given a=5, /=16i, ^=3320; find 7i and ^. 11. Given a=3, ;z=50, ^=3825; find / and ^. 12. Given «=— 45, n=Sly s=0; find /and <^. 13. Given /=49, n=19, s==50S^; find a and ^. 14. Given /=105, 72=16, ^=840; find a and ^. 15. Given n=S5, ^=2485, d=S; find <3j and /. 16. Given 7z=25, ^=—25, d=^; find ^ and /. 17. Given ^=4784, a=41, ^=2; find / and 72. 330 UNIVERSITY ALGEBRA. i8. Given ^=624, a=9, d=4; find /and n, ig. Given 5=278, d=5, /=77; find a and n. 20. Given ^=1008, ^=4, /=88; find a andn, 21. What is the sum of the first 200 natural numbers? 22. What is the sum of the even numbers from to 200? 23. What is the sum of the odd numbers from 1 to 200? 24. What is the sum of the first n even numbers ? 25. What is the sum of the first n odd numbers ? 26. Insert 9 arithmetical means between ■— |- and +|-. 27. Sum the series l/|-+l/2+3l/|-+. . . to 20 terms^ 28. Sum the series 5— 2— 9—. ..to 8 terms. 29. Insert 5 arithmetical means between 10 and 8. 30. Insert 4 arithmetical means between —2 and —16. 31. Sum (a + dy + (a^-\-d'') + (a-5y to n terms. 32. Find the sum of the first 10 multiples of 3. 33. Find the sum of the first 50 multiples of 7. 34. Find the sum of the odd numbers between 200 and 300. 35. The sum of 25 successive terms of the progression 5_|.8 + 11 + . . . is 1025 ; what is the first term? 36. The sum of 10 terms of an arithmetical progression is 15, and the fifth term is ; what is the first term ? 37. How many terms of the progression 9 + 13 + 17 + . » must be taken in order that the sum may equal 624 ? 38. Find the arithmetical progression whose sum is 500, whose middle term equals 50, and whose last term is three times the first term. 39. We must take how many terms of the progression ^l+-^W(l+-^W(l+-^W. . . in order that the sum may be 6^. PROGRESSIONS. 331 40. How many terms must be taken from the com- mencement of the series 1-f 54-9 + 13 + 17, etc., so that the sum of the 13 succeeding terms shall be 741? 41. The sum of the first three terms of an arithmetical progression is 15, and the sum of their squares is 83; find the common difference. Let ;t:=first term and;K the common difference. Then a'+(^-f-j/) + (^+27)z=15, and ^2 + (x+7)2 + (:r+2y)2=z:83. Another notation which is very convenient in a problem like this is: Represent the three terms by x—y, x, and x-\-y, whence we write {x—y) +^ -l-(-^+7) =15, and (^-/)^-4-:*:2_|.(;^^^)2^83^ 42. There are two arithmetical progressions which have the same common difference ; the first terms are 3 and 5 respectively, and the sum of seven terms of the one is to the sum of seven terms of the other as 2 to 3. Determine the progressions. 43. The sum of three numbers in arithmetical pro- gression is 12, and the sum of their squares is QQ>. Find the numbers. 44. The sum of three numbers in arithmetical pro- gression is 33, and the sum of their squares is 461. What are the numbers? GKOMKTRICAI. PROGRESSIONS. 495. A Geometrical Progression is a series of terms such that each term is the product of the preceding term by a fixed factor called the Ratio. The following are examples : (1) 3, 6, 12, 24, 48. (3) i, i, i, -^, ^\. (2) 100, ~60, 25, -12f (4) a, ar, ar\ ar\ ar^. The first and last terms are often called the Extremes and the other terms the Means. 332 UNIVERSITY ALGEBRA. 496. The Geometrical Mean of two numbers a and b is found as follows : Let G stand for the required mean. Then, by the definition of a geometrical progression, G^b_ a G Whence, G^=ab, or G=\/ab. [4] Thus 4 is the geometrical mean of 2 and 8. The arithmetical mean Q)i 2 and 8 is 5. By the geometrical mean of n positive numbers is meant the positive value of the n th root of their product. Thus the geometrical mean of 8, 9, and 24 is 12=1^8X9X24. 497. The n th Term. Let a represent the first term of any geometrical progression, and r the ratio. Then the progression may be written No, of Term: 1. 2. 3. 4. 5. Progression: a^ ar^ ar*^, ar^, ar^^ , . . We notice that, by the nature of the progression, every time the number of terms is increased by 1 the exponent ot r is increased by 1 also, and the exponent of r in any term is one less than the number of that term. Therefore, representing the n th term by /, l=ar"-i. [5] 498. The Sum of n Terms. Representing by ^ the sum of n terms of any geometrical progression, we have s=a+ar+ar'^-\-ar^ + , . .+ar''-'^ + ar''-'^, (1) Multiplying this equation through by r, we get rs=:ar+ar'^ + ar^+ar^ + , . ,+ar''-^+ar^, (2) Subtracting (1) from (2), we have rs — s=ar" — a, (3) Whence s(r—l) = ar*'—a, ar--a or ^=~i^=T" [Pi PROGRESSIONS. 333 Now, from [5] l=ar''~'^. Therefore, ar'*=r(^r''~^) = r/, and [6] may be written -?£f- [7] 499. We give a few examples of the use of formulas [5], [6], and [7]. (1) Find the 7th term of the progression 4+8+16+ . . . Here <^=:4, r=2, and «=7, whence /=4x26=r256, (2) Find sum of 6 terms of progression 13+1. 3 + . 13+ Here ^=13, ^=6, and r^^^, whence 13X(iy«-13 . 10 ■•■ 13-13000000 _ 12999 987_ that IS, ''- 100000- 1000000 ~~900000~~-^^-'*^^^'^- (3) Insert 3 geometrical means between 31 and 496. Here ^=31, /=496, and n=D, whence 496=:31X^^, or ^4 = 16, therefore, r:= ± 2. Consequently the required means are 62, 124, and 248, or —62, +124, and -248. 500. The two equations contain five letters. If any two of them are unknown numbers and the values of the other three are given, the value of the two unknown numbers can be determined by solving the system of two equations. But if r is an unknown number, the equations of the system are of a high degree, since n is usually a large number and always greater than 2 at least. In this case we shall be unable to solve the system, as it is beyond the range of Chapter XVII. Also, lin is an unknown number, we shall 334 UNIVERSITY ALGEBRA. have an equation with the unknown number appearing as an expoiient, which is a kind of equation we have not yet considered. Hence there are but a limited number of cases in which, with our present means, we can solve the above system. We give a few samples of cases readily solved. (1) Find the sum of a geometrical progression of 7 terms, of which the last term is 128, the ratio being 2. Here /==128, r—2, and w=7, whence 128=^ 2« (1) s= J (2) From {1) a=2, whence from (2) j=254, (2) Find the sum of a geometrical progression of 5 terms, the ex- tremes being 8 and 10368. Here a=8, /= 10368, and n=:5, whence 10368=:8r4 (1) r 10368 -8 '^- r-1 (2) From (1) r=Q, whence from (2) ^=12450. (3) Find the extremes of a geometrical progression whose sum is 635, if the ratio is 2 and the number of terms 7. Here j=635, r=2, and n=.7, whence /=a2^ (1) 2/-« 635=— J- (2) Substituting / from (1) in (2), we get 635=128^ -d5. Whence a=^; hence /=320. (4) The 4th term of a geometrical progression is 4, and the 6th term is 1. What is the 10th term? Here ar^=4: (1) and ar^ = l (2) Whence, by dividing (2) by (1), r«=}. Whence, ^=±|. Therefore, from (1) fl=— =±33. Then the 10th term is ±S2{±i)*=^. PROGRESSIONS. 335 EXAMPLKS AND PROBIvEMS. 1. Find the sum of 7 terms of 4 + 8 + 16 + . . . 2. Find the sum of 9 terms of 2 + 6 + 18 + . . . 3. Find the sum of 7 terms of 1+4 + 16+ . . . 4. Find the 10th term and the sum of 10 terms of 4-2 + 1-. . . _ _ _ 5. Sum the series l"' 3+ 1^6+ 1^12+. . . to 8 terms. 6. Sum the series —4+8—16 + 32—. . . to 6 terms. 'j,S\xma + a(l-{-x) + a(l+xy + . . . to 8 terms. 8. Given /= 78125, r=5, n=S; find a and s, 9. Given /= 2T» ^— i^ n=6 ; find a and s, 10. Given 5= 635, n=7, r=2 ; find a and /. 11. Find rand s; given a=2\ /=31250, n=7. 12. Find r and s; given a=S6, /=^, n=7, 13. Find r and s; given a=3, /=49152, n=8. 14. Find r and s; given a=7, /=3584, ;^=10. 15. Insert 2 geometrical means between 47 and 1269. 16. Insert 3 geometrical means between 2 and 3. 17. Insert 1 geometrical mean between 14 and 686. 18. Given a=\, /=1024, 71=14; find rand s. 19. Insert 7 geometrical means between a^ and d^, ^o. Find the sum of the first 10 consecutive powers of 2. 21. Find the sum of the first lOconsecutivepowersof—^-. 22. Sum d(l+xy-^-JrK^+xy-^ + , . . to n terms. 23. Sum x"-'^ +x*'-^y-j-x"-^j/^ +x*'~^j/^ + , .ton terms. 24. Sum x"-^ —x^-^y+x^'-'y^ —x"-^y^ + ..ton terms. 25. Sum a—ar'^+ar'^'-'ar~^ + , , . to n terms. 26. Select 6 terms from the progression ^—2+8—. . . whose sum shall equal —6536. 336 UNIVERSITY ALGEBRA. 27. The sum of the extremes of a geometrical pro- gression of 4 terms is 56, and the sum of the means is 24. Find the 4 terms. 1 3 5 1 3 5 28. Sum the series 2 + 22"^23"*"2^"^P~^2« "*"' ' ' ^^ ^ . /I , 3 , 5\ /I , 3 , 5\1 , terms, or the progression [2+2^ + 2 3 j + ^ 2 "*" 22 "^ 2 ^ 72 ^ + to three terms. 29. The 4th term of a geometrical progression is 192 . and the 7th term is 12288 ; find the sum of the first 3 terms. 30. The 6th term of a geometrical progression is 156, and the 8th term is 7644 ; what is the 4th term? 31. Prove that if numbers are in geometrical progres- sion their difierences are also in geometrical progression, having the same common ratio as before. 32. If a-j-d+c-i-d-}-. . . is a geometrical progression, prove that (a' + d''')-\-(d''+c^) + (ic^+d^) + . . . is also a geometrical progression. 33. A man agreed to pay for the shoeing of his horse as follows: .0001 cents for the first nail, .0002 cents for the second nail, .0004 cents for the third nail, and so on until the 8 nails in each shoe were paid for. How many- dollars did he agree to pay? How much did the last nail cost him? INFINITE GEOMKTRICAIv PROGRESSIONS. 501. If the ratio of a geometrical progression is a proper fraction, the progression is said to be Decreasing. Thus, 1, h hi and -J, i, 2V. sV are decreasing geometrical progressions. PROGRESSIONS. 337 502. Limit of Sum as n Increases. If we increase the number of terms in the decreasing progression 1, |-, J, . . . the sum of the terms will never equal 2, but will approach 2 as near as we please. We wish to show that the sum of every decreasing geometrical progression ap- proaches a deiSnite limit as the number of terms increases without limit. We know '-°^- (1) If we suppose r a proper fraction and n increasing without limit, we have the case under consideration. As n varies, both sides of (1) are variables, and since they are always equal, we have limit s= limit P^^J^^n (2) Since r is a proper fraction, the term r** continually approaches as a limit as n increases. Whence, taking the limit of the right hand side of (2), we may write limit s=j^' [8] (1) Find the limit of J— i + ^ — tV + - • as » increases without limit. Here «=J, andr=— J. Whence, limit .=f:^L_^j. (2) Find the limit of .3333, ... Here «=^0 and ^=^q. Whence, limit s 1-A~^- KXAMPI^KS. As n increases without limit, find the limit ot each of the following : 1. 9-6+4-. . . 5. .272727 . . . 2. .279279279 ... 6. ^--1+^^ ... 22 — U. A. 338 UNIVERSITY ALGEBRA. 3. 4+.8+.16+. . . 7. 1/8 + 1/4+1/2+. a+x a—x o ^^ r ^ . a-;r ^+^ • • • V3 + I 1/3 + 3 9. Express the number 8 as the sum of an infinite geometrical progression whose second term is 2. HARMONICAL PROGRESSIONS. 503. A series of numbers which are such that their reciprocals form an arithmetical progressiom are said to form an Harmonical Progression. The following are examples : (X)hhhh (4) i. 1,-1, -i. (2) 1, i, i, A- (5) 4, 6, 12. x—y X x+y a a+d a-\-2d ■ The first and last terms are called the Extremes, and the other terms the Means. 504. Fundamental Property. The difference between the first and second of any three consecutive terms in harmon- ical progression is to the difference between the second and the third as the first is to the third. Let a, by and c be the three terms in harmonical pro- gression. Then we have, by definition, b a c d a—b b—c whence. -W^-b^- >w*. /. ^ — b ab a Therefore, 7 — =-r =-♦ b — c be c that is, a-h\h—c—a\c. "1)^ PROGRESSIONS. 339 505. The Harmonical Mean of two numbers a and b is found as follows : IG>H. [10] 508. Relation between A, G, and H. The follow- ing relation is important. As before, we have Whence, AH^ab, But G'^=-ab. Therefore, G^V^S. [11] That is to say, the geometrical mean of any two positive numbers is the same as the geometrical mean of their arithmetical and harmonical means. MISCKI.I.ANEOUS KXCERCISKS. A. P.. G. P., and H. P. stand for Arithmetical^ Geometrical^ and Harmonical Progression, respectively. 1. Continue the H. P. 12, 6, 4. 2. Sum the series l + 2r+3r2+4r^ + . . , \.o n terms. 3. If a, b, Cy ^be in A. P., a, e,f din G. P., a, g^ h^ din H. P., then ad=ef=bh=cg. PROGRESSIONS. ' 34I 4. The sides of a right triangle are in A. P. Show that they are proportional to 3, 4, 5. 5. Three numbers are in G. P.; if each be increased by 15 they are in H. P. Find them. 6. If the sum of the first p terms in an A. P.=0, the sum of the next q terms = / 1 p—1 7. If ^, b, c, dhe in H. P., show that 8. If X, y, z be in G. P. , prove that 9. Find the difference (lf+li+f4-. . to5terms)-(l|+li+f+. . to5terms). ID. If the A. mean between two numbers equals 1, show that the H. mean is the square of the G. mean. 11. If x—a, y—a, and z—a be a G. P., prove that twice jj'—^^^ is the harmonic mean between y — x and y—z. 12. If a, b, c be in A. P, and x be the G. mean of <3J-and b, and y the G. mean of b and c, then will x", b'^ , y'^ be in A. P. 13. Find an equation whose roots will be the arith- mitical and harmonical means between the roots of x'^—px+q=0. 14. The sum of 10 terms of an A. P. is 145, and the sum of its fourth and ninth terms is 5 times the third ; determine the series. 15. The A. mean between two numbers is to the G. mean as 5 : 4 and the difference of their G. and H. means is 34; find the numbers. 16. To each of three consecutive terms of a G. P. the second of the three is added. Show that the three resulting numbers are in H. P. 342 . UNIVERSITY ALGEBRA. 17. Show that the product of any odd number of con- secutive terms of a G. P. will be equal to the n th power of the middle term, n being the number of terms. 18. The natural numbers are divided into groups, as follows: 1; 2, 3; 4, 5, 6; 7, 8, 9, 10; and so on. Prove that the sum of the numbers in the ^th group is \k{k'^ -|- 1). 19. If a, by Cy X be all real numbers in the equation prove that a, b^ c are in G. P. and that x is their common ratio. 20. AG. P., whose common ratio is V«, has the same first and second terms as an H. P. Prove that the third term of the former series will be equal tothe(«+2)nd term of the latter. 21. If ^1, ^2» -^s) ^tc, are the sums of r A. P., each to n terms, the first terms being 1, 2, 3, etc., respectively, and the difierences 1, 3, 5, etc., respectively, show that ^i+^2+'y8 + - • • '\'Sr=\nr{rir-\-V). CHAPTER XXI. ARRANGEMENTS AND GROUPS. 509. Every different order in which given things can be placed is called an Arrangement or Permutation, and every different selection that can be made is called a Group or Combination. Thus if we take the letters a, by Cy two at a time, there are six arrangements, viz. : aby ac, ba, be, ca, cby but there are only three groups, viz. : ab, ac, be. If we take the letters a, b, e all at a time, there are six arrangements, viz.: abe, aeb, bae, bca, eaby ebUy but there is only one group, viz. : abe. 510. Theorem. The number of arrangements of n different things taken all at a time equals n times the num- ber of arrangements ofn—1 things taken all at a time. Let us first take two special cases and then pass to the general case. First. If we take three things, say «, 3, Cy there are six arrangements, viz.: abCy acby bac, bca, cab, cba. Now these six arrangements may ho, looked upon as composed of three classes. In the first class the letter a stands first, in the second class the letter b stands first, and in the third class the letter c stands first. The 344 UNIVERSITY ALGEBRA. arrangements in the first class may be looked upon as obtained by arranging the two letters b and c in every possible way, and writing a before each arrangement; those in the second class may be obtained by arranging the letters a and c in every possible way and writing b before each arrangement ; those in the third class may be obtained by arranging the letters a and b in every possible way and writing c before each arrangement. By considering the arrangements formed in the manner just described it is evident that all three letters appear in each arrangement, and it is also evident that the number of arrangements in which a stands first is exactly the same as the number in which b stands first, and also exactly the same as the number in which c stands first. Hence, we see that there are three times as many arrangements of three things taken all at a time as there are of two things taken all at a time. Second, If we take four things, say a, b, c, d, then we may arrange the three letters b, c, d in every possible way and place a before each arrangement, then arrange the three letters a, a d \vl every possible way and place b before each arrangement, then arrange the three letters a, b, d in every possible way and place the letter c before each arrangement, and finally arrange the three letters a, b, ^ in every possible way and place the letter d before each arrangement. It is evident that all four letters a, b, c, d appear in each arrangement thus formed, and it is also evident that the number of arrangements in which a stands first is exactly the same as the number in which b stands first, and so bn. Hence there are in all four times as many arrangements of four things taken all at a time as there are ol three things taken all at a time. ARRANGEMENTS AND GROUPS. 345 In general, if we have n things, say the letters a, b, c, d, e, f, . . then we , may suppose all the letters but a arranged in every possible order and then a placed before each of these arrangements ; next we may suppose all the letters but b arranged in every possible order and then b placed before each of these arrangements, and so on. It is evident that all n letters appear in each arrange- ment thus formed, and it is also evident that the number of arrangements in which a stands first is exactly the same as the number in which any other letter stands first. Now the number of arrangements in which a stands first is evidently the number of arrangements of {n—V) things taken all at a time, and hence the total number of arrangements of n things taken all at a time is n times the number of arrangements oi n—\ things taken all at a time. Let us represent the number of arrangements oin things taken all at a time by A^. Then, by what has just been shown we have the formula KXAMPI.BS. 1. If the number of arrangements of 9 things taken all at a time equals x, the number of arrangements of 10 things taken all at a time equals what? 2. Using the result found in example 1 for the number of arrangements of 10 things all at a time, find the num- ber of arrangements of 11 things taken all at a time. 3. If ^6=;r, what does Aq equal? 4. If ^7 = 10;ir, what does A^ equal? 5. Express A12 ^^ terms of ^ ^ ^ . 6. Express ^ 5 q in terms of ^ 4 9 . 7. Express A^ in terms of A^_^, 346 UNIVERSITY ALGEBRA. 511. Problem. To find the number of arrangements ofn different things taken all at a tiifie. By the theorem of the last article we may write each of the following equations except the last one : A„^=nA„_iy A„_^ = (n--1)A„_2, A ^= 6 A 2 , A, = l. The last equation of this list is not written by an appli- cation of the theorem, but is evidently true, for one thing can evidently be arranged in but one way. Now multiplying these equations together, member by member, we get A^A^A^,.. A„=2A^ 3^2 • • • ^-^^-i = lx2x3...;^^l^2••• ^«-i- By cancelling the common factors, we get An=lX2xS . . n. The product of the integer numbers from n down to 1 or from 1 up to « is often represented by in or nl ^ and is read factorial n^ or n admiration. With this notation we may write EXAMPI^BS. 1. How many arrangements can be made of 5 things taken all at a time ? 2. How many different numbers can be made with the four digits 1, 2, 3, 4, using each digit once and only once to form each number? 3. The number of arrangements of four things taken all at a time bears what ratio to the number of arrange- ments of six things taken all at a time ? ARRANGEMENTS AND GROUPS. 34/ 4. How many four-figure numbers can be formed with the digits 1, 2, 3, 4, having 1 for the first digit in each number and using each digit once and only once in each number? 5. How many five-figure numbers can be formed with the digits 1, 2, 3, 4, 5, having 1 for the first and 5 for the last digit in each number, and using each digit once and only once in each number? 512. Theorem. The number of arrangements of n different things taken r at a time is equal to n times the number of arrangements of n—\ things taken r — 1 at a time. Let us first take a particular case, say the number of arrangements of five things, say the five letters a, b, c, d, ). G{'r-&- -r+l (11) (12) Multiplying both numerator and denominator of this last fraction by r and remembering that r |r--l = \r and that \n—r+l =(n—r+l) \n'—r , we get r\n <^(?-l) = 7 («-r+l)|r Comparing (13) and (8) we easily get From (8) it easily follows thBt G(X.X)= n—r \n-\ From (15) and (9) we get r-\ n—r (13) (14) (15) {n-r) \n-\ r F n—r—l + \n-l r— 1 \n--r \n-l n-1 l^ Tr \n—r \r n—r \r n—r I r I «— r which by Art. (514) equals 6^(?), hence c;(;)=^(r^)+6^(?ri). (16) We have obtained a few relations connecting arrange- ments with arrangements in equations (1), (2), (6), (7), also a few relations connecting groups with groups in equations (11), (14), (16). We now obtain a few relations involving both arrangements and groups in the same equation. We have already found in Art. 514 ^(;)=[rG(;), (17) ARRANGEMENTS AND GROUPS. 363 and as \r==A(^ we may write (17) in the form A(^}=AO GO. (18) From (7), -^(0=^(;:_i) and writing this value in (18) we get AO^^ACr.,) GO). (19) In (18) substitute the value of G(r) given in (16) and we get ■ A0)=AO lGCr-')-i-GOzl)l (20) But it readily follows from (18) that A0-')=AO G(r'). Substituting in (20), we get AO')=A(r')+AC;) GC^zl). (21) Since by Art. 518, groupsin which repetitions are allowed can be expressed in terms of groups in which repetitions are not allowed, it would be an easy matter to obtain equa- tions involving groups with repetitions, but enough has already been given to show that a great variety of relations can be obtained. KXAMPI.KS AND PROBIyKMS. 1. In how many ways can seven people sit at a round table? 2. How many different groups of 13 each can be made of 52 men? 3. How many different groups of two each can be made with the letters a, d, I, n, si 4. How many arrangements of five each can be made with the letters of the wor^ group s'>. 5. How many different products of three each can be made with the four letters a, b, c, d'- 6. There are 5 straight lines in a plane, no two of which are parallel; how many intersections are there? 7. In how many different ways can the letters of the word algebra be written, using all the letters ? 364 UNIVERSITY ALGFBRA. 8. How many different signals can be made with five flags of different colors hoisted one above another all at a time? 9. How many different signals can be made with seven flags of different colors hoisted one above another, five aj; a time? 10. How many different arrangements can be made of nine ball players, supposing only two of them can catch and one pitch? 11. How many different signals can be made with five flags of different colors, which can be hoisted any number at a time one above another? 12. In how many ways can a child be named, supposing that there are 400 different Christian names, without giving it more than three names? 13. There are n points in a plane no three of which are in the same straight line. Find the number of straight lines which result from joining :them. 14. In how many ways can a committee of 3 be appointed from 5 Germans, 3 Frenchmen; and 7 Ameri- cans, so that each nationality is represented? 15. How many different signals can be made with seven flags of which 2 are red, 1 white, 3 blue, 1 yellow when all are displayed together, one above another, for each signal? 16. On a railway there are 20 stations of a certain class. Find the number of different kinds of tickets required in order that tickets may be sold at each station for each of the others. 17. Find the number of signals that can be made with four lights of different colors, which may be displayed any number at a time, arranged either one above another, side by side, or diagonally ARRANGEMENTS AND GROUPS. 365 18. There are n points in a plane, no three of which are in the same straight line except r, which are all in the same straight line; find the number of straight lines which result from joining them. 19. A certain lock opens for some arrangement of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, taken 6 at a time, repetitions allowed. How many trials must be made before we would be sure of opening the lock ? 20. A lock contains 5 levers, each capable of being placed in 10 distinct positions. At a certain arrangement of the levers the lock is open. How many locks of this kind can be made so that no two shall have the same key? 21. There are n points in space, no four of which are in the same plane with the exception of r, which are all in the same plane, and no three of which are in the same straight line. How many planes are there, each con- taining three of the points ? 22. From a company of 90 men, 20 are detached for mounting guard each day ; how long will it be before the same twenty men are on guard together, supposing the men to be changed as much as possible? How often will each man have been on guard during this time ? CHAPTER XXII. BINOMIAIy THKOREJM. 523. The Binomial Theorem enables us to write out any power of a binomial without actually performing the multiplication. It is* a deduction from the gener- alized distributive law of Art. 86. The product of any number of parentheses is the aggre- gate of AiyL the possible partial products which can be made by taking one term and only one from each of the paren- theses. Thus : ace + acf+ ade + adf-{- bee + bcf-^ bde + bdf. 524. Binomial Formula. We are required to write out the value of {x-^ay, where x and a stand for any two numbers whatever and ;z is a positive whole number. That is, we must consider the product of the n parentheses {x-\-a^{x+a^{x-\-d) . . . (^+a). First. We may take an x from each of the parentheses to form one of the partial products. This gives the term x'\ Second. We may take an a from the first parenthesis with an x from each of the other (n—V) parentheses. This gives ax''~'^ as another partial product. But if we take a from the second parenthesis and an x from each of the other {n—X) parentheses we have ax*""^ as another partial product. I^ikewise, by taking a from any of the parentheses and an x from each of the other {n—V) parentheses we shall obtain ax*'~'^ as a partial product. Whence, the product contains n terms like ao(f'~'^, or 7iax''~^ is a part of the product. BINOMIAL THEOREM. 367 ITitrd. We may obtain a partial product like a'^x''~'^ by taking an a from any two of the parentheses, together with the x's from each of the other (;^--2) parentheses. Whence, there are as many partial products like a'^x''^'^ as there are ways of selecting 2 a's from n parentheses; that is, in as many ways as there are groups of n things taken two at a time, oj n{n—l) 1.2"* Whence, -^ — ^ <22^«-2 jg another part of the product. Fourth. We may obtain a partial product like a^xf*~^ by taking an a from ^;zj/ three of the parentheses together with the x's from each of the other (n—Z) parentheses. Whence, there are as many partial products like a^x*'~^ as there are ways of selecting 3 a's from n parentheses; that is, in as many ways as there are groups of n things n{n—l'){n--2') taken three at a time, or 1-2.3 fi(fi Y)(n 2") Whence, — z — ^^^—^ — -^^^""^^ is another part of the product. In general. We may obtain a partial product like a*^af~^ (where r is a whole number less than n) by taking an a from any r of the parentheses together with the jt's from each of the other (n—r) parentheses. Whence, there are as many partial products like a''x''~^ as there are ways of selecting r a's from n parentheses ; that is, in as many ways as there are groups of n things taken r at \n \JL _ ' a time, or -. — r= Whence, , — j a^x"* '' stands for * \r In—r \ r\ n—r any term in general in the product {x-\-dy*. 368 UNIVERSITY ALGEBRA. Finally, we may obtain one partial product like a*" by- taking an a from each of tbe parentheses. Whence, «" is the last term in the product. Thus we have proved or {x+a)'*==x*'+naa>**-^+'^^!^^!^^ .. \n The expression on the right-hand side of the equation is called the Expansion or Development of the power of the binomial. 525. The expansion of (xdtay is usually called the Binomial Formula. If in the result of the last Article, we substitute dta for a, we get the following as the expansion of (xdha)**: 2 2.3 ^ n(n-1)(n-2)(n-3) ^4^„4^ _ ^j Therefore, in any power of the difference of two num- bers the sign of the first term is +, of the second — , and so on, alternately + and — . 526. Binomial Theorem. By observing carefully the expansion of (x+a)** written above it will be seen that we may state the binomial formula in the form of a theorem as follows : I. Exponents. In any power of a binomial, x+a, the exponent of x commences in the first term with the expojient of the required power, and in the following terms contin- ually decreases by unity. The exponent of a commences with 1 in the second term of ^he power, and conthiually increases by unity. BINOMIAL THEOREM. 369 II. COKFFICIKNTS. The coefficient in the first term is 1, that in the second term is the exponent of the power ; and if the coefficient in any term be multiplied by the exponent of X in that term and divided by the exponent of a increased by ly it will give the coefficient in the succeeding term,. 527. Below we give a few examples of the use of the binomial theorem. (1) Expand («+<5)«. We may expand this at once by the theorem as follows : or we may substitute x=^a, a=b, and n~^ in the formula [1], obtaining : +2.3.4.5 ^2.3.4.5.6 which reduces to the same result as above. (2) Expand {u + Sy)^. Here x=u and a=3y. By the theorem we get «5 4-5^^4(3/) + 102^3(3^)2_|_ 10^^2(3y)3 4.5^^(3^)4 _|_ (3^)5^ Performing the indicated operations, we get u^ + 15uy^90u^y^+210u^y^+i05uy*+2idy^, (3) Expand (^2 -2)*. ^-«T lo 'r^dmuM MSM Here x=r^, a=—2, and «=4. By the theorem (r2)4_4(;.2)82+6(r2)222-4(r2)23+24. Performing the indicated operations, we get rs - 8^6 + 24^4 - 32^2 + 16. (4) Expand (3^-i)8. HeTex=db, az=—^, n=S. By the theorem ... (3^)3-3(3^)2(i) + 3(3^)(4)2-(4)8r Performing the indicated operations, we get 24 -U. A. 370 UNIVERSITY ALGEBRA. KXAMPLKS. Expand each of the following by the binomial theorem or formula : 1. (a+xy. 9. (1 + ^)'^ 17. (m-hky. 2. (d+xy, 10. (2+xy. 18. (a-xy. 3. (d+yy. II. (2-xy, 19. (ddSyy, 4. (C+Xy, 12. (i + ^)^ 20. (^3_^2)8^ 5. (-X+2ay, 13. (^2_^2)5^ 21^ (2/^2_3^3)5^ 6. (2;t;+3a)«. 14. (:r+2^)^ 22. (Sx^-iy, 7. (1-.^)^ 15. (Sa+^y. 23. (i/^4-^2'- 8. (1--^)^ 16. (2ax-x^y 24. (3^-l/;r)^ 2 3 25. (;t:_y— ^^)^. 30. (.r'5'+;»;2")6. 26. (l/^-f/^)6. 31. («-2-3^4. 27. (ia+lx+y]y. 32. (^2 4.2^;|;+^2)8^ 28. ([a+^]-2)». 33. {v^^^^y. 29. (a+^-J/)^ 34. {^+^2} • PROPERTIES OF THE EXPANSION. 528. Number of Terms. The exponents of a through the binomial formula constitute the following scale : 0, 1, 2, 3, 4, ... ^2. The number of terms in this scale is n+1. Therefore the number of terms in the expansion of (^x+aY is n+1. 529. Value of the r th Term. The general term in the expansion of (x+ay we have seen to be [n I r \ n—r BINOMIAL THEOREM. 3/1 By noting the exponent of a, this is seen to be the r-\- 1 term from the beginning, since Xh.^ first term contains the zero power of a. Therefore the r th term in the expansion will be found from the general term by decreasing the r in it by unity. This gives \n r th term in (a; + »)"= i == «;"-''+ ^ a"" ^ [2] I r— 1 [w.-r+l 530. Coefficients Equidistant from the Ends. We have just found that the coefficient of the r th term from the beginning in the expansion of {x-\-ay is |_^ I r—\ 1 7^— r+1 Since there are n-\-\ terms altogether, there must be (72 + 1)— r terms before the r th term from the end; that is, the r th term from the end is the n—r-\-^ term count- ing from the beginning. Substituting n—r-\-^ for r in the expression for the r th term, we get the following value fro the r th term from the end : \n — - ^r— 1 >y«— r-f- 1 \n^r^-\\r--r ' "" Thus we note that in the expansion of (x+a)** the coefficients of terms equidistant from the ends are equal, 531. Coefficients of Odd and Even Terms. In the expansion of {x-^rcCy put x=^ 1 and a= — 1. We thus obtain ft_^ , <^-l) , n{n^V){n^^Xn-^^ , \ V "^ 1 . 2 "^ 1.2.3.4 •+"•••; which shows that the sum of the coefficients of the first, third, fifth, . . . terms equals the sum of the coefficients of the second, fourth, sixth, . . . terms. 372 UNIVERSITY ALGEBRA. In the binomial formula the sum of the coefficients of the even terms equals the sum of the coefficients of the odd terms. The above result may be also written ^G)+6^e2)+^a)+. . .=^a)+6^(3)+^(5)+. . . where both sides of the equality are to be continued until 6^(2) occurs on one side or the other. 532. Sum of the Coefficients. In the expansion of {x+dy put x=-\ and «=1. We then have That is, the sum of all the coefficients in the expansion of (x-\-ay equals 2". 533. MidVile Term. If the expansion consists of an odd number of terms, there is a middle term. That is, there is a middle term if n-\-\ is odd; that is if ^ is even. Since the coefi&cients of terms equidistant from the ends are equal, we may write 2'*=2(l+;z+^^^-+. • . to I terms) + middle term. In order that this equation may be true, the middle term must be divisible by 2. That is, the coefficient of the middle term in the binomial formula is divisible by 2. KXAMPl^KS. I. Find the n th term of {nT—n"**)**, f 1 \i* 2. Find the 11th term of 4;i; --A ' V 2Vx/ 3. Write down the 10th term of (a—xy^. 4. Write down the middle term of (1+^)^". 5. Write down the term containing .;*:' in the expansion of(l+xy. BINOMIAL THEOREM. 373 6. Write down the two middle terms in the expansion 7. Write down the two middle terms in the expansion »K'+^) 2M-1 MUI.TINOMIAI. THEORKM. 534. The expansion of any power of a polynomial, a-{-b-\-c+ . . . can be found by successive applications of the binomial formula. Thus, (^a-{-b+c+d+ . , ,y is the same as (a+ld+c-{-d+ . . .])'*. The general term in the expansion of this latter is by Art. 529. In the same way the general term in the expansion of {l>+c+d+. . .y- or ib+[c+d+. . .])«-" is ^-l>'(c-i-d+. . . )«-'-^ ^2) \L Similarly, the general term in the expansion of ic+d+. . .y-"--' or (r+[^+. . .])'*— ^ is \n — r — s , '1 / (^+. . .)""""""', (3) and so on. Whence, the general term in the expansion of {a-^-b+c-^-d^-, . .y is the product of (1), (2), (3), etc., \n \n — r I n — r — s or, -X X Vr n—r \s n—r—s T7 n—r—s—f . oTb'c^. which, simplified, gives It is evident from the process of the formation of [3] that the numbers r, s, t, . . . may stand for any positive whole numbers whatever, provided that r-\-s+t+ . . . =«. 374 UNIVERSITY ALGEBRA. 535. General term in (a + dx-\-cx'^+dx^+ . . .y\ This can be written down at once from the result just obtained. The general term is evidently t \g\r ^ \l.. -a^bxYicx'^yidx^y. \n that is, -. — I — ri=-i-: a^b-c'd'. . . x'+^'+^'+ ' ' • if [r If li--- In this, as before, q+r+s+i+, . .=;^. EXAMPI^KS. 1. Write out the coefficient of x^ in the expansion of (2+X'-x''y. The general term is ., \- , 2^(-lK^>-+2J, \i Lrii- Now, g-{-r-{-s=5, (1) and r+2j=8. (2) Whence, we may have from (2) r=0, s=4;^r=2, J=3; r=:4, ^=2; r=Q, s=l', r— 8, ^=0. Of these, however, only r=0, J=4; ^=2, ^=3 will go with (1). Whence we say r=0, J3=4, ^=1, and r=2, ^=3, ^=0. which gives the terms L5 li The sum of these two terms is 0. Whence, the coefi&cient of x^ in the expansion of (2 + ^— :«;*)** is 2. Write out the coefficient of a^d^c^ in the expansion of (ia-\-d-\-cy^, 3. Find the coefficient of x^ in the expansion of 4. Find the coefficient of x^ in the expansion of BINOMIAL THEOREM. 375 5. Show that the coefficient of x^ in the expansion of (l + 2x+x^y is 56. 6. Write out the coefficients of a^b^, and d^cd in the expanvSion of (a+d+c+d)^. Historical Note. The Binomial Formula is engraved upon the tomb of Sir Isaac Newton. (1640-1727) in Westminster Abbey. He discovered it while he was a student at Cambridge. It grew out of the study of Wallis's investigations on the quadrature of curves. Newton gave no formal proof of the theorem. Indeed, not until the present century were rigorous proofs discovered, applicable to the case when the expansion becomes an infinite series. Near approaches to the discovery of the Binomial Formula for the case of positive integral exponents were made before the time of Newton. Vieta (1540-1603) gave the rule for obtaining the powers of a binomial. He observed as a necessary result of the process of mul- tiplication that the successive coefficients of any power of a binomial are: first, unity; second, the sum of the first and second coefl&cients in the preceding power; third, the sum of the second and third coeffi- cients in the preceding power, and so on. Vieta noticed also the uniformity in the product of binomial factors of the form of x + a, x-\-d, x-\-c, etc. But Harriot (1560-1621) independently and more fully treated of these products in showing the nature of the composi- tion of a rational integral equation. In this connection it is interesting to note that Harriot was the first mathematician to transpose all the terms of an equation to the left member. CHAPTER XXIII. THEORY OF PROBABII^ITIKS. 636. Mathematical Probability. There is a large class of events of which our knowledge is insufficient to permit us to predict, in a specified case, the precise thing that will happen, yet of which our knowledge is sufficient to enable us to express, in a quantitative way, what ought to be our belief concerning the happening of the event. Thus, if a coin be tossed, our knowledge is not sufficient to enable us to say whether it will fall ' ' heads ' ' or whether it will fall '*tails*', but our knowledge is sufficient to enable us to say that // zs just as likely to fall heads as tails. Likewise, there is another class of events in which we cannot make a definite statement concerning any one event of a group, yet concerning the group itself we may be able to predict this or that fact with considerable approximation to accuracy. For example, nothing can be more uncertain than to say that a certain man aged 30 and now in good health will be alive 10 or 20 years hence. Yet of a large number of such men we may assert, with safety, that 84 per cent, will be alive 10 years hence. If the number be 10,000 the assertion may be made with greater confidence than if the number be 1,000; thus, though each individual case is most uncertain in its out- come, yet a very large group of these uncertain cases may be treated with almost as much confidence as if dealing with the law of gravity itself. A quantitative expression which measures what our belief ought to be concerning an event of either of the above classes in which absolute knowledge is impossible, THEORY OF PROBABILITIES. 377 is called the mathematical probability of the event, and may be defined as follows : If an event is in question in a + d cases, all of which are equally likely, and if in a of these cases the event will happen, and in the remaining cases will fail to happen, then the Probability of its happening is — -r-v and the probability of its failing is —rrr Thus, suppose that an ordinary die (which has six faces numbered from 1 to 6) be thrown; what is the probability that a certain face, say 5, will be uppermost? Since the die has six faces, there are six cases in question, all of which are equally likely; that is, we have no reason to expect one rather than another of the faces to come up. In one of these cases the event (throwing a 5) happens and in the other five cases the event fails to happen. Whence the probability of throwing a 5, by definition, is \. If a die be thrown, what is the probability of getting an even number? As before, there are six cases in question, all of which are equally likely, and in three of which the event happens. Whence, the probability of throwing an even number is f or \. If a die be thrown, what is the probability of getting less than 5? Here there are 6 equally likely events in question, in 4 of which the event happens. Whence, the probability is f . ' • I prefer to say that the theory of probability deals with quantity of knowledge. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. It teaches us to regulate our action with regard to future events in a way which will, in the long run, lead to the least amount of disappointment and injury. It is, as Laplace as happily expressed it, good sense reduced to calculation. ' ' "The theory consists in putting similar cases upon a parj and distributing equally imong them whatever knowledge we may possess. 378 UNIVERSITY ALGEBRA. Throw a penny into the air, and consider what we know with regard to its mode of falling. We know that it will certainly fall upon a flat side, so that the head or tail will be uppermost, but as to whether it will be head or tail our knowledge is equally divided. Whatever we know concerning head, we know as much concerning tail, so that we have no reason for expecting one more than the other. The least predominance of belief to either side would be irrational, and would consist of treating unequally things of which our knowledge is equal." — Quoted, with omissions, from W. S. Jevons' ''Principles of Science. ' ' 537. Broader Definitions. Problems like those in insurance, where the probability must be estimated by- means of data, secured by observation covering a large number of cases, require a broadening of the above defi- nition. To illustrate the cases in which the enumeration of the number of cases in question is impracticable, Bertrand asks, ''What is the probability that the Seine will be frozen at Paris in the course of 1995 ?*' In this connection the following definitions of probability are valuable: If on taking any very large number n out of a series of cases in which an event A is in question, A happens on pn occasions, the probability of the event A is said to be p. — Crystal's Algebra. ' ' The probability of an event is the ratio of the favorable cases to the total number of cases possible; a condition is understood: all the cases must be equally likely." — Preliminary definition, Bertrand' s Calcul des Probabi life's. ' ' The probability of an event, whatever its nature may be, is said to equal a given fraction /, when he who awaits the event might exchange, indifferently, the fears or hopes, the advan- tages or disadvantages incident to the happening of the event, for the consequences (supposed identical) of drawing a ball from an urn whose composition gives rise to a probability equal to /." — Bertrand, subsequent definition. 538. Certainty. If an event happens in every case in which it is in question, without fail, then the proba- bility of the event, by definition, is — ^ or 1. Thus, certainty, in accordance with our definition, is denoted by THEORY OF PROBABILITIES. 3/9 unity. For example, what is the probability of throwing less than 10 with a common die? There are 6 events in question, in all of which the event happens. Therefore, the probability is 1 or certainty. Also, if an event happens in no case in which it is in question, then the probability of the event is, by definition, ^r — I or 0. Thus, impossibility, in accordance with our definition, is denoted by 0. 539. Complementary Events. If p stands for the probability of an event happening, then \—p is the probability that the event fails to happen; for evidently -^ the probability of the event failing to happen, equals 1-^^- Thus we note that the probability of an event is always expressed by a positive proper fraction — a num- ber between and 1 . 540. Odds. Instead of saying that the probability of an event is — — , we may say that the odds are a \,o b a-\-o hi favor of the event or that the odds are b X.o a against the event. Thus, instead of saying that the probability of throwing less than 3 with a single die is \, we may say that the odds are 1 to 2 in favor of throwing less than 3 or 2 to 1 against throwing less than 3. SIMPLE PROBABILI'TY. 541. Many problems in probability involve nothing but an application of the definition of probability and a knowledge of the theorems in arrangements and groups. (1) Two dice are thrown. What is the probabiUty of throwing twoC's ? A 5 and a 6 ? A sum greater than 9 ? A sum equal to 10? 38o UNIVERSITY ALGEBRA. When two dice are thrown the number of events in question is the number of ways of arranging the faces taken 2 at a time, repetitions allowed, or 6^. One of these is the case in which the getting of two 6's occurs. Whence the probability of throwing two 6's is ^. A 5 and a 6 may occur in two ways ; the first die may turn 5 and the second 6, or the first die may turn 6 and the second 5. Whence, the probability of a 5 and a 6 is 3% or ^. A sum greater than 9 may be made in the following ways : First die, 6 6 5 5 6 4 Second die, 6 5 6 5 4 6 Sum. 12 n 11 To 10 10 or 6 ways in all. Whence, the probability of throwing more than 9 is 3^ or ^. A sum equal to 10 can be made in three ways. Whence, the probability of throwing exactly 10 is f^ or ^. (2) Two balls are to be drawn from a bag containing 4 white and 6 black balls. What is the probability that both will be white ? Altogether there are G{^^) different pairs which may be drawn, all of which are equally likely. In G{^) of these both are white. Whence, the required probability is 4 . 3 . 10 . 9 ^ 1.2* 1 . 2 °^ 15* (3) Thirty coins are thoroughly shaken and thrown. What is the probability that all turn heads ? Thirty coms can fall in 2^® ways. The event (all turning heads) happens in but one of these ways. Whence, the required probability is p^=.000 000 000 93133. The probability is, of course, the same as getting all heads in 30 throws with a single coin. (4) A ball is to be drawn from an urn containing 4 white, 6 red, and 10 black balls. What is the probability of either white or black? The probability of not drawing white or black (or the probability of drawing red) is ^% or ^5. Whence, the probability of drawing white or black is 1— x^o or ^^. EXAMPI^KS. I. An urn contains 3 white, 4 red, and 7 blue balls. If three be drawn what is the probabilit}^ of getting red, white, and blue? THEORY OF PROBABILITIES. 38 1 The total number of cases in question is the number of arrangements of 14 things three at a time, or ^(V). The number of cases in which the event happens is the number of arrangements of 3 things taken all at a time multiplied by 3X4X7. 2. What is the chance of throwing one 6 and only one in a single throw of two dice ? The number of cases in question is 36. The cases in which the event occurs are when the 6 of the first die falls with one of the other five faces of the second die, and when the 6 of the second die falls with one of the other five faces of the first die. 3. An urn contains 3 white, 4 red, and 7 blue balls. If three be drawn in succession, what is the probabilitj'- of . getting the order, red, white, and blue? 4. What is the probability of throwing doublets with two dice? 5. If four coins be tossed, what is the probability that exactly half will fall heads ? The number of cases in question is 2* or 16. The number of cases in which the event happens is the number of arrangements of two «'s 14 and two b's or — = — , \% ^ 6. When two dice are thrown, what is the probability of not throwing an ace ? 7. A set of dominoes is numbered from double blank to double six. If one be drawn at random, what is the probability that it contains a 6 ? The number of dominoes in the set is the number of groups of seven 7 8 things taken 2 at a time, repetitions allowed, or — '- — . 2 8. If four cards be drawn from a pack, what is the probability that there will be one of each suit? 9. In a ba^ there are 4 white and 6 black balls. What is the probability, if they be drawn in succession, that 382 UNIVERSITY ALGEBRA. the white balls will all be drawn first and then the black balls? The number of cases in question is ' — -. \± [^ 10. In a bag there are 4 white and 6 black balls. If 3 be drawn, what is the probability that all will be black ? 11. Show that the probability of throwing more than 15 in one throw with three dice is less than ■^. 12. Four cards are missing from a pack. What is the probability that there is one from each suit ? 13. Thirteen persons take places at a round table. Show that it is 5 to 1 against two specified persons sitting together. 14. What is the probability, in throwing two dice three times in succession, of obtaining a doublet at least once? The number of cases in question is 36^ or 46656. At each throw 6 of the 36 possible cases gives doublets. Whence, the number of cases which give no doublets in three throws is 30^ or 27000. The number of cases which contain at least one doublet is then 363-303 = 19656. Whence, the required probability is i?555=.4213 .. 46656 15. If 4 «'s and 3 ^'s be placed in a row at random, show that the probability of the first and last letters being both ^'s is \, TOTAI. PROBABII^ITY. 542. Probability has been defined as the ratio of the number of cases in which an event happens to the total number of cases in question. If the cases in which the event happens are divided into several classes, the proba- bility of the event will be the sum of the probabilities pertaining to each of the classes. THEORY OF PROBABILITIES. 383 Thus, the probability of throwing more than 14 with three dice is the sum of the probabilities of throwing 15, 16, 17, and 18. Likewise, if A's probability of taking a prize is \, and B's probability of taking it is -|-, and C's probability of taking it is |, then the probability that either A, B, or C takes the prize is i+-|-+| or ^. The probability of some one else taking it is then ^. 543. The division into classes, as supposed above, is arbitrary, with the implied condition that the division must be made so as to include all of the cases in which the event happens, without including any case more than once. To express this condition, we say that all the events belonging to the different classes must be Mutually Exclusive; that is, the supposition that any one of the events happens, must be incompatible with the supposition that any other happens. Thus, the probability of throwing a 3 or a 4 with two dice is noi the sum of the probabilities of throwing a 3 and a 4. The number of •events in question is 36. The number of cases in which a 3 is thrown is 11 and the same for a 4. But in estimating the cases in which a 3 occurg we counted the throws 3, 4 and 4, 3. Evidently these should not be recounted in estimating the number of cases in which a 3 or a 4 is thrown. Containing 3, * * xxxxxx * ^^ ^^ -r^.^xxro J 1st die, mill 222222 333333 444444 555555 666666 inrows, -j 2nd die, 123456 123456 123456 123456 123456 123456 •Containing 4 * * * xxxxxx * ^^ The required probability is |J or | and not |J, The events sup- posed are not mutually exclusive, since the supposition that a 3 turns is not incompatible with the supposition that a 4 turns, for they may both turn at once. KXA.MPI,ES. I. If the probability that a shot aimed at a target hits the bull's eye be ^, the probability that it hits the first ring be |, and the probability that it hits the outer ring he \, what is the probability that it hits the target at all? 384 UNIVERSITY ALGEBRA. The rivents are mutually exclusive and the event of hitting the target can be decomposed into these three groups of events. There- fore, the required probability is 1*5+ J+ J or J. 2. If the probabilty of A winning a race be -|-, and the probability of B winning be ^^ what is the probability that either A or B wins? 3. A bag contains 4 red, 8 black, 12 white, and 16 blue balls. What is the probability, if one be drawn, of get- ting either red, white, or blue ? 4. If a ticket be drawn from a set numbered from 1 to 30, what it its probability that its number will not be a multiple of 6 or of 7 ? 5. When two dice are thrown, what is the probability that the throw will be greater than 8 ? 6. If the probability of A taking a prize be ^, and the probability of B taking it is I-, what is the probability that neither takes it? COMPOUND PROBABIIvlTY. 544. Several single events occurring simultaneously or in succession may be considered as constitutifig a Compound Event. Thus, the drawing of a red, a white and a blue ball in succession may be considered a compound event composed of the three simple events mentioned. Events are said to be Dependent or Independent according as the occurrence of one does or does not affect the occurrence of the others. 545. The probability that two independent events will both happen is the product of their respective probabilities of happening. Let a be the number of ways in which the first event may happen, and b the number of ways in which it may THEORY OF PROBABILITIES. 385 fail, all these ways being equally likely to occur. Also, let a^ be the number of ways in which the second event may happen, and dj^ the number of ways in which it may fail, all these ways being equally likely to occur. Each case out of the a + d cases may be associated with each case out of the ^1 + ^1 cases. Thus, there are (a + ^)(«i+<^i) compound cases which are equally likely to occur. In auj^ of these compound cases both events happen, in ddj^ of them both events fail, in ad^ of them the first event happens and the second fails, and a^^d of them the first event fails and the second happens. Thus, we have THE FRACTIONS. THK RKSPKCTIVB PROBABII^lTi:^ ^^1 That both events happen. That both events fail. That first event happens and second fails. That first event fails and second happens ab^ {a-^b){a^-^b^) a-^b (a + ^)(^, + ^) If p and />! stand for the probabilities of the happening of the two events, we have from above : The probability that both events happen equals pp-^^. The probability that both of the events fail equals (i-/)(i-/0. The probability that first event happens and second fails equals /(I —/>i). The probability that first event fails and second hap- pens equals iX~P)P\' 25 — u. A. 386 UNIVERSITY ALGEBRA. (1) The probability that A can solve a certain problem is J, and that B can |solve it is |. If they both try, find the probability (1) that they both succeed ; (2) that A succeeds and B fails ; (3) that A fails and B succeeds ; (4) that both fail ; (5) that the problem will be solved. Here we have p =i ^—p =f- Probability both succeed =:pp^-=^. Probability A succeeds, B fails =/{! -/i)=i^. Probability A fails, B succeeds ={\~p)p^=:^. Probability both fail =(1 -/)(! -p^ )=\. i+i^ + i + i=H==l as it should, The probability that the problem is solved is the probability that one at least succeeds in solving it, or 1 — (1— /)(!— /i)=|. (2) A man draws a ball from each of two urns. The. first urn con- tains 4 white and 7 black balls and the second contains 7 white and 3 red balls. What is the probability of drawing two white balls ? The events in question are independent and their respective proba- bilities are ^ and /g. Whence, the probability of drawing two whiteisi^X/oorif. (3) In the same problem, what is the probability of drawing black and red? The probabilities of the (independent) .events are respectively ^ and y%. The probability is then ^. 546. If /i, ^2» /a* • • • A Stand for the respective prob- abilities of n independent events, it is evident that we may extend the results of the previous article and say Pip2pz ' ' -A is the probability that a// the events happen, (1— /i)(l— /2)(l-/3) . • . (1— A) is the probability that all the events fail to happen, /^ (1 —p5 + . . . ; Cs=i + ^ + . . . ; or, A's=f; B's=f; C's=f. MATHEMATICAI^ EXPECTATION. 548. If p is the probability of the occurrence of a cer- tain event and m the sum of money which a person will realize in case the event occurs, then the sum of money expressed by pm is called his Mathematical Expecta- tion. Thus, suppose a man is to receive $5 if a die falls ace. Then his mathematical expectation is $5 multiplied by the probability (|) of the event, or $.83^. The mathematical expectation represents the average sum which would be realized from each event provided a very large number of trials be made in which the given event is in question. For, in n trials (^n being a large number) the event will occur, on the average, pn times. The amount realized each time the event happens being %in, the whole amount realized will be %pnm^ an average of %pm realized from each trial. There is an important distinction between tke equitable value of a trial and the mathematical expectation. If the probability of the success of a venture be ^gi and the sum to be realized, if successful, be $1,000,000, the mathematical expectation is $100,000, but one could not prudently give this amount for the given chance. If a person's capital be small, it might be imprudent to give $500 for the chance. Even though the person were sure he would be able to take advantage of identical opportunities on many occasions, he must bear in mind that there is sojue probability that he will lose every time. The com- putation of this margin, in connection with the known size of his capital to start with, will determine what a man might equitably give for a given chance, which in every case would differ from the mathematical expectation, unless his capital or credit be unlimited in amount. THEORY OF PROBABILITIES. 389 KXAMPI^ES. 1. What is the probability of throwing one ace and only one if four dice be thrown ? 2. If four dice are thrown, what is the probability that two faces at least turn alike ? 3. If four dice are thrown, what is the probability that exactly two faces turn alike ? 4. What is the expectation when drawing a bill from one of two boxes, one box containing five $1 and three $5 and the other containing two $1 and four $5? 5. A person undertakes to throw 5 or 6 points in a single throw with two dice. Find the probability of success. 6. If, on an average, 9 ships out of 10 return safe to port, what is the chance that out of 5 ships expected, at least 3 will arrive ? 7. A teetotum of 8 faces, numbered from 1 to 8, and a common die are thrown. What is the probability that the same number is turned up on each ? 8. A person goes on throwing a single die until it turns up ace. What is the probability that he will have to make at least ten throws ? that he will have to make exactly ten throws? 9. Out of a bag containing 12 balls, 5 are drawn and replaced, and afterwards 6 are drawn. Find the proba- bility that exactly 3 balls were common to the two drawings. 10. A purse contains four silver dollars and one eagle. A second purse contains ten silver dollars. If two coins be taken at random from the first purse and placed in the second, what is t^e probable value of the contents of each purse? 390 UNIVERSITY ALGEBRA. 11. A has three tickets in a lottery where there are 3 prizes and 6 blanks. He also has a ticket in another lottery where there is but 1 prize and 2 blanks. Find the probability of his drawing at least one prize. 12. In a bag there are 5 black and 4 white balls. If they be drawn out in succession, what is the probability that the first will be white, the second black, and so on^ alternately, white and black ? 13. A bag contains 6 black balls and 1 red. A person is to draw them out in succession, and is to receive a dollar for every ball he draws until he draws the red one. What is his expectation ? 14. If four cards be drawn from a pack, what is the probability that they will be marked one, two, three, four? If they be drawn in succession, what is the probability of the order one, two, three, four ? 15. A letter is chosen at random out of each of the words musical and a musing. What is the proba- bility that the same letter is chosen in each case ? If the letters are both consonants, what is the probability that they are the same ? 16. A has three dollar pieces and B has two. They agree that each shall toss his money and the one who gets the greater number of heads shall have all the money. Is this a fair arrangement? What are the respective expectations (1) if they make but one throw? (2) if they throw until one of the two wins ? 17. A person undertakes in two throws with two dice to make 7 points the first throw and 9 points the second throw. Find the probability (1) that at least one throw is successful, (2) that both throws are successful, (3) that only the first throw is successful, (4) that both throws fail, (5) that one throw (and no more) succeeds. THEORY OF PROBABILITIES. 391 18. A purse contains four silver dollars and an eagle, and a second purse contains five silver dollars. If two coins be taken from the first purse and put in the second, and then if three coins be taken from the second purse and placed in the first, what is the probable value of each purse? SUCCKSSIVB TRIAI^. 549. Two Trials. Let there be an event which must turn out in one of two ways, W or B, as in drawing a ball from an urn containing white and black balls only. Let the probability of W^ happening be p and of B hap- pening be q, so that p+q must equal unity. If two trials be made (as drawing from an urn, restoring the ball, and drawing again) the four possible cases which may occur are WW, WB, BW, BB, the respective probabilities being /^2, pq, qp, q'^, by the theory of compound events. In other words, the probability of drawing two white balls is p'^\ the proba- bility of drawing white and black in the order WB is pq\ the probability of drawing black and white in the order BW is likewise pq; the probability of drawing two black balls is q^. The probability of drawing white and black, irrespective of the order in which they are drawn, is plailny 2pq, Thus, we note 550. n Trials. Instead of two, letthere be T^trials made. Then the probability of W happening every time is p"". The probability of I^happening n--l times and B once in an assigned order is p**~^ q. If the order is indifferent, the probability of W happening n—1 times and B once is the sum of the probabilities of B happening on the first trial and PF" happening on the other trials, of B happening on the second trial and Won the other trials, etc., or P'*''^q+P*'~^q+P"~'^q+' . .ton terms ^^np^'-^q. 392 UNIVERSITY ALGEBRA. The probability of W happening n—2 times and B twice in an assigned order is />'*~^$'^. If a specified order is not required, the 2 trials on which B happens can be selected from the n trials, in ~ — ^- ways. Whence, the probability of W happening n—^ times and B twice, if n(n — 1") the order is indifferent, is the sum of — :j — x- terms each fi(fi '\\ equal to p**~^q^i and is therefore equal to —i — ~ />**"" ^^^. In general, the probability of PP' happening ^—r times and B happening r times in an assigned order is P'~''q''. If a specified order is not required, the r trials on which B happens can be selected from the n trials in \7i—r ivays. Whence, the probability of W happening n^r times and B r times, if the order is indifferent, is the sum \n of -, — ^= — terms each equal to P" ''^'', and is therefore L- n — r \n equal to -. — r= — P'^ ^—^> r—^. The probability required may be decomposed into the probabilities of winning all 3, and of winning exactly 2 out of 3 games. Whence, we have (l)^ 4-3(|)2 |=p, (3) The odds are 2 to 1 in favor of A winning a single game against B. A lacks 8 games and B lacks 5 games of winning a match. Find A's probability of winning the match. A must win his 8 games out of the next 12 games, for otherwise B will win. Whence, A's probability of success is composed of the probabilities of his winning exactly 8 games out of the next 8, of his winning exactly 8 games out of the next 9, and of his winning exactly 8 games out of the next 10, ... , and of his winning exactly 8 games out of the next 12. Altogether this gives 10 . 9 11 . 10 . 9 l, and indeterminate if x=l. 558. Series are often used to replace certain expres- sions the values of which are given to closer and closer degrees of approximation by taking more and more terms of the series. By means of series we are often enabled to calculate the numerical values of expressions more easily and more CONVERGENCE AND DIVERGENCE. 39/ rapidly, or to study the properties of functions more easily than could be done without the use of series. But in order that this simplification be permissible it is necessary that the series substituted should have the proposed func- tion for a limit, i. etc., in each case the subscript being the same as the number of the term; and let -^j be the remainder after the first term, R^ the remainder after the second term, R^ the remainder after the third term, etc. ; in each case the remainder after any number of terms are taken is represented by R with a subscript equal to the number of terms already taken ; and further, let the sum of any CONVERGENCE AND DIVERGENCE. 399 number of terms be represented by S with a subscript equal to the number of terms taken, t. e,y the sum of two terms will be represented by 6*2, the sum of three terms by 6*3, and so on. 561. With the notation just explained, the sum of a series which has a limited number of terms will be repre- sented by Sg+Rg, whether ^ is 1 or 2 or 3 or any other number not exceeding the whole number of terms of the series. In an infinite convergent series Sn approaches a limit as n increases without limit, and the value of this limit is Sg-\-Rq, where q is any positive whole number whatever. It is easy to see in this case that R^^^ 2,sn increases without limit. In an infinite divergent series Sn does not approach any limit neither does R^ approach any limit, and^S^H-^^ has no definite value at all. 562. It is evident that a series cannot be convergent unless, after a certain number of terms are taken, the successive terms decrease in absolute magnitude, or, in other words, unless ^^«^ as n increases without limit. But while this is necessary it is not sufiicient, for a series may be divergent and still 2^^^ as n increases without limit. Take for example the harmonic series in which the n th term is — » which evidently approaches zero as n increases without limit. If the terms of this series be grouped thus : 1 4.00 UNIVERSITY ALGEBRA then in no group is the sum less than ^ and as there is an unlimited number of groups, the series evidently does not approach any limit, but increases without limit as the number of terms increases without limit. Therefore the series is divergent. 563. Before proving the theorems, by means of which the convergence or divergence of series is usually deter- mined, we desire to show how the convergence of some series may be established by comparing with some standard series. I^et us take the series where the only restriction placed upon the successive terms is that u^':: as « increases indefinitely. With this restriction it is plain that as n increases indefinitely the sum of the first n terms appoaches nearer and nearer the value u^.or, as we usually express it, the sum of the series equals u^. Therefore, with this restriction, the series is convergent, and we are permitted to write In this equation we may make any substitution we please consistent with the above restriction, that Un'^ 0> and the series obtained will be convergent. Let us take u^ = l, u^=x, u^=x^ etc. Wherein x+2)d term by B, etc. We thus form a new series 2^1+2/2+ • • • -\-Up-\- AUfj^^-^ Bup^^-\- ... (2) We are to prove that the series (2) is an absolutely convergent series. From the series (1) form another series whose terms are all positive. Represent this new series by z;i+z;2+^3+^4+ • • • (S) Plainly, the terms of (1) and (3) have the same absolute values, and the only difference between these two series is that the terms of (3) are all positive, while the terms of (1) are positive or negative at pleasure. Also, from the series (2) form another series whose terms are all positive, and represent this new series by ^1+^2+ • • • +^/ + ^^/-M+^^/+2+- • • (4) Plainly, the only difference between (2) and (4) is that the terms of (4) are all positive, while the terms of (2) are not necessarily all positive. 406 UNIVERSITY ALGEBRA. Now suppose K to stand for some positive finite num- ber as great as the numerically greatest of the chosen multipliers A, B, C, etc., then But by theorem I approaches some definite limit. Therefore, ^(z/^i +^/+2 + • • • ) approaches some defi- nite limit. Therefore, Avp^-^ +^i>+2+ * • • approaches some definite limit. Therefore, v^+V2+ ••• +v^+Avj^i+Bv^2+ ' — which is the series (4), is convergent. Since (4) is a convergent series, all of whose terms are positive, it follows that (2) (which becomes (4) when all its terms are taken positively) is an absolutely convergent series by definition, Art. 571. 573. Theorem V. Any series is absolutely convergent ify after some particular term, each of its subsequent terms is numerically less than the corresponding term of a series which is known to be absolutely convergent. This follows immediately from the preceding demon- stration by taking the multipliers A, B, C, etc., any fractions numerically less than unity. 574. Theorem VI. A series is absolutely convergent if, after any particular term, each of its subsequent terms bears a finite ratio to the correspondifig term of a series which is known to be absolutely convergent. This theorem is an immediate consequence of theo- rem IV. For, after the/> th term of each series, the ratios CONVERGENCE AND DIVERGENCE. 407 of the terms of the second series to the corresponding terms of the first series are Ay B, C, etc., which by- supposition are finite numbers. 575. Theorems IV, V, VI, hold for semi-convergent as well as absolutely convergent series, provided, in each case, the terms of the second series have the same signs as the corresponding terms of the first series. 576. A series all ol whose terms are positive must be either divergent or absolutely convergent. Since all the terms are positive, the more terms there are taken the greater is the sum of those terms, that is, the sum of the first n terms continually increases as n increases. Now, in this continual increase, the sum of the first n terms must increase indefinitely or else must approach some definite limit as n increases indefinitely. In the first case the series is divergent, and in the second case the series is convergent, and, having all its terms positive, is absolutely convergent. 577. Theorem VII. If, after any particular term of a divergent series all of whose terms are positive, all the subsequent terms be ?nultiplied by any positive finite numbers whatever, the resulting series is divergent. Let us represent the given divergent series all of whose terms are positive by 2^1+2^2+^3+^4+ • • • (1) Then taking A, B, C, etc., to represent any positive finite numbers, let us multiply the (/>+l)st term by A^ the (/+2)d term by B, etc. We thus form a new series «l+«2+^3+- • •+^/ + ^^/+l+^2^i4.2+ • • • (2) all of whose terms are positive. We are to prove that the series (2) is divergent. 408 UNIVERSITY ALGEBRA. Suppose K to stand for some positive number as small as the smallest of the chosen multipliers A, B, C^ etc., then But the given series (1) being divergent ^(^/+l+«/+2+^/+3H ) increases indefinitely as the number of terms increases indefinitely. Hence ^?^i»+i +Bup^^ + Cup^^ + • • • increases indefinitely as the number of terms increases indefinitely. Therefore, the series (2) is divergent. 578. Theorem VIII. A series, all of whose terms are positive, is divergent if, after any particular term, each sub- sequent term is greater than the corresponding term of a series all of whose terms are positive and which is known to be divergent. This follows immediately from the preceding demon- stration by taking the multipliers A, B, C, etc., any positive numbers greater than unity. 579. Theorem IX. A series, all of whose terms are positive, is divergent if, after any particular term, each of its subseque7it terms bears a finite ratio to the corresponding tervi of a series all of whose terms are positive and which is known to be divergent. This is an immediate consequence of theorem VII. For after the / th term of each series the ratios of the terms of the second series to those of the first series are A , B^ C, etc., which, by supposition, are finite numbers. 580. Theorem X. If, after any particular term of a series all of whose terms are positive, the ratio of each term to the preceding is less than some fixed number which is itself less than unity the series is absolutely convergent. CONVERGENCE AND DIVERGENCE. 409 Let the series all of whose terms are positive be repre- sented by and suppose that after the n th term the ratio of each term to the preceding is less than k where k is some fixed number less than unity. We then have !^!f±i<;^, "^^O^, '^^<^, etc. From these inequalities we readily obtain ^u+ 2 "^ ^^^H+ 1 • * • ^«+ 2 *^ ^ ^ ^«' Thus we see that, after the n th term, each term of the series 7^l4-^^2+ • • • -{-^^n + ^^n+i+^u+2+ ■ • • is less than the corresponding term of the series But this last series is easily seen to be convergent. For when k<,l the series is convergent and its sum is j— -- (see first example tinder Art. 563). Therefore the series is convergent and its sum is t^H • Therefore, the series ^i+?^2+ • • • 4-^„+^^^«+/^2^n+ • • . is convergent. Therefore, by theorem V, the given series u^-^U2-\-u^-\- • . . +^^«+^^«+l4-^^„+2+ • • • is convergent, and, having all its terms positive, is abso- lutely convergent. 4.IO UNIVERSITY ALGEBRA. 581. Theorem XI. If ^ after any particular term of an infinite series all of whose terms are positive, the ratio of mch term to the preceding is greater than some fixed num- ber which is itself greater than unity, the series is divergent. Representing by k some fixed number greater than unity and otherwise adopting the notation of the preced- ing article, we have ^«+l^^ ^«+2 >k, ''^+^»^, etc. U^ ^ ^' z^„+i ^■^n+2 From these inequalities we readily obtain u,,^^>ku„. Z^,+ 2>>^^«+l .-. u„^^>k''-u„. ^«+ 3 ^ ^^«+ 2 .'. u,+ s>k^u„. Thus we see that after the n th term each term of the series U^-^U2+ ' • • +«n + w«+i+ • • • is greater than the corresponding term of the series u^+U2-\- • • • +u„-{-ku,,-\-k'^u„-i- . . . But this last is easily seen to be a divergent series. Por when >^> 1 the series 1+^+^2 +>^*+- • - is evidently divergent. Therefore, the series Un-\-kUn-\-k'^Un+k'^Un+ • • • is divergent. Therefore, the series 2^1+2^2 + ^3+ • • • -\-Un'\'kUn-\-k'^U^-\- ' . • is divergent. Therefore, by Art. 678, the series ^1+^2+^8+ • • * +2^«+2^n+l+ • • • is divergent. CONVERGENCE AND DIVERGENCE. 4II 582. Theorem XII. In an infinite series all of whose terms are positive if, as n increases indefinitely^ the ratio -^^-^ approaches a definite limit a, the series is absolutely convergent if a r^^«+2 . •. ^«+2 > ;rV2 . Vn-\- 2 ''w Now, since it is given that the series is divergent, therefore by theorem VII the series u u u is divergent, therefore the series is divergent, therefore the series u u %c Vn ^n *^n is divergent, therefore by theorem VIII the series Is divergent. 587. In all the discussion of series given thus far it has been assumed that the terms of a series are arranged in a given order and this order has not in any case yet CONVERGENCE AND DIVERGENCE. 415 given been changed at all. As a matter of fact, however, the order 01 the terms of a series is sometimes quite important, for it has been shown by Lejeune-Dirichlet that, for a certain class of series, a change in the order of terms may change the sum of a series or may even change the character from semi-convergent to divergent. 588. To show the effect of changing the order of terms of a semi-convergent series, we shall take the series 1 l_Ll l_Ll 1_L and show that it is semi-convergent and therefore has a definite sum. Then we shall show that when the terms are written in the order i+i-i+i+4-i+ • • • the sum is not the same as before. 589. Let us now study the character of the series i-i+i-i+i-i+ • • • (1) We may write 1_^ 1_1 1_ 1 1_ 1 2~1 .2' 343.4' 5 6"~5 . 6' ^*'^- and thus the given series (1) may be replaced by the series ^ + J^ + J_ + ^g_+ . . . (2) But the terms of the series (2) are less than the corres- ponding terms of the series 1 , JL_ 4. _i_ . _J__ , which has already been shown to be convergent. (See second example under Art. 563.) Now since (3) is convergent, therefore by theorem V (2) is also convergent, and, since (2) is made by simply grouping the terms of (1) in sets of two, therefore (1) is convergent. 4l6 UNIVERSITY ALGEBRA. If all the terms of (1) are raade positive, we get the harmc ^ series which has been shown to be divergent. Therefore (1) is a convergent series which becomes divergent when all terms are made positive, /. e,, (1) is a semi-convergent series. 590. Since the series i-*+i-i+i-i+ • • • (1) is semi-convergent it has a definite sum which we may represent by 5*. Let us represent the sum of the series l+i-i+i+|-i+ • • • (2) by 5', and if we can prove that 5' has a definite value it will follow that the series (2) is convergent, and if this definite value is different from 6* it will follow that a change in the order of terms has an effect on the sum of the series. Now let us subtract the series (1) from the series (2), arranging the work in such a way that the order of terms in each series shall remain unchanged, but at the same time certain gaps shall be left in series (2) to allow the terms having odd denominators in the series (1) to come immediately under the terms of (2) which have the same denominators, and letting terms having even denom- inators come where they will. We have, then, ... 5'-5= \ -\ +i -i +... By inspection we see that the right-hand member of the equation below the line is exactly one-half of the series (1), from which it is evident that Hence, ^'=f^. CONVERGENCE AND DIVERGENCE. 417 Thus we see that S^ has a definite value differe"^ from S, from which we conclude that the series (2) is conver- gent, and that the change in the or-^^-^ of the terms of (1) has changed the sum of the series. 591. Theorem XVI. In an absolutely convei^gent series the order of the terms may be changed in any manner with- out affecting the sum or the character of the series, provided each term, whose positio7i is changed is removed only a finite number of steps from the position it originally occupied. Let any absolutely convergent series be represented by u^-{-u^-\-u^-\- ... (1) and let the series obtained by making all the terms of (1) positive be represented by v^-{-v^+v^Ar ... (2) and finally, let the series obtained by changing the order of terms of (1) be represented by w^-k-w^^rw^-k- . . ^ (3) Let Wr represent the sum of the first r terms of (3) and Un the sum of the first n terms of (1). Plainly, then, we may take r so great that Wr will include the sum of the first n terms of (1) and the sum of r—n other terms of fl) after the first n terms. Plainly, also, we may take p to represent a number so large that the r—n other terms of (1) will be found scattered around between the terms Un and UnJ>^p. Hence, the sum of these r—n other terms cannot be greater than Hence, ^r— ^«>*^«+^«+i+z'„+2+ ■ • • +z^„+^ *The symbol ^ is used to express the fact that the expression written on the left of the symbol is not greater than the expression written on the right. Similarly, the symbol 5C shows that the expres- sion on the left is not less than the expression on the right. 27~U. A. 41 8 UNIVERSITY ALGEBRA. Now, if we let n and therefore r increase indefinitely, the expression to the right of the sign > approaches zero and therefore Wr— U^ also approaches zero. Hence, limit ( Wr— ^«)=0. Hence, limit ^^— limit 17„=0, Hence, limit W^^= limit 17„. Hence, the sum of the series (3) equals the sum of the series (1). Therefore, changing the order of terms has not changed the sum. ' If now we represent by 2-1+^2+^8+- • • (4) the series obtained by making all the terms of (3) positive, then plainly (4) is obtained from (2) in exactly the same way that (3) is from (1). Hence, by the demonstration just given, the sum of the series (4) is the same as the sum of the series (2). Therefore, (4) is a convergent series all of whose terms are positive, and since (4) is the series obtained by making all the terms of (3) posi- tive and since (4) is convergent, therefore (3) is an absolutely convergent series. Therefore, a change in the , order of the terms Of (1) has not changed either the sum • or the character of the series. 592. Theorem XVII. If the terms of a series are alternately positive and negative and after some particular term, each term is numerically less than the preceding one^ and the nth term approaches zero as n increases indefinitely ^ the series is convergent. Let the series be and let the sum of the series be represented by S\ then, with the notation of Art. 560, we may write either 6'=5^+(a^^-fi-^^+2) + K+8-^^+4)+ • • (1) or ^= Vi-K+2-^^+3)-K+4-«i^+5) C2) CONVERGENCE AND DIVERGENCE. 419 After a certain number of terms, say k, each term is less than the preceding one, so if q be chosen larger than k, each parenthesis in (1) and also in (2) is positive, and therefore from (1) S^Sq, and from (2) S<^Sq^^. Thus we see that kS* is intermediate in value between Sq and Sqj^^, which two quantities differ by ^^+1. Similarly, whatever positive whole number be repre- sented by Ty we get ►S>6'^+2^ and ►S' . .) =Aq-\-A^x+A2X^+ . . . +^«_l^''-l+i?l^^ where ^j^ is of course the difference between the limit of the sum as the number of terms increases without limit and the actual sum of the first n terms. AQ.A^y' . 'A„_-^ are constants, buti?i is not a constant, for if it were the series would terminate. In fact jRj^x** approaches zero as n increases, for if it did not the series would not be convergent. An inspection of the series shows that every term after the first contains x, every term after the second contains x^^ every term after the 428 UNIVERSITY ALGEBRA. third contains x^, and so on; hence every term after the nth will contain the factor ^", and hence it is natural to assume the remainder after n terms are written to be of the/orm R^x*". Instead of writing limit of Aq+A^x+ - • -as the num- ber of terms increases without limit, we write Aq+A^x-j-A'2X^+ . . -A„_-^x''-^+R^x'' and in the same way we write Bo-hB^x+B^x^+ . . ■i-B^.^x^-^+R^x' instead of writing limit of Bq-\-Bj^x+B2x'^+ . . -as the number of terms increases without limit. Using this notation we may write Ao+A^x+ . . . -{-A„_^x»-''+R,x^ =^0 +^i^+ ■ • • +B^_^x^-^ +i?2^. (1) If now we consider x as sl variable approaching zero we have here two variables which are always equal, and therefore by Art. 420, their limits are equal. By Art. 598 the limit of the left hand member equals Aq and the limit of the right hand member equals Bq; hence Aq=Bq. Subtracting Aq from the left hand member and Bq from the right hand member of (1), we get A^x-i-A2X^+^ . .+^1-^" =B,x+B^x''+ . . ■ +i?2-^ (2) Dividing both members of (2) by x, we get A^+A2X+ . . . +A„_^x^-''-^jR,x''-^ ==B, +B^x+ . . . -{-B^x^'-K (3) As before, we have two variables always equal, hence their limits are equal. But as X approaches zero the limit of the left hand member equals A ^ and the limit of the right hand member equals B^. Hence, by Art. 420, A,=B,. UNDETERMINED COEFFICIENTS. 429 Repeating the reasoning, we may show successively that -^Z — ^Z) etc. 600. The theorem of the last article enables us to change the form of a function. The method of doing this consists in assuming a func- tion of the required form with unknown coefficients and then determining the coefficients so that the function assumed shall be identical with the function proposed. The unknown coefficients are determined by placing the proposed function equal to the assumed function, reducing to the rational integral form, and equating the coefficients of like powers of the variable on the two sides of the equation. If the proposed function can be placed in the assumed form it will be found that there are as many independent compatible equations as there are unknown numbers to determine. 601. A function is said to be Developed or Expanded when it is expressed in the form of a series, the sum of whose terms when the number of terms of the series is limited, and the limit of the sum when the number of terms is unlimited, equals the given function. 602. The development! of functions is one of the most common applications of the method described in Art. 600. The process is usually referred to as the Method of Undetermined Coefficients. We will illustrate the method by working an example. Let us develop the fraction zj— ^ j— 2 ' 430 UNIVERSITY ALGEBRA. ALSSume Multiplying both sides of (1) by 1— 3;ir+4;«r2 we obtain 2+Zx=A^ + (Ai-ZA^')x+iA^-ZA^+4:A^-)x^ + (As-SA^+4A,)x»+. .. + CJ?-SA„^^+4A„_2)x" + (-SJi+4A^.^-)x"+^+4J?x"+-'. We see that in the left hand member the coefficient of each power of x, beyond the first power, is zero. Hence, equating coefficients, we get ^0=2. ^^-3^0=3. •. ^1 = 3+3^0- ^2-3^1+4^0=0. . ,-. ^2=3^1-4^0, ^3—3^2+4^1=0. , .-. ^3=3^2-4^1 etc. etc. From these equations the law of the series is so evident that we can write as many more equations as we please without further calculation. From the second column of equations it is evident that each coefficient after A ^ equals three times the preceding one minus four times the second preceding one. Now since Aq=2 and ^1=3 + 3^0=9, we may substitute these values in (1) and determine other coefficients by the law just stated. Hence, we obtain ^ ^ + ^-^^ ^ -=2+9;ir+19;i;2+21.y«-13.a:^-123;i;g-317^«... As we usually determine only a few of the coefficients, and then discover if we can the law of the series, so it is usual in the assumed series with undetermined coefficients, to write only a few terms and indicate the others includ- ing the remainder by dots, thus : ^■^^^ .^A^+A^x+A^x^+A^x^+ . . . Instead of using the method of undetermined coeffi- op^!. ^ Develop il=^. 604. Not only fractions but some irrational expres- sions may be developed by the method of undetermined coefficients. 28 — U. A. 434 UNIVERSITY ALGEBRA. I^t US develop Vl—x. Assume Squaring each side, we get l—x=-Ao^ + 2AoA^x+(2A^A^+Ai^)x' + (2A^At + 2AiA^-)x'^ + (2A^A^+2AiA^+A,'')x* + (2AoA, + 2A^A^+2A^As)x^+ . ■ Equating coefficients of like powers of x, we get ^0 = 1. 2A,A, = -1, 2A^A^+A^^=0, 2^0^4 + 2^1-48+^2^=0, 2^0^5 + 2^1^4+2^2^8=0, 2A,A,+2A^A^+2A^Ai+Ai^'='0, etc. From these we get A,=l, A, = - A=- 2^0 A,^ ^8=- A^=- A,=- 2Ao 2A,A^ ' 2A, ' 2A^A^+Aj^ 2A, ' 2AiAi + 2A,Ag 2A, A, = - 2A,As + 2A^A^+At* 2^0 etc. From these the law of the series can be seen. UNDETERMINED COEFFICIENTS. 435 Taking these equations in order, we find the numerical values of the undetermined coefficients to be as follows : A 7 A 21 ^6 — ^S'B"? -^6— 1024* Making these substitutions in the assumed development, we obtain X x^ x^ 5x^ ' 7x^ 21;r« ^ "^^ 2""'8"""l6~128"~256""i024" KXAMPI^KS. VX^ / 1 1 \T- l + 1x—-^' 4. Develop (-gH h^j 2. Develop l/^T^- 5- Develop (1+^x'')^ 3. Develop (1 + ^)^. 6. Develop ;t:(l+:ir+ ;i;2)ir- 605. It is interesting to note that the development of an irrational expression may turn out to be a series of a limited number of terms. Suppose, for example, we wish to develop Vl—2x+x^ and do not recognize that l—2x+x'^ is a perfect square, then assume as before V1—2x-^x^=Aq+A^x+A^x'^+^ . . Squaring both sides, we obtain l-'lx+x^=^A^^ + 2A^A^x+(,A,'' + 2A^A^)x^ + (2AoA^+2A,A^')x^. Equating coefficients of like powers of x, we get Aq =1, .*. AQ = ly 2AoA, 2.'.A^ = -1, ^1^+2^0-42 = 1 •••^2=0, 2A^A^+2A^A^=0.-.A^=0, Ai-'+2A^A^+2A,A^=0 .: A^^O, etc., etc., 436 UNIVERSITY ALGEBRA. and each, of the subsequent coeflScients will turn out to be zero, hence we get 606. In developing irrational expressions it sometimes happens that we should deg-zn our assumed developments with some negative power of x. An inspection of the proposed example will show with what power of x the development should begin ; for the assumed series must be such that, when the equation obtained by putting the given function equal to the assumed series is reduced to the rational integral form, then the lowest power of x on the side which contains the undetermined coefficients must be as low as the lowest power on the other side of the equation. Thus, to develop -i/lH — ^ we begin the assumed series with a term containing ^"^ , for when this is squared the lowest power of x is x~^ and when both sides are multiplied by x^ to reduce to the integral form then the series on the right side of the equation will begin with an absolute term as it should. 607. If we wish to develop the algebraic sum of two or more radicals it is best to develop each one by itself and then find the algebraic sum of the results. KXAMPI^KS. I. Develop yi~-2. ^^+j_+T/r 2. Develop -i/^+-2 + 1/1+^^. 3. Develop Vl+4x+6x^ + 6x^+5x^ + 2x^+x\ UNDETERMINED COEFFICIENTS. 43/ 5. Develop ^^-^)^~^,. 6. Develop (;«r2+Ar-'2)T+(2;tr2+:r-2)T 7. Develop (l+;r)2+jq- _1 8. Develop i •^■^ — 2" X 9. Develop 1/1 + 2jr+3;r2+2:r3+jt-*. 10. Develop (^r^— 4^7+4^6+e^4_i2^3+9)| CHAPTER XXVI. SUMMATION OF SKRIKS. 608. No general method can be given for finding the sum of series which will be applicable to series of all kinds, but in certain cases the sum may be found and it is the object of this chapter to explain how to proceed in some of these cases. SKRIKS REDUCIBI^B TO THE) FORM 609. In Art. 563 it was shown that an infinite series is convergent when it is capable of being expressed in the form u^'-U2-\-U2—u^+Uq—u^+ ... (1) provided only that limit tc„=0 as n increases indefinitely. But since the sum of the series (1) is u^ it follows that we can find the sum of any infinite series which is reducible to this form. For example, the series ' ■ ' +^^^+.-- (2) xiix+l) ' (x+lXx+2) ■ (^+2)(^+3) may be expresed in the form (1) The nth term of (2) is lx+(n~l)']\_x-hn'] which is easily seen to be equal to 1 1__ x+(n—l) x+n SUMMATION OF SERIES. 439 Now, as the series (2) is obtained by taking all the terms obtained by giving to n in the expression .= — -7 3-r=rT — ; — ^ all positive integral values, it follows that a series equivalent to (2) may be obtained by giving . . . 1 1 ,, . . . n\VL the expression —— r^ -— - all positive integral values. Hence, (2) may be written By the form of (3) it is evident that the sum is — > and 1 "^ hence the sum of the series (2) is also — X 610. It is well to notice that the sum of a limited number of terms of a series reducible to the form (1) may be easily obtained. For example, the sum n terms of the series (2) is evi- dently the sum oil n terms the series (3); /. =1 .-. D=\, 3Z>+2C=2 .-. C=^. D^C^-B=1 .-. ^=|. Substituting the values here found in equation (1), we have 12+22+32+42=^+1+^+^. Since this equation is to hold for. all values of n, we may determine A by assigning to 7i any value we please. Making ;?=!, we obtain Therefore ^4=0, and we get as our final result 2 n n" ^ n 12+22+32+ . . +^2=-+-_+:__ ^n^ -^Zn'' -{ -n _ n{n + \X^n + r) 6 " ■" 6 KXA.MPLKS. 1. Show that 12 + 32+52+. . .+(2;^— 1)2 3 2. Show that 22+42+62+ ... +(2;^)2 _ 2;^(7^ + l)(2;^ + l) 3 3. Show that 18+23+33+ |-;^3^^y(^ + l)l^ 4. Show that 13 + 33 + 53 + . .. + (2;2—l)3=^2(2^2__x) 5. Show that 23 +43 + 63+. . . +(2;^)3 = 2;^2(^ + l)2^ Show that 1.2 + 3.4 + 5.6+ . . . +(2;^— 1)(2;^) _ ;^(;^ + l)(2;^~l) 3 6 442 UNIVERSITY ALGEBRA. METHOD OF DIFFKRKNCES. 612. If, in any series, each term be subtracted from the succeeding term, the various remainders form a new- series called the First Order of Differences of the given series. If, in this new series, each term be subtracted from the succeeding term, the various remainders form still another series called the Second Order of Differences of the given series. In the same manner from this last series a new series may be formed called the Third Order of Differences of the given series, and so on. For example, if the given series be 6, 11, 23, 45, 80, 131, 201, we have the following series : Given series: 6, 11, 23, 45, 80, 131, 201; 1st order of differences: 5, 12, 22, 35, 51, 70; 2d order of differences :(:■ 7, 10, 13, 16, 19; 3d order of differences: 3, 3, 3, 3; 4th order of differences : 0, 0, 0. 613. The method of finding any term of a given series, or the sum of any number of terms, by means of its successive orders of differences, is called the Method of Differences. 614. To find any term of a given series, Let the terms of the given series be represented by ^1, z^2> ^3' ^4. etc. Forming the successive orders of differences, we have 1st order z^ 2 — ^u ^3 — ^2j ^4"~^3> ^5 — ^4* etc. 2d order 2^8—22^2 + ^^1' u^ — ^u^-^-Uc^, u^—2u^+u^, etc. 3d order 2^4 — 3^3 + 02^2 — z^i, 2^5— 3^^4^-3^^3— z^2> etc. 4th order u^— 4:21^ -\-6u.^—4iU 2 + u^y etc. SUMMATION OF SERIES. 445 Let the first terms of these successive orders of differ- ences be represented by d^, d^, d^, d^, etc., then d^ = U2 — ^1. (^2 = ^3 — 2z^2+^l- = 2^1+2^1+^2- .*. 2^4 = 2^1— 32^2 +^^^3 +^3- = 2^1-3(^^1 +^l) + 3(ZiJl +2^1 +<)+^3. =xZ^l+3^1+3^2+^3. We notice in the value of 2^3 the numerical coefficients are the same as in the expansion of (a + xy and in the value of the u^^ the numerical coefficients are the same as in the expansion of (a+;r)^, and from this we are led to think that probably the same law holds good generally ; that is, that ^ (.:.l)(.-2)(.-3) ^^_^ _^ (1) 615. We will now prove by mathematical induction that this law is true generally. Let us' assume for the moment that this law holds for the ^th term of any series, then it must hold for the first order of differences of that series. Writing down the :^th term of the first order of differences by this law, remem- bering that the nth term equals u„+^—u„, and that d^j ^2, <^3, etc., for the first order of differences correspond to u^y d^, d^, etc., for the given series, we have I {n-l){n-2-)(n-^) T IQ "4+ •• 444 UNIVERSITY ALGEBRA. Adding this equation to the preceding, we obtain , f in-lXn-2Xn-3) , (n-iyn- 2)\_^ +[ [3 + [2 /»+••• Hence, «^i=»«i+«^iH — ^-r^-d^ ^x.1\x^—^x'^.Vdx'^, or -123;tr5 = 3^(-13;f4)-4;tr2.21;r8, or any other equation obtained from the given series by the same law as that used in obtaining the equations here written. Of course if, in any of these equations, all the terms were transposed to one member the resulting equa- tion would be called the identical relation. If all the terms of the identical relation here written be transposed to the left member and if we then write the sum of the coefl5cients of those terms of the series, which appear in the identical relation, we obtain the scale of relation 1—3.^+4^2 no matter which of the above equations is taken for the identical relation. SUMMATION OF SERIES. 447 621. To find the scale of relation of a recurring series. If the series is of the first order its scale of relation contains two terms ; if of the second order the scale of relation contains three terms ; if of the ;'th order its scale of relation contains r-\- 1 terms. The identical relation of course contains the same number of terms as the scale of relation. Let the terms of the series considered be represented t>y ^i> ^2) ^3' ^t^M ^^^ l^t ^^ suppose first that the series is of the first order, then the identical relation is plainly 2/«+M.-i=0 (1) where p is some coefficient at present unknown. The scale of relation in this case is, by definition, zc From equation (1) it follows th.2Ltp= — and hence the scale of relation is 1 - Next, let us suppose the series is of the second order, then the identical relation is u„-j-pu„_^+gu„_2=0 ' (2) and the scale of relation in this case is, by definition, 1+p + q. From the identical relation we have u„-^pu„^^-\-gu„_2=^y • (2) and u„^ 1 +pu,, -f qti„_ ^ = 0. (3) Equations (2) and (3) are sufficient to enable us to find values ofp and g and when found the scale of relation is obtained by substituting these values in the expression 1+p-^g. Similarly, if the series is of the third order we have three equations from the identical relation, viz: u„+pu„_j^ +gu„_2+ru^_j^=0 ««+ 1 -^P^n + qun^ J + ru^- 2 = ««+ 2 +j^^«+ 1 + ^«« + ^/^«- 1 = 44^ UNIVERSITY ALGEBRA. and from these three equations we can find the values of p, q, r, which substituted in the expression give the scale of relation. In a similar manner, if the series is of the rth order, the scale of relation may be written down containing r unknown numbers and the identical relation gives us r equations from which to determine these f unknown numbers. 622. If a series is known to be recurring but its order is unknown we first make trial of a scale \-\-p and by- applying this scale to two consecutive terms of the series we write down the identical relation and then applying the scale to two other consecutive terms we write down the identical relation again, and so on as many times as we please. From any one of the identical relations thus written we determine the value of p and if the value thus found satisfies all the other equations written, the true scale of relation is obtained by writing this value in place of p in the expression 1 +/. If the value of p determined as just explained does not satisfy the other equations we have not found the true scale of relation of the series and we assume another scale, viz: 1+^^+^, and applying this to three consecu- tive terms of the series we write the identical relation, and then applying the scale to three other consecutive terms of the series we write the identical relation again, and so on as many times as we please. From any two of these equations we may determine p and q, and if the values thus found satisfy all the other equations written, the true scale of relation is obtained by substituting the values of;i> and q thus found in the expression \'\-p-\-q. If the values of p and q found as just explained do not SUMMATION OF SERIES. 449 satisfy all the subsequent equations we have not yet found the scale of relation of the series and must assume another scale l+p-\-q+r and proceed in a similar man- ner and so on until finally a scale is found which leads to no inconsistencies. KXAMPi^KS. 1. Find the scale of relation of the series Assuming the scale of relation of the series to be 1-1-/, we have From the first of these equations /= — _, but this value of p does 2 not satisfy either of the other equations. Hence the scale of relation is not of the form 1 -f /. Assuming the scale of relation to be of the form 1 -\-p -\- q, we have — 3a;5 ^px^ + 2qx^ =0 2x6— 3/;«:5 + ^^4— %7 + 2/^6-3^^s=0. From the first two of these equations we find p^=x, q—x^, and as these values of / and q satisfy the other two equations we conclude that 1 + ;*: 4-^:2 is the scale of relation of the given series. 2. Find the scale of relation of the series x—x^-i-x^—x'^-{-x^—x^^-\- . . . 3. Prove that 2+4x+14:X^ + A6x^ + 152x^+ ... is a recurring series of the second order. 4. Prove that l + 6x-lbx^ + 57^'-159x^ + 48dx^-'U55x^+ . . . is a recurring series of the second order. 623. To find the sum of a Recurring Series. The method of finding the sum of a recurring series will be clear if we give the explanation for a series of the second order. 29 — u. A. 450 UNIVERSITY ALGEBRA. Let Ui, «2» ^3' ^4' ^^c., represent the terms of the series and let Sn represent the sum of the first n terms, and if the series be an infinite converging series, let kS represent the sum of the series, and let 1+p+gr represent the scale of relation, then we have pSn== pu^ ■\-pu^ -\-pu.^-^ . . . +pu„_^ ^-pUn + (u„-\-pUn-^ ■\-qUn-.'2) + {^pUn-\-qUn-^ + qu^. As the terms here grouped (each expression in a paren- thesis being considered as one term) it is evident that the identical relation makes each term of the second member except the first two and the last two, equal to zero. Hence, Sn-\-pSn+qSn=u^+{u^+pUy)^(^pu,,+qu„_^)-{-qu„ . ^ _ (ui+u^+pu^) + {pu„+qu„_^-{-q u,;) 1+P^-q If the series proposed is an infinite convergent series the expression {pu,,-{-qu„_^+qu^ approaches the limit zero as n increases indefinitely, and therefore for an infinite converging series, we have u^+u^+pu ^ 1+p+q 624. In case of a recurring series arranged according to increasing powers of x^ as for example the series a^+a^x+a<2,x'^-\-a^x^+ • • ., the two formulas of the last article become _ (aQ-^a^x+paQx)-\-x''{pan_^+qa„^o,-^qan-i X) ^'^ 1+px+qx'' a^+a^x+pa^x l^px-\-qx^ SUMMATION OF SERIES. 45 1 Now it must be remembered that the expression 1+px+gx^ is the sum of the series a^-j-a^x+a^x^ -\-a^x^ + . . . when and only when this series is convergent. But if the fraction aQ-\-a^x-{-paQX 1+px+qx'^ be developed according to ascending powers of x that development will be the series aQ—a-^x+a2x'^-\-a^x^+ • • . whether this series is convergent or not. For this reason the fraction aQ + a^x-\-paQX l-j-px+qx^ is called the Generating Function of the series aQ+a^x-\-a2x'^-^a^x^+ • . • Thus we see that the generating function of a recur- ring series is the sum of the series only when the series is convergent. KXAMPI^ES. 1. Find the sum of six terms of the series l+2x+Sx'^+4:X^+bx^+. . . 2. Find the generating function of the series in ex- ample 1. 3. Find the generating function of the series l+x^-x^^^+2x^Sx^-i-5x^Sx'^+ . . . 4. Find the generating function of the series 5. Find the generating function of the series l + 2x^-2x'^+4x^-6x^ + 10x^—16x'^-i- • . . 452 UNIVERSITY ALGEBRA. 6. Find the generating function of the series . 7. Find the generating function of the series 8. Find the generating function of the series i + i^-|^2+|3^3_2 9^4+ . . . 9. Find the generating function of the series 10. Find the generating function of the series l+x+2x^ +^x^ +bx^ + ^x^ + . . . CHAPTER XXVII. BINOMIAI. THEOREM FOR FRACTIONAI. AND NEGATIVE EXPONENTS. 625. In chapter XXII it was shown that when n is any positive integer we have n{7i — V) „ . n(n — ])(^^— -2) _ , (l+;^)«=l+;^Jr+-^-|^^;^:2 + -^ )^ L^z j^ . . . We shall prove in this chapter that the same equation holds for certain values of x when n is any negative integer or any positive or negative fraction. 626. Exponent a Negative Integer. By division, we get 1+x 1+x If X is intermediate between +1 and —1, that is, if l>;r> — 1, then as n increases indefinitely the remainder approaches zero, and we may write j^-=l-x+x^-x^+.... (1) If !>;»;> — 1 this series is absolutely convergent. We may, therefore, by Art. 593, raise both members to any power by multiplication without liability of error. We get = l—2x-\-Sx'^'-4:X^+ ... (2) = l~3;r+6;i:2_ 10:^3+ ... (3) 454 UNIVERSITY ALGEBRA. The products (2) and (3) may be expressed as follows: -2(-2-l>2 (l^x)-^ = l + C-2)x- L2 2— nr — 9— 9> -2(-2-l)(-2-2) _^ -3(-3-g(-3-2) ^3^^^^ (4) We see from this that when the exponent ;e= — 2 or —3 the law of coefficients is the same as when n is sl positive integer. Does this law hold for all negative integral values of the exponent ? Suppose it holds n=—r, then we have (l+^)-''=l+/i^+/>2-^'+ • • • +/X+ • • • (5) where /i = — ^, ^2 = To ' ^^^ ^^ ^^* Multiply the [^ members of equation (5) by the members of (1), and we get + (A-A-i+- ..+1K+... (6) But A-1^ r^U, A"-/i + l= ^~'^""^|2""'' ~' Thus the coefficients of x and ^ir^ in (6) are seen to be the regular binomial coefficients. Is this true of all the coefficients in (6)? Suppose the law is true for the coefficient of x^~'^. Then that coefficient will be (-r~l)(~r~2). . .(~r-^+l) ^-1 Observe that the co- efficient of x^ in (5), minus the coefficient of x^ ^ in (6), BINOMIAL THEOREM. ANY EXPONENT. 455 equals the coeflScient of x" in (6). Hence the coefficient of x' in (jo) is equal to -^(_^_l)..(-^-5+l) (-r-lX-r-2)..(-r-s+l-) s-1 (-r-l)(-r-2)..(-r-^) = j > s which is the regular form for the coefficient of x^ in the binomial formula. Hence, if the coefficient of x^~'^ obeys the law, then does also that of x^. But the coefficient of x^ does it; therefore, that of x^ does it, and so on. We see, therefore, that the law of coefficients holds for all the terms in (5), and, therefore, holds when the expo- nent n= — r—l if it holds for n= — r. But it has been shown above to hold for n= — S; therefore, it holds for n^—4, and so on, for any negative exponent. Thus, we have found the formula (l+xr=l + nx-h^^^^x^+ . . . If when n is any negative integral exponent, provided that 627. W^hen the Exponent is Fractional. I^et P n=~^ p and q being integers, and q positive. Consider the series + • • • (1) This series is seen to be absolutely convergent when l>j;>— 1; itisso also when .ar=±l and ->0, but it is 4S6 UNIVERSITY ALGEBRA. somewliat difficult to prove this. When the series is absolutely convergent we may multiply it by itself. If we square it, we get q \l f (f-o- ^ (f-O ^^^ ' -^ -^'■+... (2) f(?-)-(?-0 + -^ ^ ^ ^^^ - " ^-+--- (3) Observe that in (2) and (3) the numerical coefficient of p is equal to the exponent of Y. Is this generally true? If it is true when the coefficient is 5, then we can show that it is true for (^+1). For, multiply the members of q [2 ^q\q ) \q 1^^ ^^^ by the members of (1), and get -f 1^ ^ + . . . But, since the coefficient of p is the same as the expo- nent of y^ when we aibe the members of (1) it is when we BINOMIAT. THEOREM. ANY EXPONENT. 457 raise them to the fourth power, and so on. Hence, to raise the series (1) to any power, a, we need only substi- tute ap for p. Raise the members of (1) to the q th power, then the denominators of the fractions disappear, and we have \r But we know that (l+^y=l+/-^+^^^5 — ^x'^^-^ .. Hence, Y is one of the ^th roots of {X-^xy. That is, y=(l+:r)^. It follows, therefore, from (1) that -(--1) q |Z provided that l>;t->— 1 or that Ji;=dbl and — >0. 628. To expand any power of any binomial, say i^+yTy we write and then expand (1 + -J as above. KXAMPIvKS. I. (1-^)^ 5. (2+^r» 2. (1-3ji:)-3 6. (l-:r2)-s 3. (l^Sxy 7. (1 + 2^)"^ 4. (1—4;^;)"^ 8. (Aa-Sx)-^ 9. Show that only two terms in the expansion of (1—^)"^ have a positive sign. 45 8 UNIVERSITY ALGEBRA. 10. The coefficients of the third term in (l—x)^** is -J. Find n and the coefficient of the fifth term. II. Find the rth term of (-5)-' 12. Find the coefficient of :r^ in the expansion of 13. Ifx is very small, show that 1—^x is an approxi- mate value of ^- ' l + ;t:+l/l— :i; CHAPTER XXVIII. CONTINUED FRACTIONS. 629. Continued fractions have already been mentioned in Art. 179, example 23, but in this chapter we desire to study this kind of fractions further and to find out some of their principal properties. 630. Starting with any given positive number, say n, and representing the integral part of this number by a^ we evidently have n=a-\ — wherein x^l, • In the same way, representing the integral part of x by b, we have x=b-\ — » y wherein j^/>l. Similarly we have y=c+—y u Replacing x^y^ z- - - by their values, we have '+7+., From the way in which this expression i? obtained it is evident that a may be a positive whole number or zero, but b, Cy d' ' ' must be positive whole numbers at least as great as unity. 460 UNIVERSITY ALGEBRA. 631. Two cases of continued fractions may present themselves: First, one of the numbers x^ y, z,- - - of Art. 630, may be an integer, in which case the continued fraction termi- nates and we have the exact expression for the number n in the form of a continued fraction. Second, it may be that, however far the above operation is carried, none of the numbers jr, j/, ^, • . . are integers. In this case the continued fraction will never terminate, but if one of the fractions— ' — > — > ... of Art. 630, be X y z neglected the resulting fraction will be an approximate value of the number n, and the approximation will be closer and closer as the neglected fraction is farther and farther removed from the beginning. 632. The successive approximations found as just described are called the successive Convergents of the continued fraction. In the above continued fraction a, the integral part of n, found by neglecting the fraction-, is called the First \ X 1 Convergent; « + 7> found by neglecting the fraction -» o 1 ^ is called the Second Convergent ; a-\ r, found by 1 ^ neglecting the fraction -, is called the Third Conver- z gent, and so on. 633. It is well to notice that when a number is expressed as a continued fraction, the first approximate value of that number is not necessarily the first conver- gent. For example, if we express .29 as a continued fraction, we readily find CONTINUED FRACTIONS. 461 .29=^ 2+i In this case we would naturally call 5 the first approx- imation to the number .29, but in this case the first convergent is 0; z, e., the integral part of .29, but the second convergent is ^• It will now be readily seen that in case of any positive number less than unity, the first convergent is 0. 634. The expression of any negative number as a con- tinued fraction will be readily understood from a special case: -1.54=~2 + .46=~24-— -j 2 + ^ 5+^ Evidently any negative number may be treated in a manner similar to that in which —1.54 is treated in this illustration. Hence the expression of any negative number as a continued fraction may be made to depend upon the expression of some positive number as a continued frac- tion, and it is therefore not necessary to consider negative numbers in our discussion of continued fractions. For this reason we shall, throughout the present chapter, consider all the numbers with which we deal as positive numbers unless the contrary is expressly stated.* 635. Theorem I. Every commensurable number corresponds to a contmued fraction which terminates and conversely, every continued fraction which terminates represe?its some commensurable number. 462 UNIVERSITY ALGEBRA. m Let — represent any commensurable number. Divide m hy n and let a represent the quotient and r the remainder. Then the integral part of — is a, and we have m r . \ ... n n fn\ ^ ^ Now divide n by r and let b represent the quotient and Ti the remainder. Then, evidently, r r / ; \ k^) (r,) Next, divide r by r^ and let c represent the quotient and ^2 the remainder. Then But from what has already been given, it is evident that the successive operations in this process are pre- cisely those in the process of finding the H. C. F. of m and n, and if these operations are continued long enough there will come a time when some division is exact (even in the case in which m and n are prime to each other, in which case the last divisor is unity) and the process comes to an end. If now, in the equation (1) we substute for — its value T ^ found in (2), and then for — its value found in (3), and so on, we get the commensurable number — expressed n as a continued fraction which terminates. That a continued fraction which terminates represents a commensurable number is at once evident when the indicated operations are performed, for in this case a CONTINUED FRACTIONS. 463 fraction is finally reached whose numerator and denomi- nator are whole numbers. To fix the ideas let us take a numerical example. Suppose we have the continued fraction 1 2 + 4+i ^4 2+^=f. Therefore the given fraction equals 4+^==V'- Therefore, the given fraction equals 2+ 1 -2+^-^^. 636. Theorem II. Every incommensurable number corresponds to an unlimited continued fraction and con- versely, every unlimited continued fraction represents some incommensurable number. The process of converting a number into a continued fraction (Art. 630) applies to an incommensurable as well as to a commensurable number, but if an incommensur- able number be converted into a continued fraction the fraction will not terminate, for if it did terminate it would represent a commensurable number, by theorem I. Again, as every commensurable number corresponds to a continued fraction which terminates ; therefore, a con- tinued fraction which does not terminate represents a number which is not commensurable ; /. ^., which is incommensurable. 4.64 UNIVERSITY ALGEBRA. 637. Theorem III. In every continued fraction the values of the convex gents are alternately less and greater than the value of the continued fraction itself ; the first, third, fifth, etc., convergents being less, and the second, fourth, sixth, etc,, convergents being greater than the con- tinued fraction. In Art. 630 we used the equations ^=^+^ (^>1) (1) ^=^+i (y>l) ■ (2> y-'-^\ (^>1) (3) z=d-v\ {u>V) (4> u and from these we obtained the equation n=a^ 1^- bV--^ (5) Evidently since - is positive a<^a-\ — , i.e., a<^; therefore --,-^, therefore — t<— » c + - 2 therefore, b-{ — — - is less than ^-| — ; c-\ — 2 therefore is greater than :: b+- c therefore a-\-- ^+- • 2 is greater than a H r c 30 — U. A. 466 UNIVERSITY ALGEBRA.^ But the first of these last two expressions is 7i\ i.e., the value of the continued fraction itself, and the second is the third convergent. Therefore, the third convergent is less than the value of the continued fraction itself. Similar reasoning carried to any convergent desired shows the truth of the theorem. Since the first, third, fifth, etc., convergents are each less than the value of the continued fraction itself, and the second, fourth, sixth, etc., convergents are each greater than the value of the continued fraction itself it follows that the contiyiued fraction is intermediate in value between any two successive convergents. 638. Theorem IV. In any contimied fractio7t any convergent is intermediate in value betweeji the two immedi- ately preceding convergents. Any convergent, say the rth convergent of any con- tinued fraction, is itself a continued fraction, all of whose convergents up to the ;th are exactly the same as the corresponding convergents of the given continued fraction, and by theorem III the value of this continued fraction is intermediate between any two of its successive conver- gents. Therefore, this continued fraction is intermediate between the (r— 2)d and the (r— l)st convergents, and therefore the rth convergent of any continued fraction is intermediate in value between its (r— 2)d and (r— l)st convergents as stated in the theorem. Illustration. The continued fraction ■ 1+^ 3+-^ (1) 4+ ^-\ .M CONTINUED FRACTIONS. 467 has for its fifth convergent 2+-^ (2) Now it is evident by inspection that the first, second, third, fourth, and fifth convergents of (1) are respectively the same as the first, second, third, fourth, and fifth con- vergents of (2) (the fifth convergent of (2) being (2) itself). Since by theorem III (2) is intermediate in value between its third and fourth convergents, therefore the fifth convergent of (1) is intermediate in value between its third and fourth convergents. By calculating the values of the convergents of (l),we readil}^ find 10 291G0 third convergent =--=^g^ f ,u ^43 29197 fourth convergent =3^= 20370 ... . 139 29190 fifth convergent =-- = ^^^ 639. Theorem V. Iti any continued fraction the odd convergents taken 171 order form an increasi7ig series and the even convergents taken in order forj?t a decreasing series. By theorem III the odd convergents are less, and the even convergents greater, than the continued fraction itself. Hence, any odd convergent is less than any even convergent. Now let us consider any odd convergent, say the (2r+l)st convergent. This is less than the 2/th con- vergent (since the 2rth is an eveji convergent), and being 468 UNIVERSITY ALGEBRA. by theorem IV intermedate between the 2rth and the (2r— l)st convergents it must be greater than the (2r— l)st convergent, and, since this is true for any value of r it follows that the odd convergents taken in order form an increasing series. Similarly the 2rth con- vergent (being an eve7i convergent) is greater than the (2r — l)st convergent, and, being intermediate between the (2r— l)st and (2r— 2)d convergents, is necessarily less than the (2r— 2)d convergent, and this being true for any value of r it follows that the even convergents taken in order form a decreasing series. 640. Since the even convergents are always greater than the continued fraction itself and taken in order form a decreasing series, therefore the successive even convergents approach nearer and nearer to the value of the continued fraction itself. Also, since the odd convergents are always less than the continued fraction itself, and, taken in order, form an increasing series, therefore the successive odd convergents approach nearer and nearer to the value of the continued fraction itself, 641. Law of Formation of the Convergents. In the equation n=a-\ 7 e+, ^ a, 5y c, d, etc., are called the first, second, third, fourth, etc.. Incomplete Quotients. By actual calculation the first four convergents are found to be as follows : CONTINUED FRACTIONS. 469 Theyfr^/ convergziit is a a or :r- The second convergent is , 1 ab-V\ The third convergent is ,1 , c abc-^c+a c The fourth convergejit is A ,1 , ^^+1 ^-{ 7- = a-\ = a-\- , , 1 ' d bcd^b-^d ^+;~1 ^^7dV\ ^__ abcd-\-cd-\-ad^-ab-V\ ~ bcd-\'b-\'d ^' In the third convergent we notice that the numerator is obtained by multiplying the numerator of the second convergent by the third incomplete quotient and adding the numerator of the first convergent. The denominator also ot the third convergent is obtained in a similar manner, viz: by multiplying the denominator of the second convergent by the third incomplete quotient and adding the denominator of the first convergent. Also, in the fourth convergent, we notice that the numerator and denominator can be obtained by the same law, and we are led to think that this same law is prob- ably general. We shall now show by induction that this law is general. n n o n Let -7- » -7*^ J -- be three consecutive convergents, the last of which is derived from the others by the law under 470 UNIVERSITY ALGEBRA. consideration, and let q^, q^, q^ be the three correspond- ing incomplete quotients ; /. e. , if -^- represents the kWi convergent, then q^ represents the /^th incomplete quotient. Because the law holds for the third of the above con- vergents, therefore ^3=^3^2 + ^^! ^3^ ^3^^2+^l ^3 ^3^2 + ^1 Now an inspection of the convergents of any continued fraction will make it plain that we may pass from -- to to the next convergent, say — - by changing ^3 into 1 ^4 ^sH — where q^ stands for the next incomplete quotient ^4 after q^. Making this change, we obtain (^<4H F2+^l ^ . ^ . ^4 ^ V q^l ^ (^3^2+^^l)g'4+^^2 _ ^4^3 +^^2 ^4~/ , 1\, , ., "(^3^2+ ^1)^4 +^2" ^4^3 +^2 ' (^3+-y2+v/, Hence the law holds for the next convergent after -—• Hence if the law holds for the formation of some conver- gent it also holds for the formation of the next convergent. But the law does hold for the formation of the fourth convergent, therefore it holds for the fifth convergent, therefore for the sixth convergent, and so on; hence, the law is general. 642. Theorem VI. The difference between any two successive co7ivergents is equal to unity divided by the product of the denomi7tators of the tivo convergents. The difference between the first and second convergents ab+1 ab-j-1 — ab 1 __ a = ^ J CONTINUED FRACTIONS. 47 1 The difference between the second and third conver- gent s is adc+c-\-a ab-\-l __^ ab'^c-{-bc+ab — ab'^c—ab—bc — 1 bc+l 'b~ b{bc-\-V) bibc^-Y) We notice in this difference that the numerator is — 1, but this we ought to expect for we know that the odd convergents are less than the even ones. The theorem supposes that in taking the difference between two suc- cessive convergents the less is subtracted from the greater, /. e. , the odd convergent is always the subtrahend if the numerator of the difference is unity. If we indicate the difference between two successive convergents, not know- ing which is the odd convergent, the numerator of the difference is -f 1 or —1 according as the subtrahend is an odd or an even convergent. Now suppose the theorem true for some two successive convergents, and let them be represented b}^ -—- and -~ di d^ respectively, then -^ -7- = d-^ d^ d^d^i ,'. n^d^ — n2,d-^ = d[zl. Now let -7- represent the next convergent after --- d^ «2 and let q represent the last incomplete quotient which forms part of -7-' then by Art. 641 d^ qd<2,-\-d^ ' ' d^ d.^ d^ qd^-\-d^ qn^d^+n^d^ — qnc^dc2, — n^d^, 722^1 — ^i-^d^, ^2^1 — ^1^2 -F-^ 472 UNIVERSITY ALGEBRA. Therefore, if the theorem holds for the difference between some two successive convergents, it holds for the difference between the next two successive conver- gents, therefore the theorem holds for the difference between any tw^o successive convergents. 643. Theorem VII. All the convergents are irre- diicible fractio7is. Let -— and — - be any two successive convergents, then by the preceding article 7i^d^—n^d^^=±\, Now if it were possible for n ^ and d^ to have a common factor other than unity, that factor would be a factor of the whole left number. But the left member being equal to ±1 can have no factor other than unity. Therefore, ^1 and d^ have no common factor, therefore the fraction — - is irreducible. 71 But -— ■ is a7iy convergent, hence any convergent is an irreducible fraction, hence all convergents are irreducible fractions. The first convergent is an integer number or zero, but we may consider either case as a fraction whose denomi- nator is unity, and hence the first convergent is no exception to the theorem. 644. Theorem VIII. Any convergent is nearer the value of the continued fraction than any previous conve7gent. In Art. 630, the expressions a-f-? b-\ — > etc., are X y called 'Complete Quotients. Plainly, in any conver- gent, if the last incomplete quotient be replaced by the corresponding complete quotient, the result is the con- tinued fraction itself. CONTINUED FRACTIONS. 473 ^7 7Z 7Z Now, let -,-' 3-> -,— be any three successive conver- gents, and let q be the last incomplete quotient which n 1 forms part of -%— and q-\ — the corresponding complete a.^ r no q7i1. Also, if q' represents the last incomplete quotient which forms part of ~ and -^ represents the convergent immediately preceding -— , then by Art. 641, d2 = q' d^+d\ and as all these numbers are positive, and at least as great as 1, it follows that ^2^^i- Now, as {q-\ — )>1 and as d2~>dy, plainly, on both these accounts the second member of (2) is less than the second member of(l). Therefore, n— ^<~ —n, d<2. «i Therefore, the convergent —^ is nearer the value of the continued fraction than -7^ is. But -~ is any conver- gent, therefore any convergent is nearer the continued fraction than any preceding convergent. In Art. 640, we found that the odd convergents are less than the continued fraction, and, taken in order, approach the value of the continued fraction. Also, that the even convergents are greater than the continued fraction, and, taken in order, approach the value of the continued fraction. Article 640, taken in connection CONTINUED FRACTIONS. 475 with this article, shows that all the convergents taken in order are alternately less and greater than the continued fraction, but continually approach the value of the con- tinued fraction. KXAMPI^KS. 339 1. Convert the fraction ^^7^-^ into a continued fraction. Zoo 236 2. Convert the fraction 7^7^ into a continued fraction. ooU 126 . 3. Convert the fraction :j-^^ into a continued fraction. lol 4. Convert 3.1416 into a continued fraction. 5. Find the successive convergents of the continued 30 fraction which is equal to jo* 6. Find the successive convergents of the continued fraction which is equal to t^t^- 7. If V> -r' -j~ b^ ^^y three successive convergents ^1 "2 ^3 of a continued fraction show that (71 3 —n^dc^^ — {d^—d^n o . 8. Prove that the numerators of any two successive convergents of a continued fraction are prime to each other. Prove also that the denominators are prime to each other. 9. If-/' ^-' —- be the first three convergents of a ^1 ^2 d^ , n. 71^ 1 1 continued fraction show that -/- — -,-=-rT "~ TT ' 10. If--) ~-y -7-' -;- be the first four convergents of di ^2 ^3 ^4 a continued fraction show that n^ __ n^ _ _1 1_ , _1 d^ d^ ^1^2 ^2^3 ^3^4 476 UNIVERSITY ALGEBRA. 645. We have seen that any commensurable number may be expressed as a continued fraction which termi- nates, and that an incommensurable number may be expressed as a continued fraction which does not termi- nate. We desire now to consider those continued fractions which result from a particular kind of incommensurable numbers, viz : quadratic surds, and, as the subject offers some difficulty, we will take first a particular quadratic surd, express it as a continued fraction and notice the form of the result, after which we shall be better able to understand the general case. Let us then express jL„ as a continued fraction. Since the integral part of this expression is 1, we have V6-1 2 _l-(/6-l)(v^6+l) 6-^5"/ =i+2_v|::3==i+_L_ (2) 21/6-3 5 _ 5(2t/6 + 3) _ 10l/G + 15^ 2K^6 + 3 2i/6-3~ (2l/6-3) {2\/&+S) 1^ 3 =2+(^^^-2)=2+2J^^=2 + -4- (3) 2J/6-3 CONTINUED FRACTIONS. 477 3(21/6 + 3) 61/6 + 9 21/6 + 3 (5) 2T/6-3 (2V/6-3) (21/6 + 3) 15 5 = l + (^_l)=l + ?l^=l^^ (4) 21/6-2 5 ^ 5(21/6 + 2) ^ 101/6 + 10 ^ 1/6 + 1 21/6-2 (2/6-2) (21^6 + 2)"" 20 "~ 2 Now, the last expression in (5) is exactly the expression we started with. Hence, to find its value in the form of a continued fraction we would, of course, go through with the work over again and, of course, finally arrive at the same expression again. It is therefore unnecessary to go through any more numerical work, but from the results already obtained, we can write as many terms as we please of the continued fraction which represents the quadratic surd — ^ To write this continued fraction we substitute for the denominator of the last expression in (1) its equal ob- tained from (2), and then for the denominator of the last expression in (2) its equal obtained from (3), and so on. Thus we obtain ^-h 2+? 1 + -. In this continued fraction it is to be noticed that the 478 UNIVERSITY ALGEBRA. incomplete quotients repeat themselves over and over again, always in the same order, 1, 1, 2, 1. 646. If we represent the quadratic surd — ^r— by x and notice that the last expression in equation (5) of the preceding article is x we may write 1 + — 1 X Reducing the second member to a common fraction, we 7;tr-f5 have ^ 4x+3 Hence, ^x''-\-Zx=lx^h .'. Ax'^—Ax=b. Solving we get l±l/6 From this w^e see that the above continued fraction is one of the roots of a certain quadratic equation. We shall see presently that the same thing is true of a whole class of continued fractions. 647. When, in an unlimited continued fraction, a certain number of the incomplete quotients continually recur in the same order, the fraction is called a Periodic Continued Fraction. The incomplete quotients which recur constitute the Period. When the period begins with the first incomplete quotient the fraction is called a Simple Periodic Con- tinued Fraction, but when one or more incomplete quotients occur before the period begins the fraction is called a Mixed Periodic Continued Fraction. CONTINUED FRACTIONS. 479 For example : is a simple periodic continued fraction whose period is formed from the incomplete quotients a, b, c. b+ is a mixed periodic continued fraction whose period is formed from the incomplete quotients e. f. , and in which the incomplete quotients a, b occur before the period begins. 648. Theorem IX. Every simple periodic continued fraction is a root of a quadratic equation with rational coeffi- cients whose roots arc of opposite signs. Let -p^ and -- be the last two convergents of the first dy_^ d^ period of a simple periodic continued fraction, and let x be the value of the continued fraction. Then by Art. 641 ___xnr'\-nr^-^ from which we readily obtain drX'^ + {dr-^—n^x—Ur-x^^. Now, as dr, dr^^, Ur, n^-i are all rational, the coeffi- cients of this equation are rational, and as the coefficient of ;t:^ is positive and the absolute term negative, the roots of this equation are of opposite signs. (See Art. 290). ^SO UNIVERSITY ALGEBRA. EXAMPLES. 1. Convert — ^ — into a continued fraction. 2. Convert p — into a continued fraction. 5 3 3. Convert 7= into a continued fraction. 4 + 1/3 4. Find the quadratic equation one of whose roots is the value of the simple periodic continued fraction whose period has the incomplete quotients 1, 2, 3, 4. 5. If— and ~ represent respectively the third and fourth convergents of the continued fraction in example 4, express — and ~ as continued fractions. 7Z^ d^ 6. Find the quadratic equation one of whose roots is the value of the simple periodic continued fraction whose period has the incomplete quotients 4, 3, 2, 1. 7. Find how the roots of the equation found in ex- ample 6 compare with those of the equation found in sxample 4. 8. Find the quadratic equation one of whose roots is the value of the mixed periodic continued fraction 2- 5+1- 2-fl- ^+-1 ^+3— CONTINUED FRACTIONS. 48 1 9. Find the quadratic equation one of whose roots is the value of the mixed continued fraction 1 5 + 3+- 2+' 3+. 10. Do the roots of the equation found in example 8 have the same or opposite signs ? 11. Do the roots of the equation found in equation 9 have the same or opposite signs ? 12. Convert V 5 into a continued fraction. 13. Convert 1^10 into a continued fraction. 14. Convert 1^2 into a continued fraction. 15. What kind of continued fractions are obtained in examples 12, 13, 14? 31 — U. A. CHAPTER XXIX. DERIVATIVES. 649. Notation. A definition of a function of a quan- tity was given in Art. 358. To represent a function of a quantity we enclose in a parenthesis the letter which represents the quantity, and write/ or F or some other functional symbol before the parenthesis, e. g, f{x), F(^x), F^{x) denote functions oi x. /OO' ^(j)y/i(y) denote functions of y. f{x-\-Ji), F{x+h'),f{x-\-]i) denote functions of ji:+/2. /(^), Fid), ^\f{a) denote functions of a. The student must be careful not to look upon the ex- pression fix) as meaning / times x. The symbol /, as used here, is not a multiplier at all, but simply an abbreviation for the words function of It frequently happens that, in the same discussion, we wish to refer to different functions of x, in which case we use different functional symbols, as F(^x), f{x~), f^{x), fix), FXx), <^(^), lAW, etc. It also frequently happens that, in the same discussion, we wish to refer to the same function of different quanti- ties, in which case we use the same functional symbol before the parenthesis but different letters within the. parenthesis, e. g . \if{x) denotes x'^ -\-\ then /{a) denotes «^-fl, f{z) denotes ^^-fl, etc., and if F^x) denotes V x-\-Z, then F(^y) denotes Vjz-fB, F{x-\-1i) denotes V x-\-h'\-Z, etc. DERIVATIVES. 483 A function of two quantities is any expression in which both the quantities appear. To represent a function of two quantities, say ;r andjj^, we enclose x and _y in a parenthesis, separated by a comma, and write the letter / or /^ or <^ or some other functional symbol before the parenthesis, e. g. f{x, j/), F{x,y), ^{x,y), etc. 650. In such an expression as f{x, y), the x and y are entirely unrestricted in value and independent of each other; but if we have an equation like/(jf, jr)=0, then x and y are to some extent restricted; any value may indeed be given to one of the quantities but then the equation fixes the value of the other, or in other words, either one of the quantities x or y depends upon the other one. For example: if /(^, jv) stands for ;ir— j/4-2, then when this is not put equal to anything there is no rela- tion between x and y. We may let x=^S and j/=5 or 7 or 10 or any other number. But if we put this same function equal to zero, f/ien there is some relation between x and y and they are to so77te extent restricted in value. We may let ;t:=3, but then j/=5 and nothing but 5. 651. If the equation F{x, >')=0 can be solved for y, we can express y in terms of x, or y can be determined as a function of x. If w^e thus determine y we have In this equation, y'=f{x), we may look upon Jtr as a variable, and of course if x varies y will also vary. We may consider x to vary in any way we please, but then the equation determines the way in which j/ varies. For this reason ^is called the Independent Variable, and j^, which is the function of x^ is called the Dependent Variable. 484 UNIVERSITY ALGEBRA. 652. In the equation of jr=/(-^), if a value be given to X, then y will have some corresponding value, and if X be given another value different from the first one then y will have some value different from the one it had at first. Moreover, the amount by which y thus changes in value will depend in some way upon the amount by which X changes, or, in other words, there is some rela- tion connecting the change in the value of y with the change in the value of x. This relation we shall examine, and it will be found to be a very important relation in all that follows. 653. Suppose /(;i;) to stand for 2x + 4, then putting this equal to y^ we have y=2x+4:. Let us now give to ;r a series of values, say the success- ive integers from 1 to 10, and in each case compute the corresponding value of y. The results may be expressed in the form jK I 6 8 10 12 14 16 18 20 22 24 ^1 123456789 10 where any number in the lower line is one of the values of X and the number immediately above it is the corres- ponding value of y. If x= 2 the corresponding value of j/ is 8, and if ;t:=10 the corresponding value of j^ is 24, and if :t: be considered to increase from 2 to 10 then at the same time y will increase from 8 to 24, or, starting Sitx=2, \i X increases by 8, y will increase by 16, or if the increase of x is 8, the corresponding increase of _y is 16. Still starting at x=2, let us increase x by various amounts and determine the corresponding increase oiy. The results may be arranged in the form increase of jK I 16 14 12 10 8 6 4 2 increase of ;f I 87 6 54321 DERIVATIVES. 485 If we had started with some other value of x than 2 we would have obtained similar results. In every ob- served case we see that the increase oiy is just twice the increase of x, or in every observed case increase of j/ _^ increase of x It is easy to see that this is necessarily the case what- ever the value of x with which we start and whatever the amount b}^ which x is increased, for if x increases by any amount, 2x will increase by just twice that amount and the change in the value of x does not affect the 4, therefore 2;t:+4, or jk, will increase twice as much as x increases, or increase of j/ _ ^ increase of ;t: ' 654. Notation. In what follows we deal largely with equations formed by putting y equal to a function of X, and as we shall make extensive use of the increase in the value of x and the corresponding increase in the value of y it is well to have a convenient notation by which these amounts of increase are denoted. So in future we shall use A:r to denote the increase in the value of X and Aj to denote the corresponding increase in the value of y. In this notation the fraction at the end of Art. 653 would be written The student is cautioned not to think ot A;t; as being A times x for the symbol A as here used does not stand for the coefficient of x at all, but simply for the words in- crease of. 486 UNIVERSITY ALGEBRA. 655. Let us now consider the equation j/=^'-i-f 1. In this equation give x the successive integer values from —3 to 7 and compute the corresponding values of j/. We may arrange the results as in Art. 653. j^l 10 5 2 1 2 5 10 17 26 37 50 ^1-3-2-1012 3 4 5 6 ~~7 If x=^\ the corresponding value of jr is 2, and if x=7 the corresponding value of y is 50, and if x be supposed to increase from 1 to 7, at the same time y will increase from 2 to 50, or starting at x=\, \i x increases by 6 then y will increase by 48, or when A;t;=6, Aj/=48. Still starting at x—\, let us give t^x various values and determine the corresponding values of l^y. The results may be arranged in the form Aj 48 35 24 15 8 3 ^x 6 5 4 3 2 1 Ai/ Here we have a case in which the ratio ~- is not al- ^x ways the same as was the case in Art. 653, but at one time it is -V-, or 8, at another time it is ^f-y or 7, etc. Ai/ As can be seen by the above scheme, the fraction ~- ^x takes successively the values 8, 7, 6, 5, 4, 3 as A;r takes the successive values 6, 5, 4, 3, 2, 1. We now give to x values intermediate between 1 and 2 and compute the corresponding values of _>'. The results may be arranged in the form y I 2 2.0000200001 2.00020001 2.002001 2.0201 2.21 x\ 1 1.00001 1.0001 1.001 1.01 1.1 As before, let us start at :r=l and give to t^x various fractional values and determine the corresponding values of Aj/. DERIVATIVES. 487 The results may be arranged in the form y I .21 .0201 . 002001 .0002 0001 .0000200001 x\ .1 .01 .001 .0001 ;ooooi An examination of this scheme shows that whenAjr=.l, then ~^= ~ =2.1 A;t- .1 when A;r=.01, then ^= :^?i =2.01 A;t- .01 when A^=.001, then ^ = •™-^ =2.001 A^ .001 1, A nnm .t, ^y .00020001 ^^^^, when A;i:=.0001, then -^ = — =2.0001 t, A nmm .-u ^J .0000200001 ^ ;,^^^, when A^-=. 00001, then -f- = ^^^^^^ — = 2.00001 A;r .00001 From the first part of the article it appears that — is a variable, and from w^hat we have just obtained it fur- ther appears that as ^x is taken smaller and smaller the Aj/ fraction -^- approaches nearer and nearer the value 2, or Ay in other words, it appears that the fraction ~- approaches 2, i. e., 2 times 1, as ^x approaches zero. In obtaining the result it is to be noticed that we con- sider X to increase frotn the value 1, but if we let x increase by various amounts, beginning to count the increase in x from the valued, reasoning exactly as we have just done would lead to the conclusion that ~ ^ A.V approaches 4, i. c, 2 times 2, as t^x approaches zero. Again, if we begin to count the increase in x front the value 3, reasoning as above would lead to the conclusion Ai/ that -^ approaches 6, /. ^., 2 times 3, as A^ approaches zero. 488 UNIVERSITY ALGEBRA. 656. In general, if a be taken as the value of :r from which we begin to count the increase of x analogy would lead us to expect that the fraction -~ approaches 2<^ as A;r approaches zero, or, using the notation of Art. 432, limit /Aj^/\ This we will now prove. Since j/=x2 + l, (1) whatever value be assigned to x the equation will enable us to compute the corresponding value of y. First, let x—a and represent the corresponding value of J/ by ^, then b^a'^ + X, (2) Now let x=a-\-^x then, representing the corresponding value oiy by b-\-L^y, from equation (1) we get ^+A^=(« + A^)2 + 1, (3) or simplifying, ^ + Aj^/=.^2^2aA;t:+(Ax)2 + l. .(4) Subtracting (2) from (4), we get A_>/=2aA:r+(A;r)2. (5; Dividing (5) by A;t:, we obtain ^=2^ + A;r, (6) As ^x varies, of course the two sides of equation (C) are variables, and, indeed, they are two variables that are always equal, and as A:r approaches zero these two vari- ables approach limits. Hence, by chapter XVIII, Art. 420, their limits must be equal. limit r.^)^2a. Hence, A;^- OVA;^. Aj If we attempt to find the limit of — numerator and denominator separately we are led to ths form A7 If we attempt to find the limit of -r— by finding the limit of the DERIVATIVES. 489 Av 7: which is indeterminate. We must therefore take the fraction -— A;c as a whole and not try to find the limits of numerator and denomin- ator separately. Ay 657. The value of the fraction -r- when that frac- ^^ Ay tion is constant, or the limit of the fraction —- as Ax Ax approaches zero when that fraction is a variable, is called the Derivative of j' with respect to x, and is repre- dy sented by the notation — - where j^ is a function of x. dy Although the expression ~r is written in the form of a fraction the student should not attempt at this stage to assign any meaning to numerator and denominator separately, but should look upon the expression as a whole, as though it were represented by a single symbol. When the subject of Differential Calculus is reached, the student will doubtless find a meaning for dx and dy. 658. The general method of finding the derivative of y with respect to x is that used in Art. 656, viz: give to X some value, say a, arid find the corresponding value of y; then give to x a new value, a + Ax, and again find the corresponding value of y. Subtract the first of the equations thus obtained from the secofid and the result will be the value of Ay. A y Divide both sides by Ax and the ; esult will be the value of — ^ • •^ -^ Ax Finally, fifid the limit of this fraction as Ax approaches zero. 659. We will now illustrate the method by a few examples. First. Find^ whenj/=4.r2-f5. (1) Let x=a and represent the corresponding value of y by ^, hence ^=4^2+5. (2) 490 UNIVERSITY ALGEBRA. Now let :r=a + A;r and the corresponding value of j/ will be the value b plus the amount by which y has been increased, or b + t^y, hence ^+Ajj/=4(a + Ajr)2 + 5. (3) Expanding, b-\-l^y=\a'^ + %ab.x-\-^{l^xy -^h, (4) Subtracting (2) from (4), we get A_y= 8aAjr + 4(A;r) ^ . (5) Dividing (5) by ^x, we obtain ^=8« + 4(A^). (6) Taking the limit of each side as A,r approaches zero we get |-8». (8) dy Second. Find-f^- when ^=^^2+^. (1) Let ^=<2 and represent the corresponding value oi y by b, then b=ca''-' -{-e. (2) Now, letting x=a-^^x we get b + Aj/= ^(a + A;t:) 2 + ^, (3) or, expanding, ^+ Ajv=^a2 + 2«/=4jt:2+8;rAx+4(A;t')2 + 5. (3) Subtracting (1) from (3), we obtain A_y=8;t:A;r+4(A;i;)2 (4) Dividing both sides of (4) by ^x, we get ^=8;t-+4A;r. (5) A;r ^ ^ Taking the limit of each side as ^x approaches zero, we g=8.. (7) We notice that the result is exactly the same as equa- tion (8) in the first example, under Art. 659, except that X appears here where a appeared before. We shall hereafter proceed as we have just done. Usually-;^ will be a function of Xy but occasionally, ^^ dv as in Art. 654, -^ will turn out to be a constant. dx 494 UNIVERSITY ALGEBRA. 663. Derivative of a Constant. I^et j= a constant. Then, as x does not appear in the expression for y, any change in the value of x does not affect y, or ^x may have any value whatever, but Aj/ is always zero. Hence T^=0. t^x dy ^ therefore '3~=0. ax 664. Derivative of the Sum of two Functions of X, Let one function of x be represented by ti and the other by V, and let their sum be represented by y\ then y=7i-\-v (1) When X is increased by Ax suppose that 7c is increased by A?^, V is increased by Az;, and j^ is increased by Aj^/, then after x is increased by Aji^ we have y-{-Ay—7c-\-A7C-\-v-i-Av, (2) Subtracting (1) from (2), we get Ay=A7i + Av. (3) Dividing both sides of (3) by A;t:, we get ^=^ + ^. a^ A;t; A:r Ajt: ^ ^ Taking the limit of each side of (4) as Ax approaches zero, we get limit (^\^ limit (^\_i. limit (^_^\ .rx Ax :: 0\AxJ Ax:^0 \AxJ "^ Ax ^ 0\AxJ ^^ dy __du dv dx dx dx In the same way if y=u—v we would get dy du dv dx dx dx The result may be expressed thus: The derivative of the algebraic sum of two f7cnctio7is of x equals the sum of their separate derivatives. DERIVATIVES. 495 665. Derivative of the Sum of any Number of Functions of x, lyCt there be any number of functions of .r represented by u, V, w, etc., and let their sum be represented by jf; then y=u-\-v-\-w-\- • • • (1) Increase x by the amount A.r and suppose that u, v, w, etc., are increased by the amounts t^tt, Az;, Aze', etc., respectively and y is increased by Aj/, then, after x is thus increased, j/+^y=2i-{-A2i-j-v-{-Av-\-w-\-Aw-{- • . . (2) Subtracting (1) from (2), we get Aj/=A?^ + A'z^H-Aze;-f ... (3) Dividing both sides of (3) by ^x, we have Ay^A^ A^ A^ /S.x Ax^Ax ^ Ax^ ' ' ^ ^ Taking the limit of both sides of (4) as Ax approaches zero, we have dy _du dv dzv dx ~dx dx dx If some of the signs in (1) had been minus, the same process could have been applied and the result would have had minus signs in the same positions as they appeared in the original functions. The result may be expressed thus: The derivative of the algebraic sum of several finictions of X eqjials the algebraic sum of their separate derivatives. EXAMPLES. Find the derivative with respect to x of the following expressions: 1. Ix^-Y^x^^+x, 4. x'-' + Zx—^ 2. X^+x'^+X+1, 5. X'^—X-{~1. 3. x^ — l. 6. x^ + 1. 496 UNIVERSITY ALGEBRA. 666. Derivative of the Product of two Functions of X. Let 2i and v be the two functions of x^ and let y be their product, then y=2iv. (1) Now increase x by ^x and suppose the corresponding amounts by which y, u, and v respectively increase are Ay, Az^, and Az;, then j|/4-Aj/=(z/-fAzO(^^+Az;). (2) Expanding (2) we get y-\-^y=2iv + 7i^^v-\-v^u + ^.u^.v, (3) or j/ + A_y=2^^' + 2/Az;+(z;+Az;)Az^. ^ (4) Subtracting (1) from (4), we get Aj/=z/A2;+(z^+Az;)Az/. (5) Dividing both sides of (5) by A;tr, we get Ajt: A;i; ^ ^ ^x ^ ^ Taking the limit of each side of (6), remembering that the last term of the right hand member is the product of two variables, hence its limit equals the product of their separate limits, and that Az; approaches zero as A.r ay^- proaches zero, hence the limit oi v-\-l^v=v, we get limit fM=^ limit ^ + z. limit ^. (7) Aj^:: ^\^xl i^x A^ "• ^ dy ^ (^,^du^ -.o>. dx dx dx 667. Derivative of the Product of any Number of Functions of x. First, take three functions of x, say u, v, and Wy and let y be their product, then we have y=uvzv, (1) Let v'w^=v\ then y=zw\ and hence dy __ ,du dv' ^^^ dx dx dx DERIVATIVES. 497 ^ dv' dv , dw ,^. But —-^w-^-\-v--' (3) dx dx dx hence by substitution in (2) we have dy dii ' dv , dw ... U'.\' U/.X' Vt/^V' U/.A' Now, if we have any number of functions of x, say u, Vy w,' ■ ' and if we let y be their product, we have y=-uvwz • • (1) I^et the product of all the functions after the first be represented by a single letter, that is, let v'=vwz- . . then y=uv' . d V Find -J- as the product of two functions. Then -T-=v'- — Vu-z-' (2) J I dx dx dx Find -7— by letting v'=v2v\ where w' represents the product wz - ■ ' Substitute the value thus found in (2). The result will contain one term involvino^ -— • ^ dx Find the derivative by considering w' as the product of two factors. Continue this process until finally the product of the last two factors of the expression with which we started is^ reached. The result may be stated thus : The derivative with respect to x of the product of miy number of functimis is equal to the sum of all the products obtained by multiplying the derivative of each factor by the product of all the other factors . If the equation here described be divided through by the product of all the given functions, the result may be. represented in quite a convenient form, viz : 1 dy ^\ du 1 dv 1 dw 1 dz y dx ti dx V dx w dx z dx ' ' ' 49? UNIVERSITY ALGEBRA. KXAMPLKS Find the derivative with respect to x of the following expressions without performing the multiplications in- dicated : 1. (;t:— l)(:r— 2) compare with example 4, Art. 665. 2. {x'^—x+V){x-^V) compare with example 6, Art. 665. 3. (jr^+;r+l)(;r— 1) compare with example 3, Art. 665. 4. {x'^ + \Xx^-\-ax+b'). 668. Derivative of the Quotient of two Func- tions of X. Let u and v be the given functions of x^ and let y be their quotient, then y-- (1) From (1), by multiplying by v, we get zc=vy, (2) du dy dv .^. ^^''''^ 'di'^'^dx^^Tx ^^^ U/.\/ fAfJ\^ lA/.A' du _ dy u dv dx dx V dx " Multiplying both sides by v, we get du^ ^dy dv_^ , dx dx dx Transposing and dividing by z'^, we get du dv dy dx dx du dv .^. dx v'^ Expressed in words this is The derivative of a fraction equals the deno7ninator into the derivative of the mcmerator minus the numerator into tfie derivative of the denominator all divided by the square of the denominator. DERIVATIVES. 499 KXAMPI^KS. Find the derivative with respect to x of the following expressions : 1. compare wath example 5, Art. 665. X^ —^X'^ -\-\\x—^ . . 1 ^ A i. nnr^ 2. _ compare with example 4, Art. 665. x''-\ 3. 4. x + 1 x'^ + l 669. Derivative of a Function of Another Func- tion of X. Suppose J/ is some function of z, and ^ is some function of X, then ultimately j/ is a function of Xy hence it has a derivative with respect to x. But as y is directly a function of ^ it has a derivative with respect to ^. Moreover, as -sr is a function of x it has a derivative with respect to x. We have identically Ax Az ' Ax Taking the limit of each side as Ax approaches zero, remembering that the limit of the product of two vari- ables equals the product of their limits, we have Hmit (^\^ limit {^\ limit /^\ (2) Ax^ 0\AxJ Ax :; OVA-a'/ * Ax ^ 0\AxJ Now z being a function of x, we may write and if x be increased by Ax, we have 2-\-Az=/(x-\-Ax'), 5o6 UNIVERSITY ALGEBRA. and from this it is evident that, as t^x approaches zero, A^ must also approach zero. Hence limit /^^\ limit l^\ (3) A^ - OVaW ^x :: OVA^y Substituting from (3) in (2), we have limit {^^ limit /^\ limit l^_l\ (4) A;r :: OVAaV A^ :; V A^ y * A;t: - OV'A;^/ d V The left member of (4) is -y-^ the first factor of the dy ^^ right member is -~^ for it is just the same as the left member except that z everywhere takes the place of x ; dz and the second factor of the righi member is -^ — Hence ?=?? (5) dx dz dx li j/=z2^ and z=x'^-i-2, then dz dx Hence by equation (5) ^^=2^. 2x=4zx=4x(x'^ +2)=:4:x^ +8x. It is easy to see that this result is correct, for in the equation jy=z^, substituting the value of z we have y:=(x'^ +2y=X^ +4:X^ +4:, ^-=4:X^^SX. dx KXAMPLKS. Find the derivative with respect to x of the following expressions: 1. (x'^+ax+dy. 2. (;r2 + l)2 4-3(;i;2 4-l). 3. (x+ay + 2(x+a-). DERIVATIVES. 5OI 4. (2x+Sy+5(2x+S)+4:. 6. 2(jtr2-l)3 4-4(;i:2-l)2 + (;r2-l). 670. Derivative of any Positive Integral Power of X. Let y=x\ (1) Give to X the value x-h^x, then j/ + Ay=(jt:+A;t:)". (2) Expanding the right-hand member of (2), we get Subtracting (1) from (3), we get ^y=.nx'--^Ax+ ''^'\~^K ''-'-(r^xy-i- ■ • . +(Axy\ (4) Dividing both sides of (4) by A;r, we get ^=nx''-^ -f '^fc^^^"-'Ax+ . . . +(A;r)"-i. (5) Taking the limit of each side as Ax approaches zero, we have ^=nx"-\ (6) Reasoning exactly as above we could show that when y=ax'\ ax This formula may be expressed in w^ords thus: T/ie derivative with respect to x of ax" is equal to the t>roduct of the expojient, the coefficient, and x with the expo- nent diminished by one^ It is to be noticed that this formula applies to the derivative with respect to jr of a power of x. Of course any other letter could be used as well as x to denote a variable. Thus, when r=«y, -^-^naz**''^ . "^ dz 502 UNIVERSITY ALGEBRA. But we must be careful not to use this formula to find the derivative with respect to some quantity, of a power of some other quantity, or, in other words, in order to be able to use this formula the quantity which is raised to a power must be the same as that with respect to which the derivative is taken. 671. Derivative of any Negative Integral Power of X, I.et _;,=^-=l. (1) g=^^;^ by Art. 668. (2) Simplifying, remembering that the derivative of a con- stant is zero, we get dy. nx''~^ „ , dx x^'' It may be objected to this method that we have used the formula for the derivative of a fraction w^hose numer- ator is 1 when that formula supposed that numerator and denominator were each functions of x. X We may, then, take y——;^^x and now using the formula of Art. ^^'^, we get, as before dx It easily follows that if y:=ax~**. then -~ = —nax "" ^. dx Here, as in Art. G70, in order to use the formula, the quantity raised to a power must be the same as the one with respect to which the derivative is taken. DERIVATIVES. 503 We may express this formula in words thus: The derivative with respect to x of ax"'' is equal to the prodzict of the exponent, the coefficient, a7td x with its expo- nent diminished by one, 672. Derivative of a Fractional Power of x. p Let . z=x"^ (1) then let y^Z^^X^, (2) Then J'=$'.J-. Art.669. dx dz dx • (3) But g=^-. (4) hence dy ^ . dz dx ^ ' dx (5) But from (2) dx ^^ ' (6) hence from (5) and (6) - # (7) Dividing by qz'^ ', which is the same gx^, we get ,dz p , dx a (8) Multiplying the left member by z and the right member by p x'q, which is the equal of z, we obtain ^dz p l-i — - = -X9 dx q t The same reasoning would show that \i y=^axq, then dy ap ^ -■[ dx q Hence, as in the two preceding articles, The derivative with respect to x of axg is eqiial to the product of the exponent, the coefficient, and x with its expo- nent diminished by 07ie, 504 UNIVERSITY ALGEBRA. BXAMPIvKS. Find the derivative with respect to x of the following expressions: I. (a-2-^+l)-f2(jr2-:t:+l). (x^-\\ , Jx^-\\ - i^H"^) Jx'^^1 i>V-i 5. (x+vT^^Y- Vi+x-^vT^x vY+x—Vr^x ( ^__ y 8. {a^+J)^a^ + x^^ Va-i^x 9. 10. Va+ }/x 11. -xla-i h--^- \ X x^ 12. {a-^xy.(d-\-xy. CHAPTER XXX. INCOMMKNSURABI.K KXPONKNTS AND LOGARITHMS. 673. We have found before that whatever commen- surable numbers be represented by 7i and r, the following laws of exponents hold : (^«)-=^'- \b) ~~y but no meaning has yet been given to numbers with incommensurable indices. A number raised to any power, as a above, is called a Base. The present discussion is confined to the case in which the base is positive. 674. A given base, affected by an exponent, may have more than one value, as for instance: (25)2"=db5, but a base affected by any commensurable index has among its values one which is positive. For any integral power of a positive base is evidently positive. Any root of a positive base has, among its values, one which is positive, and since any power of this i;oot is positive, therefore a positive base affected by a positive fractional index has among its values one w^hich is positive. Again, since a positive base affected by a negative index is the reciprocal of the same base affected by a positive index, therefore a positive base affected by a negative fractional index has among its values one which is positive. This positive value is all that is con- sidered in the present discussion. So that whenever we 506 UNIVERSITY ALGEBRA. deal with an expression like a'' in the present chapter both a and a'' are positive. These restrictions must not be lost sight of. 675. What meaning must a base affected j with an incommensurable exponent have and be consistent with the meanings already assigned to commensurable expo- nents ? For example : what is the meaning of 10^ ^ ? The exponent V 2 does not show how many times 10 is used as a factor, for t/2 is not a whole number, and it does not indicate a power of a root of 10, for l/2 is not a commensurable fraction._ While we do not know the meaning or value of 10^^, we do know the meaning of 101-^,— it is the 14th power of the 10th root of 10. Likewise 10^*^^ has a meaning, — the 141th power ot the 100th root of 10. It would involve tedious work to compute these powers and roots of 10, but if they be so computed we would find, to seven places 101-4 ^ 2.511887- . . 101-4 1 ^ 2.570396 . . 101-414 ^ 2.594179. . . 101-4 14 2 = 2.595374. . . 101.41421 ^ 2.595434. • . 101-414213 =2.595452... 101-4142135 _ 2.595455. . . Now we have purposely selected closer and closer approximations to ]/2 as the exponents of 10 in the left members of these equations. Whence we conclude that the right members of these same equations are closer and closer approximations to 10 ". Thus we observe that 10 is an incommensurable number of which 2.595455 is a commensurable approximation to seven places of decimals. LOGARITHMS. 507 It is not hard to see that we could get an approximate value of any incommensurable power of a base in a similar way to the above. Whence we make the following defi- nition of an incommensurable power of a number: If n is incommensicrable, a'' is the limit of a"", in which x is an always cornmensurable variable appj'oaching n as a limit. In symbols this statement is seen to be a"=limit of a\ or a "-'*°f-= limit of a\ [1] As a further illustration of a number affected by an incommensurable exponent, consider the expression a'^ where tt represents the ratio of the diameter of a circle to the circumference. It can be shown that the incommen- surable number tt can be expressed as the limit of the following infinite series: ^ — 4 4(4 44_ 4 That is, the incommensurable number tt is expressed as the limit of the commensurable variable in the right member. Then we have 4_4 I 4_4 I 4 . . , ^^=hmit a Thus a'^ is the limit of a^, in which x is the commen- surable variable 4— f + f— ^+ • • • which approaches the incommensurable limit tt. 676. Since in the expression a^ the :r appears in such an unrestricted form (having just provided for incom- mensurable values) it is common to speak of the expression as an Exponential Function of x, intending to call attention thereby to the fact that x may be considered a continuous variable as in any ordinary algebraic expres- sion. If in the et^uation ci'—y, we assume x to pass from one extreme of the algebraic scale to the other, taking in succession every possible S08 UNIVERSITY ALGEBRA. value, then we are able to give a meaning to this equation in two variable; because for every possible value of x, ^^, that is, jj/, has a definite meaning and value. It must be remembered that when we speak of d^ we mean that a is a positive number, a?id by the value of a"" we mean that one of its values which is positive. Hence in the. equation a^'—y we are to think of but one value oi y resulting when any particular value is assigned to x. Thus in 10-^=j^/ we are to understand jf= + 1^10 and not jj/=--l/lO or any other possible value oi y. Of course the very restrictions just mentioned prevent y from having a riegative value. Moreover, it is not evi- dent that y can have every positive value w^e please. For example, it is not plain that a value of x exists which satisfies the equation 10^= tt. In general, while it is easy to see that in the equation a''=y there always exists a value oi y for any value assigned to X, it is far from evident that there exists a value of x corresponding to every value which may be assigned to y. Whence the necessity for the following theorems: I 677. The expression a"" can be Tnade to differ from 1 by less tha7i any assigned number ij x be sufficiently in- creased. Suppose it be required to increase x so that a-a, or, by the binomial theorem, x must be large enough to give i4-;^^+f.^+i)^2+ . . .^^ (2) LOGARITHMS. 509 It is plain that 1 + xd can always be made greater than a by taking x sufficiently large. In fact, the inequality l+xd>a will hold if , xd^a—1, or xf ^>—J — a Now, since l + xdy-a if :t:>— -7— ? then much more is the left member of (2) greater than the right member if Hence, to make w less than 1 + d take a-1 ^>-d~ (1) Find X such that loi<1.0001. Here ^==.0001 and ^ = 10; whence 9 678. Laws of Incommensurable Indices. In this article x and y stand for two commensurable variables which approach the incommensurable limits n and r re- spectivel}^ Therefore we know, x and j/ being commensurable, a-a^=^"+'\ (1) Taking limit of each side, Art. 420, limit a-*'^^= limit ^-^+-^, (2) and by Art. 424, limit a"-' limit ^■^= limit «*'"^-^. (3) But by [1] limit a''= « '"""''=«'', and similarly for any o*:her variables as y or {x-\-y). Therefore, a"a^=^"+'', [2] or tile law of exponents in multiplication \% proved for incommensurable exponents. 5IO UNIVERSITY ALGEBRA. Starting with a''-^a^=a''-^ it can be shown in an iden- tical way that holds for incommensurable exponents. Because x and y are commensurable, we have {a^y=^a^\ (4) Whence, by Art. 420 and [1], limit (^")-^= limit a''^=a''\ (.5) Now let x—ii-^tt\ whence 21 is a variable whose limit is zero. Then we have {af'y= (^"+'0''= {a^'a^y^ {cCy^a'y (6) Whence limit (^T= limit [(^"^(^'0^] = limit {ary limit {cCy. (7) But limit ?^^0, therefore limit {cCy—\ by Art. 677. Hence, comparing (5) and (7), we have a""= limit {cCy. By [1] limit {cCy^ia'y, whence we have proved {a:y==a^^\ [4] for incommensurable exponents. Starting with {abcy^^a'^b^c^ and gy= ~ it is easy to prove the formulas {abcy=^a''y'd\ [5] for incommensurable exponents. 679. The expressio7i a^ is a contijitwics function of x. Suppcse a''=y and let x take on any increase, ^^ and suppose the corresponding value of y be j/+A so that a^^'=^y-^t. (1) We are to prove that as x passes cofitiritionsly from x to x-\-s then jK passes contimioiLsIy from jj' toj/+/; that is, as X changes from x X.Q x-\-s by passing over every iiitervie- LOGARITHMS. 51I diate value that j/ changes from jk to y-\-t by passing over every intermediate value. The equation a''^'=^y-\-t may be written a^a'=.yj^t, (2) and since a'—-y, this may be written a'^a'^a'-^t, (3) or a\a'—V)^t. (4) Now., by Art. 677, by taking ,y small enough <2'"maybe made to differ from 1 by an amount as small as we please. Hence in equation (4) t may be made as small as we please by taking ,? small enough. That is, the difference between two successive values of ^^'^ can be made as small as we please. Therefore it is a continuous function of .r. 680. It follows directly from the above article that for every positive value which may be assigned to y in the equation a^'^y^ a corresponding value of x exists which will satisfy the equatiori. For the last article shows that if x is increased contin- uously from the value without limit then y increases continuously from the value 1 withoutlhiiit. Thatis, ymay have every value greater than 1. It is also seen that as X is decreased continuously from the value without limit that y decreases continuously from the value 1. That is, y may have every fractional value. The above states that if any value be assigned to y in the equation a*==_y that a value of x exists which will satisfy it, but it! does i^ot explain how to find that value. Thus it does not show how to find x in the equation 10"'= 5. The method of finding this will be explained later. I^OGARITHMS. 681. In the equation 10''=_>', x is called the Common Logarithm oiy. That is, the common logarithm of any 512 UNIVERSITY ALGEBRA. number is the exponent of the power to which 10 must be raised to produce the given number. Thus 2 is the common logarithm of 100, since 10- = 100 ; likewise. 1.301030 is the common logarithm of 20, since 10^ •301030—20. 682. In the equation a''=y, in which a is a positive number not 1 : The constant a is called the Base. The number y is called the Exponential of x to the base a, and may be written jk=<3:''' orjF=exp^.r. The number x is called the LrOgarithm of y to the base a, and is written x=\og^y. The use of the word logarithm may be kept in mind by remem- bering this sentence : In the equation «'^=y, x is called the Exponent of the power of a or the Logai'itJijji oi y. Of course the two equations a-^y (1) x=\o%ay (2) express the same truth respecting the relation between x and y. The second equation uses the logarithmic notation and is always to be interpreted by means of the first equation, 683. Systems of Logarithms. If in the equation a'^^y, where a is any positive constant not 1, different values be assigned to y and the corresponding values of x be computed and tabulated, the results constitute a System of Logarithms. Since any positive value except 1 may be chosen for the base a, the number of different possible systems of logarithms is unlimited. As a matter of fact, however, only two systems are now used ; the Natural or Nape- rian or Hyperbolic System, whose base is approxi- mately 2.7182818 + , and the Common or Briggs' System, whose base is 10. The Natural logarithms of all numbers from 1 to 20,000 have been computed to 17 places of decimals. The common logarithms of all numbers from 1 to over 200,000 have been found. They are usually printed in tables, to four, five, six, cr seven decimal places. 8. 10-25 = i.7782 9. ^"+^'=a''<2''. 10. J^Q. 301030 — 3^ II. a^—a. 12. a\ogay=y. 13. 101ogio/=j/. 14. e^ = a LOGARITHMS. 5 I 3 In the following pages the common logarithm of any number, n^ will be denoted by the symbol log n, and not by logio?2. Thus the base is supposed to be 10 unless otherwise indicated. KXAMPIyKS. Write the following equations, using the logarithmic notation: I. 10"= TT. 2. e^=y. 3. 112 = 121. 4. 10^ = 1000. 5. 16-25 = 2. 6. 10^ = 1. 7. 10-3 = . 001. 15. 10^ ' = 2.595455. Express the following, using the exponential notation : 16. log27(i)=-.3333-f 20. Iog2l024=10. 17. log io4=. 602060. 21. log,^=l. 18. log 10 10000= 4. 22. \og,b'=b, 19. logi 0.00001 = -5. 23. \og,a==B, 24. Iog,ol^l00=ilog,ol00=f. PROPKRTIKS OF I^OGARITHMS. 684. Inasmuch as logarithms are merely the exponents of a fixed base, the properties of logarithms are entirely dependent upon the properties of exponents in general, which have already been established. Among the fundamental properties of logarithms are these : 33- u. A. 514 UNIVERSITY ALGEBRA. . The logarithin of the base itself iyi any system is 1. For a^-=a, that is, log,,a=l. The logarithm of uiiity in all systems is 0. For a<> = l, that is, logj=0. Negative numbers have no logarithins. For in the equation a''=y, a is positive, and by the value of a"" we mean that one of its values which is posi- tive, because of the restrictions we previously imposed. Hence j^ cannot be negative. See Art. 674. If we understand the same system of logarithms to be used throughout, then the following four theorems hold: 685. Logarithm of a Product. I,et 7i and r be any two positive numbers and let logji=x and log^r=j/. (1) ITow, from (1) and definition of a logarithm n=a^ r=a^. Multiplying nr=a''a''=a''+^' by Art. 678. Therefore, by definition of a logarithm, logjir=x-{-y as by (1) log.nr=log.n+log.r. [7] Hence, the logarithni of the pjvdtict of tivo numbers is equal to the sum of the logarithms of those 7iumbers, In the same way, if log««y=<3', then that is, \o%jirs=\oZan-\-\o%,,r-\-\o%aS EXAMPLES. 1. Given log 2=0.3010 and log 3=0.4771; find log 6. 2. Given log 5=0.6990 and log 7=0.8451 ; find log 35. 3. Given log 12 = 1.0792 ; find log 144. LOGARITHMS. SI 5 4. Given log 739=2.8686 and log 642=2.8075; find logarithm of product. 5. Given log 22;i;=1.9445 and log 22 = 1.3424; find log X. 6. Given log 20=1.3010; find log 200. 686. Logarithm of a Quotient. I^et n and r be any two positive numbers, and let a log,^7^=-r and log.^'^i^xy (1) From (1) and definition of a logarithm, 71= a"" r=a^. Dividing -=^--i-a^'=^"-^ by Art. 678. Therefore, by definition of a logarithm, 1 ^^ loga-==-^— Ti r or by (1) log,^ =log.ri-log.r. [8] Therefore, the logarithm of the quotient of two tuunhers equals the logarithm of the dividend minus the logarithjn of the divisor, KXAMPI^ES. 1. Given log75=1.8751andlog 15=1.1761; find log 5. 2. Given log 60=1.7782 and log 12=1.0792; find log 5. 3. Given log84=1.9243 and log 12=1.0792; find log 7. 4. Given log 435=2.6385 and log 317=2.5011; find 43.5 3 1 7 • 5. Given log y\x= 1.7761 and log 6=.7782; find log x. log -^-^^ 687. Logarithm of a Power. I,et 71 be any posi- tive number, and let \o'2«^). 10. log (I'-^ol^ + v'^^T). 6. log (3l/2^x5l'/272). II. logCl^aVT^^a^). 7. log (f/ 9-4-|/3). 12. log (1 ^8^1/2^). 8. log 1^" 12-- t/ 6). 13. log(2lKi^-T-l'V). 14. loe: — 7=. 15. loo: — . . - . 10. loo" i • 17. log[(^2#(^^^)*]. 20. log(^-^yY „ , /jrO'2^ 2\3 10^ /n , VaV?xi 19. logf-^^^YJ 22. log- , - 3 23. Prove that loga^= log.« CHARACTERISTIC AND MANTISSA. 689. For reasons which will appear later the common logarithm of a number is always written so that it shall consist of a positive decimal part less than 1 and an integral part which may be either positive or negative. When a logarithm of a number is thus arranged, special names are given to each part. The positive or negative integral part is called the Characteristic of the logar- ithm. The positive decimal part is called the Mantissa. Thus, in log 200=2.3010, 2 is the characteristic and .3010 the mantissa. 5i8 UNIVERSITY ALGEBRA. 690. Since 10^ = 1 and 10^ = 1, that is, since log 1=0 and log 10=1, it follows from Art. 679 that any number lying between 1 and 10 has for its logarithm a number lying between and 1; that is, a proper fraction. Thus, log 2=.3010, log 9=.9542, log 1.56=. 1931. Starting with the equation log 1.56=. 1931 we have logl5.6=log(l. 56x10) =logl.56 + logl0 =.1931 + 1. logl56.=log(l. 56x100) =logl.56+logl00 =.1931+2, Iogl560=log(1.56xl000)=logl.56+log 1000=. 1931+3, etc., etc., etc. lyikewise log. 156 =log(l. 56-10) =logl.56-logl0 =.1931-1, log.0156 =log(l. 56-100) =logl 56-loglOO =.1921-2, log.00156=log(1.56-1000)=logl.56-logl000=.1931-3, etc., etc., etc. We observe that log 15.6, log 156, log 1560, etc. have for their characteristics 1, 2, 3, etc. respectively, the mantissa of each of these logarithms being .1931. I^ike- wise log .156, log .0156, log .00156, etc., have for their characteristics —1, —2, —3, etc. respectively, the man- tissa being .1931, as before. Thus we see that the value of the characteristics of these logarithms is dependent merely upon the position of the decimal point in the number. 691. To Find the Characteristics. 1. The charac- teristic of the common logarithm oj a?iy number greater than unity is one: i,KSS than the number of figures preceding the decimal point, 2. The characteristic of the common logarithm of a num- ber less than 7mity is negative and numerically on:^ more than the number of zeros immediately following the decimal poi7it. LOGARITHMS. 519 To prove these statements we have merely to generalize the illustration given above. Thus, instead of 1.56, let us use a to stand for any number between 1 and 10, and let m be its logarithm. Then, since log 1 = and log 10=1, 7n must be a proper fraction, as .1931 in illustration above. Then we have log a=7n^ log 10^=;;/ + 1, log -^-^a — fn — l, log 100a=?;^^-2, log ^^-^a^m—^, log 1000^=;;z + 3, etc. log -^-^^-^a=m—o, etc. In this case the mantissa ot each logarithm is m, and the characteristics are respectively 0, 1, 2, 3, —1,— 2, —3. But, sin^e a is a number with one figure to the left of of the decimal point, 10^ is a number with two figures to the left of the decimal point, 100^ is a number with three figures to the left of the decimal point, and so on. Then we see No. of figures preceding Characteristic of decimal point in number. Logarithm. 1, 2, +1 3, etc. +2, etc. Thus we observe that the characteristic of the logarithm is always one less than the number of figures in the num- ber preceding the decimal point. Similarly, since « is a number with one figure to the left of the decimal point, -^^a is a number less than 1 with no zero immediately following the decimal point, -^^^a is a number with one zero immediately following the decimal point, y-^Vir^ ^^ ^ number with two zeros im- mediately following the decimal point, and so on. Then we have No. of zeros following Characteristic of decimal point in number. ■ Logarithm. 0, -1, 1, -2, 2, etc. —3, etc. 520 UNIVERSITY ALGEBRA. Thus the characteristic of the logarithm of a number less than unity is negative and numerically one more than the number of zeros immediately following the decimal point. Negative characteristics are sometimes written as in log .00156=3.1931, instead of as in log .00156=. 1931-3, but preferable to either of these is the notation log .00156 = 7.1931 — 10, made by adding 10 to the negative char- acteristic and then subtracting 10 to preserve the value unaltered. Some prefer to use the following statements in determining charac- teristics. The only difference from the above being in counting from units'' place instead of from the decimal point, which is to the right of units' place. The characteristic of the common logarithm of a numhS equals the number of places the first significant figure of the number is removed from units' place, and is positive if the first significant figure stands to the left of units' place and is negative if it stands to the right of units' place. Thus, log 1.56=0.1913, since first significant figure is removed places from units' place; log 1500=3.1931 since first significant figure stands 3 places to the left of units' place; and log .000156=171931 since first significant figure stands 4 places to the right of units' place. KXAMPI^KS. 1 . What numbers have for the characteristic of their logarithms? What numbers have for the mantissa of their logarithms ? 2. Find by inspection the characteristics of the log- arithms of the following numbers: 5123, 647152, 41.4, 257.752, 5, 5.5, .5. .0000089, .0010089. 3. Given log 1235=3.0917, write the logarithms of the following numbers: 12.35, 1235000, 1.235, .001235, 123.5, .1235, .00001235, .01235000. 4. Given log 476=2.6776, write down the numbers which have the following logarithms: 1.6776, 6.6776, LOGARITHMS. . 521 10.6776, 0.6776, 3.6776, 3.6776, 1,6776, 5.6776, 4.6776, 8.6776. 5. Given log 2=0.30103 and log 3=0.47712, find log 64; also log 182. 6. Show that log ii+log ffr-^ log |=log 2. , , 62x53x(75)^x4x3 .. ^ i k 7. Showthatlog ^^3 x20^ i-=i-log3-log 5. 14^x30 8. Show that log ^^^o""""^^^^""^^^^- g. Given log 2=0.30103, find Jog 1/1725. Hint: V1.25=V|=VJ30. 10. Given log 2=0.30103, find log (3.125)^ 11. Given log 1331 = 3.1242 and log 539 = 2.7316, find log 7 and log 11. Solution: 1331 = 11x11X11 = 1^ and 539r=llX7X7:=llX72. Hence. log 1331=3 log 11 and log 539=log 11 + 2 log 7. Hence we have 3 log 11 = 3.1242 and log 11+2 log 7=2.7310. Hence, log 11 = 1.0414. Hence. 2 log 7=1.6902. Therefore. log 7=0.8451. 12. Given log 144=2.1584 and log 324=2.5105, find log 2 and log 3. 13. Find an expression for the value of x from the equation 3''=567. Solution: Take the logarithm of each member and we have X log 3=log 567. But 567=7X44, therefore, x log 3=4 log 3 f-log 7. 4 log 3 + log 7 Hence i. e., x=4-\- log3 log 7 log 3 522 UNIVERSITY ALGEBRA. 14. Find an expression for x in the equation B"" =405. 15. Find an expression for x in the equation 3" X 2^+ 1 = 1/612. 16. Find an expression for x in the equation 5^+2x23 = 6"-ix2"-^i. 17. Given log 2=. 30103 and log 3=. 47712, find how many digits to the left of the decimal point in (f)^ ^ ^ ^. 18. Given log 2=. 30103 and log 3=. 47712, find how many digits to the left of the decimal point in 6^^. 19. If jK=10i^^^^^and<3'=10i-^«&-^show that.r=10i-^«g •^. 20. If the logarithms of a, b, c be respectively x, j/, 2, prove that a-^-"<^^-". 1 i^>j,- «i ao. log. d. 110. log. d. no. log. d. pp. 600 501 502 503 6990 8 9 9 8 550 551 552 553 7404 7412 7419 7427 8 7 8 8 600 601 602 603 7782 7789 7796 7803 7 7 7 7 650 651 652 653 8129 "873-6 8142 8149 7 6 7 7 6998 7007 7016 504 505 506 7024 7033 7042 9 9 8 554 556 7435 7443 7451 8 8 8 604 605 606 7810 7818 7825 8 7 7 654 655 656 8156 8162 8169 6 7 7 507 508 509 610 511 512 513 7050 7059 7067 9 8 9 8 9 8 q 557 558 559 560 561 562 563 7459 7466 7474 7 8 8 8 7 8 8 607 608 609 610 611 612 613 7832 7839 7846 7 7 7 7 8 7 7 657 658 659 660 661 662 663 8176 8182 8189 8195 6 7 6 7 7 6 7 7076 7084 7093 7101 7482 7490 7497 7505 7853 I 2 3 4 y 0.9 1.8 2.7 3-6 S 0.8 1.6 2.4 3-2 7860 7868 7875 8202 8209 8215 514 515 516 7110 7118 7126 8 8 Q 564 565 566 7513 7520 7528 7 8 8 614 615 616 7882 7889 7896 7 7 7 664 665 666 8222 8228 8235 6 7 6 7 8 9 4-5 5-4 6.3 7.2 517 518 519 620 521 522 523 7135 7143 7152 8 9 8 8 9 8 8 567 568 569 570 571 572 573 7536 7543 7551 7 8 8 7 8 8 7 617 618 619 620 621 622 623 7903 7910 7917 7924 7 7 7 7 7 7 7 667 668 669 670 671 672 673 8241 8248 8254 7 6 7 6 7 6 7 7160 7559 8261 7168 7177 7185 7566 7574 7582 7931 7938 7945 8267 8274 8280 524 525 526 7193 7202 7210 9 8 8 574 575 576 7589 7597 7604 8 7 8 624 625 626 7952 7959 7966 7 7 7 674 675 676 8287 8293 8299 6 6 7 527 528 529 630 531 532 533 7218 7226 7235 8 9 8 8 8 8 8 577 578 579 580 581 582 583 7612 7619 7627 7 8 7 8 7 8 7 627 628 629 630 631 632 ^33 7973 7980 7987 7 7 6 7 7 7 7 677 678 679 680 681 682 683 8306 8312 8319 ^25_ 8331 8338 8344 6 7 6 6 7 6 7 7243 7634 7642 7649 7657 7993 7251 7259 7267 8000 8007 8014 534 535 536 7275 7284 7292 9 8 8 584 585 586 7664 7672 7679 8 7 7 634 635 636 8021 8028 8035 7 7 6 684 685 686 8351 8357 8363 6 6 7 I 2 7 0.7 1.4 6 0.6 1.2 537 538 539 540 541 542 543 7300 7308 7316 8 8 8 8 8 8 8 587 588 589 690 591 592 593 7686 7694 _77oi 7709 8 7 8 7 7 8 637 638 639 640 641 642 643 8041 8048 8055 7 7 7 7 6 7 7 7 6 7 7 6 687 688 689 690 691 692 693 8370 8376 8382 8388 6 6 6 7 6 6 7 3 4 5 6 9 2.1 2.8 3.5 4.2 4-9 1.8 2.4 1.6 n 5.4 7324 8062 "806^ 8075 8082 7332 7340 7348 7716 7723 7731 8395 8401 8407 544 545 546 7356 7364 7372 8 8 8 594 596 7738 7745 7752 7 7 R 644 645 646 8089 8096 8102 694 695 696 8414 8420 8426 6 6 6 548 549 7380 7388 7396 8 8 597 598 599 7760 7767 7774 7 7 647 648 649 8109 8116 8122 697 698 699 8432 8439 8445 7 6 650 7404 8 600 7782 660 8129 / 700 8451 ]LOGARITHMS OF NUMBERS. no. log. d. no. log. d. uo. log. d. no. log. d. pp. 700 701 702 703 8451 8457 8463 8470 6 6 7 6 760 751 752 753 8751 5 6 6 6 800 801 802 803 9031 5 6 5 6 850 851 852 853 _9?94 9299 9304 9309 5 5 5 6 8756 8762 8768 9036 9042 9047 7 704 705 706 8476 8482 8488 6 6 6 754 755 756 8774 8779 8785 5 6 6 804 805 806 9053 9058 9063 5 5 6 854 855 856 9315 9320 9325 5 5 ■1 I 2 3 4 I 7 8 9 0.7 1.4 2.1 2 8 707 708 709 710 711 712 713 8494 8500 8506 6 6 7 6 6 6 6 757 758 759 760 761 762 763 8791 8797 8802 8808 8814 8820 8825 6 5 6 6 6 5 6 807 808 809 810 811 812 813 9069 9074 9079 9085 5 5 6 5 6 5 857 858 859 860 861 862 863 9330 9335 9340 9345 9350 9355 9360 5 5 5 5 5 5 5 3.5 4.2 4.9 5.0 6.3 8513 8519 8525 8531 9090 9096 9IOI 714 716 8537 8543 8549 6 6 6 764 765 766 8831 8837 8842 6 5 6 814 815 816 9106 9II2 9II7 6 5 864 865 866 9365 9370 9375 5 5 5 6 717 718 719 720 721 722 723 8555 8561 8567 6 6 6 6 6 6 6 767 768 769 770 771 772 773 8848 8854 8859 8865 6 5 6 6 5 6 5 817 818 819 820 821 822 823 9122 9128 9133 9138 9143 9149 9154 6 5 5 5 6 5 867 868 869 870 871 872 873 9380 9385 9390 5 5 5 5 5 5 5 I 2 3 4 I 9 0.6 1.2 1.8 2.4 3.6 4.8 5-4 8573 9395 9400 9405 9410 8579 8585 8591 8871 8876 8882 724 725 726 8597 8603 8609 6 6 6 774 775 776 8887 8893 8899 6 6 5 824 825 826 9159 9165 9170 6 5 874 875 876 9415 9420 9425 5 5 5 727 728 729 730 731 732 733 8615 8621 8627 8633 6 6 6 6 6 6 6 m 778 779 780 781 782 783 8904 8910 8915 6 5 6 6 5 6 5 827 828 829 830 831 832 833 9175 9180 9186 9I9I 9196 9201 9206 5 6 5 5 5 5 6 877 878 879 880 881 882 883 9430 9435 9440 9445 9450 9455 9460 5 5 5 5 5 5 5 1 2 3 4 5 6 5 0.5 I.O 1.5 2.0 2.5 3-0 3-5 8921 8639 8645 8651 8927 8932 8938 734 735 736 8657 8663 8669 6 6 6 784 785 786 8943 8949 8954 6 5 6 834 835 836 9212 9217 9222 5 5 5 884 885 886 9465 9469 9474 4 5 9 4.0 4.5 737 738 739 740 741 742 743 8675 8681 8686 8692 6 5 6 6 6 6 6 787 788 789 790 791 792 793 8960 8965 8971 8976 5 6 5 6 5 6 5 6 5 6 837 838 839 840 841 842 843 9227 9232 9238 9243 9248 9253 9258 5 6 5 5 5 5 5 6 5 5 887 888 889 890 891 892 893 9479 9484 9489 5 5 5 5 5 5 9494 9499' 9504 9509 1 2 3 4 It 1.2 8698 8704 8710 8982 8987 8993 744 745 746 8716 8722 8727 6 5 794 795 796 8998 9004 9009 844 845 846 9263 9269 9274 894 895 896 9513 9518 9523 5 5 K 4 I I 1.6 2.0 li 3.2 747 748 749 750 8733 8739 8745 8751 6 6 6 797 798 799 800 9015 9020 9025 5 5 6 847 848 849 850 9279 9284 9289 5 5 5 897 898 899 900 9528 9533 9538 5 5 4 9 U 9031 9294 9542 ( /530 LOGARITHMS OF NUMBERS. no. log. d. no. log. d. no. log. d. no. log. d. pp. 900 901 9542 5 5 5 5 950 951 9777 1000 lOOI 0000 1050 105 1 0212 9547 9782 0004 0216 902 9552 952 9786 1002 0009 1052 0220 903 9557 953 9791 1003 0013 1053 0224 904 9562 954 9795 1004 0017 1054 0228 905 9566 4 955 9800 1005 0022 1055 0233 906 9571 5 5 956 9805 1006 0026 1056 0237 907 9576 957 9809 1007 0030 1057 0241 908 9581 5 958 9814 1008 0035 1058 0245 909 910 911 9586 5 4 5 959 980 961 9818 1009 1010 lOII 0039 1059 1060 1061 0^49^ 0253 9590 9823 0043 9595 9827 0048 d257 5 912 9600 5 962 9832 1012 0052 1062 0261 913 9605 5 4 963 9836 1013 0056 1063 0265 z 2 0.5 I.O 914 9609 964 9841 1014 0060 1064 0269 3 1-5 915 9614 5 965 9845 1015 0065 1065 0273 4 5 2.5 916 9619 5 5 966 9850 1016 0069 1066 0278 6 7 3-0 3-5 917 9624 967 9854 1017 0073 1067 0282 8 4.0 918 9628 4 968 9859 1018 0077 1068 0286 9 4.5 919 920 921 9633 5 5 5 969 970 971 9863 1019 1020 1021 0082 1069 1070 1071 0290 0294 0298 9638 9868 0086 0090 9643 9872 922 9647 4 972 9877 1022 0095 1072 0302 923 9652 5 5 973 9881 1023 0099 1073 0306 924 9657 974 9886 1024 0103 1074 0310 925 9661 4 975 9890 1025 0107 1075 0314 926 9666 5 5 976 9894 1026 01 1 1 1076 0318 927 9671 977 9899 1027 0II6 1077 0322 928 9675 4 978 9903 1028 0120 1078 0326 929 930 931 9680 5 5 4 979 980 981 9908 1029 1030 103 1 0124 1079 1080 io8i 0330 0334 9685 9912 0128 9689 9917 0133 0338 932 9694 5 982 9921 1032 0137 1082 0342 933 9699 5 4 983 9926 1033 0I4I 1083 0346 ^ 934 9703 984 9930 1034 0145 1084 0350 I Yt 935 9708 5 985 9934 1035 0149 1085 0354 3 1.2 936 9713 5 4 986 9939 1036 0154 1086 0358 4 5 1.6 2.0 937 9717 987 9943 1037 0158 1087 0362 6 2.4 938 9722 5 988 9948 1038 0162 1088 0366 9 2.8 939 940 941 9727 5 4 5 989 990 991 9952 . 1039 1040 1 041 0166 1089 1090 1091 0370 3.2 3.6 9731 9956 0170 0374 0378 9736 9961 0175 942 9741 5 992 9965 ^ 1042 0179 1092 0382 943 9745 4 5 993 9969 1043 0183 1093 0386 944 9750 994 9974 1044 0187 1094 0390 945 9754 4 995 9978 1045 0I9I 1095 0394 946 9759 5 996 9983 1046 0195 1096 0398 947 9763 997 9987 1047 0199 1097 0402 948 9768 5 998 9991 1048 0204 1098 0406 949 950 9773 5 4 999 1000 9996 1049 1050 0208 1099 1100 0410 9777 0000 0212 0414 LOGARITHMS OF NUMBERS. 631 log. log. d. logr. log. pp. 1100 IIOI 1 02 1 103 104 1 106 1 107 108 1 109 1110 [III [112 [II3 [II4 [II5 [16 [I18 [II9 1120 1121 122 [123 1 124 125 1126 1127 128 [129 1130 1131 1135 136 1137 1138 1139 1140 1 141 1 142 143 [144 [145 146 [147 [148 [149 1150 0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 0453 0457 0461 0465 0469 0473 0477 0481 0484 0492 0496 0500 0504 0508 0512 0515 0519 0523 0522 0531 0535 0538 0542 0546 0550 0554 0558 0561 0565^ 0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 0607 1150 151 152 153 154 155 156 157 158 159 1160 161 162 163 164 165 166 167 168 169 1170 171 172 173 174 175 176 177 178 179 1180 181 182 183 184 185 186 187 188 189 1190 191 192 193 194 195 196 197 198 199 1200 0607 061 1 0615 0618 0622 0626 0630 0633 0637 0641 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 0682 0686 0693 0697 0700 0704 0708 07 1 1 0715 0719 0722 0726 0730 0734 0737 0741 0745 0748 07^2 0751 0759 0763 0766 0770 0774 0777 0781 0785 0788 0792 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 0792 0797 0799 0803 0806 0810 0813 0817 0821 0824 0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 0864 0867 0871 0874 0878 0881 0885 0S92 0896 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 ^± 0938 0941 0945 0948 0952 0955 0959 0962 0966 -0969 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 0969 0973 0976 0980 0983 0990 0993 0997 000 004 007 on 014 017 021 024 028 031 035 038 041 045 048 052 055 059 062 065 069 072 075 079 082 086 092 096 099 103 106 109 "3 116 119 123 126 129 133 _i36 139 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3-2 3.6 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 5352 ANTII.OGA RITHMS. log:. no. d. log. no. d. log:. no. d. log. no. d. log. no. d. pp. .000 .001 1000 .050 .051 II22' 3 2 .100 . lOI ^259^ 1262 3 3 3 3 .160 .151 I413 3 .200 .201 1585 4 3 1002 ^ 1 125 1416 1589 .002 1005 3 .052 II27 3 2 .102 1265 .152 I419 3 .202 1592 .003 1007 2 .055 1 130 .103 1268 .153 1422 3 4 .203 1596 .004 1009 .054 II32 3 . 104 1271 3 .154 1426 3 .204 1600 .005 IOI2 3 .055 II35 .105 1274 • 155 1429 .205 1603 . .006 IOI4 2 2 .056 II38 3 2 .106 1276 3 .156 1432 3 3 .206 1607 2 .007 IOI6 .057 1 140 .107 1279 .157 1435 .207 161I I .og8 IOI9 3 .058 1 143 3 .108 1282 3 .158 1439 4 .208 1614 2 .009 .010 .oil 102 1 2 3 .059 .060 .061 1 146 3 2 3 . 109 .110 .III 1285 3 3 3 .159 .160 .161 1442 3 3 4 1 .209 .210 .211 1618 1622 3 4 5 6 7 1023 1 148 1288 1445 1026 II51 1291 1449 T6^ .012 1028 2 .062 I153 2 .112 1294 3 .162 1452 3 .212 1629 8 2 .013 1030 2 3 .063 I156 3 3 .113 1297 3 3 163 1455 3 4 .213 1633 9 2 .014 1033 .064 II59 .114 1300 .164 1459 .214 1637 .015 1035 ^ .065 I161 .115 1303 3 .165 1462 3 • 215 1 641 .016 1038 3 2 .066 1 164 3 3 .116 1306 3 3 .166 1466 4 3 .216 1644 .017 1040 .067 I167 .117 1309 .167 1469 ,217 1648 .018 1042 ^ .068 1 169 ^ .118 1312 3 .168 1472 3 .218 1652 .019 .020 1045 1047 3 2 .069 .070 II72 3 3 .119 .120 1315 3 3 3 .169 .170 1476 4 3 .219 .220 1656 1660 I175 1318 1479 .021 1050 3 .071 I178 3 .121 1321 .171 1483 4 .221 1663 .022 1052 ^ .072 1 180 .122 1324 3 .172 i486 3 .222 1667 3 .023 1054 3 .073 1 183 3 3 .123 1327 3 3 • ^73 1489 3 4 .223 I67I I .024 1057 .074 1 186 .124 1330 .174 1493 .224 1675 2 3 4 I .025 1059 ^ .075 1 189 3 .125 1334 4 .175 1496 3 .225 1679 I .026 1062 3 2 .076 II91 2 3 .126 1337 3 3 .176 1500 4 3 .226 1683 5 6 2 2 .027 1064 .077 II94 .127 1340 .177 1503 .227 1687 7 2 .028 1067 3 .078 II97 3 .128 1343 3 .178 1507 4 .228 1690 8 2 .029 .030 .031 1069 2 3 2 .079 .080 .081 1 199 1202 2 3 3 .129 .130 .131 1346 3 3 3 .179 .180 .181 1510 3 4 3 .229 .230 .231 1694 9 3 1072 1349 1514 1698 1074 1205 1352 1517 1702 .032 1076 2 .082 1208 3 .132 1355 3 .182 1521 4 .232 1706 .033 1079 3 2 .083 I2II 3 2 .133 1358 3 3 .183 1524 3 4 .233 I7I0 .034 I081 .084 I213 .134 1361 .184 1528 .234 I7I4 .035 1084 3 .085 I216 3 .135 1365 4 .185 1531 3 .235 I7I8 .036 1086 3 .086 I219 3 3 .136 1368 3 3 .186 1535 4 3 .236 1722 .037 1089 .087 1222 .137 1371 .187 1538 .237 1726 .038 IO9I 2 .088 1225 3 .138 1374 3 .188 1542, 4 .238 1730 4 .039 .040 1094 3 2 .089 .090 1227 3 .139 .140 1377 3 3 .189 .190 1545 3 4 .239 .240 1734 I 2 3 4 I 1096 1230 1380 1549 1738 .041 1099 3 .091 1233 3 .i4i 1384 4 .191 1552 3 .241 1742 2 .042 I 102 3 .092 1236 3 .142 1387 3 .192. 1556 4 .242 1746 5 2 .043 1 104 2 3 .093 1239 3 3 .143 1390 3 3 .193 1560 4 3 .243 1750 3 .044 1 107 .094 1242 .144 1393 .194 1563 .244 1754 9 3 .045 1 109 2 .095 1245 3 .145 1396 3 .195 1567 4 .245 1758 .046 III2 3 2 .096 1247 2 3 .146 1400 4 3 .196 1570 3 4 .246 1762 .047 III4 .097 1250 .147 1403 .197 1574 .247 1766 .048 in7 3 .098 1253 3 .148 1406 3 .198 1578 4 .248 1770 .049 .060 1119 2 .099 .100 1256 3 3 .149 .150 1409 1413 3 4 .199 .200 1581 3 4 .249 .250 1774 1778 1122 , 3 1259 1585 I ANTILOGARITHMS. 533 log. no. d. log. no. d. log. no. d. log. no. d. log. no. d. pp. .260 .251 .252 .253 1778 4 4 5 4 .300 .301 .302 .303 1995 5 4 5 S .350 .351 .352 .353 2239 5 5 5 .400 .401 .402 .403 2512 2518 2523 2529 6 5 6 6 .450 .451 .452 .453 2818 - 7 6 7 6 1782 1786 1791 2000 2004 2009 2244 2249 2254 2825 2831 2838 T 4 .254 .255 .256 1795 1799 1803 4 4 4 .304 .305 .306 2014 2018. 2023 4 5 5 .354 .355 .356 2259 2265 2270 6 5 5 .404 .405 .406 2535 2541 2547 6 6 6 .454 •455 .456 2844 2851 2858 7 7 6 2 3 4 5 I T 2 2 .257 .258 .259 .260 .261 .262 .263 1807 181I 1816 4 5 4 4 4 4 ■i .307 .308 .309 .310 • 311 .312 .313 2028 2032 2037 4 5 5 4 5 5 .357 .358 .359 .360 .361 .362 .363 2275 2280 2286 5 6 5 5 5 6 .407 .408 .409 .410 .411 .412 .413 2553 2559 2564 6 5 6 6 6 6 6 .457 .458 •459 .460 .461 .462 .463 2864 2871 2877 7 6 7 7 6 7 7 7 8 9 2 3 3 4 1820 782^ 1828 1832 2042 2046 2051 2056 2291 2570 2576 2582 2588 2884 1 2296 2301 2307 2891 2897 2904 1 5 .264 .265 .266 1837 I84I 1845 4 4 4 .314 .315 .316 2061 2065 2070 4 5 5 .364 .365 .366 2312 2317 2323 5 6 5 .414 .415 .416 2594 2600 2606 6 6 6 .464 .465 .466 2911 2917 2924 6 7 7 2 3 4 i z 2 2 2 .267 .268 .269 .270 .271 .272 .273 1849 1854 1858 5 4 4 4 5 4 4 .317 .318 .319 .320 .321 .322 .323 2075 2080 2084 5 4 5 5 5 5 5 .367 .368 .369 .370 .371 .372 .373 2328 2333 2339 2344 2350 2355 2360 5 6 5 6 5 5 6 .417 .418 .419 .420 .421 .422 .423 2612 2618 2624 2630 6 6 6 6 6 7 6 .467 .468 .469 .470 • 471 .472 .473 2931 2938 2944 7 6 7 7 7 7 7 I 9 4 4 4 1862 2089 2951 1 1866 I87I 1875 2094 2099 2104 2636 2642 2649 2958 2965 2972 1 6 I .274 .275 .276 1879 1884 1888 5 4 4 .324 .325 .326 2109 2113 2118 4 5 .374 .375 .376 2366 2371 2377 5 6 .424 .425 .426 2655 2661 2667 6 6 6 .474 .476 2979 2985 2992 6 7 7 2 3 4 I 2 2 3 4 .277 .278 .279 .280 .281 .282 .283 1892 1897 I90I 5 4 4 5 4 5 4 .327 .328 .329 .330 .331 .332 .333 2123 2128 2133 5 5 5 5 5 5 5 .377 .378 .379 .380 .381 .382 .383 2382 2388 2393 6 5 6 5 6 5 6 .427 .428 .429 .430 •431 .432 .433 2673 2679 2685 6 6 7 6 6 6 6 .477 .478 .479 .480 .481 .482 .483 2999 3006 3013 3020 7 7 7 7 7 7 7 9 4 5 5 1905 2138 2399 2692 1 I9I0 I9I4 I9I9 2143 2148 2153 2404 2410 2415 2698 2704 2710 3027 3034 3041 I 7 1 .284 .285 .286 1923 1928 1932 5 4 4 .334 .335 .336 2158 2163 2168 5 5 5 .384 .385 .386 2421 2427 2432 6 5 6 .434 .435 .436 2716 2723 2729 7 6 6 .484 .485 .486 3048 3055 3062 7 7 7 3 4 5 6 2 3 4 4 .287 .288 .289 .290 .291 .292 .293 1936 I94I 1945 5 4 5 4 5 4 .337 •338 .339 .340 •341 .342 .343 2173 2178 2183 2T88" 5 5 5 5 5 5 5 .387 .388 .389 .390 .391 .392 .393 2438 2443 2449 5 6 6 5 6 6 s .437 .438 .439 .440 .441 .442 .443 2735 2742 2748 7 6 6 7 6 6 7 .487 .488 .489 .490 .491 .492 .493 3069 3076 3083 7 7 7 7 8 7 7 7 8 9 6 1950 ^55_ 2460 2466 2472 27S± 2761 2767 2773 3090 1 1954 1959 1963 2193 2198 2203 3097 3105 3112 I ? 8 I 2 .294 .295 .296 1968 1972 1977 4 5 5 .344 .345 .346 2208 2213 2218 5 5 5 .394 .395 .396 2477 2483 2489 6 6 6 .444 .445 .446 2780 2786 2793 6 7 6 .494 .495 .496 3119 3126 3133 7 3 4 5 6 2 3 4 5 6 6 7 .297 .298 .299 .300 1982 1986 I99I 4 5 4 .347 .348 .349 .350 2223 2228 2234 5 6 5 .397 .398 .399 .400 2495 2500 2506 5 6 6 .447 .448 .449 .450 2799 2805 2812 6 7 6 .497 .498 •499 .600 3141 3148 3155 7 7 7 8 9 1995 2239 2512 2818 3162 ANTIXOGARITHMS. log. no. d. log. no. d. log. no. d. log. no. d. log. no. d. pp. .600 .501 .502 .503 3162 8 7 7 R .550 .551 • 552 .553 3548 3556 3565 3573 8 9 8 8 .600 .601 .602 .603 3981 9 9 10 q .650 .652 .653 4467 4477 4487 4498 10 10 II TO .700 .701 .702 .703 5012 ^^ 78 3170 3177 3184 3990 3999 4009 S023 ^^ 5035 [I 5047 ;: III 212 322 4 3 3 .504 .506 3192 3199 3206 7 7 8 .554 .555 .556 3581 3589 3597 8 8 q .604 .605 .606 4018 4027 4036 9 9 10 .654 .655 .656 4508 4519 4529 II 10 10 .704 .705 .706 5 44 6 4 5 7 5 6 8 6 6 967 9 .507 .508 .509 .510 • 5" .513 3214 3221 3228 3236 3243 3251 3258 7 7 8 7 8 7 8 .557 .558 •559 .560 .562 .563 3606 3614 3622 3631 8 8 9 8 9 8 8 .607 .608 .609 .610 .611 .612 .613 4046 4055 4064 4074 9 9 10 10 9 Q .657 .658 .659 .660 .661 .662 .663 4539 4550 4560 4571 II 10 II 10 11 II TO .707 .708 .709 .710 .711 .712 .713 5093 ,, 5105 11 S117 5129 1 I 2 2 3639 3648 3656 4083 4093 4102 4581 4592 4603 3 3 4 4 5 4 6 5 .514 .516 3266 3273 3281 7 8 8 •564 .565 .566 3664 3673 3681 9 8 f) .614 .615 .616 41 1 1 4121 4130 10 9 TO .664 .665 .666 4613 4624 4634 II 10 TT .714 .715 .716 lilt - 5200 ;: 7 6 8 7 9 8 .518 .519 .520 .521 .522 .523 3289 3296 3304 7 8 7 8 8 7 8 .567 .568 .569 .570 • 571 .572 .573 3690 3698 3707 8 9 8 9 9 8 q .617 .618 .619 .620 .621 .622 .623 4140 4150 4159 4169 4178 4188 4198 10 9 10 9 10 10 q .667 .668 .669 .670 .671 .672 .673 4645 4656 4667 XI 11 10 11 II 11 TT .717 • 718 .719 .720 .721 .722 .723 5212 5224 " 5236 " 10 1 I 2 2 3311 3319 3327 3334 3715 4677 4688 4699 4710 5248 " 3 3 3724 3733 3741 5260 " 5272 " 5284 ;^ \ I 7 7 3 8 .524 .526 3342 3350 3357 8 7 8 .574 .575 .576 3750 3758 3767 8 9 n .624 .625 .626 4207 4217 4227 10 TO .674 .675 .676 4721 4732 4742 II TO TT .724 .725 .726 5297 ,^ 5309 11 S32I 11 3 9 11 .527 .528 .529 .530 .531 .532 .533 3365 3373 3381 8 8 7 8 8 8 8 .577 .578 .579 .580 .582 .583 3776 3784 3793 8 9 9 9 8 9 .627 .628 .629 .630 .631 .632 .633 4236 4246 4256 10 10 10 10 9 TO TO .677 .678 .679 .680 .681 .682 .683 4753 4764 4775 4786 4797 4808 4819 II 11 11 11 11 11 T? .727 .728 .729 .730 .731 .732 •733 5333 5346 3 5358 " , S370 - , 5395 " ' 5408 ;3 <. r" 1 2 2 3 3 ^ 4 3388 3802 4266 4276 4285 4295 \ 6 3 7 7 8 ^ 9 ) 10 3396 3404 3412 3811 3819 3828 .534 .535 .536 3420 3428 3436 8 8 7 .584 .585 .586 3837 3846 3855 9 9 • 634 .635 .636 4305 4315 4325 10 10 TO .684 .685 .686 4831 4842 4853 II 11 TT .734 .735 •73^ 5420 5433 5445 ;; ; 12 I I 2 .537 .538 .539 .540 .541 .542 .543 3443 3451 3459_ 3467 3475 3483 3491 8 8 8 8 8 8 8 .587 .588 .589 .590 .591 .592 • 593 3864 3873 3882 9 9 8 9 9 9 9 10 9 9 9 9 9 .637 .638 .639 .640 .641 .642 .643 4335 4345 4355 10 TO TO 10 10 10 .687 .688 .689 .690 .691 .692 .693 4864 4875 4887 4898 4909 4920 4932 11 12 II 11 11 12 TT .737 .738 .739 .740 .741 .742 .743 5458 ,, : 5470 , j 5483 " I 4 5 6 > 7 3890 3899 3908 3917 4365 5495 " ; 5521 5534 ]l 8 4375 4385 4395 11 13 I 3 4 .544 .546 3499 3508 3516 9 8 8 .594 •595 .596 3926 3936 3945 .644 .645 .646 4406 4416 4426 10 10 TO .694 .695 .696 4943 4955 4966 12 11 TT .744 .745 .746 5546 , 5559 \\ ^ 5572 3 .547 .548 .549 .550 3524 3532 3540 8 8 8 .597 .598 .599 .600 3954 3963 3972 .647 .648 .649 .650 4436 4446 445_7_ 4467 10 II 10 .697 .698 .699 .700 4977 4989 5000 12 11 12 .747 .748 .749 .750 5585 \ 5598 \\ « 5610 " 7 5 6 8 9 3548 3981 5012 5623 '^ 12 ANTirOGARITHMS. log. no. d log. no. d log. no. d. log. no. d log. no. d. pp. .750 .751 • 752 .753 5623 13 13 13 .800 .801 .802 .803 6310 14 15 14 .850 .851 .852 .853 7079 7096 7112 7129 .900 17 7943 19 18 18 19 .950 .951 .952 .953 8913 13(14 ^° I I I 21 2 3 3 20 3 4 4 Ji 4 5 6 5636 5649 5662 6324 6339 6353 7962 7980 7998 8933 , 8954 . 8974 . .754 .755 .756 5675 5689 5702 14 13 T3 .804 .805 .806 6368 6383 6397 15 14 T5 .854 .855 .856 7145 7161 7178 I -906 8017 8035 8054 18 19 18 .954 • 955 .956 8995 . 9016 ; 9036 ; .1 6 -I n 6 7 8 8 9 10 II •757 .758 .759 .760 .761 .762 .763 5715 5728 5741 5754 13 13 13 14 13 13 Trj .807 .808 .809 .810 .811 .812 .813 6412 6427 6442 15 14 .857 .858 .859 .860 .861 .862 .863 7194 7211 7228 r7 •9°7 .909 ^9I3 8072 8091 8110 T9 19 18 19 19 19 TO .957 .958 .959 .960 .961 .962 .963 9057 . 9078 ; 9099 ' I ^ 15 16 I I 2 2 I 2 3 3 3 4 5 I 4 6 7 1588 I 6 9 10 6457 7244 8128 9120 5768 5781 5794 6471 6486 6501 7261 7278 7295 8147 8166 8185 9141 , 9162 ■ 9183: .764 .765 .766 5808 5821 5834 13 13 .814 .815 .816 6516 6531 6546 15 T5 .864 .865 .866 7311 7328 7345 , , .914 , -915 1 .916 8204 8222 8241 18 19 TO .964 .965 .966 9226 ^^ 9 ,4 ,4^ 9247 ,, .767 .768 .769 .770 .771 .772 .773 5848 5861 5875 13 14 13 14 14 13 .817 .818 .819 .820 .821 .822 .823 6561 6577 6592 14 15 15 15 15 16 T5 .867 .868 .869 .870 .871 .872 .873 7362 7379 , 7396 7413 , -917 7 .918 ^ .919 ^ .920 ^ .922 8 -9^3 8260 8279 8299 19 20 19 19 19 19 7.0 .967 .968 .969 .970 .971 .972 .973 9268 9290 ^ 9311 9333 ' 2 17 18 J I 2 2 234 2355 14 7 7^ . 5 8 9* 2 6 10 II ^ 7 12 13 2 8 14 Id 5888 6607 8318 5902 5916 5929 6622 6637 6653 7430 7447 7464 ^ 8337 8356 8375 9354 1 9376 9397 : .774 .775 .776 5943 5957 5970 14 13 .824 .825 .826 6668 6683 6699 15 16 15 .874 .875 .876 7482 7499 7516 ; .924 ; .925 I -926 8395 8414 8433 19 19 .974 .975 .976 9419 9441 ^ 9462 ^ 9 1 2 I 7 11 5!i6 9 20 .777 .778 .779 .780 .781 .782 .783 5984 5998 6012 14 14 14 13 14 14 Ti] .827 .828 .829 .830 .831 .832 .833 6714 6730 6745 16 15 16 15 16 16 .877 .878 .879 .880 .881 .882 .883 7534 7551 7568 ' 7586 1 7603 ' 7621 ^ 7638 ; .927 7 .928 7 .929 ' .930 8 -931 .932 I -933 8453 8472 8492 19 20 19 20 20 19 .977 .978 .979 .980 .981 .982 .983 9484 ^ 9506 1 9528 ^ I 2 2 2 3 ^ 51 2 6 I 2 7 I 8 I 2 9 I 2 2 4 4 6 6 8 8 10 1 12 3 14 5 16 7I18 6026 6761 8511 9550 ' 9572 1 9594 1 9616 ^ 6039 6053 6067 6776 6792 6808 8531 8551 8570 .784 .785 .786 6081 6095 6109 14 14 T5 .834 .835 .836 6823 6839 6855 16 16 .884 .885 .886 7656 7674 ' 7691 ' .934 ' -935 \ .936 8590 8610 8630 20 20 .984 .985 .986 9638 9661 ^ 9683 1 5 2 2 I 2 2 1[22 2 2 .787 .788 .789 .790 .791 .792 .793 6124 6138 6152 14 14 14 14 14 15 .837 .838 .839 .840 .841 .842 .843 6871 6887 6902 16 15 16 16 16 16 t6 .887 .888 .889 .890 .891 .892 .893 7709 7727 ^ 7745 ' 7762 ' « -937 .938 ^ .939 B ''' R -941 .942 I -943 8650 8670 8690 8710 20 20 20 20 20 20 .987 .988 .989 .990 .991 .992 .993 9705 9727 1 9750 ^- 9772 ^ 3 ^ 4 5 5 I 2 ^ ^ 7 1 5 8 r z 9 ^ 5 7 B 9 D II 3 13 5 15 7 18 3 20 6166 6180 6194 6209 6918 6934 6950 6966 7780 ' 7798 ' 7816 ' 8730 8750 8770 9795 ^' 9817 : 9840 ^; .794 .795 .796 6223 6237 6252 14 15 .844 .845 .846 6982 6998 7015 16 17 16 .894 .895 .896 7834 7852 ' 7870 ' , -944 •945 ] .946 8790 8810 8831 20 21 .994 .995 .996 9863 9886 ^- 9908 - . I 2 '- 3 23 2 5 7 .797 .798 .799 .800 6266 6281 6295 6310 15 14 15 .847 .848 .849 .850 7031 7047 7063 16 16 16 .897 .898 .899 .900 7889 7907 ^ 7925 ' , .947 •948 •949 .950 8851 8872 8892 21 20 21 • 997 .998 .999 1000 9931 ^. 9954 ^' 9977 '- 0000 ^~ 4 5 6 9 9 12 14 16 18 21 7079 7943 ' 8913 536 UNIVERSITY ALGEBRA. COMPUTATION BY I^OGARITHMS. 695. Cologarithms. In order that any piece of work involving multiplication, division, involution, and evolu- tion may be performed by the addition of a single column of logarithms, the Cologarithm, instead of the loga- rithm, of a divisor is written down. The cologarithm, or complementary logarithm, of a number 7i^ is defined to be (10— log;^)-10. The part (10— log;e) can be taken from the table just as easily as log n, by begin7iing with the characteristic and subtracting in order all the figures of the logarithm from 9, except the last figure which must be taken from 10. Thus log 256 is given in table as 2.4082, whence •colog 256=7.5918-10. It is plain that the addition of {10— log ^^) — 10 is the same as the subtraction of log n. The convenience arising from this use may be illus- trated as follows : Suppose it is required to find x from the proportion 1193 : ;r=749 : 1^^977 We then have log 1193 == 3.0766 ^ log .697 = 9.9216-10 colog 749 = 7.1255-10 \ozx =0.1237 Whence .r= 1.329. EXAMPLES. Compute the values of the following expressions by use of logarithms: I. 256x311x451. 2. 704 x .21 x .0649. 7643x12.82 3. 864 LOGARITHMS. 53/ Instead of adding log 7643 and log 12.82 and subtracting log 864 from the sum, the work should be done as follows: log 7643 =: 3.8833 log 12.82 = 1.1079 olog 864 = 7.0635-10 2.0547 antilog 2.0547 = 113.4. 4. 61'^-r-17^ 6. 158#^0.39. 8. (^J|) 515\4 5. ^4158. 7. 4in/613. g. ^(^-|)'- In example 6, log 0.39-9.5911-10. Instead of ^ log 0.39 =1(9.5911-10) write ^ log 0.39 = 1(49.5911 -50)=9.9182-10. 10.^ (0.0641)o-o«4i^ 11^ 8.31-0-27. 12. (-0.412)~t. Solve the following exponential and logarithmic equa- tions : 13. 5^=10. 14. 3^-2 = 5. 15. 53-2-^35^+4^ 16. 23-+2^=5 17. log.,36=1.3678. 18. Iog,2=logio4.933. 19. Find the amount of $550 in 15 years at 5 per cent, per annum compound interest. 20.' What should be paid for an annuity of $100 a year for 40 years, money being supposed to be worth 4^ per cent? 21. A corporation is to repay a loan of $100,000 by 20 equal annual payments. How much will have to be paid each year, money being supposed worth 5 per cent ? 538 UNIVERSITY ALGEBRA. 22. A man pays $5 a month into a building associa- tion which nets 6 per cent, tor 8 years. What is the value of his stock at the end of that period ? 23. The population of the United States in 1790 was 3,930,000 and in 1890 it was 62,620,000. What was the average. rate per cent, increase for every decade of this period ? 24. Find the volume of a cone the radius of the base being 16.471 feet and the altitude 8.644 feet. 25. Find the volume and surface of a sphere whose radius is 11.927. 26. What is the weight in tons of an iron sphere whose radius is 11.927 feet if the weight of a cubic foot of water is 1000 ounces and the specific gravity of iron is 7.21 ? KXPONKNTIAL AND LOGARITHMIC S:e:rIKS. 696. The expression (IH — j may be expanded by the binomial theorem no matter what value x may have, provided only that — is less than unity; that is, provided n^\. Such expansion gives 1 7ix{nx—V) 1 7ix{rix -X){71X —^ 1 '^^n^ 172 '7^'^ TT.-g ^"^'^ , nx{n x—X)' ' '{jix—r-^X) 1 . + Yr_ ; n^^ ' ' ' ^^^ which may be put in the form (1+-) =n-^+_^-2-+ (■+s"-- 1.2.3 <^-J)- -(^-V) \r LOGARITHMS. 539 This being true for all values of x, we may put ;r= 1 . This gives v^n) -^+^+t:2 + — rxs — + • ■ • ■^ ""' ^ jlJ+... (2). Hence, from (1) and (2) xix ) xix ). . Ax I = 1+^+ -^2 + • • • "^ \? +••• (^> Since this equation is true for all values of n greater than unity, it is true if n increases in value without limit. 1 2 Whence, since the limit of — or — > etc, is as ;2 increases 71 n without limit, we have, by taking the limits of both members of (3) as n increases, (Art. 420), / 11 1 \'' (i+i+[2 + L3 + ---+jy+---)- The convergent series in the left member of (4) is usually represented by e, and is called the Natural or Naperian Base. Thus we have, for all values of x, This is the Exponential Series for the base e. S40 UNIVERSITY ALGEBRA. In equation (3) above it is plain that the numerator in the rth term M Yl J . . .M ~—] approaches 1 as a limit if r is finite. As r increases without limit we do not know from anything yet con- sidered that this infinite product approaches 1. But this is a matter that need not concern us, for as the series is convergent for all the values of //>1, its terms are known to ultimately decrease in value without limit. So we shall leave the question open whether the numerators are ultimately 1 or not, knowing that the terms ultimately decrease without limit. The value of e may be readily computed as follows : 2 1.000000 1.000000 3 0.500000 4 0.166667 5 0.041667 6 0.008333 7 0.001389 8 0.000198 9 0.000025 0.000003 Adding ^=2.718282 This happens to be correct to 6 places, as calculation using more places would show, but an error in the last place or two may be usually expected owing to the neglected figures. 697. Any Base. In equation [11] above substitute ex for X. We then have Let a=e^, so that also c=\og,a. We then have, by- substitution, x\\ogAy .^=l4-;.log.a+?-'(^"^^"^' • •+- [12] |2 r which is often called the Exponential Series or Theo- rem. LOGARITHMS. 54I 698. The Logarithmic Series is the expansion of logX^+x) in terms of the ascending power of ;r. Take the exponential series a^=l+ylog.a+^-^^^+^-^'^+ .. (1) Whence, transposing the 1 and dividing through byjc, _-=log.a+^-^- + -—+ . . . j (2) Therefore, since these variables are always equal, Whence it is easy to see Now put l+;t: for a. Then we have limit/O+fVjzi^ Expanding (l+xY by binomial formula, +^^=^^^-=^''+ ■ ) (=) in which x must not be greater than 1, since the binomial formula has been used. The limit of the right member as j^/^ can be plainly seen; whence we obtain the equation logXl+^)=^-^2-+ 3" 4-+ • • • + — ~^ + - ■ -[13] This is the Logarithmic Series. 699. Convergency of the Series. The above series is not convergent for values of x greater than 1 , and hence cannot be used for computing the logarithm of any integral , ..^ , limit/ (l+^)--l \ \ozl\-x)=-x — -- — o— — J- (2) 542 UNIVERSITY ALGEBRA. number but 2. The following scheme will give a series which is available for computing the logarithms of all integers. 700. Logarithmic Series Convergent for Integral Values of x. In the logarithmic series logXl + x)=x-^2' + ~3~-T+ ■ ■ ■ (^) substitute —x for x and we shall have "2" 3"~~4 Subtracting (2) from (1), observing that \-\-x log/l + -^0~^ogXl~--^)=lo§'^T3~' we obtain 1 , , ^ 2^+2 . 2^ Now put x—-^ — —T) whence !+;»;= 5 — -—,\-x=r: — — r> and:; = Therefore we obtain \—x z l+^_^/ 1 1 1 \ ^^^ ^ V2^+l'^3(2^+l/"^5(2^+iy"^' * J ^^ Whence, since log^ =log/l4-'2')— log^-s'. by substitut- z ing and transposing log^-a- we have logXl+^)=log^+2(^j + 3^J^^^3 + g^J^^^, + ..)(5) This series converges rapidly for integral values of z. Its use in computing the logarithms of numbers will now b)e explained. 701. To Compute the Natural Logarithms of Numbers. The logarithm of 1 is in all systems. To compute log^2, put z= 1 in equation (5) above. We then ^obtain LOGARITHMS. 543 l°^'2=2(l+3-L + 5-l,+^4-,+,4,+ . . .)=.6931472 Now put ^=2 in equation (5). Then we have log.3=.6931472+2(^+3^+g^ + ^+. . .)=1.0986123 To find log,4 we know log^4=log,22 = 2 log^2; whence log,4= 1.3862944 To find log^5, put ^=i in equation (5). We then have log,6=1.3862944+(i+gi^3 + g^+^+ ■ ■ .)=1.6094376 In like manner the logarithms of all numbers may be found. The logarithms of composite numbers need not be computed by the series, since the logarithm of any composite number can be found by adding the logarithms of its component factors. 702. Logarithms of a Number in Different Sys- tems. Consider the systems whose bases are a and e. Then, if n is any positive number, we wish to find the relation between log^^^ and logji. Let x=logen and y=\ogji. Then 7t=e^ and n^=^a^\ whence, e''=^a^. (1) X Therefore, a^e~y. (2) If we write this in logarithmic notation, we have log,^=y , (3) or, substituting the values of x and y, we obtain log^;^ ^ ^ Therefore loga^=:j log,;^, (5) 703. Modulus of Common Logarithms. If, in equation (5) above, we understand e to represent the 544 UNIVERSITY ALGEBRA. Natural base and a the common base, then equation (5) becomes log n=- rr^log^^^. (1) But log40=log,2+log,5=(by Art. 701) 2.3025851 and — j^=. 43429448. Therefore, representing .43429448 by M, we have log n=Mlog,n, (2) The decimal represented by M is known to 282 decimal places and is called the Modulus of the system of com- mon logarithms. Equation (2) is seen to express the important truth that ^ke common logarithm, of any ftumber can be obtainei by multiplying the natural logarithm of that number by thL modulus of the common system. 704. Computation of Common Logarithms. We can now compute the common logarithms of numbers. We merely need to multiply each of the Natural loga- rithms already found by the modulus .43429448- • . In this manner we find log 2=0.3010300 log 3=0.4771213 log 4=0.6020600 log 5=0.6989700 etc. etc. How can you find log 6 ? Historical Note. — The almost miraculous power of modem calculation is due to three inventions : the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the 17th century was admirably timed, for Kepler was then studying planetary orbits and Galileo had just turned his telescope toward the stars. It has been said that the invention of logarithms, "by shortening the labors, doubled the life of the astronomer." The honor of invention belongs to a Scotchman, John Napier, Baron of Merchiston. His first work, the Mirifici logarithmoru?n canonis LOGARITHMS. 545 descriptio, 1614, contains a table of natural sines and their logarithms to seven decimal places. Napier's logarithms were not the same as the Natural Logarithms. The base of his logarithmic number was not e, but nearly ^-i. The base required by his reasoning is exactly ^~i. It must be remembered, however, that Napier never connected logarithms with the idea of a base. This concept was introduced later. To us who know how naturally logarithms flow from the exponential symbol, it seems curious, indeed, that logarithms should have been invented before exponents had come into general use. Retnemher that integral exponents (as we now have them) ivere first used by Decartes, 1637, while negative and fractio7ial exponents luere first used by Joh?i Wallis in 1656, and literal exponents by Nezvton in 1676. Napier's* conception of logarithms was quite different from ours, and is contained in the meaning of the term itself, which comes from two Greek words meaning themunber of the ratios. This idea of a loga- tithm may be thus explained : Suppose the ratio of 1 to 10 be divided into a large number of equal ratios (or factors), say 1,000,000. Then it is true that the ratio of 1 to 2 is composed of 801030 of these equal ratios, and 301030, the nwjiber of the ratios, is the logarithm of 2. In the same way, the ratio of 1 to 3 is composed of 477121 of these equal ratios, and the logarithm of 3 is hence said to be 477121. His method of computing them, though ingenious, was tedious. In fact, the great work of computing logarithmic tables was completed before the discovery of the logarithmic series. Napier made log 10''=0 and caused his logarithmic figures to increase as the numbers themselves decreased. This is shown by the following arithmetic and : geometric progressions: Napier' s logaritlwis, 0, 1, 2, ... n. Numbers m, lo^(i-iif), lo'(i-i^) • ■ • io'(i-i[.)"- The relation between Napier's and natural logarithms is expressed by the following formula: 10^ log;v^.y=10^ \og — The news of the wonderful invention of logarithms induced Henry Briggs, professor of Gresham College, London, to visit Napier in 1615. Briggs suggested to him the advantages of a system in which the logarithm of 1 should be and the logarithm of 10 should be 1. Napier, having already thought of this change, encouraged Briggs to. 35 — U. A. 546 UNIVERSITY ALGEBRA. compute a system of new logarithms and made many important sug- gestions, among which was that of keeping the mantissas of all loga- rithms positive, by using negative characteristics. In 1G17 Briggs published the common logarithms of the first 1000 numbers, and in 1624 the common logarithm of numbers from 1 to 20000, and from 90000 to 100000, to 14 decimal places. The gap between 20000 and 90000 was filled up by Adrain Vlacq, of Gouda in Holland, who pub- lished 1628 the logarithms of numbers from 1 to 100000 to ten places. Vlacq' s table is the source from which nearly all the tables have been derived which have been published since. The first calculation of logarithms to the base of the natural sys- tem was made by John Speidell in his New Logarithms, published in London in 1616. Problem. — Prove that the base of Napier's logarithmic numbers is nearly ^- l , by taking (in the arithmetical and geometric progressions given above), ;?— lO'', dividing each term in both progressions by 10^ and calculating the value of the term in the new geometric progression, whose logarithm is unity.- Then deduce the formula, given above, expressing the relation between Napier's and natural logarithms. CHAPTER XXXI. COMPI.KX NUMBERS. 705. Arithmetical and Algebraic Numbers. Let us notice what is meant by Algebraic Number, and how the notion of it may originate. The primitive conception of number is used when we enumerate the marbles in a box, and say: 0, 1, 2, 3, 4, etc. Our simple scale is atithnietical number^ and it runs down to a definite nothing, or zero, and stops. But let us attempt to apply this scale in the measurement of other things. Suppose we are estimating time, where is the zero from which all time is to be measured in one ** direction" or sense? There is no such zero, as in the case of marbles ; for we can conceive of no event so far past that no other events precede it. We are forced to select a standard event, and measure the time of other events with refer- ence to the lapse before or after that. The zero used is an arbitrary one, and there is quantity in reference to it in two opposite senses, future and past, or, as is said in algebra, positive and 7iegative. We are likewise obliged to recognize quantity as extending in two opposite senses from zero in the attempt to measure many other j things; in locating points along an east and west line, no point is so far west that there are no other points west of it, hence could not be located in the arithmetical scale ; the same in measuring force, which may be attractive or reptdsive; or motion, which may be tozvard or fro ?n, etc. Thus our notion of algebraic quantity, as we name this kind of quantity, has arisen. 548 UNIVERSITY ALGEBRA. Because of the peculiar analogy between our notion of time and algebraic quantity, algebra has been called the science of pure time. All quantities are measured exactly as a past and future, or, graphically, along a line, in both directions from a zero point. Fixing our attention on any event, time exists in one sense (future) and in exactly the opposite sense ^past) and in no other sense at all. Likewise with algebraic numbers, we never get <9z^of the line. This kind of quantity, although more general than arithmetical number, is really quite restricted. We ob- serve, at once, that there is an opportunity of enlarging our conception of quantity if we can only get out of our line, or ''one-way spread'' as some say, and explore the region without. We may seek, then, an extension of our notion of quantity which will enabJe us to consider, along with the points of our line, those which lie without. 706. Numbers as Operators. We usually distin- guish in algebra between symbols of number or quantity and sy7nbols of operation. Thus a symbol which may be considered as answering the question "how many?" or *'how much?" is called a number or quantity, while a symbol which tells us to do something and which may be read as a verb in the imperative mood, is called a symbol of operation or simply an Operator. Thus, in V 2, when read "take the square root of two' ' , we distinguish readily the S3^mbol of operation from the symbol of number. Likewise in log 21 we may look upon "log," the symbol for "find the logarithm of", as a symbol of operation and 21 as a symbol of number or quantity. It is interesting to note that any number may be re- garded as a symbol of operation; and that thereby some original conceptions may be very conveniently extended. Thus, 10 may be regarded not only as ten, answering COMPLEX NUMBERS. 549 the question ''how many?", but as denoting the ^/(?r^- tion of taking unity, or whatever follows it, ten times; to express this, we may write 10 1, in which 10 may be called a teyisor, (that is, ''stretcher''^ or a symbol of the operation of stretchiyig a unit until the result obtained is ten fold the size of the unit itself. In the same way, the symbol 2 may be looked upon as denoting the operation of doubling unity, or whatever follows it; likewise, the tensor 3 may be looked upon as a trebler, 4 as a qiiadrupler^ etc. With the usual understanding that any symbol of operation operates upon what follows it, we may have com- pound operators like 2.2.3. Here 3 denotes that unity is to be trebled, 2 denotes that this result is to be doubled, and 2 denotes that this result is to be doubled. Thus representing the unit by a line running to the right, we have the following representation of the operators : The unit — > 3.1 —> — >—> 2.3.1 > > 2.2.3.1 > > Notice now the extended significance of an exponent. It means to repeat the operation designated by the base; that is, the operation designated by the base is to be performed, and performed again on the result, and per- formed again on this result, and so on, the number of oper- ations being denoted by the exponent. Thus 10 ^ means to peform the operation of repeating unity ten times (indicated by 10) and then to perform the operation of repeating the result ten times, that is, 10(10-1). Also, 10^ means 10[10(10 • 1)]. Then, of course, the exponent zero can only mean that the operation on unity denoted by the number is not to be performed at all; that is, unity is to be left unchanged; thus 10^ or 10^ • 1 = 1. The expression V 4, looked upon as a symbol of oper- ation, denotes an operation which must be performed 550 UNIVERSITY ALGEBRA. twice to qnadrnple; that is, such that (1^4) 2 = 4. lyikewise, f' 4 denotes an operation which must be performed three times in succession in order to be equivalent to quad- rupling. We know that the operation denoted by 2, if performed twice, is equivalent to quadrupling, therefore 1^4=2, etc. Just as 42, 4^, etc.^may be called stro7ige7 tensors than a single 4, so l/4, 1^4 may be called weake? tensors than the operator 4. The expression —1, looked upon as a symbol of oper- ation, is not a tensor, as it leaves the size unchanged of what it operates upon. But if applied to any quantity it will change the sense in which the quantity is then taken to exactly the opposite sense. Thus, if -f 6 stands for six hours after, then (— l)(-f 6) stands for six hours before a certain event, and —1 is the symbol of this operation of reversing. Also if (4-6) stands for a line running six units to the right of a certain point, then ( — 1)( + 6) stands for a line running six units to the left of that point, so that (—1) is the symbol which denotes the operation of turning a straight line through 180°. As 2, 3, 4, etc., were called tensors when looked upon as symbols of operation, we may conveniently designate the operator —1 as the reversor, 707. Imaginaries. We r em embe r that such expres- sions as 3-f 1/— 5, c+V^—d, j/ — 24, etc., were forced upon our notice in the solution of quadratic equations. It is customary to call such Imag inar ies, because of the presence in them of a term like V —a, which, evidently, does not correspond to any algebraic number whatever. But, remembering the restricted nature of algebraic num- ber, it is possible that such expressions are unreal only in an algebraic sense; that it the restriction can be re- COMPLEX NUMBERS. 551 moved by an extension of our conception beyond a mere linear or past and future notion of quantity, the expres- sion may, perhaps, become as ''real" as algebraic numbers now are. Although V — 1 cannot consistently with the meanings of V and —1 be looked upon as answering the question ''how many?" or "how much?" and therefore is not an algebraic number, yet if we consider it as a symbol of ope7'ation, it can be given a meaning consistent with the operators already considered. For if 2 is the operator that doubles and t/2 is the operator that when used twice, doubles, t hen if —1 is the operator that reverses, the expression V — 1 should be an operator that when used twice, reverses. So, as —1 may be defined as the .symbol which operates to turn a straight line through an angle of 180°, in a similar way we define the expression v — 1 ^^ that symbol which denotes the operation of turning a straight line through aji angle of 90° in the positive directio7i. It is customary in mathematics to consider rotation opposite to that of the hands of a watch as positive rota- tion. The restriction of positive rotation is inserted in the definition merely for the sake of convenience. 708. Graphic Representation, In figure 16, let a be any line. Then a operated on by l/ — 1, that is, V — \'a is a turned tip, or pos itiv ely, through 90°, which gives OB, Now, of course, V —\ can operate on V —\'a just as well as on a. Then l/— l(l/— 1 • a) or OC is is V —\ a o r OB turned posi tivel y through 90°. y ^[l/^(y — 1 • a)] is l/— l(l/—l . a) turned through 90°, etc. As we are at liberty to consid er tw o tur ns of 90° the same as one turn of 180°,.*. V —\{V —\ • d)=(^—X)a. 552 UNIVERSITY ALGEBRA. Also OD=i-V)OB, .■.On=-0/-l.a), but>^-l(-a) = 0B, .-. -(i/-l.«) = i/:^(_a). Thus the student may show many like relations. B I I _ \ \ <8 I Fig. i6. The operator v—1 is usually represented by the sym- bol / and will generally be so represented in what follows: KXAMPI^KS. Interpret each of the following expressions as a symbol of operation : 1. 2, 3, 4, -1. 2. 32, 23, 40, (-1)2, (-1)5. 3. i/2, 1/3, V^, i/2, r^. Select a convenient unit and construct each of the fol- lowing expressions geometrically, explaining the meaning of each operator: 4. 2-3-51. 7. (-1)2. 1/^1. 5. 23(-l)-l. 8. 22(-l)3(i/Zi)o.i, 6. 3t/^-2 1. 9. Sl/^(-l)l/^l. ' COMPLEX NUMBERS. 553 709. Laws. It is implied in the definition above that the operator i must follow the ordinar}^ laws of algebra as set forth in Arts. 107-113. The requirement of these laws completes the definition of i given above. The following are illustrations of each law: Commutative Law : c-\- di-\- a + bi= c-\- a + di-\- bi= di-\- c+ bi+ a, etc. az=ia^ iai=ua=au^ etc. The equation lOj/ -l-j/-! • 10, or better lO]/^-!-! = |/' — 1 . 10 • 1 may be said to mean that the result of perfonnUtg the operation of turning unity through OO'^ and performiiig upon the result the operation of takijzg it ten times is the same as the result of perform- ing the operation of takittg tinity ten tifues and performing upon this result the operation of turnitts^ through 90®. Associative Law : (c-\-di')-{-{a-\-bi^ = c+(di+d) + bi, etc. (ab')i==a(bi) = abi, etc. Distributive Law: (a-\-b)i=ai+bi, etc. 710. The special relat ion V~^'' = a\/-1, [1] which follows necessarily from the above, is very im- portant. In other symbols it may be written in which form it is seen to be an application of one of the index laws. It may be deduced from the laws above, as follows. (z^a") 2"= (iiad)'^=^ (m/a) 2"= \_{id) (z^)] 2"= ia. By me ans of this relation, we put expressions like V"'— 3, 1/— 4, V—b, etc., in the forms zl/3, 22, z|/^ etc. In what follows it is presupposed that all such ex- pressions are reduced to this form. The relation "l/-4 =2 j/ — 1 may be interpreted as follows: (—4) is the operator that quadruples and reverses; then j,/ — 4 is an opera- 554 UNIVERSITY ALGEBRA. tor which used twice quadruples and reverses. But 2v^ — 1 is an operator such that two such operators quadruple and reverse. That is, 711. Typical Form. It will be shown in the follow- ing theorems that any expression containing both real numbers and imaginaries may be put in the form a + bi, in which both a and b are real. The expression a + bi is therefore said to be the Typical Form of the imaginary. An expression of the form a-{-bi is also called a Com- plex Number, since it contains a term taken from each of the following scales, so that the unit is not single but double or complex : ._3, -2. -1,0, +1, +2, +3,... - . — 3^, —2/, — /, 0, +^; +2/, +3/, . . . It is important to note that the only element common to the two series in this complex scale is 0. 712. Graph of a Complex Number. Any real number, or any expression containing nothing but real numbers, may be considered as locating a point in a line. Thus, suppose we wish to draw the expression 2 + 5. Let O be the zero point and OX the positive direction. Lay off 0A = 2 in the direction OX and at A lay off AB=5 in the direction OX. Then the path OA + AB is the geometrical representation of 2-f 5. O A B X Any complex number may be taken as the representa- tion of the position of a point in a plane. For, suppose c-\-di\s the complex number. Let O be the zero point and OX the positive direction. Lay off OA=-\-c in the direction OX and at A erect di in the direction OY, instead of in the direction OX as in last example. Then c-\-di defines the position of the point P with reference to COMPLEX NUMBERS. S5S (9, and the path OA+AP, or OP, is a geometrical repre- sentation of c-\-di. In the same manner C'—diy —c—di and —c+di may be constructed. Y Fig. 17. 713. The meaning of some of the laws of algebra as applied to imaginaries may now be illustrated. I,et us construct c-^-di-^-a-i-bi, a c Y F\ G P '^ + I h\ B a c b ^ ^ E\ d D X J C A F IG. 18. The first two terms, c-\-di, give OA + AB, locating B. The next two terms, a + di, give BC+CP, locating P. 5S6 UNIVERSITY ALGEBRA. Hence the entire expression locates the point P with reference to O. Now if the original expression be changed in any manner allowed by the laws of algebra, the result is merely a different path to the same point. Thus: c+a + di-\-bt is the path OA, AD, DC, CP, (^+^)+ {d+b)i is the path OD, DP. a + dl-\-c^-biis> the path OE, EH, HC, CP. a + dz-{-bi+cis the path OE, EH, HE, EP, etc. The student should consider other cases. Are there any methods of locating P with the same four elements, which the figure does not illustrate? 714. Powers of i. We shall now interpret the powers of z by means of the new significance of an exponent and by the commutative, associative and other laws. First: i^ or /o • 1 = + 1. i^ or i^ ' 1 = /. z2 = — 1. i^ = pt = — /. i^=.iH'' = +1. i^ = iH = i. i^ = i^z = — 1. V = iH = — /. i^^Vi = + 1. etc. etc. Whence it is seen that all even powers of / are either + 1 or —1, and all odd powers are either / or —i. The student may reconcile this with figure 16. 715. Two complex numbers are said to be Conjugate if they differ only in the sign of the term containing V — 1. Such are x+ry and x—ry. COMPLEX NUMBERS. 557 716. Co77jugate iiuaginaries have a real sum ayid a real product. For {x +yi) + {x —yi) =x-j-yi+x—yi, by associative law. =x+x+j/z—j/i, by commutative law. = 2x+(yi—j/iX by associative law. = 2x-\-(j/—y')i, by distributive law. = 2x. Likewise (x +yi') (x—yi) =x(x—yz)-\-yi(x—yi), by distributive law. =x^'—xyi-\-yzx—yiyi, by distributive law. =x'^—-y'^P+xyz — xyz\ by commutative law. . =x'^+y'^-\-(xy — xy')z, by distributive law and by substituting P = — l. - =x'^-\-y'^. It is well to note that l/ze prod zed of two co7zjzigate cozn- plex numbers is always positive and the sum of two squares. 717. The sum, prodzid, or quotient of two coznplex numbers is, i7z geiieral, a co7nplex ?zzi7nber of the typical for7n a + bi. Let the two complex numbers be x+yi and zc-[-vi. (1) Their sum is {x-\-yi) + {zi-\-vi) . =x-\-yi-{-ti + vi, =x-\-u-\-yi+vi, = (x-i-u') + (y+v)i, by the laws of algebra. This last expression is in the form a-\-bi. (2) Their product is (;t:+j'0(?^ + 2;/) = x(u + vi) -\-yi(zi + vi) , = xu + xvi+yiu -\-yivi^ = xu -i-yvi'^ + xvi+yui, = (xu —yv) 4- (xv -\-y2c)i, by the laws ot algebra. This last expression is in the form a + bi. 558 UNIVERSITY ALGEBRA. (3) Their quotient is X -\-yi (x -\-yi ) (ji — vi) u -f vi (ii-[- vi) (jc — vi) B}^ the preceding-, the numerator is of the form a'-^b'i. By Art. 716, the denominator equals ti'^-\-v'^. Then the quotient equals a'-\-b'z ^' _L ^' • by distributive law. This last expression is of the form a H- bi. KXAMPI.ES. Reduce the following expressions to the typical form a-\-bi: , 3. (;^_[2 + 3z])(x-[2-3/]). 4. (_5+l2l/:I:l)^ 6. (v'i+^-)(i/i^*). 5. (3«4i/i:i)2. 7. (i/7_ 1/1:7)2^ Q ^ 1 2 1-23 9. ;rTT7=^- ^3 10. 3 + 1/--2 (1-0' I-1/-7' ^"^^ l+2i/^* _ a+xz a — xz 10. : ; ;• a — xz a+xz COMPLEX NUMBERS. 559 718. If cin imaginary is equal to zero, the imaginary and real parts are separately equal to zero. Suppose x-^yV — 1=0 then x= — yV —1. Now it is absurd for a real number to equal an imagin- ary, except they each be zero. Therefore x=0 and y=0, 719. If two imaginaries are equal, then the real and the imagi?iary parts must be respectively equal. For if X'\-yi=:^u-\-vi then {x-^ii)-\-(^y—v)i=0. Whence, by Art. 718, x—u-=0 and j/— z/=0. That is, x=u and y=v. MODUIvUS AND AMPI^ITUDK. 720. lyet the complex number x-\-yi be constructed, as in figure 19, in which OA=x and AP=y. Draw the line OP, and let the angle A OP he called 0. y p "T Fig. 19. 721. The numerical length of OP is called the Mod- ulus of the complex numbe r x-\-yi. It is algebraically represented by -\- V x'^ -\-y^ , in which the sign + is placed before the radical to show that merely the nu- merical value of the sq uare root is called for. Thus, mod(3 + 40= + 1/9 + 16=5. 560 UNIVERSITY ALGEBRA. The student can easily see that two conjugate complex numbers have the same 'modulus^ which is the positive value of the square root of their product, \iy—^, the mod {x-^yi) — \/x^ = ^x. Thus the modulus of any real ^umber is the same as what is called the niunerical or absolute value of the number. Thus, mod (—5) =5. 722. In Fig. 19 the angle A OP or is called the Argument or Amplitude of the complex number x+yi. Putting r= + Vx'^-\-y'^ = VLioA(^x-\-yi), we have sin6'=-, cos^=— r r Therefore, x+yi=r cos 0-{-tr vSin ^=r(cos O + i sin 9) in which we have expressed the complex number x+yi in terms of its modulus and amplitude. To put 3—4/ in this form, we have .y^ 4- ^^./3 ^ o mod (3-40 = '/9 + 16=5; sin e=^=-~\ cos r b r o Therefore, (3-40 = 5(f-4)/. The amplitude of all positive numbers is 0, and of all negative numbers is 180'^. The unit expressed in terms of its modulus and amplitude is evidently l(cos 0+^ sin 0). 723. The point P, located by OA+AP or x+yi, may also be considered as located by the directed line OP-, that is, by a line starting at O, of length r and making an angle with the direction OX. A directed line, as we are now considering OP, is called a Vector. When thus considered, the two parts of the compound operator r- (cos ^+/sin 0) - 1 receive the following interpretation: The operator (cos + i sin ^), which depends upon alone, turns the unit through an angle 0. The operator r is a tensor, which stretches the turned unit in the ratio r: 1. The COMPLEX NUMBERS. 561 result of these two operations is that the point P is located r units from 6> in a direction making the angle ^ with OX. Thus, the operator (cos B-\-i sin ^) is simply a more general operator than /, but of the same kind. The operator i turns a unit through a right angle and the operator (cos 0-\-i sin &) turns a unit through an angle B. If be put equal to 90°, cos 0-\-i sin reduces to /. For ^=0, cos ^+/sin reduces to 1. ^=90°, cos ^+2* sin B reduces to /. ^=180°, cos <9-j-2 sin B reduces to —1. ^=270°, cos B+i sin B reduces to — /. Since 3— 4^=5(f— fz) the point located by 3 — 4^' may be reached by turning the unit an angle ^=sin~^(-f) =cos~^ f and stretching the result in the ratio 5:1. 724. If d complex numbef vanishes, its modulus van- ishes; and, conversely, if the modulus vanishes, the complex number vanishes. 1{ x+yi=0, then x=0 andj/=0, by Art. 718. There- fore, l/x^-i-y'^=0. Also,if y ^^M^=0, thenx'^+j;^=0, and since x andj/ are real, neither x^ nor j/^ are negative,, and so their sum is not zero unless each be zero. 725. 1/ two complex numbers are equal, their moduli are equal, but if two moduli are equal, the complex 7ium- bers are not 7iecessarily equal. If x-\-yi=u-\-vi, then x^u and y-=v by Art. 719. Therefore, V x'^^-y'^ = V u''-'-\-v'^ . But if l/.a;2+_y2==|/^2_j_^2^ ^2 jieed not equal tc'^ nor y'^=zv fi-U. A. 562 UNIVERSITY ALGEBRA. 726. The modulus of the sum of two or more complex niiiubers is never greater than the sum of their moduli. If ^^ = Xi-j-y^i and ^2=-^2+>'2^* t)e any two complex numbers, we are to prove that mod (-2'i-f 2'2)>mod -s'l+mod ^2> or+l/(^l+-^2)V(J^l+J^'2)^> + ■^■^l^4-JKl^ + ^/:r2 24-J^^2^ but since the square roots are all positive it is sufficient to prove that or that ^1^2 +^1 ^2 > ^'^1 ' -+-j^ ' X "^^-^2 ' +JK2 ^ or, again squaring, we must prove that x^ 2^2 ^ +2^1-^2:^1 JK2 +yi V2 '^ >^i 2^2 ^ +-^2 ^^1 ^ +-^1 ^^2 ^ +J^i Ij^2 ^ or that 0y>X2^yi'^—2xj^X2yiy2+^i^J^2^f or that 0>Cr2jri— ^lJK2)^ which is obvious, since x^, x^, yi and y^ are real. Likewise, mod (^2^+22+2z)>mod 2^+mod {s^+^z) >mod ^1 +mod <2'2 +mod 2:.^, 727. The product of tzvo or more complex numbers is a complex number whose Tnodulus is the product of the moduli and whose ajnplitude is the sum of the amplitudes of the complex numbers. Let the complex numbers be 2^=^x^-\-y^i=r-^(sos 0^-\-ism 0^) ^2=;i;2+jK22'=^'2(cos ^2+^* sin 0.f), etc. We are to prove that mod -a' ^-s* 2= mod 2^ mod 2 2, and that amp 5'i2'.j = amp ^i+amp 2^. COMPLEX NUMBERS. 563 By actual multiplication, we get 2'^^^-rr[(cos^^cos^^— sin^^sin^^) + (sin^^cos^^+cos^^sin^J/] = ^iV2[cos(^i+^2) + ^* sin(^i+^2)]. Whence it is seen that r^ r 2 is the modulus of the product and (^1 + ^2) is the amplitude. Also mod ^i2'2-s'3=mod ^1 mod -3'2'2'3, but mod ^2'2'3=inod ^2 ^^d ^3. Therefore/ mod z^2,^2^=mo6. 2^ mod 22 mod 2^, etc. Likewise amp 2'i^2'2'3 = amp 2 ^+2im^ 2^,2^, but amp ^2-^3==^^? -^2 + ^^? '^'3. Therefore amp -3'i^2'2'3=amp -s'^+amp ^2+^^P '2'3,etc. 728. The quotient of two complex numbers is a complex number whose modulus is the quotient of the moduli and whose amplitude is the difference of the amplitudes of the two complex numbers. Let the complex numbers be ^j=;trj+jKii5*=^i(cos ^i+/sin B^ 22=X2+y2^===^2(^os O^ + i sin O^) we are to prove that ^^1 mod 2^ mod— = :; -y 22 mod 22 and that amp— = amp ^i —amp . 3*2 we have ^^__ri(cos ^1 4-2 sin ^i)(cos ^2~^*sin ^2) ^2 ^2 (cos ^2+^ sin ^2)(cos ^2~^ sin ^2) ^ r,[cos (^,-^2) + / sin(^i-6>2)] r2(cos2^2+sin2(92) " =^[cos((9i-^2) + ^*sin((9i-6>2)], ^2 Whence it is seen that — is the modulus ol the quotient and (^1—^2) is the amplitude. 564 UNIVERSITY ALGEBRA. 729. De Moivre's Theorem. As a special case of Art. 727, consider the expression (cos 0-^z sin Oy This being the product oi .n factors like (cos 0+z sin (?), we write, by means of Art. 727, (cos 0+z sin ^)(cos 0-\-i sin 0) • - - = [cos(<9+(9-|- . . .)-f ^' sin((9+(9+ . . .)] or (cos ^+2 sin ^)"=(cos >^^H-/ sin ?iO) [3] which relation is known as De Moivre's theorem. DeMoivre's theorem holds for fractional values of n. For, first consider the expression 1 (cos ^+/sin oy A Put ^=^<3^, SO that <^=-- 1 1 Then (cos ^+/sin ^)^ = (cos i)' = [(cos -\-i sin )^ ' by [3] =cos +i sin «=cos -+2 sm -• Next consider the case in which n=-' We know (cos 0+ i sin B) \ = [(cos B-^i sin ^)^] ' «=(cos ^^ + /sin^^)' sB ^ , , sB = cos — +2 sm-- I^ikewise, the theorem may be proved for negative values of n. COMPLEX NUMBERS. 565 KXAMPI.KS. 1. Find value of (-1-f l/^)^ + (-l-l/^)^ by De Moivre's theorem. 2. If ;tr^= cos^ + / sin^>prove that limit [^i^r 2^3... :r J as r increases = cos it. 3. Find the value oi x'^—2x-\-2 for x=\-\-i, 4. If yi = — 4-+|^l/+3 and y2 = —i— i^/^, show inai 7i — i, J2 — ^y J\ --V2> 72 yi) yi 72 — -"^j y 3«+i — y CHAPTER XXXII. TH^ RATIONAI, INTKGRAI, FUNCTION. 730. Variable and Constant. A Variable is a number or quantity existing under such a law or suppo- sition that it has an unlimited number of values. A Constant is a number or quantity existing under such a law or supposition that it has a fixed value. , Thus the volume of mercury in a thermometer is a variable and its weight or mass is a constant. 731. Function of a Number. A Function of a number is a name applied to any mathematical expres- sion in which the number appears. Thus, •^ x^—S * are all functions of x. In the same manner we speak of functions of several numbers. The second expression above may be called a function of x and y. Obviously, a function of a number might be otherwise defined as any expression which depends upon the number for its value. 732. Rational Integral Function. A Rational Function of a number is one in which the number is not involved in a radical or is not affected with a fractional exponent. An Integral Function of a number is one in which the number does not appear in the denominator of a fraction or is not affected with a negative exponent. X — ^ Thus x^ '-'2x'^y~^ -\-h—^ — 6>/ ^ is a rational function of x, an integral function of x, an irrational function of jj/, a fractional function of ^. RATIONAL INTEGRAL FUNCTIONS. 56/ A function may be both rational and integral, in which case it is called a Rational Integral Function. If n is a positive whole number and a^, a^, a^y . . . «« stand for any real numbers whatever, then is a general expression representing any rational iJitegral function ofxof the nth degree, if we confine ourselves to the case of real coefficients. Such expressions 2,s function of x, function of a, func- tion of x-\rh, etc., are abbreviated into F{x), F(a), F{x+h)y or fix), f (a), f(^x+h), or a similar expression. It must be kept well in mind that F, f etc., are not coefficients, but abbreviations for the words ''function of.'* If/(;r) and/(^), or 7^(^) and F{a), occur together in the same discussion, f{a) stands for what /(jt) becomes when a is put for x. Thus, \^ f{x) is x^—lx'^-{-2x-\-4:, then/(a)is ^'^—T^^ ^2^+4, and /(2) is 2=^ --7 -22 + 2 -2 + 4 or —12. In the vSame way, /(;»;+/2) stands for what/(jr) becomes when {^x-\-h) is put for x. 733. Notation. The symbol /(jr) will be used throughout this chapter to stand for a rational integral function of x. A function which is not both rational and integral must be represented by one of the symbols F{x), ^{x^y etc. If we suppose f{x^ to be divided through by the coeffi- cient of the highest power of x, then the following will represent any fix) : If none of the above coefficients are zero, the function is said to be Complete. The term ^„, or />„, is called the Absolute Term. 568 UNIVERSITY ALGEBRA. 734. Continuous and Discontinuous. A Contin- uous Variable, in passing from one value to another, passes over every intermediate value. Thus, if x^ in passing from 2 to 7, passes over every intermediate value, then X is said to be continuous between the limits 2 and 7. The repeating decimal .33333- • . is an illustra- tion of a Discontinuous Variable, as in passing from .33 to .333333 it skips all intermediate values except .333, .3333, and .33333. 735. Identities. Two expressions that are equal for all values of the letters involved are said to be Iden- tically Equal or Identical. See Art. 214. The symbol for identity is =. Thus, we write (x-f<2)(^—^) = ^2—<22. A very convenient use of the symbol = is, in writing ''let /(x') = x^—4:ax + a'^j'' instead of "let /(x) stand for x'^-4ax-{-a\'' 736. Roots. Any real number or imaginary, which substituted for x in/(x) makes /(:r) vanish (that is, equal to zero) we shall call a Root of /(:r). Thus, 1, 2, and 3, are roots of x^ — 6x'^ +11x^6 and 2, 2+3/, 2— 3/ are roots of^3-6:r2-5;r-26. 737. A Rational Integral Equation containing one unknown number is one which can be placed in the form /(;t)=0; that is, in the form af)X"-ha^x''-'^ -{-a^x"-'^ +a^x''-^ ^ • • • +a,,_i^+a„=0. (1) A rational integral equation may also be represented by x^+p,x'^-^ 4-^2^-2 +^3^«-3 + . . . +^_^^+^^ -0 (2) since the equation (1) is unchanged if we divide through by the coefficient of x"". When the equation /(x) = is written out in either of above forms it is commonly spoken of as the General Equation of the 7zth degree. RATIONAL INTEGRAL FUNCTIONS. 569 " 738. The difference between f{x) and f(a) is exactly divisible by x—a. We are to prove -^—^ — ^^-^ = a quotient without a re- mainder. Now f(x) = aQX" + a^x''~'^+aci^x''~'^+ - • • +a„_^x+a„ and y*(a) = (2Qa" + ^ia"~i+<^2^"~^+ • • • +<^«-iOt+^«. Therefore, - ^ ^ ^ -^ ^ ^ = X — a Jf— a equals some quotient without a remainder, since the dif- ference of like powers is divisible by the difference of the numbers themselves, by Art. 136. 739. When fix) is divided by x—a, the first remainder that does not contain x is equal to /(a). As an illustration, divide x^ — ^x'^+llx—Q by x—a, as follows : x^—6x^ + llx-6 X 3-_n X—a ^2 + (a~6>"+(a2-6a-fll) (a—6)x'^ — (a'^—6a)x (a2-6a+ll>-6 • (a2-^6a+ll).r-(a^— 6a2 + lla) " a3_-.6a2 + lla— 6 Thus, the first remainder which does not contain x is 1^— Ga^H-lla— 6, which can be made from x'^ — ^x'^ + llx — 6 by putting a for x. TTo prove the theorem, we have, from Art. 738, fix) -/{.-)_ x—a - = quotient, no remainder. That is, A^_) _/(.«)_ =q,,otient. X—a X—a ^ Therefore -^-^ = quotient +*^-^ X — a X — a Therefore /(a) is the remainder. 570 UNIVERSITY ALGEBRA. 740. Any fix) is exactly divisible by x mi7iiis a root, . Let the root be a. Now from Art. 739/-^-^^ — ^^-^— = quotient, no remainder. X — a But since a is supposed to be a root,y*(a)=0 by Art. 736. Therefore, ^= quotient, no remainder. We know 1, 2, and 3 are roots of x^ —^x^ -\-\\x — 6. Therefore, by this theorem, x^ —^x^-\-\\x-^ is exactly divisible by Jt — 1, x — 2, and x — ^. 741. Conversely, if xny f(x) is exactly divisible by X — a, then a is a root of f{x). Since f{x) is exactly divisible by x—a, we know that x—a. is one factor of/(;t:). Representing the other factor by <^(;r), we have f{x)~{x—a) (x). Substituting a for x, we get /(a) = (a-a)(a) = 0. Since f(x) becomes zero when a is put for :r, a is a root of /(;r) by definition. 742. The above theorem is sometimes useful for indi- cating the factors of an expression, as in the following examples: (1). Factor {d-c){d-\-c)^ + {c—a){c-\-ay -\-{a—l>){a-\-d)^. Put^=<:. The expression becomes (^ — rt!)(<:+«)2 + («--^){rt!4-^)^, or 0. Since the original expression vanishes for d:=^c, it is by Art. 740, divisible by ^—c. Likewise it will be found to be divisible by c-^a and d — a. Whence, we have Since each side of this identity is of the third degree, Z is an undeter- mined number independent of a, b, and c. On the left side the coeffi- cient of b'^c is —1. On the right side the coefficient of b'^c is — Z. Therefore. Z=:l. RATIONAL INTEGRAL FUNCTIONS. 57 1 (2). Factor {d-c)^-\- {c-a)^-t {a- d)^. This expression is zero for d=c, c—a, a—b. Therefore, by Art. 740, it is divisible by {b—c){c—a){a—b). But the expression being of the fifth degree it must have a remaining factor of the second degree, which maybe represented by La^-{-Lb^ ■\-Lc'^-\-Mbc-\- Mca-\-AIab, in which Z«and J/ are undetermined coefficients. Therefore, we have {b—a)^ + {c—a)^-{-{a- b)^ ={b—c){c—a){a-b){La^-^Lb^-\-Lc'^ + Mbc + McaA-Mab). The coefficients of «* on the respective sides of this identity are seen to be given in {^c-bb)a^--L{b-c)a'^ Whence Z=:4-5. The coefficients of b^c^ are likewise seen to be given in + lQb^c^ = {-M+L)b^c^, whence —M-^tL^^O, or M= — ^. Therefore the factors of the original expression are b{b-c){c—a){a-b){c^ + b'^ + c^ — bc-ca—ab). EXAMPLES. Show by Art. 740 that the following expressions are divisible by (b—c)(c—a)(a — b). Remaining factors may be found by the principle of undetermined coefficients, if desired. 1. bc{b—c) + ca(c—d) + ab(a—b), 2. (^b—cy -{-(c—ay -^{a—by . 3. (b—c)(a — b+c)(a-{-b—c)-^(c—-d)(a^b—c)(^-a-\-b-\-c) + (a--b){'-a-\-b^c){a — b+c). 4. a^{b-c)-^b^(c—a) + c^{a—b). 5. a{^b-cy ^b{c-ay +c(a-by\ 6. (^b-'C)(ib-\-cy + (c-a)(ic+ay-^{a-^b){a + by. 7. b''c''(ib-c)-\-c''a''(jr-a) + a''b''(a-b), 8. a^{b—c')-^b\c-a)-{-c^(^a—b). 9. {b-cXb-^-cY^^c-a^tic+ay + ^a-bXa + by. 572 UNIVERSITY ALGEBRA. 10. a(d—c)^ + 5(c—a)^+c(a'-d')^. 11. a^(d—c)-i-d^(c--a) + c^(ia—d). 12. d^ c^ (d—c) + c^ a^ (c—a) + a^ d^ (a — d"). Show, by Art, 740, that the following expressions are divisible by (<^+r)(<;+<^)(<^ + ^): 13. (a + d+c)^—a^ — d^ — c^, 14. (a + d+c)^—a^—d^—c^. 15. Show that a4(^^-^') + <^^(^^-^^)+^^(^'-~^^) 16. Show that (a + d+cy — (id + c-'ay — (ic+a-d)^ — (a-{-b—c)^ is divisible by abc, 743. Synthetic Division. We shall now explain a short method of dividing any f(^x) by x—a. For con- venience suppose the f{x) to be for it will be plain that the process to be explained will be applicable to a polynomial of any degree whatever. It is known that the quotient oi f{x)-h-{x—a) will be of one lower degree than /(.r), therefore we may assume. a^x^ ■\-a^x^-{-a^x^-\-a^x'^-\-a^x-\-a^ X — a X — a in which A, B, C, etc., are undetermined coefficients. Put R for Remairider. Then a^x^ '\-a^x^-\-a.^x^ +a^x'^-\-a^x+a^ = {x--a\{Ax^ + Bx^ + Cx- -\-Dx + E + -^ \ ^Ax^+(B—aA')x'-j-(iC-aB)x^ + iiD'-aC)x'+(£'-aD)x+R-aB, RATIONAL INTEGRAL FUNCTIONS. 573 Equating coefficients of like powers of x, by Art. 595, we get A =^0 A= a^ B—aA=a^ B=aA-{-a^ C-aB^a^ Therefore C=^aB+a^ Z7-aC=«3 ^^eretore, D=aC+a^ R—aE=a^ R=aE-\-a^ Arranging the right hand column of equations horizontally : Coefficients in dividend, a^ -f^^ +^2 +^3 +^4 +(1^ + aA -^aB 4-aC -^aP -\-aE Coefficients || || || ~|| || || in quotient, A B C D E R From this we observe the following law of coefficients in the quotient: The first coefficiejit in the quotient is the same as the first coefificient in the dividend. The second coefficient i7i the quotieyit equals the second coefficient in the dividend, plus a times A, the one just fo2ind. The third coefficient in the qjwtient equals the third coefficient in the dividend plus a times By the one just found. And any coefficient in the quotient equals the correspo7iding coefficient in the dividend plus a times the preceding one in the quotient. The process of finding the coefficients in the quotient, and the remainder, is more apparent in particular cases : (1). Find the quotient of Ix^+^x'^-^x^ -\^x^—%x-{-^ by ^-2* Coefficients in dividend, 7 +8—6 —15—8 +4 (2 14 44 76 122 228 Coefficients in quotient, 7 22 38 61 114 232 Hence the quotientis, 7;*r4 + 22jr3 + 38;ir2 + 61x + 114 + -^ x—% (2). Find the quotient of x^ — '^\ by ^-3: Coefficients in dividend, ,1 -81 (3 3 9 27 81 Coefficients in quotient, 1 3 9 27 Quotient, x3 + 3;t^ + 9;«^+27, no remainder. 574 UNIVERSITY ALGEBRA. This method of obtaining a quotient is called syjithetic division. It will evidently apply whatever the degree of the dividend. What change will there be if the divisor is ;^; + a? 744. The short method of division together with the theorem of Art. 739 furnishes a short way of finding the value of 2iViy f{x) for a given value of its variable. Thus, suppose we wish the value of x^ ^^ x^ -^-^x'^ + Vdx-\-*l^ when x=h\ that is, we wish/(5). Now, by Art. 739, the remainder when /(x) is divided by x—6 is /(5). Whence, 1 -7 +6 4-10 4-70 (5 5 -10 -20 -50 1 -2 -4 -10 4-20 That is, the value of ;i:*— 7:^3 4- 6.r 24- 10.^4-70 when x=h is 20. Such examples should be done in this way, as it is a shorter process than direct substitution. KXAMPI.KS. 1. Divide^t:*— 5;»;3 4-12;r2 4-4;t:— 8by ;»:— 2. 2. Divide:tr3 4-ll-^^ + 36;t:4-15by .r4-5. 3. Divide ;r^ 4- 6;t:4-10;i:3-112;i;2-207.r-110 by ,v4-5. 4. Divide^i;* — 12;r3 4-47:r2— 72ji;4-36by ;r— 5. 5. Prove that 3 is a root oi x^--^x'^ + llx—^. Art. 741. 6. Prove that 6 is a root of ;r* — 12;r3 + 47ji:2— 72^4-36. 7. Prove that 4 is a root of .^i;*— 55;f2 4-30;r4-'504. 8. Prove that —11 is a root of x^-^lOx^'-Zbx'^—ZOOx -396. 9. Provethat— 6isarootof;i:*— 4.r3— 29.^-2 4.l56;^;— 180. RATIONAL INTEGRAL FUNCTIONS. 575 10. mndva\ueo^x^ + 16x'^—ix+li{x=2. Art. 739. 11. Find value of jr^ + 9;t:2 — 19;r--76 when :r= 5. 12. Find value oi x^ — Sx'^ +ox-{-178 when x=^4. 13. Show that (l + a)\l+c^)-(l + dy(l+c^) is ex- actly divisible by a—d. Art. 738. 14. Show that/(«2)— /(/^^) J3 exactly divisible hya + d. 745. /f ^ is any complex number^ then f(£) is a com- plex fiicmber. Lfet/(^) = aQ2:''+a^3''"'^-\- • • • +a,^_-^2:+a„2ind2=x+yi. Now in the expression ^o(-^+j'0"+^i(-^+j^O""^+ • • • +^«-i(-^+j^O+^« (1) let the terms (jt+jkO") C-^+J^O"""^) etc., be expanded by the binomial theorem. Then all the terms containing yi to an odd power will be imaginary, and their sum may be represented by Vz. All other terms will be real and their sum may be represented by X, Thus, we have 746. ///ix+yi') = X+ Vz, lken/(x-yz) = X- Vz, For, changing the sign of y will change the signs of all the odd powers of yz, and hence V will be changed in sign, but not otherwise altered. X, being the sum of terms independent of y and of terms containing only even powers of yz, will not be affected by changing the sign of y. Hence, if f(x+yz) = X-^Vz then A^—y = ^— ^i 747. To Express f(x+h) in Powers of h. Suppose f{pc) = aQx''+a-^x''~'^+a2x"~'^-\- - - • +«„_i^+^«. Then we have /(X'^k) = ao(x+/iy+a^(x-{-/iT''^ -^a^(ix+ky-^+ . . +a,,_^(x+/i')+a„. 576 UNIVERSITY ALGEBRA. Expanding (x+k)", (x-}-ky'~'^, etc., by the binomial theorem, and collecting in terms of powers of k, we have aQx" + aiX"-^+a^x"-'^+ • • • +a,,_-^x+a,, + /i[?iaQX"-'^ + (n—l')a^x"-^ + (i7i—2)a.2x''-^+ • . .' -^^[?i(i7i-l)a^x"-'' + (in-'l)(7i-2')a^x"-^ + (/^~2)(;^-3>o-r«-4+ . . . +2a,_2] + • • • It is observed that the portion of the result independent of h isy(;i;), and that the coefficients of the successive powers of k are functions of x of degrees decreasing by unity. These coefficients are known respectively as the First Derivative, Second Derivative, etc., of/(.:r) and are represented by the symbols f'{x), f'\^)^ etc., or the symbols /i(^), f^)-/(^)+/'(^)/^+/"(^)j^+ • • • +a,h\ [2] Since x-^h—h-^-x, we know f(x-\- h')=f{h + x). Then if we interchange x and h in the right member of [1], we obtain 2 /(^+/^)=/(;^)+/'(/^)^..;-/-(/0J72+ • • • +^o^"> [3] in which f{x-\-K) is expressed in powers of .r. RATIONAL INTEGRAL FUNCTIONS. 57/ (1). Find the result of substituting x-\-/i ior x in x^ - Qx^ +llx—Q. /{x)=x^-Qx^ + 11^-6 /'{x)=3x^-12x+n /"(;^)=G^'-12 /"'{x)=Q. Therefore. /{x^/i)=/{x)-\-/' {x)/i+/"{x)y-^^-{-/' "{x)t^ =x^-Qx'^-j-nx-Q + {^x^-12x-\-U)/i+{Qx-12/^-\-Qk^. (2). Find the result of substituting :r+5 iovx'm jt* — 6^2 + 20x;- 14. f{x)^x'^ — Qx'^ + 20x-l4: f'[x)=ix^-\2x-^2Q f"{x)=\2x'^-l2 /"'{X) = 24:X /iy[x)=24:. Therefore. /{x-\-5)=/{x)+/' {x)5+/"{xy%^-]-/' "(x)H^-l-/iv(x)-Vf- =x^ + 20x3-\-U4.x^-hi(j0x+5Gl. • 748. Continuity of f(oo). If x changes from the value a to the value d, we wish to show that /(a) changes to /(d) by passing over all the intermediate values. That is, we wish to show that /(x) is a co7iti7iuotcs variable, and not a variable that skips values like the variable .3333. . . or .27272727. • •, etc. Let a be any value of ;t:, and let a-\-h be another value of X, Thus, f{a) is one value of f{pc) and f(a+h) is another value of /(;r). Then we have Whence, f^a-Vh')--f{a^^f\ayi+f\d)^^-^ . . . j^a,h\ Now, the right hand member of this equation may be made as small as we please by taking h small enough, since every term contains h. Thus, the difference be- tween f(a-\-Ii) and /(a) can be made as small as we please. . But a and a-\-h are two values of ;f an^fia-^h) 2nd/(«)) 37 — u. A. 573 UNIVERSITY ALGEBRA. are two values oi f{x). Thus we have shown that the difference between successive values of f(x) can be made as small as we please ; that is, /(x) is continuous. 749. A7iy term in f{x) can be made greater than the sum of all the terms of lower degree if x be sufficiejitly increased, and any term in fix) can be made greater than the stem of all the terms of higher degree if x be sufficiently diminished. Let f{x)^a^x''-\-a^x''~'^^.. , + a^''~''+, ,,+a„^^x+a*'. We shall show that if X be taken large enough. Since x is to be taken large, ^«-^-i ig ^\^Q greatest of the numbers x"~''~'^, x'"*"^^ etc. Let a^ be the greatest of the coefficients a^-^i, ^^+2* • • * ««_i, a„. Then, since there are ;z--r terms in the right member of (1), we may write (n^r)asX''-''-'^> a^+ix"-''-'^ + • • • +a„_-^x+a„. (2) Thus, it sufficient to prove that ^^"-^> (n — r^a^''-*" ^ , (3) for much more, then, will the left member of (1) be greater than the right member. But dividing the members of (3) by x"-"-^, we have a^>(n'-r)asy (4) which is true if you take a, f-^ f^ d Thus, if you take x>-^ ^—^y the term a^*'-'' will be a, greater than the sum of all the terms of lower degree. Let us next show that a,x''-''>a^x"+a^x''-'' + • • • ^-a,_l^''-''+^ (6) RATIONAL INTEGRAL FUNCTIONS. 379 if X be taken small enough. Since x is to be made very small, x"~''^'^ is the greatest of the numbers x"", x"~'^ , . • . Let a^ be the greatest of the coefficients ^o> ^i^ ' • * ^r-i- Then, since there are r terms in the right member of (6), we may write ra,x"-''+^>aQX*' + aiX*'-'^ + f-^r-i-^""'"^^ . (7) Thus, it is sufficient to prove that a^"-''>ra^""'-^^ (8) Dividing the members of (8) by x*'~'', we have a^^ra^x (9) which is true if you take Thus, if you take ;r<— ^> the term «^"~'' will be greater than the sum of all the terms of higher degree. 750. If 2 a7id h are a7iy two complex numbers, h may be giveji such a value that mod f{z-\-li) < mod fiz), provided, of course, that mod f{2)^0. We know that /(^+/0==/(^) +/'(^)/i+/"(^)j;^+ • ■ • +a,h". (1) In this equation, any of the functions /'(^), f'\2), etc. may be zero, but a^h'' is not zero, since a^ and h are not zero. Dividing both sides of (1) hyf{z), we have Smce 2' IS a complex number, •^-^^-e^)-^^) etc., are complex numbers by Arts. 745 and 717. 58o UNIVERSITY ALGEBRA. Representing these by ^^(cos O^-^-z sin ^i), etc., and k by p(cos a-f ^ sin a), we get r. . =1 + ^1 (cos ^1 4-2* sin ^i)p(cos a+/sin a) / w + r2(cos ^2 + ^* sin ^2)p^(cos 2a+z sin 2a) + . . . + r„(cos 0„-\-z sin ^„)p"(cos ?ia+z sin no), (3) in which p^(cos 2a+? sin 2a), etc., are written for /z^, etc., by De Moivre's theorem. Any of the moduli r^, rg, etc., may be zero, except r„ is not zero. By Art. 727, we write ^j^=l + ^l/> [C0s(^,-|-a) + 2Sin(^,+a)] + r2p2[cos((92 + 2a)+2sin(i92 4-2a)] 4- . . . + / „ p" [cos(^,, + na) + z sin (0,, + ;za)] (4) Since ^, and therefore p and a, is at our disposal, take a so that ^i+a=180°; but if ri=0, take a so that ^2 + 2a=180°, and so on. If ?i=0 note that all of the numbers r^^, rg, • • • are not zero, for r,,^0. Then, taking ^iH-a=180° and letting the resulting values ot ^2+2a, ^g + Sa, etc., be represented by <^2» S^3> etc., we have •^-^--^=l--;iP-f^2p^[cos ^2 + ^'sin <;f>2] + ^3,o^[cos <^3 + /sin t^g] + • • + r„ p" [cos <^„ + z sin <;(> J (5) in which p is still at our disposal. Take p so small that i— r^p is positive; that is, take p<— » then mod (1— z'lP) =1— r,p. Hence, by Art. 726, we derive RATIONAL INTEGRAL FUNCTIONS. 58 1 Now, by Art. 749, r^p can be made greater than tbe sum of all the terms of higher degree by taking p small enough. Then (6) becomes mod/(2') in which ^ is a positive number. /TA1 r mod/(^4-/^) .-, Therefore, • ^r7rT^<^- mod/(^) That is, mod/(2'+//)x+b', l-2l/^. Another root is 1 + 21/"^ and a factor of f{x) is therefore j;(;^_1)2+4] or x'^-2x^^. (;r3_;»;24.3^4-5)-=-(^2_2^ + 5)=^ + l. Therefore, the roots are 1-2^, 1 + 2/, and —1. A factor is x^ ^_i^ and Since ^2 + 4^^.5^[(^ + 2)2 + l], the roots are t/-1. -t/-1, 8. x^ + x^-2bx''+Alx-{-m', 3-t/^. g. ;»;4 + 5;^3__15^2_97^_110; 1 + V\2. 10. x^-x^^-^x^'-^x-lb] 1-21/^, l/3. 11. :r4-5^3-12x2_i3_7. _l±lzi?. 757. I^i ci'iiy f{^)^ tJ^^ coefficient of the highest powe7' being unity, the coefficients of the other powers a?'e functions of the negatives of the roots. Since we know it is evident that the values of /i, /s' /yj ^tc, can be expressed in terms of the roots by forming the product of the binomial factors^ as in Art. 91. Such product is : ;^"+(— a,-a,— . . . —a,:)x"-'-\-{aa^ + aa^+ , . . +a„_,a„)^"-^ + ( — aja2a3 — a^aga^ — a^a^o^^ — • • • — OL,^_^aj^_^a,^x^~^ H- . . . +( — I)"aia2a3 • • • a„. p^, or the fi7'st parenthesis, is the sum of the negatives of the roots. S86 UNIVERSITY ALGEBRA. /a, or the second par eiithesis, is the sum of the products of the negatives of the root taken two at a time. />3, or the third parenthesis, is the sum of the products of the negatives of the roots taken three at a time, and so on. /„, or the nth parenthesis, is the product of the negatives of all the roots. Using the 2 notation, we may express this by writing + • • • +( — I)"aia2a3 . . .a„. Thus, in x^ — ^x'^A-Wx -^, —6 is the sum of the negatives of the roots, -1-11 is the sum of the products of the negatives of the roots taken two at a time, and —6 is the product of the negatives of all the roots. The roots being 1, 2, and 3, we verify by writing -6^-(l-f2+3) ll = (1.2-t-1.3 + 2.3) -6-(-l)(-2)(-3). But in 2x3 — llx2 + 17>r— 6, ^y^^ sum of the negatives of the roots is not -11, for the coefficient of the highest power of f{x) is not unity. The sum of the negatives of the roots is, in fact, —^-. 758. If the signs of all terms containing the odd powers, or the even powers, of x be changed, f{x) is t?'ansformed into f{ — x) or —f{—x) respectively , the roots of either of the latter being negatives of the roots of f(x). = (^— a J(^— a2)(.^— ag) • . • {x—a,;). (1) Putting —X for x throughout the equation, we shall have f(x) with the signs of the odd powers changed, equal to the product of 7i factors of the form i—x—a), or I x''—px''-^ +A-^"~''"~ • • • ^Pn-^^ -^Pn, if n is even = (— ^--a,)(— ^— a2)(— ^-^3)- • '{—X—a,^ = (_l)«(^ + a,)(^ + a2)Cr + a3) • . . (.T+aJ (2) RATIONAL INTEGRAL FUNCTIONS. 587 Multiplying this through by —1, we shall have f{x) with the signs of the even powers changed, or _/r- ^ = l""^" +/x-^"~'-A-^"~' + • • • •\-pn-X'-pn, if n is even. ^(_l)«+i(^4-aj(:r + a2)(^ + a3) • . . (^ + a,) (3) Now, by changing the signs of the odd powers in/(;r) we must obtain what is contained in the { in (2), which equals /(—^), and from the latter pait of (2), namely: (— l)"(^+ai)(^ + a2)(^ + a3). . • (^ + a„), it is evident that the roots are — a^, —ag, —o.^, • • • — ««, or the negatives of the roots (1). By changing the signs of the even powers in f(x) we must obtain what is in the { in (3), which equals —f{—x), and from the latter member of the equation, it appears that the roots are the negatives of those of (1) or f\x). Cha7iging the sig7is of all the terms of fix) does not affect the roots. Changing all the signs is multiplying by -1. Thus, the roots of x^ — Qtx'^ — llx — Q, being 1, 2, and 3. we know that the roots of either —x^—Qx^ — llx — Qovx^-\-QiX^-\-llx-\-Q are -1, -2, and -3. 759. If nojie of the coefficients in a?iy fix) are fractio7is , and the coefficient of the highest power of x is unity, the f(x) cannot have an irreducible fraction for a root. Let such f{x) be x'' -\-m^x"~^ -\r7n^x"~'^ + • • . +vi,,_^x-\-7nn, in which 7n^, 7712, 77i^, etc., are integral. If this /(;f) can vanish when x is an irreducible fraction, let that fraction be ^- Then rm- on+^^h-Q^i+^2o;;=:2 + • • • +^n-iQ+m„ 588 : UNIVERSITY ALGEBRA. If the value of this is zero, it will be zero when multiplied by/5"-i. But ^-+;;^la"-l+^^2/^«""^+ • • • +^„-iiS'^"2a+^'^-iw«^0,for -- is an irreducible fraction, since | is ; and the rest of the function does not contain a fraction, since none among the numbers a, /?, m^, vi2, m^, etc., are such; and a fraction must be combined with a fraction to make zero. (X Therefore, it is impossible for ^ to be a root. Note that a f{x) satisfying the conditions of the theorem need not necessarily have integral roots, for some or all of the roots may be incommensurable. Thus. x'^—4:X-\-2 has no fractional roots, but on the other hand the roots are not whole numbers, being, in fact, the incommensurable numbers 2+|/2, 2 — j,/2. 760. Any f{pc) can be transformed ifito a fu7ictio7i of y, in which no coefficients shall be fractions and the coeffi- cierits of the highest power of y shall be unity. Suppose /(:r) in the form ;r"+/>i^"-i 4-/2-^""' + • • ' +A-2-^' +A-i-^+A- (1) If any of the coefficients, p^, p^, /g, etc., are fractions, y let their common denominator be q. Put jj;= -» then fix) becomes 4;+^^;^!+^^+ . . . +A^^ ^« q'' 1 q'' ■- q ^ Multiplying through by q", we obtain in which none of the coefficients are irreducible fractions, since each is multiplied by q, the common denominator. Representing (2) by the symbol (jl/) less than those of /(x), it is sufficient to assign a positive value to h. Thus, toincreasa the roots of x^ — Gx~-fllj\;— G by 2, aput //=— 2 and write /(-2) = (-2)3-6(-2)2 + ll(-2)-6=-60 /'(-2)=3(-2)2-12(-2)-|-ll = 47 /"(-2)=:6(-2)-12^-24 /"'(-2)-6. Then the result is /(^)=/(j-2)= -60 + 47;'— ^-^jj/^ + fj/^ 762. Horner's Method of finding the coefficients in <^(jk) above. Changing the roots oi f{pc) by an as- signed number is such an important transformation that a shorter process than the tedious method given above is of great value. We shall first explain the new process by means of a particular example. Take the example given in the last article. We are required to increase the roots of x^ — Qx'^ -i-llx—6 by 2. By the preceding article we know that the result of sub- stituting jk — 2ior xinx^ — Qx'^' + llx—Q will be a function of the form in which we shall proceed to determine A-^^ A^^ and A.^. Putting x=y—2 in fix), we have Since y=^x+2, we may write this RATIONAL INTEGRAL FUNCTIONS. 591 Dividing both sides of the equation by x-{-2, we have jr+2 ^ ^ -^ ^ ;t:+2 Since the dividends and divisors are equal, the re- mainders are equal. Therefore, ^3 = _60. -60 Subtracting the equals —~k and -^-^ from each side, x-^'2 we get ;tr2-8;t:+27==(^+2)2+^^(;r+2) + ^2- Dividing both sides by ;t: + 2 again, we have x-\-2 x+2 Since the dividends and divisors are respectively equal, the remainders are equal, and hence ^2=47. 47 A Subtracting — — -. and --]\ from both sides, 'we have x—\0=x-\-2 + A^. Again dividing both sides by x-\-2, we have ^;r+2 ^x+2 Whence, ^i = ~12. We have found the values of ^3, A,^, and A^. There- fore, <^(j/)=jK^ — 12j/2+47j/--60. Noticing that A^, A^, and A^ are the successive re- mainders as f{x) is successively divided by x-^2, we can perform the successive divisions by the short method of division and for convenience arrange the works as follows: 1 -6 -fll -6 (-2 -2 -fl6 -54 1 -8 -2 + 27 + 20 -60 II 1 -10 -2 + 47 II A, ^a 1 -12 592 UNIVERSITY ALGEBRA. The above is called Horner's Method of computing the coefficients of (y). We must now consider the general case. Suppose and that we are to find the result of substituting y+k for X. By Art. 761, we have a^x"-}- - . . +a,,_2x'^ +a,,_^x+a,, =A,r+ . . . +A,,_,y^+A,,_,y+A,^, (1) in which we are to explain a short way of finding the values of ^0' ^i> ^^c- SincQ x=jy-j-k, y=x—k, and making this latter sub- stitution in right member of (1), we have =Ao(x^hr+ • • ~ +A^,_,(x^/iy+A^,_,Cx-/i) + A,, (2) Divide both sides of (2) by {x — k) and call the quotient in the left member Q-^ and the remainder J^^ . We then have +^„_2(-r~/0+^«_i+^|- (3) Since the dividends and divisors are equal, the re- mainders are equal, and we have That is, the last undetermined coefficient in <^(jO equals the remainder when /(x) is divided by x—/i, R A ' Subtracting ^ and — — from the members of (3), we have ^1=^0(^+^0""'+ • • • 4-^„-2(^-/0+^.-i. (4) Again dividing by x—h and calling the quotient in the left member Q^ and the remainder Rc^^, we have RATIONAL INTEGRAL FUNCTIONS. 593 Whence it is seen that ^4„_i , the next to the last undeter- mined coefficient in f(^a-\-h) and also /(^)>/(«-/^). li f(b) be a minimum value of /(;i-), we must have /(^)/(^ + /0 and f{a)>f{a-^Ji), in which case f{a) is plainly a maximum value oi f{pc), (1). Find the max. and min. values of ^:X^ — 15^^^ ^12^— 2. /"(x)=:24;*:-30 The roots of /' {x) are \ and 2. /"(i)=-18 .-. /(i)=|=max. value of /(^). /"(2)=-f 18.-. /(2)=-3:=min. value of /(^). (2). Find the values of x for which 3^^5 — 125^'' + 21 60.r is a max. or min, /'(x)==15x4-475;c24-2160 /"(;^)=G0;tr3-950^-. RATIONAL INTEGRAL FUNCTIONS. 597 Roots of f'{x) are -4, -3, 3, and 4. Use 6^3 _95;^ for f"{x) and X* - 35^2 _|_ 144 for /'(x). /"(_4)=-4 .-. /(-4) = max. /"(_3)=rl23 .♦. /(-3)=min. /"(3)=-123 .-. /(3)=: max. /"(4)= 4 .-. /(4)= min. Therefore, —4 and 3 are the values of x that render f{x) a max., and — 3 and 4 are the values of x that render f{x) a min. 766. ^f f{p) ^^ ^ maximum or a minimum of f{x)^ f\a) va7iishes. This follows at once from (1) and (2) above, for when k is very small the left members of (1) and (2) cannot have like signs unless /'(^)=0. 767. Between two consecutive real roots a and b of f{x), there is at least 07ie real root of f\x). Since a and b are consecutive roots, fix) is not again zero, as x changes from a to b. Therefore, as x varies from a to b, fix) must either at first increase and then decrease, in which case fix) has a maximum value; or fix) must at first decrease and afterwards increase, in which case fix) has a minimum value. In either case, the value of x which renders fix) a maximum or a min- imum, is a root of /'(•^). 768. Constitution of /'(aj), in terms of factors of /(;r). We know fix) = ix—a^ix—a^iX'-a^. • • (:r— a„). We seek an expression for fix) in terms of these same factors x—a.^, x—o.^^ etc. Put h-\-x for X, we then have /(>^ + ^) = [/^ + (^-ai)][/^+(ji;-a2)]+. • -C/^ + C^-a,,)] 598 UNIVERSITY ALGEBRA. in which, by Art. 91, we know ^i = Gr— ai) + (;i;— a2) + (;r— a3)-f ■ • .+(^~aj . . 4- • • • +(.^-^«-i)(^-—a,,) + . . . +(^— ai)(;r— ag) • • • (x—a,_^) But by [1], Art. 747, we have whence we have, equating coefficients of powers ol // in the two expressions for /(/i+x), fCx') = (_X-a^X^-^s)"-(.^-<^n') + (x-a^)(x-a.^).,.(^-^n) + • • • +(^--tti)(;r--a2.) . . . Cx—a,,_^) [4] etc. etc. The expression for /'(^) may be written in the useful form X — a^ x—a^ ^—oLn Thus, if /(x)^x^-Qx^ + nx-G=(x-l){x-2){x-^) then /' {x)^{x-2)lx-3)-\-{x-l){x-3) + {x-l){x-2) _ /(^) , /(^) , /(^)_ 769. Equal Roots. // /(x) has r roots equal to ti. f'{x) has r—1 roots equal to a. This can be seen from the form of /'(-''^) found above: /'(^) = (^-a2)(:f-a3)...(;j;-aj4-(^-ai)(^-a3)...(;r-a„) + • ' • +(-r— ai)(;r— a2) • • • (^^~a«._ J. For if ai=a2, then f'(x) is seen to be divisible by x—a^ and hencea^ is a root of/' (;t:). l{ a^—a^—a.^^ then f\x) RATIONAL INTEGRAL FUNCTIONS. 599 is seen to be divisible by (x—a^y-, and so on. Thus, if /(x) is divisible by (^— a)'', then /'(x) is divisible by Note that if /(x) has equal roots, /(:r) and /'(^) have a common divisor. (1). Thus, if /(x)=;^3_5^2_^7^_3=(^_l(^_l)(^_3) then /'{x)={x-l){x-:\) + {x-l){x-3)i-{x-l){x-l), and f'{x) is seen to be divisible by x — 1. (2). The expression x^ - Ix'^-hlQx - 12 has equal roots; find them. /'{x):^3x^-Ux-^lQ. The H. C. F. of /(x) and /' (x) must now be found. 3x^-2lx^ + ASx-m I 3x^-Ux-^lQ ^x^-Ux^-hlGx I x + 7 - lx'^-{-'S2x- 36 21^2 -96;r+ 108 2U=^-98x + 112 2x— 4, etc. whence X— 2 is a common factor. Therefore,/(x)=(;t: — 2)(^— 2)(jr:— 3), in which the factor {x — 3) is found by dividing /{x) by {x-2)^. BINOMIAIv COEFFICIENTS. 770. In the theory of the rational integral function it is often advantageous to represent /(x) as follows: 7l(fl — 1^ a^x''+7ia^x''~'^-\ — z — ^a etc. CHAPTER XXXIII. SPKCIAL EQUATIONS. 774. The treatment of rational integral equations is based upon the properties oi f{x), as considered in the last chapter. The present chapter is devoted to a few important special classes of rational integral equations, namely: reciprocal equations, binomial equations, cubic equations, and biquadratic equations. RKCIPROCAIy KQUATIONS. 775. If an equation is unaltered by changing x into -> it is called a Reciprocal Equation. Thus, x^+5x^ --10;i;2 + 5;i;+l=0 is a reciprocal equation. 776. Classes of Reciprocal Equations. Consider the equation x"+p^x"-^+p^x''-^+ . . . +A-2-^''+A-i^+/«=0. Putting — for x, and clearing the result of fractions, or x"+^x"-'-h^'x"-^+ . . . +l^x^+llx+^=0. Pn Pn Pn pn Pn If the original equation is unaltered by this substitution, we must have pn pn pn Pn pn From the last equation we have /„==fcl, and thus we have two classes of reciprocal equations : SPECIAL EQUATIONS. 603 I. If /!,= !, we have P\=Pn-\, Pl=Pn-1y • • • , Pz=pn-Z, etC. That is, the coefficients of terms equidistant from the ends are equal. II. If jz^„=—- 1, we have Pl^^—Pn-I, P'i = —pn-2^ Pz='-Pn-Z^^^^' That is, the coefficients of terms equidistant from the ends are numerically equal but of opposite signs. If the equation happens to be of an even degree, say 2r, then pr= —pr, or j2^^=0. Thus, if an equation of this class is of an even degree, the middle term is wanting. 777. Roots Reciprocals. Since a reciprocal equa- tion is unaltered by changing x to — » it follows that if a is a root of a reciprocal equation, — must also be a root. a 11 Thus, roots enter in pairs of the form a,, — ; a^, — , etc. If the equation be of an odd degree it must have a root which is its own reciprocal, and as +1 and —1 are the only numbers of this kind, one of these must be a root. 778. If a pair of reciprocal roots, or any root that is its own reciprocal, be removed from the equation by dividing its left member by the proper expression, then the resulting equation still has its roots in reciprocal pairs, and therefore is a reciprocal equation. 779. Reduction to a Single Class. A reciprocal equation of class I, say or Gr"+l)+/>i^(-^-"+l)+/>2-^'(-^" + l)+ • • • =0, of an oda degree has a root —1, because its left member 604 UNIVERSITY ALGEBRA. is divisible by :r+l. If ^(;r) be the quotient, then <^(;j;) = is a reciprocal equation (Art. 778) of an even degree, having its last term positive. A reciprocal equation of class II, say or (:r«-l)+/i^C;i:«~l)+/2-^'(^"-l)+ • • • =0, of an odd degree has a root +1, because the left member is divisible by .:r— 1. If ^{x) be the quotient, then <^(x)=0 is a reciprocal equation (Art. 778) of an even degree, with its last term positive. A reciprocal equation of class II of an even degree has a root 4-1 and a root —1, because its left member is divisibleby ^2— 1. \i<^{x) be the quotient, then (^(:r)=0 is a reciprocal equation (Art. 778) of an even degree, with its last term positive. Thus, we have shown that any reciprocal equation may be reduced io one of even degree^ with its last term positive. This may be called the Standard Form of the reciprocal equation. 780. Any reciprocal equation in the standard for7n may be reduced to an ordinary equation whose degree is half that of the standard form. Let the equation in the standard form be ^2^+i^i^'"-'+ • • • +A^''+ • • • +/i^-f 1=0. Dividing by x'', and grouping terms equidistant from the ends, we have Now, x*-\-—^ may be expressed as follows : SPECIAL EQUATIONS. 605 Giving p the values.2, 3, 4, • . • , in succession and rep- 1 1 resenting x-\ — hy u and x^-\ — ^ by V^, we get F2 = 7^2_2 or :r2+-4 = ^^'-2. 2 x^ Vr=u Vr-i — Vr-2 y ^nd SO on. Substituting these values above, we get an equation of the rth degree in 7c. From the values of u the values of X may be found by solving X-] =71. X (1). Solve x4- 10x3 4- 26x2 -10;c4- 1=0. Dividing by x"^, we get (.^.+l^^_lo(x+i] + 26=0. Substituting the values of Fg and V^ in terms of u, we get «2_ 10^4. 24=0. Whence, «=4 or 6. Then, x+— =4 or 6, X or jr3 - 4^^+1=0 and x2_ (3;^; 4. 1—0 Therefore, x=z2±\/T, 3±2v/2: (2). Solve 2;»:5-15;»r4+37x3-37;c2 + 15-r-2=0. Dividing by ^ — 1, 2 -15 +37 -37 4-15 -2 (1 2 -13 24 -13 2 2 -13 24 -13 2 we get 2^4_i3;^3_|.24x;2-13;«:+2=0. Dividing by x^, we get 2(.»+l)-13(.+j) + 24=0 Substituting the values of V^ and F"^ in terms of u, we get 2z<2-13z/+20=0. Whence, «=4 or f . 6o6 UNIVERSITY ALGEBRA. Then, from x-\-~=4:Ot% X ^ ' we get x=2, i 2±\/W. Therefore, in the original equation x=\, 2, 1, 2±\/3: (3). Resolve x"^ + 1 into quadratic factors. We know x'^ + l—O is a reciprocal equation. Put it in the form or w2-2=:0. Therefore, u=±Vr. That is X X or ^2-v/^.r+l=r0and^2 + \/2'x4-l=0. Hence, ^4 + 1^(X2 - v/ ^^--M)(;t:2 ^. ^2 ;c + l). EXAMPLES. Solve the following equations : I. 2;^4__5^3^6^-2_5^_^2=0. 3. :^;4+5^3 + 2;t:2+5;r+l=0. 4. x^ +4:ax^+2x^ +Aax^ + 1=0. 6. 2;i:<5+Ji:'^ — 13;i;4 + 13;i;'^— ;tr— 2=0. 7. Two roots of x^^llx''+iSx^-79x^+79x'^—i7x+'i:0=0sire 2 and 5 Find all the other roots. 8. Show that _ x^ + l = (x'^ + l)(x'^ + VSx+lXx^ --l/Sx+l). BINOMIAI, EQUATIONS. 781. An equation of the form ;r'*=fc^=0, in which a^O, is called a Binomial Equation. This equation has n roots by Art. 752. SPECIAL EQUATIONS. 607 782. Roots all Different. A binomial equation cannot have equal roots. For, since f(^x)=^x"^a, we \i2N^ f {x) — nx''-'^ , But x^'^^a andnx"~^ have no com- mon divisor. Therefore, of the n roots of a binomial equation, no two can be alike. 783. Roots of a Complex Number. The demon- strations in Arts. 751 and 752 hold whether the coefficients ofyC^) be complex numbers or not. Therefore, if a be a complex number, the equation x''—a=0 or x"=ahas n roots, which we have just seen are all different. But every root of the equation x''—a must be a value of i^' a. Therefore, every coinplex member has n differejit nth roots and no more. Since a positive real number is a complex number ot amplitude 0, and every negative real number is a complex number of amplitude 180°, it follows that every positive or negative real nu^nber has n nth roots a7id no m^ore. 784. Suppose the complex numbers to be represented by r(cosO + / sin^) . "Then the following n expressions are The n nth Roots of the Complex Number a, V 6> . ^\ ?" cos--f ^sm- (1) \ n n) ^ ^ V 27r+^ , . . 27r-f (9\ r'\ cos h t vSm ) (2) \ 71 n I ^ ^ r "( cos h I sm j (3) r«(cos h^sm ) (s-\-Y) \ n 71 J ^ r"[ Qos- h z sm^ --] (71) \ n n ) ^ ^ 6o8 UNIVERSITY ALGEBRA. For, take the nth power of any of these 7z expressions, as the(^+l)st. Then r«(cos f-2sm — J / s27r + e , . . s2'7r + e\" = r[ cos h z sm ) \ 71 71 I „(, cos-^ — --\-i sm-^ — -j by De Moivre's theorem, = r[cos(527r+(9) + / sin(^27r+l9)] = r(cos 0-\-i sin &) whatever integral value ^ may have. Since the 7^th power of any one of the n expressions written above is r(cos ^+/sin ^), and since no two of these expressions are alike, therefore these 7i expressions are the 7i Tith roots of r(cos ^+/sin 0) or a, which in Art. 783 were shown to exist. 1 Note. — Remember that r" in the above stands for the positive or arithmetical value of the n'Oa. root of r. 785. In the 71 expressions written in the last article, put ^=0 and we shall have The n nth Roots of a Positive Number, r. ^" , . (1) 1/ 27r , . . 27r\ r'\ cos h I sm — ) (2) \ n 7t J ^ ^ 1/ 47r , . . 47r\ ^^^ r'i cos 1- 1 sin — ) (3) r« cos Vi sm — U+1) \ ^ n ) ^ ^ 1/ (;^-l)27r , . . (>^~l)27r\ r «( cos-^ h I sm- — I (ft) \ 71 71 J ^ SPECIAL EQUATIONS. 609 The first of these is real. If n is an even number, say 2^, then the (^+l)st expression reduces to \_ ^ 1 r''(cos TT+z sin 7r) or — r" That is, if n is even there are two real roots of opposite signs, but all the remaining roots are imaginary. 786. In the n expressions of Art. 784, put ^=7r, and we shall have The n nth Roots of a Negative Number —r, r"( cos- + / sin- ) (1) \ n n) ^ ^ r"\ cos V I sm — ) (2) r" cos Vt sm — (3) \ 71 n I ^ ^ r« cos-^^ ^ — \-t sm- — (^+1) V 71 71 J ^ 1/ (2;z~l)7r , . . (2n-V)iT\ r'icos- — [-1 sm- ) (n)^ \ 71 71 J ^ ^ If 71 is an even number, all of these expressions are imaginary. If 71 is an odd number, say 2^+1, then the (5+l)st expression reduces to —r" and all the rest are imaginary. That is, if 7i is odd there is one real root of — r; namely: the negative number, — r«. 787. If in the 7i expressions in Art. 785 we put r=l, we shall obtain the 7i ?ith. roots of + 1. The first imagin- ary 7zth. root of + 1 is called a Primitive nth Root of + J, and is represented by the symbol w; that is, 27r , . . 27r COS \-i sm — =0). • 71 71 6lO UNIVERSITY ALGEBRA. Then, by DeMoivre's theorem, 47r 47r cos ^z sm — =0)2. ' n n cos Yz sm — =0)3. and so on for the other ;?.th roots of +1. The n power of the primitive root gives us cos Vi sin — ) =cos 27r+/sin 27r= + l, \ n 7zJ ' the first ni\\ root of+1. Therefore, the following is true: If 0) represents a priyiiitive 7ith root of +1, then all the nth roots of -\-\ are represented by the expressions o), o)^, O)^ . . ., 0)". Since the ;^th roots of +; differ from those of +1 1 merely by the presence of the power of the modulus r'\ we may say: If r is a positive number arid w a primitive nth root of -f 1, then the n nth roots of r are represented by the expres- 11 1 sions r"o), r"o)2, • • • r"ni^. 788. If, in the n expressions in Art. 786, we put r=^l, we shall obtain the n n\h roots of —1. The first imaginary ;^th root of —1 is called a Primitive nth root of — 1, and is represented by o>'; that is, TT . . TT , COS— + 2 sm— =0) . 71 71 Then it is easy to see, by De Moivre's theorem, that the following is true: If o)' represent a primitive nth root of — 1, then all the nth roots of — 1 are 7'eprese7ited by the expressio7is o)', w'^^ 0)'3, . . ., 0)'". Since the ?eth roots of —r differ from those ot —1 merely by the presence of the power of the modulus r**, we ma}^ say: SPECIAL EQUATIONS. 6 1 1 If —r is a negative number and ' a primitive 7ith root of —1, then the n 7ith roots of — r are represented by the expressions r" oj', ^« co'^^ . . .^ ;^« o)''*. These same roots may, however, be obtained by multi- plying any one of them by the 7i different 72th roots of +1. Therefore, we may say the n nth roots of any mmiber, pos- itive or negative^ may be obtained by i7iidtiplying any particular 07ie of the nth roots by the different nth roots of +1. That is, by w, w^, w^, - . • w"". 789. As applied to the binomial equatio7i x" + a=Oy we have shown: I. In case a is imaginary, there are n different im- aginary roots. II. In case a is positive, (1) there is one negative real root and (71 — 1) different imaginary roots if n is odd, and (2) there are n different imaginary roots if n is even. III. In case a is negative, (1) there is one positive real root and (71 — 1} imaginary roots if n is odd, and (2) there is one positive and one negative real root and (ji—2) imaginary roots if n is even. 790. If r and t be prime to each other, the equations x'' — \=^^ a7id x^ — 1 = have no com7non root except -f 1. The amplitudes of the roots of ;»;''— 1 are seen, from Art. 787, to be ^ 2ir 47r (r— l)27r (J — , — , . . . .. r r r Likewise, the roots of ;t:^— 1=0 are ^ 27r Aw {t—iyiir The common amplitude indicates the common root -f 1, but no fraction in the first series can equal a fraction in the second series, since r and / are prime to each other. 6l2 UNIVERSITY ALGEBRA. Since the amplitudes of the roots of ;t:'' — l=0, except 0, differ from the amplitudes of the roots of x'--l=0, the equations have no roots in common, except +1. 791. If, in the equation jr"— -1 = 0, 7i is the product of two numbers s and /, which are prime to each other, theji the roots of ;r"— 1 = can be found from the roots oj ar^-l = Oand;r^-l=0. In fact if a be a primitive root of ^'■—1=0 and P a primitive root of :r'— 1 = 0, then the 7t roots of .r"— 1 = will be the n ter^is in the product (a4-a2+a3+. . . -^aOC/^ + i^^ +^3 _^ . . . ^^.) (^y For, let a^/3^ be any term in this product. Now (a^)'"=l, since a^ is a root of ;tr^— 1=0; also (yS^)^=l, since P^ is a root of ;r'— 1=0. Since (aO'=l, (aO''=l, and since (^^)^= 1, (^^)-^= 1. Therefore, io^y(.P'y or (a^l^'^y or (a^ySO"=l. Since (a^P'y=l, a^p^ is a root of jr"— 1=0. Therefore, any term given by the product (1) is a root of ^"—1=0. The product (1) gives 7t roots of jt''— -1 = 0, and hence (1) gives all the roots of ;i:"— -1=0 unless two roots given by (1) are equal. Suppose that two roots given by (1) are equal, if possible. For example, suppose . a^^^=a«^^ then a^-^=p'-'. The expression on the left side of this equation is a root of ^—1=0, and the expression on the right side is a root of ;t:'— -1=0. But by Art. 790 these expressions cannot be equal. Therefore, (1) gives all the roots of jj;"— 1 = 0. In a similar way we show that if n is the product of />^r^^ prime numbers, r, s, and /, that the roots of ;r"— 1=0 can be found from the roots ;v''— 1=0, x'—l=0, and :xf—l=^0, and so on. SPECIAL EQUATIONS. 6 1 3 792. In general, by a Primitive nth Root of +1 is meant any root of the equation ^i:''— 1=0 that is not also a root of a binomial equation of lower degree. Thus, of the roots of ;i;6-l=0, ~|+i-i/:i3, -i-^i/ITs, and +1, are also roots of the equation x^ — 1=0, while + 1 and ~ 1 are also roots of the equation .r^ — 1 = 0. This leaves -|-+^l/— 3 and ^—^V —S as primitive roots of :r^ — 1=0. Likewise we speak of the primitive roots of x"-{ 1 = 0. Plainly, all the roots of ;t:"— 1=0 except 1 are primitive if 71 is a prime number. For, n being a prime number, it is prime to all lesser numbers, and therefore, by Art. 790, ;*:"— 1 = cannot have a root except 1 which is also a root of a binomial equation of lower degree. Whatever value n may have, it is plain that the second of the 72th roots of -f 1, if arranged as in Art. 785, is always a primitive root of ;t'" — 1=0. 793. A general solution of the binomial equation is given by Art. 784. But since the equations ;i;"+l=0, and :t:"— 1 = are reciprocal ^(^dXxoxvs, they maybe solved algebraically if not of too high degree. \in is a large num- ber, but composed of several factors prime to each other, then the binomial equation may often be solved by re- ducing its solution to the solution of several equations of lower degree, as indicated in the last article. Below are given a few examples of the solution of binomial equations: (1). Solve A'3-- 1=0. Removing the factor x — \ from this reciprocal equation, we have Solving this quadratic, we get Thus x = l,-| + ^\/-3 and —J-jV^. Notice that if we call -^i^rW~^^-^* then --^-jv'^rrwz ^nd l=a)». 6l4 UNIVERSITY ALGEBRA. (2). Solve .r6 -1=0. We may reduce this to the standard forjji of a reciprocal equation by dividing by x"^ — \, and then solve; or, we may proceed thus: Write out all the terms of (a + a^)(^ + /5'^ + ^^), in which a is a primitive root of jc^ — 1=0 and p is a primitive imaginary root of x^ — 1=0. This gives these six roots of Jir6_l— : i-4x/:=B. i+ix/^ -i+i^z:^. -i-iv/33, _i+i. (3). Solve ^9-1=0. Write the equation (;.')' -1=0, in which the values of x^ are w, 0)3 and ^^ . Whence, the roots of .^^ — 1=0 are the roots of the three equations ;tr3_o)=i0, ;>:3-3 = 0. 9. Show that the sum of all the n ;zth roots of + 1 equals zero. 10. Show that a^+b^-\-c^-'Zahc = (a + ^-f r) (« + o)<^4- («^^^) (^ + («^^ ^+ <*>^). 11. Resolve x^ ■\- x"^ ■\- x'^- -\- x -^\ into real factors of the second degree. SPECIAL EQUATIONS. 615 CUBIC KQUATIONS. 794. We have seen that any equation can be trans- formed into another in which next to the highest power is wanting; so that any cubic equation can be transformed into another in which there is no term of the second degree. We will therefore confine our discussion of cubic equations to the form x^-\-ax-\-b=0. The solution given below is usually called Cardan's Solution. 795. Algebraic Solution of the Cubic. In the equation x^ + ax+b=0, (1) let x—y+2, then K^y+zY'Vaiiy-\-2) + b=0. Hence, y^ + z^ + {^yz+a)(iy+z)-\-b=0. (2) The only restriction placed upon the two unknown numbers y and z is that their sum equals x, i, e. , a root of the given equation (1), and as two restrictions not inconsistent with each other can be placed upon two un- known numbers, we will place upon z the further restric- tion that Syz + a=0. Introducing this restriction into equation (2), we obtain ^3+^3 + ^=^=0. (3) But from Syz+a=0 it follows that a Substituting in (3), we have or _y6 + 5^3_g=0. (4) Whence, since (4) is a quadratic in terms of y'^, = -^-7" = — 2=FY 2T__ b a" and ^' = -^-7" = — o=FA/4- + 2f 6l6 UNIVERSITY ALGEBRA. In these values of j/^ and z^ we must take the upper signs or else the lower signs in both equations. Select- ing the upper signs in both eq uations, we have ( b ^ ld\ a^ W / b lb'' , a^ W 796. If we could give to each of the cube roots occur- ring in the value of x any one of the three values of which it is capable, then there would be in all nine values of x, whereas there should be only three values of x since an equation of the third degree has only three roots. Evi- dently then, these different cube roots must be combined in some way so as to give only three instead of nine values. From the equation oyz-\-a=0, we get y2:= — ^• Hence, the cube roots above must be so selected that their product equals —-x- If now we represent one of the cube roots of ( — ^-f-y/ T + 9^ ) ^y ^^^ ^^^ other cube roots will be mo) and ?n(t)'^ by Art. 788, and if we represent one of b Ib'^ a^ the cube roots of ■~o"~a/t'^27 ^^ ^^ ^^^ other cube roots will be thh and ;zo>2, and if nt and n be so selected that mn= — 77* then the cube roots in the value of x must o go together in pairs, thus : m goes with n, moi goes with nm'^, niiii^ goes with ;z(o, because in each of these cases the product is the same as the product mn. We may then write x=^ni-\-n, or X'=m(ii-\-n(ji'^ ^ or .r=Wa)2-f;^w. SPECIAL EQUATIONS. 617 797. Let us now examine a little closer into the character of the roots of a cubic equation. The values of y^ and ^^ given in Art. 795 are both real i^ -j-+^ is or any positive number, and imaginary if ^ +07 is neg- ative. First. Suppose -|- +07 is positive. Then y^ and 2^ are both real but jk and 2 may be either real or imaginary. From the equation y2=^ —-^ it follows that these quantities o y and 2 are either both real or both imaginary. Let 771 and 71 denote the real cube roots ofy^ and 2^ respectively, then the three roots of the given equation are m-}-7i, m(t)-\-7i(o'^y moi'^ -{-7i(D^ but since a>=— i+i]/— 3, and co2 = -^-^t/-3, therefore the three roots of the given equation are m-i-7i, — i(77t + 7l')-\-^(77Z-'7t')V — S — ^lm + 7l) — 1(771 — 7l)l/ — 0, b"- a^ Therefore, when ^ +9^ is positive, one of the roots of the given cubic is real and the other two are imaginary. Second. Suppose that ^ + ^^=0. Inthiscase theform of the roots of the given cubic may be deduced from the preceding case ; for now ;?2=/z, each being equal to the real cube root of -^« The three roots of the given cubic in this case are therefore, 2/;^, — 771, —771. Hence, when ^2 a^ -J +27 = all three roots of the given cubic are real, but two of them are equal to each other. 6l8 UNIVERSITY ALGEBRA. Third. Suppose -j- + ^^ is negative. In this case, be- cause y^ and z^ are imaginary, y and z must also be imaginary, and since y^ and z^ differ only in the sign of the imaginary part one of the three possible values of y is the conjugate of one of the three possible values ot z, Let the three possible values of y be represented by r-\-is, then the three possible values of z are r^is, (r — is)oi, (r — is)oi'^ . The values of y and z must be selected in such a way that their product equals — k-' which is real. Hence, if r+is be selected as the value of j/, then r—is must be taken as the value of z. Since this is the only one of the three possible values of z, which, multiplied by the selected value of y, makes the product real. Hence, as in the preceding article, the three roots of the given cubic are r-\-is+r — is or 2r, _ (r-\-is)(o-\-(r—is)o>'^ or — r— ^F 3. (r+is)(ii'^ + (r — is^di or —r-{-sv^. Thus we see that when ^^ + 07 is negative the three roots of the given cubic are real and unequal. 798. We have found that when -7 + 07 is positive, the equation x^ +ax + b=^0 has one real and two imagin- ary roots. In this case the formula enables us to find the real root, after which the equation may be reduced to one of the second degree which may readily be solved. ^2 ^3 We have also found that when -r + :7-; is negative, the 4 27 SPECIAL EQUATIONS. 619 equation x^-\-ax-\-d=0 has three real roots, two of which are equal to each other. In this case the formula enables us to find all three roots. We have also found that when ^ + ^— is negative, the equation x^-{-ax-i-d=0 has its three roots real and un- equal. In this case the expressions above given for y^ and z^ are each imaginary, and although we know that these expressions have cube roots, there is no arithmet- ical method of finding them and no algebraic method of finding them exactly. In this case, therefore, although the roots are real and distinct, they are presented to us in a form which is very inconvenient for use. For this reason this case is often referred to as the ' ' irreducible case, ' ' 799. We have stated that there is no algebraic method of finding exactly the cube root of an expression of the form p+iq. But in such an expression as (,p-{-iq)^, if q<.p w^e ma}^ expand by the binomial theorem into a converging series arranged according to the ascending i_ powers of iq and find an approximate value of {p-\-iq^^ in the form P-\- iQ, where P is the sum of the real terms of the expansion, and iQ is the sum of the imaginary terms of the expansion. If, however, ^>/>, then, since p^-iq-=i{q—ip), therefore {p-\-iq)'^—i'^{(j—ipy^, and since i^-= — i .'. (—i)^ = i.'. —z=t^ therefore (p+iq)'^=—iCq—ip)'^. Now, —i(q—ip')'^ can be developed by the binomial theorem into a converging series arranged according to ascending powers of ip, and from this development an approximate value of —iiq—ip)^ can be found as in the preceding case. 620 UNIVERSITY ALGEBRA. KXAMPI^KS. 1. Given (2 + 111/^)^=2 + 1/^ and (2-111/^) 3 = 2-1/^ solve the equation ;r^ — 15jt:— 4=0. 2. Given (10+ V'108) "^=1 + 1/3 and (l0-l/l08)^=l-l/3 solve the equation jr-^ + 6x— 20=0. 3. Solve the equation x^ — 24iX+72 = 0, 4. Show that if a is any positive number, the equa- tion x^-i-ax-i-d=0 has two imaginary roots. 21/ "3 5. Show that the equation x^ — c'^x-\ ^^=0 has two roots which are equal to each other whatever be the value of c. 6. Show that the equation .r^ — 12;r+3=0 has its three roots, real and unequal, when d is numerically less than 16, and has two imaginary roots when d is numeri- cally greater than 16. 7. Show that the equation x^+ax+d=0 has two im- aginary roots when a is any positive number whatever. BIQUADRATIC EQUATIONS. 800. Since an equation can always be deprived of the term of next to the highest degree, we confine our dis- cussion of biquadratic equations or equations of the fourth degree to the form x^-\-ax^- + dx+c=0. The solution which follows is called Descartes' solution. Suppose x'^+ax^-j-dx+c- (x'^ -j-Ax-{-B')(ix^- + Cx + B) where A, B, C, and D are undetermined coefficients. SPECIAL EQUATIONS. 62 I Then x"^ +ax'^-\-dx-\-c =x^-i-(A + C)x^ + CB+AC+D')x^ + (iAD+BC)x+BD. Equating coefficients : A-hC=0 whence A^-C B+AC+B=a '' B+D-A^ = a An+BC=d '' A{^D-B) = b BD=c *' BD=c Therefore AD-\-AB=A^ + aA] AD-AB^b; A'^BD=cA'^. Finding AB and AD from the first two equations and substituting in the third, {A^-\-aA-^)(iA^-\-aA-b) = 4.cA'^. Multiplying out, recognizing a product of sum and dif- ference, A^ +2aA^ -\-(a'' -ic)A'^—b''=0, which is a cubic in terms of y^^. Therefore A can be found and then B, C, and D from the equations above. Finally, four values of x can be found by solving the two quadratics. x'^+Ax-^B=0 and x'^ + Cx+D=0. 801. We notice that the equation for finding A is of the sixth degree, but that this is what we ought to ex- pect may be readily shown. If we represent the roots of the given equation by r^, r^, ^3, ^4, we know that x^-\-ax'^-\-bx-]-c=(^x—r^'){x—ro)(x—'r^^{x—r^) (1) also x'^^-ax'^-{-bx^-c=(^x''--\-Ax+BXx'^^Cx+D) (2) From (1) and (2) it is evident that x'^+Ax+B is the product of two of the linear factors of the second membe^ of (1), and as a7ty of these four factors may be taken together, and as two things can be selected from four things in six ways, therefore x'^+Ax+B may stand for any one of six quadratic expressions. But from the solution given above it is evident that when A is fixed B is also fixed, and hence when A is fixed the whole 622 ' UNIVERSITY ALGEBRA. expression x'^+Ax+B is fixed. Now, if x'^ -]-Ax-\-B may stand for any one of six quadratic expressions, and if this expression is fixed when A is fixed, A ought to have six values; /. e., A ought to be found from an equation of the sixth degree. Moreover, it is readily seen from the solution given above, by computing the values of ^, C, and D in terms of -^, that changing the sign of A merely interchanges the tw^o quadratic factors in the second member of (2); so that when one quadratic factor corresponding to a certain value of A is found, another factor may be found by simply changing the sign of A. Hence, of the six values of A, three are simply the negatives of the other three. Thus we see why the equation from which A is found is a aidic in terms of A"^. This cubic is called the Auxiliary Cubic. 802. Let us now notice the relation between the roots of the given biquadratic and those of the auxiliary cubic. If the roots of the biquadratic are all real, then, since A is the sum of fwo of these roots with their signs changed, A is real; therefore, A'^ is real; therefore, the roots of the auxiliary cubic are all real. Again, if the roots of the given biquadratic are all im- aginary, since their sum equals zero they must have the forms a+?yS, a— /y8, — a+zy, — a— /y, and as A is the sum of hi^o of these roots with their signs changed, therefore the only possible values of ^ are db2a, db/(/3+y), =h2*(/5— y), therefore the only values of A ^ are 4a2, -(^+y)^ -(/?-y)^ which are all real. Thus we see that when the roots of the given biquadratic are all real or all iviaginajy the auxiliary cubic falls ujider the irreducible case unless this cubic has equal roots. SPECIAL EQUATIONS. 623 If the given biquadratic has two real and two imaginary roots, then, since the sum of the roots equals zero, the roots must have the forms a + //3, a—//?, —a + y, — a— y ; therefore, the only possible values of ^ are d=2a, d=(y+2^), db(y— //?); therefore, the only possible values of A"^ are 4a2, (a^—l3'^) + 2z/3y, a'^ — p'- —2iPy ; and if y^O two of these values are imaginary and one real, but if y=0 all three values of ^^ are real but two of them are equal to each other. But when a cubic equation has three real roots, two of which are equal to each other, or one real and two imaginary roots, the equation does not fall under the irreducible case. Therefore, whe7i the given biquadratic has two real and two imaginary roots^ the auxiliary cubic does not fall under the irreducible case. 803. The roots of the biquadratic may be expressed in terms of the roots of the auxiliary cubic. Let a^, /32^ yi represent the three roots of the auxiliary cubic, then we have a2/?2^'^ = ^^ (1) a24-;S2-fy2 = -.2^. (2) Now, x''-\-Ax^B^x''-^Ax-^\[A''-^a-^' Substituting herein the values of a and b found from (1) and (2) and taking a for the value of y^, we obtain x'' +Ax-\-B=x'' +a;ir+i[a2 -^(a^ +^2 _^y 2)_^^] = ;^2_|.^^^^(-^2_^2_^2_2/3y) Placing this last expression equal to zero and solving, we get ^=-i(a+iS+y) orK~«+^+r). Treating the vSecond quadratic factor x'^-]-Cx+D in a similar manner, we obtain ^=-K^-/5+7) or x(a+^^y). 624 UNIVERSITY ALGEBRA. Hence, the four roots of the given biquadratic equation are —K^+P+y), K-^+^+y), K^-P+y). K^+P-y)- EXAMPLES. 1. Solve the equation x"^ — 5;r" + 10;i;— 6=0. 2. Solve the equation x^—x'^-{-2x+2, 3. Solve the equation jr^ — 9:^2-— 4%'4- 12=0. 4. Solve the equation x'^—6x'^+Sx—S=0. 5. Solve the equation x"^ — 4x^ -\-16x—16=0, 6. From the relation between the roots of a biquad- ratic equation and its auxiliary cubic equation, express the roots of the cubic in terms of those of the biquadratic. 7. If a biquadratic equation has one root equal to zero and has no term in ^r^, show that the roots of the auxil- iary cubic are the squares of the three remaining roots of the biquadratic. 8. If a biquadratic equation has three equal roots and has no term in x^, show that the auxiliary cubic has two equal roots. 9. If a biquadratic equation has three equal roots and has no term in x^, show that the auxiliary cubic has three equal roots and hence is a perfect cube. 10. If a biquadratic equation has three of its roots proportional to the numbers 1? 2, 3 and has no term in x'^^y show that the roots of the auxiliary cubic are propor- tional to the numbers 9, 16, 25. SPECIAL EQUATIONS. 625 Historical Note. — Of the few discoveries in pure mathematics by the Arabs, the most creditable is their geometric solution of cubic equations. At the beginning of the eleventh century, Abul Gud made much progress in the mastery of this problem. Soon after, it was brought to a more complete solution by Omar al Hayyami. The roots were determined by the intersection of conic sections. Only positive roots of the equations were recognized in these constructions. The algebraic solution of cubics was discovered in Italy in the six- teenth century. The first to attack the problem was Scipio Ferro, but his mode of solution was never made public. It was the practice in those days to keep discoveries secret, to secure by that means an advantage over rivals by proposing problems beyond their reach. The first solution of cubics handed down to us is that of Tartaglia. (died, 1557). In a contest with a pupil of Ferro, he beat him by solving thirty cubic equations in two hours, while Ferro' s pupil could not solve any of those proposed by him. Tartaglia' s victory became known to Hieronimo Cardano (1501-1576) of Milan, who, after giving the most sacred promises of secrecy, succeeded in obtaining from Tartaglia a knowledge of his rules. But Cardan broke his solemn pledge by inserting the much sought for rules in his A7's Magna. Tartaglia had intended himself to publish a work of which his solu- tion of cubics should be the crown. Cardan's treachery made Tar- taglia desperate. A long and acrimonious controversy between the two parties followed, in which Tartaglia demonstrated his superiority as a mathematician. Modern writers have done Tartaglia great injustice by attributing the solution of cubics to Cardan. To one of Cardan's pupils, Ferrari, belongs the honor of the general solution of equations of the fourth degree. The solution was published by Bombelli, in his treatise on algebra, in 1579. A solution known as Simpson's, which is not essentially different from Ferrari's, was pub- lished about 1740. In 1637, Descartes' work appeared. Besides his solution of the biquadratic equation it contained many important ad- ditions to algebra, especially the recognition of negative and imaginary roots of equations and the " Rule of Signs" which is given in the next chapter. Descartes attempted to obtain a general algebraic solution of equa- tions but, of course, failed. Euler attempted the same problem with the same result, and throughout the eighteenth century many math- ematicians busied themselves with this problem, but all attempts failed for equations above the fourth degree. On account of the failure to solve general equations beyond the fourth degree, it was natural that mathematicians should query whether such solutions were possible. Demonstrations have been given by Abel and Wantzelof the impossibility of ^oWm^ algebraically a general equation above the fourth degree. A transce^tdental solution of an equation of th.3 fifth degree has been given by Hermite in a form, involving elliptic integrals. 40 - U. A. CHAPTER XXXIV. SKPARATION OF ROOTS. 804. As we have already seen, equations of the first, second, third and fourth degrees can be solved algebraically. By this we mean that when an equation has literal coeffi- cients and is perfectly general in form, there is a straight- forward process of solution which will enable us to find the roots, and that the roots will be expressed in terms of the coefficients in such a manner as to involve only the ordinary operations in algebra, viz : addition, subtraction, multiplication, division, and involution and evolution to commensurable powers and roots. Abel, a Norwegian mathematician, has proved that a general equation above the fourth degree cannot be solved algebraically; that is, the roots cannot be expressed by means of the ordinary symbols of operation alone, but the function of the coefficients, which expresses the roots, is not within the range of algebraic analysis. Although we cannot solve, algebraically, a general literal equation above the fourth degree, still methods are known which-enable us to find the roots of nuuiericat equations above the fourth degree, and these methods it will be the object of this and the following chapter to explain. 805. The solution of numerical equations embraces two distinct problems ; first, the separation of roots, and second, the numerical calculation of the roots. The first of these two problems is the only one we shall consider in this chapter. SEPARATION OF ROOTS. 627 8C6. Any of tlie roots of an equation are said to be separated from the other roots when two numbers are found between which these roots and no others lie. Our attention at first will be confined to real roots, and of these we shall first consider the separation of the positive from the negative roots. 807. A Superior Limit of the positive roots of an equation is any positive number greater than the greatest of the positive roots. An Inferior Limit of the positive roots of an equation is any positive number less than the least of the positive roots. A sitperiof limit of the negative roots of an equation is any negative number numerically greater than the numeri- cally greatest of the negative roots. An inferior limit of the negative roots of an equation is any negative number numerically less than the numeri- cally least of the negative roots. 808. If tivo numbers be substituted in turri for x in f{pc) , giving results with opposite signSy there is an odd 7iumber of real roots of fix) between the numbers substituted. Suppose ai^"-'+/'2-^"~'+ • • • +A=0, if — N represent the 7iumerically greatest negative coefficient^ N-\- 1 is a superior limit of the positive roots. lyet the first member of the given equation be repre- sented by f{x'). Then, evidently, any positive value of X which makes x''—N{x''-'^-Vx''-'^^- . . . +1) positive will also make f(^x) positive. But ;r«-A^(;r'^-i+;t:«-2+ . . . +l)=x''-A^ «_1 x-\ -1 ... ,.^"-1 and x''-Kf^^—^>x''-\-N- also x—\ x—1 Therefore, any positive value of x which makes positive will also make f(x') positive. / N \ But (;t;"— 1)( 1 —TT ) is evidently positive if x—l';>JV; that is, if ;t:>iV^+l. Therefore, f^x) is positive for any value of x which is greater than N+ 1. Again, if x=N+l, then X** — 1 x''--N^^—i=l. x—1 Hence, x*'—N(ix*'~'^+x''-''^-{' • • • +1) is positive, and therefore /(;»;) is also positive when x=N+l. Therefore, when X is equal to or greater than A^+l,/(-^) is positive. Therefore, N+1 is a superior limit of the positive roots of/(;t)=0. 630 UNIVERSITY ALGEBRA. 811. In the equation if pr be the first 7iegative coefficient^ ajid if the nnmerically greatest negative coefficient be represe?ited by — N^ theJi 1 + V N is a superior li?nit of the positive roots. Let us represent the first member of the given equation by f{pc), then for any positive value of x each term before the term piX''~'' is positive. Hence, for any positive value of X /(:r)>;^«+/^^^-''+A+i-^''~''"' + ' ' * +A. (1) .-. /(;r)>:t:"-iV(;r«-''+;»;"-'-i+ . . . +1). (2) i. e. , /(^)>^«~i\^^!^^. (3) Hence, for any value of x greater than 1, Therefore, f{x) will be positive for any value of x that satisfies the two inequalities x>\ and;r"(^— 1)— iV;«^""'"^^>0. The second of these inequalities may be written ;i;''-i(:r-l)-A^>0, which is evidently satisfied when {x-\r-N>^, that is, when x—V>1/N, that is, when x> 1 + V N. Therefore, f{x) is positive for any value of x which is greater than _ \^Vn. Also if ;r= 1 + "!/ N the inequality x"-! (:*:■- 1)~A^>0 is still satisfied, for, x being; positive, ;f''-i(^-lj>(^-iy, SEPARATION OF ROOTS. 63 1 and since in this case A^=(;tr— 1)'', therefore Therefore, fix) is positive when x^=^\-\-V N. Therefore, any value of x equal to or greater than \-\-V N, makes /(;t) positive, and hence \-\-V N is a superior limit of the positive roots of/(;i:)=0. 812. If^ 2;^ a ratio7ial integral function of x^ s(^y f(.^^y each negative coefficient be divided by the sum of all the pos- itive coefficients which precede it^ and if the nuTnerically gjxatest quotient thus formed equals — iV, then N+\ is a superior limit of the positive roots of f{x)=^0. Suppose f(^x) = a^x'' + a^x''-^ + a^x"-^--a,^x''-'^+ • . . +^„, where a^, a^, a^, «3, etc., are all positive numbers, and a negative coefficient whenever it occurs is indicated by the presence of a minus sign. x—1 .'. x''-l = (_x-l)(x'-'^ +X'-'' -{-x"-^ + . . . +1). x''=lx'-l)Cx''-^ ^x"--^ +x''-^ + ■ . .+1) + 1. Now let us use this equation as a formula by which to transform each term of f(x') in which the coefficient is positive, and leave the terms in which the coefficients are negative without change. We thus obtain an ex- pression for /(jir) arranged as follows : aQ(^x—l)x"~^+aQ(x^l)x"-''^ + aQ(x—l)x''~^-j- • . . -j-a^ a^lx—l)x"-'^ +a^(x--l)x''-^ + ■ . . +a^ a2(x—l)x''-^-i- . . .+^2 ^a^x''"^ + . . . In this arrangement it is to be noticed that all the terms in any one horizontal line come from a single term of/(;i;), the first horizontal line coming from the first term of/(jr), 632 UNIVERSITY ALGEBRA. the second horizontal line from the second term oif{x), etc.; and since the fourth term of /(:t:) has a negative coefficient and therefore is left unchanged, the fourth horizontal line in this arrangement consists of but a single term, and the same is evidently true of any other hori- zontal line which comes from a term of /(:r) in which the coefficient is negative. Moreover, a little reflection upon the way in which the above expression is arranged will make it clear that when a negative coefficient appears, it must occur at the bottom of some vertical column and that only one negative coefficient can occur in any one column. Now let us consider the successive vertical columns in the above arrangement. When no negative coefficient occurs in a column, the value of that column is evidently positive when x^\. To insure a positive value to the third column, we must have (^o+^i+«2)(-^"— l)>^a) 0+^1+^2 a. i. e.y x>li-- . Similarly, ii f{x) contains a term — ^;,:r''~'' this term appears in the above arrangement at the foot of the rth vertical column and the terms above this in the same column contain onl}^ the positive coefficients of/(-r) which precede the term —arX"~*\ Therefore, that the value of the ;'th column may be positive we must have, by reason- ing as in the case of the third column, <^0+^l+^2+<^4+ • • • where the denominator is the sum of the positive coeffi- cients which precede the term — a^""''. SEPARATION OF ROOTS. 633 Now there may be several terms in f{pc) which have negative coefficients, and therefore several columns in the above arrangement in each of which a negative coefficient appears at the foot of the column, and we now see how x can be taken in such a way as to render the value of any one of these columns positive. Therefore, if x be taken greater than the greatest of the expressions of the form 1-1 ; -, , the value of each column in the above ^o+<^i+ • • • arrangement is positive, and hence the sum of all these columns or f{^x) is positive. If x be taken equal to the greatest oi Wi^ expressions of the form 1-| — — ^- j the value of one of the columns in the above arrangement is zero, but each of the others is positive, therefore the sum of all the columns or fix) is positive. Therefore, f{pc) is positive for any value of x equal to or greater than the greatest of the expressions of the form \-\ — — "— Therefore the greatest of the exjjressions of the form 1-1 "L is a superior limit of the positive roots a^-\-a^-\- • • • of/(x)==0. 813. Newton's Method. If a number, say /;, be chosen so that f{x) and its successive derivatives are all positive when x=h, then h is a superior li?nit of the positive roots of the equation f(x)=0. Let f(x) = x''+p,x''-'+p^x''-'-+p^x''-^+- . . +/,,and let the equation f(x')=0 be transformed by substituting jy-{-h {or X. By Art. 747 /(^)=0 becomes L_ L L_ Now, if the coefficients of the various powers of y are positive, it is evident that there can be no positive value of jK that will satisfy this equation, and My cannot be 634 UNIVERSITY ALGEBRA. positive X cannot be greater than h, therefore ^ is a superior limit of the positive roots of the equation /(^)=0. The problem now is to find the number h which, when put for X, will render f{pc) and its various derivatives positive. The method of finding h will be understood from a special case. Let us take the the equation ;t:5__5^.4_5^3_^25jt;2+4;r— 20=0. Here f{^x)^x^-'hx^ — hx'^-\-1hx'^-\-A:X—%) /'(x') = 5x^—20x^-16x^-{-50x+4: /^\x) = 20x'''-60x''-S0x+50 /'^'(x) = mx''-120x /^X-^) = 120;»;-120 /\x) = 120. • f^^x) is positive independent of the value of x. f'^X^) is positive when x=2. f^'ipc) is negative when :r=2. f'ipc) is positive when .r=3. f"(x) is negative when ;r=3. f"{x) is positive when jr=4. f\x) is negative when x=4:. f'{x) is positive when ;r=5. fix) is zero when x=^h. fix) is positive when jt:=6. Now, if the successive derivatives be examined when ^=6, it will be found that they are all positive; therefore, 6 is a superior limit of the positive roots of the equation /(^)=0. In this work it will be noticed that we began with the last function containing x) viz.: f^ix), and found the smallest positive integral value of x which makes this function positive, then substituting this value in the next preceding function, viz., f"\x) and finding in that case SEPARATION OF ROOTS. 635 f"\x) is negative we increased the value of x by unity. If the value of x, thus increased, had rendered f"\x) negative, we would have increased x by unity again, and so on until a value of x is found which renders f'"(x) positive. When this value of x is found, we pass to the next preceding function. In this way we pass back from one function to another until we finally arrive at the original function, and in the process we increase the value of X as often as necessary to make each of the successive functions positive, and when the original function is reached and a value of x obtained by this process which renders that original function positive, it is unnecessary to examine the functions already passed to see whether they are positive or not, for, as we shall presently prove, the process insures that all the functions passed over will be positive for the value of x reached in this way. The important fact just stated depends upon this prin- ciple; viz. : if a rational integral function of x and all its derivatives are positive when x^=^a^ tJieji this fu7iction of x is positive for a7iy value of x greater than a. Let ^{x) be a rational integral function of the rth degree, and let its successive derivatives be represented by ^'{pc), ''(-^), • • • ¥{j^^, and suppose that ^{x) and all its derivatives are positive when x=a\ that is, ^(«), ^th sign of M, while immediately under this is the (/^— l)st sign of b, which is of the opposite kind, so that the >^th sign of the product P is ?. Second, suppose the (>^— l)st and y^th signs of M are different;! then, as before, the >^th sign of a is the same as the y^th sign of M, but the sign immediately under this is the (>^— l)st sign of b, which is the (/^— l)st sign of M re- versed, and hence is the same as the /^th sign of a. Therefore, the y^th sign of P is the same as the k\h sign of M, unambiguous and unchanged. Passing along the signs of M and P from left to right, it is evident that the first s\%r). of P is the same as the first sign of y^and that the following signs of P are all 9, as long as the sign in M remains the same, but as soon as the sign in M changes from -f to ~ or from — to -f, the second sign of this change appears in P in the same position as in M. * lu this case k would be either 2, 3, 5, 6, or 8. f To illustrate this reasoning, k may be taken as either 4, 7, or 9 in the poly- nomial considered. 640 UNIVERSITY ALGEBRA. Therefore, in this portion of M and P, the signs begin alike and end alike, and as there is one change of sign in M there must be at least one change (indeed, some odd number) in P, and, as this holds good every time the sign of J/ changes, therefore there are certainly as many changes of sign in P as in M. (2). We will show next that there cannot be the same number of changes of sigji in P as in M. If the first sign of the polynomial is like the last, the number of changes of sign between the first sign and the last must be even\ but if the first sign differs from the last sign, then the number of changes of sign must be odd. Now the last sign of P is necessarily different from the last sign of M. Therefore, if the first and last signs of i^are alike, the first and last signs of P are unlike ; and if the first and last signs of M are unlike, the first and last signs oi P are alike. Hence, if there is an even number of changes in M, there is an odd number in P, and if an odd number in M, then an even number in P. Hence P cannot have the same number of changes as M, Now we have shown : (1.) that P has at least as many changes as M\ (2.) that P cannot have the same number as M. Therefore P has more changes of signs than M. That is, the result after multiplying by x—a, or intro- ducing a positive root, contains more changes of sign than the original polynomial. Since at least one addi- tional change is brought in for each positive root which may be introduced, no f{x) can have more positive roots than it has changes of sign from -f to — - and — to +. 818. Corollary. No f{x) can have 7nore negative roots than there are changes of sig7i from -{-to — and — /{? -f , after the signs of all the odd or even powers have been changed. See art. 758. SEPARATION OF ROOTS. 64I The above theorem and corollary constitute what is known as Descartes' Rule of Signs. KXAMPivKS. 1. Show that x^ — 1 has one positive root and no other real root. This function, being the difference of like powers, if divisible by x — \', whence -|-1 is a root. There is only one change of sign, hence it can have no more than one positive root, hence none other than -f- 1. Changing the signs of all the odd or even powers of x, there are no changes of signs, hence no negative roots. 2. Show that x^—1 has two real roots only, one positive and one negative. 3. Show that x^ + a has no real roots 4. Discuss the roots x"—a when n is odd, and also when n is even. 5. Discuss the roots of x^ + a when n is odd, also when 91 is even. STURM'S THKORKM. 819. In the previous articles of the present chapter we have found the limits of the roots of equations, and from these it is easy to separate the positive roots from' the other roots. We have also seen by Descartes' rule of" signs that the nurnberoi positive roots cannot exceed some- number easily found. We now proceed to a theorem which enables us to separate from all the other roots those real roots which are between any two given or selected numbers, and also enables us to determine the number and situation of the real roots of an equation. 820. Notation. Let f{pc) be a rational integral function of x, and let/(:tr) = be an equation with no equal roots, and let /'(.r) be the first derivative oi f{pc) and let the operation of finding the H. C. F. oifipc) and. 642 . UNIVERSITY ALGEBRA. f\x) be performed with the modification that when any remainder is obtained its sign is immediately changed, and let the process be continued until a remainder is found which does not contain x, and let this remainder, like the others, have its sign changed. lyCt the quotients in succession be represented ^^y q^, q^^ • • • ^«-i> and the modified remainders be represented by <^2(-^)> ^sW) <^4(;r.), . . . ^,ix). The whole series of functions fix), fix), cjy.ix), . . . cl^Xx), we will call Sturm's Functions. 821. Preliminary Propositions. I. T/ze last one of Sturm's functions cannot equal zeyv. This is easily seen, for <^;,(-r) is, by supposition, independent of x, and if it could equal zero, f{x) and f\x) would have a common measure and f{x) would have equal roots, which is con- trary to the hypothesis. II. In Sturm's functions, no two consecutive functions can va?iish simultaneously. From the process described in Art. 820 and the notation there used, we have the equations fix) = q^^(ix)-^.^ix) Now, if any two consecutive functions, for example: fix) and <^2(-^0 vanish together, the second equation shows that ^zi^^ ^^^^ vanishes, and then <^2(-^) and 4»'^ix) vanishing together, the third equation shows that ^{x) have opposite signs. 822. Sturm*s Theorem. If any two members^ a and P, of which a<)S J)e substituted in turn for x in Sturfn's functions^ the excess of the number of changes of sign when x=a over the number when x==l3 is equal to the number of real roots of f(x)=0 which lie between a and p. When X is given the value a, Sturm's functions, taken in order, present a certain array of signs and no one of these functions can change sign except when x passes through a value which makes that function vanish. First. lyCt che 3, value of x which makes one of the functions /'(■^), <^2W, <^3W> • • • <^^X-^) vanish, say <^2('^)=^- ^Y II> ^^ ^^^ preceding article, neither <^^_i(0 nor ^+i(0 can equal zero, and by III, of the preceding article, ^r-i(^) and r+i(c) have oppo- site signs. Hence, of the three functions ^_,(^(<:), <^r+i (0> the first and third have opposite signs for a value a little less than or a little greater than c, and no matter which sign cf^^Cc) has, the sign must be like the one just before it or just after it. Therefore, these three functions, taken in order, present one change of sign whether the value of ;i: be a little less or a little greater than c; that is, the number of changes of sign i?i the whole series of func- tions is not changed at all by x passing through a value which makes ^t^r^x) vanish. fix+K)=Ax)^hf\x)^'^f'{x)^ . . . +yJ\x) (1) 644 UNIVERSITY ALGEBRA. Second. Let f be a value which makes f{x\ vanish; Now, by Art. 747, h'' ^- Changing the sign oih, we have fi^x-K)=Ax)-hf{x)+~f"{x)+ . . . ±^/«W (2) Replacing x by ^ and remembering that f{c)—^, we have Ac-^h)= +k/Xc)+~f'(.c-)+ ■ ■ ■ +nJ/"W (3) Ac-h)=^-hf'{c)+~f"(c)- ■ ■ ■ ±?/"C^) (4) Now, k may be taken so small that in each of the equations (3) and (4), the second member has the same sign as its first term. Hence , for a small value of k , /(c+ Ji) has the same sign as f'{c) and f(c—Ji) the opposite sign from f\c)\ i. e,, for a value oi x 2. little less than c, fix) and fix) have opposite signs and for a value of x a little greater than c, f{pc) and f'{x) have the same signs. Therefore, the number of changes of sign in the whole series of Sturm^ s functions is decreased by one wheji X passes over a value which makes f(jx:)=0\ i. ^4(-^)> • • • ^«(-^)) and represent the quotients of the various divisions hy q^, q^, q^^ * * * ^«- Of course in this case <^,j(^) is not independent of Xy but we still have the equations Fix)=q^F\x-)^^^ix) F'(ix') = q^^2(x)-^^(x) Let D represent the H. C. F. of F(x) and F'(x), then the first of these equations shows that D is also a factor of ^2(-^)» a^^ thence the second equation shows that D is also a factor of ^3(^), thence the third equation shows that D is also a factor of ^4^(x'), and so on. Hence, D is a factor of each of the functions Fix-), F\x), ^2(^), ^aW- • -^.W. Now let us divide each of these expressions by Z>, and represent the various quotients by wherein ;X^) is independent of x. From the set of equations last written we derive the following : <^2W = ^33W-^4(^) Now, it is easy to see that these functions /(x), f{x)^ ^2W» ^sW- • -^wWi (still called Sturm's functions), 646 UNIVERSITY ALGEBRA. possess precisely the properties of Sturm* s functions in Arts. 821 and 822, and hence the argument of those articles applies without modification. Even the verbal statement of Sturm's theorem for the case of an equation v^ith equal roots does not differ at all from the statement for an equa- tion without any equal roots, and the only difference in the two cases is, that in case of an equation with equal roots, Sturm's functions have a slightly more general significance than in the case of an equation without equal roots. 824. In the operation of finding Sturm's functions we are at liberty to multiply or divide any dividend or divisor by any positive number we please, for this does not affect the signs of the functions, and it is only with the signs of the functions that we are concerned. To explain the application of Sturm's theorem we will work out an example in detail. Let us find the number and location of the real roots of the equation Here f{x)^x^-Zx''-^x-\-\Z /Xx) = 3x^-6x-4: 2(;r)^2;t;—5 ,(x) = l. We may now substitute any two numbers we please in place of X in these functions. Suppose we take -—10 and -flO; then, arranging the signs under the functions to which they belong, we have /(x) f\x) ,(x) ,(x-) when;c= — 10 — + — + (3 changes) when;r=10 + + + + (no changes) SEPARATION OF ROOTS. 647 As the number of changes of sign when jtr— — 10 ex- ceeds by three the number of changes when ;t:=10 we conclude that there are three real roots of /(;r) between -10 and +10. Now let us substitute some number intermediate be- tween — 10 and +10, say 0, then we have Ax) f\x) ,(X) cl>,(x) when x=0 + — —t + (2 changes) We now see that the number of changes of sign when x= — 10 exceeds by one the number of changes when x=0; hence, we conclude that there is one real root of /(^x)=0 between —10 and 0. Also the number of changes of sign when x=0 exceeds by two the number when ;t:=10; hence, there are two roots of /(;r)=0 between and 10. To find more nearly the location of the negative root, let us substitute some number intermediate between —10 and 0, say —5, then we have when;t:=— 5 — + — + (3 changes) As the number of changes when x— —5 exceeds by one the number of changes when :r=0, we conclude that there is one root between — 5 and 0. Substituting other numbers, we find fix) f\x) ,(x) ct>,(x) when;r=— 3 — + — + (3 changes) when:r=— 2 + + "~ . + (2 changes) As the number of changes when x= —3 exceeds by one the number of changes when x= — 2, we conclude that there is one root of /(;t:)=0 between —2 and —3. As there is only one negative root, we have located it between two consecutive integers, and hence have found its approximate value. /W when x=l + when x=2 + when x=S + 648 UNIVERSITY ALGEBRA. Proceeding now to find approximate values of the two positive roots, we will substitute different numbers for x and note the number of changes of sign in Sturm's functions. /XX) CI>,(X) c^3(^) — — • + (2 changes) — — + (2 changes) -1^ + + (no changes) As the number of changes of sign when x=2 exceeds by two the number of changes when x=o, we conclude that there are two roots of y(jr) = between 2 and 3. To separate these two roots which are between 2 and 3, we must substitute numbers between 2 and 3 in Sturm's functions, and from the number of changes of sign we can finally find numbers between which these roots lie singly. when x=2^ — — + (1 change) Whether we consider to be 4- or — is immaterial, for in either case there is only one change of sign. Hence as the number of changes of sign when x=2 exceeds the number of changes when x=2^, we conclude that there is one root of /(^) = between 2 and 2|, and as the num- ber of changes when x^=2\ exceeds by one the number of changes when x=S, we conclude that there is one root of y(.r)=0 between 2^ and 3. THEOREMS OF FOURIER AND BUD AN. 824. From a theoretical standpoint, Sturm's theorem leaves nothing to be desired ; but in practice the labor of obtaining the various functions called Sturm's functions is often so great as to discourage the most patient calcu- lator. For this reason the theorem is not used much ex* cept as a last resort. SEPARATION OF ROOTS. 649 The theorems of Budan and Fourier are theoretically much less perfect than that of Sturm, but are so easy of application that one or the other of them is often used in preference to Sturm's theorem. The theorems of Budan and Fourier are essentially the same, but the verbal statements as ^iven by these two mathematicians, are different. Before giving these theo- rems we give two preliminary propositions to prepare the way. 825. Preliminary Propositions. I. If f{_x) repre- sents a rational integral function of x, and f^ipc) its first derivative, a7id if a is a root of f{x) occurrijig r times, then for a value of x a little less than a, f{pc) and f^(x) have opposite signs and for a value of x a little greater thaji a, f{x) and f^ipc) have like signs. From Art. 747, we have Replacing ;r by a and remembering thaty(a) = 0, we have A—h)- — A/iC«)+|-/2(«) - ■ • • ±?/X«) .•./(«-/o=(-/o/i(«)+^-/2(«)+ ■ ■■+^^y.k^) Similarly, Now, if a is a root occurring in f{x) r times, /(a), fx^^^f'i^^ • * ' f-i(p) each vanish, and in the two series just written the first terms which do not vanish are re- spectively 6SO UNIVERSITY ALGEBRA. and since the exponent of the power of (— >^) is even in one case and odd in the other, it is plain that these terms have opposite signs. But when h is taken small enough each of the two second members above has the same sign as the first term which does not vanish. Therefore, f(a—Ji) and/i(a— /2) have opposite signs for some small value of h. If, now, throughout the discussion just given, the sign of h be changed, then the first terms in the two series which do not vanish have like signs, and therefore /(cL-^-Ii) and /i(a + /2) have like signs. II. If a is a root occurring iji f(^x) r times then for a value of X a little less thaii a the series of functions f{x), fx(x), ' . • fr(x:), have signs alternately + a7id — , and for a value of x a little greater than a each of these fu7ictio7is has the same sign as fix). From I, f{x^ andyj(x) have opposite signs for a value of ;t: a little less than a. Also, as f^i^x') is the derivative of/i(jr) the argument of I shows that f^ix) and f ^^ch function has the opposite sign from the preceding function; that is, the signs of these functions are alternately + and — for a value of ;r a little less than a. Again, the argument of I applies to any two consecutive functions of the series/(:r),/i(;r),/2(^), • • • fr(x) when x has a value a little greater than a, and shows that any two consecutive functions of this series have like signs; hence, for a value of jr a little greater than a all these functions have like si^ns. But since a is not a root of /^(;r) = 0, it follows that fr{x) does not change its SEPARATION OF ROOTS. 651 sign when x passes through the value a. Hence, when x is a little greater than a, each of the functions /W) /iW, /2W, • • • /.W has the same sign as fripc). 826. Fourier's Theorem. If two numbers, a and P, (a Z.+ lW, • • • f stripe) have signs alternately + and — when :t: is a little less than c and all have the same sign when Jt: is a little greater than c, and this sign is the same as the sign of yS+;,(;»;). Now, for a value of a little less than c, the sign ofyS+^_i(;tr) is opposite to the sign of j^+^(;tr) .*. the sign ofyS+^_2(-^) is the same as the sign of/i+^(jt:) .'. the sign of j^^.^. 3 (:r) is opposite to the sign ofy^+^(:r) and so on. Hence, since r is an eve7i number the sign of fsipc) is the same as the sign of /^^.^.(jr). Now, X still being a little less than c, the sign oifs-^{pc) may be the same as the sign of /^^^(jt), and hence the same as the sign of /X-^)> or it may be opposite to the sign oi fs^r^P^^i and hence opposite to the sign oi fsKx), First, if the sign oi fs-^ipc) is the same as the sign of fsix) there is no change of sign between y^_i(^) and/^(:i:) when ;r is a little less than c. Therefore, by Art. 825, II, when ^ is a little less than c, the series of functions fs-^{x), fix), /.+ i(-r),- • • /.+.(-:r) present r changes of sign, and, since when :r is a little greater than c, this same series of functions all have like signs, therefore, as ^ passes through c, there are r changes of sign lost. Second, if, when :r is a little less than c, the sign oif,_^ {x) is opposite to the sign of yS(^), then there is one change of sign between fs-^{pc) and fs{pc), and therefore when x is a little less than c, the series of functions fs-xix). fsipc). /.+ i(-^), • • • /.+.(-^) present r+ 1 changes of sign, and since when ;t: is a little 654 UNIVERSITY ALGEBRA. greater than c this series of functions, with the exception ofyS_i(-^), all have like signs, therefore the series of functions /.-iW fs{pc). Z+iW, • • • /.+.(-^0 present one change of sign when ;r is a little greater than c. Therefore, as x passes through c there are r changes of sign lost. Hence, we conclude that whether the sign of y^_i(^) is the same as or opposite to the sign of /i_^^(;f) there are r changes of sign lost in the series of functions as X passes through the value c. It is true that these are only part of the whole series of functions we are dealing with in Fourier's theorem, but of the whole series from f{pc) to fnix) the only ones that change sign when x passes through c are some of those from/X-^) \.o fsJ^r-^ix) inclusive. Therefore, r being an eyen number, as x passes through c there are r changes of sign lost in the whole series of functions Let us next take the case in which r is an odd number. As before, the sign of/,.+^_i(:t:) is opposite to the sign of/^4.^(x) .*. the sign of/^+^_2(^) is the same as the sign of/^+^(x) .-. the sign oi fsJ^r-zipc) is opposite to the sign of/^+;.(;t'), and so on. Hence, since r is an odd number, the sign oifsipc) is opposite to the sign of/^+^(x). Now, X still being a little less than c, the sign oifs-^ {x) may be the same as the sign oif stripe) and hence opposite to the sign oi fs{pc), or it may be opposite to the sign of fs\r{x^, and hence the same as the sign oifs{pc). First, if the sign of yS_i(:r) is the same as the sign of fs^r^pc) and hence opposite to the sign oi fXx) there is SEPARATION OF ROOTS. ' 655 one change of sign between yS-i(-^) and fs^x) when x is a little less than c. Therefore, by Art. 825, II, the series of functions present r+l changes of sign, and, since when ;t: is a little greater than c this same series of functions all have like signs, therefore, as x passes through c there are r+1 changes of sign lost. Secojid, if the sign oi fs-^^pc) is opposite to the sign of f{pc) and hence the same as the sign of //.r), there is no change of sign between /^_i(;t:) and /X-^) when ;r is a little less than c. Therefore, by Art. 825, II, the series of functions fs-xipC). fsix). fs4-l{x),- . . fs+r(x) present r changes of sign when x is a little less than c. Also, by Art. 825, II, this same series of functions, with the exception of fs-\{x), all have like signs w^hen ;i: is a little greater than c. Therefore, this series of functions present one change of sign when Jt: is a little greater than c. Therefore, as x passes through c there are r—1 changes of sign lost. Hence, when r is an odd number, there are either r+1 or r—\ changes of sign lost as x passes through the value c. But when r is an odd number r+1 and r—l are both even numbers. Thus, we see that in this fourth case whether r is an even or an odd number there is always an even number of changes of sign lost as X passes through the value c. Reviewing now the four cases of the demonstration just given, we conclude I. That as x increases from a to /? there is never a gain in the number of changes of sign. II. That each time x passes through a single root of y*(;i;)=0 one change of sign is lost. 6S6 UNIVERSITY ALGEBRA. III. That each time x passes through a root which occurs r times in f(x') = 0, r changes of sign are lost. IV. That in no case except when x passes through a root of /(jr) = can an odd number of changes of sign be lost. Hence, from these results it follows that the whole number of changes lost during the change in x from a to p must be equal to the number of real roots of /(.r)=0 between a and yS, or must exceed the number of these roots by some eveti number. 827. Budan's Theorem. If the roots of the equation f(x')=Q be diminished first by a a?zd then by )8(a'+/2(«)|^+ • • • +/<(«)£=0- and from the preceding article the truth of this theorem is now evident. Budan's statement of the theorem is rather easier to apply to numerical equations than Fourier's because the transformation of the given equation is so easily accomplished by the method of Art. 762. 828. It is well to notice that Fourier's theorem really includes both Descartes' rule of signs and Newton's method of finding the limits of the roots of equations. SEPARATION OF ROOTS. 657 Descartes^ rule of signs. If we take ioxf{pc), then we have the following identities: f^(x^ = na^x^-'-^(n-l)a,x^-'^ + (n-2')a^x^-^ + ,..+a„_^ f^(x) = n(n—l)aQX''-^ + (n—l){n—2)a^x''-^-^ h«„-2 From these equations it is easily seen that when x=0 the series of functions reduce to the coefficients in /(x) taken in reverse order. But when x is taken equal to some sufficiently large number, say d, all these functions have the same sign and are all positive or negative according as a q is positive or negative. Therefore, the number of changes of sign in the coeffi- cients of /(x) is equal to the excess of the number of changes of sign in the functions when x=0 over the number of changes when x=d. Hence, by Fourier's theorem, the number of real roots of /(;i:)=0 between and d is not greater than the num- ber of changes of sign in /(-^). But d is supposed to be a number greater than the greatest positive root of/(x)=0, therefore the number of roots between and d is the number of positive roots of the equation. Hence, the number of positive roots of /(x)=0 is not greater than the number of changes of sign in /{x)y and this statement is Descartes* rule of signs. 42- U. A. 658 UNIVERSITY ALGEBRA. Newton^ s method of finding limits of roots. Suppose h is some number which, substituted for x, renders each of the functions positive. But, as was shown in Art. 813, each of these functions is also positive when x=h-\-k. Therefore, it follows by Fourier's theorem that there are no roots be- tween h and ^+>^ however great k may be. Therefore, ^ is a superior limit of the positive roots of the equation y(jtr)=0. KXAMPI^KS. 1. Show by Sturm's theorem that the equation ;t:^ + 6:1;^ + 10;r— 1=0 has only one real root and that this root is less than unity. 2. Apply Sturm's theorem to the equation 3. Apply Sturm's theorem to the equation .r4 + 2;t;2— 4;^-fl0=0. 4. Apply Sudan's theorem to the equation ;t:4-f3;i;3+7^2_j.i0;r+l-=0. [5. Show that the equation ;^;5__3^4_24;r3+95;t;2_4e^_101=0. has all its real roots between —10 and 10. 6. Find the integral part of the positive root of the equation ji:^— 4;>;— 12=0. 7. Show by means of Sturm's theorem that the equa- tion ji:^ + ll;i;^ — 102;i;+181=0 has two roots between 3 and 4. 8. Apply Sturm's theorem for equal roots to the equa- tion ^*-5;<;3+9;j;2_7^_|.2=0. CHAPTER XXXV. NUMKRICAIy EQUATIONS. 829. Although a general solution of the general equation of the nth degree does not exist, yet it is possi- ble to find the real roots of an equation of any degree, provided the given equation has numerical coefficients. The process of solution depends upon the properties of f{pc) already established, and is satisfactory in all respects, giving the value of the roots exactly, if commensurable, or to any desired degree of approximation if incommen- surable. 830. We begin with an illustration of the general method by solving a particular example, and shall after wards summarize the process in a general statement of advice for any case. Suppose it is required to solve the equation I. Put x-=\y, which transforms the given equation into the following (Art. 760), which has no fractional roots (Art. 759). <^(_y)=j,5_3j/4_i6^3_^28j/2+72j/+32=0. (2) The real roots of this must be either whole numbers or incommensurable numbers. II. In seeking for integral roots, we need only search among the factors of the absolute term (Art. 757), which are: zhl, ±2, ±4, ±8, ±16, ±32. We may now test 66o UNIVERSITY ALGEBRA. these by dividing (2) by y minus each of them (Art. 741). Thus: 1 -3 -16 +28 +72 +32 (+1 + 1-2 -18 +10 +82 1 -2 -18 +10 +82 +114 + 1 is not a root. 1 -3 _16 +28 +72 +32 (+2 + 2-2 -36 -16 +112 1 -I -.18 - 8 +56 +144 +2 is not a root. 1 _3 _16 +28 +72 +32 (+4 + 4+4 -^48 -80 -32 1 +1 _12 -20 - 8 +4 /^ a root. This last quotient gives us an equation • y +_^3 _i2y2 _90y~8=0. (3) which is of one lower degree than (1) and which contains the remaining roots. Now we need try only factors of 8, of which we have already tried +1, +2, +4. By trial it is found that +8 is not a root. In general it is better to test the small negative factors before testing the large positive factors. Dividing by J/+1, corresponding to a root —1, we have 1 +1 _12 -20 +8 (-1 -1 ^ +12 -8 1 -12 - 8 —1 is a root. The equation containing the remaining roots is ^3_i2y-8=0. (4) Using the untried factors of 8, we find no more integral roots. Hence, the equation must have three incommen- surable roots, or one incommensurable and two imaginary. NUMERICAL EQUATIONS. 66l III. It is now well to test for eqtial roots, for if two or three of these incommensurable roots are equal, we can find them very readily by Art. 769. But y^ — Vly—Z and the derivative Zy'^—Vl have no common divisor, hence there are no equal roots. IV. We next attempt to locate the incommensurable roots by assigning different values to y and calculating the corresponding values of <^( jj') (Art. 808). If we put any value, a, for y\n <^(jk), we can compute the value of {ct) by the short method of division ; for ^{a) is the re- mainder when ^(y^ is divided by j/— ^. (Art. 739). Thus, from (-2) etc., etc., etc., etc. In like manner we find the values of <^(jy) when y has other values and tabulate them as in the mar- gin. The first column contains the values JL. assigned to y and the second contains the cor- ~^ responding values of ^{^y). It is then noticed _^ that there is a root between +3 and +4 and — i between and —1 and between —3 and —4. (Art. 808\ +1 V. The roots thus located can be deter- J"^ mined to any degree of accuracy in the manner , ^ following. Since equation (4) has a root be- tween + 3 and +4, the first figure of the root is 3. Then <^(J') -21 + 1 + 8 + 3 - 8 -17 -24 -17 + 8 662 UNIVERSITY ALGEBRA. transform the equation, by Horner's method, into one whose roots are 3 less. The work is as follows : 1 +0 -12 -8 (3 + 3 +9-9 -1-3-3 + 3 +18 +6 + 3 + 15 -17 (5) 1 I +9 The resulting equation is j)/8+9j/2+15j/-17=0. This must have a root between some of these values : .0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0. By trial it is found to lie between .7 and .8, thus : 1 +9.0 +15.00 -17.000 (.7 + .7 + 6.79 +15.253 1 +9.7 +21.79 1.747 +9.0 +15.00 -17.000 + .8 + 7.84 +18.272 (.8 1 +9.8 +22.84 + 1.272 The first figure of this root of (5) is therefore 7, which is the second figure of a root of (4). Now depress the roots of (5) by .7, by Horner's method. The work is as follows : 1 +9.0 +15.00 -17.000 (.7 + .7 + 6.79 +15.253 +9.7 + ,7 +21.79 + 7.28 1 +10.4 .T_ 1|+11.1 +29.07 - 1.747 NUMERICAL QUATIONS. 663 The resulting equation is jj/3 + ll.iy+29.07>/-l. 747=0. (6) This must have a root between some of these values : .00, .01, .02, .03, .04, .05, .06, .07, .08, .09, .10. By trial it is found to lie between .05 and .06, thus : 1 +11.10 +29.0700 -1.747000 (.05 + .05 +.5575 +1.481375 1 +11.15 + 29.6275 - .265625 1 +11.10 + 29.0700 -1.747000 (.06 + .06 + .6696 + 1.784376 1 +11.16 +29.7396 + .037376 Hence, 5 is the first figure of a root of (6), which is the second figure of a root of (5), which is the third figure of a root of {^)) therefore, this root of (4), correct to three figures, is 3.75. We now depress the roots of (6), by .05. The work is as follows : 1 +11.10 +29.0700 -1.747000 .05 .5575 1.481375 (.05 1 + 11.15 .05 +29.6275 .5600 - .265625 1 +11.20 .05 +30.1875 1 I +11.25 The resulting equation is jj/3 4.ii.25y+30.1875y-.2656250=a (7) By trial this equation is found to have a root between .008 and .009. Hence 8 is the first figure of a root of (7), the second figure of a root of (6), the third figure of a root of (5), or the fourth figure of a root of (4). Hence a root 664 UNIVERSITY ALGEBRA. of (4), complete to four figures, is 3.758. It is evident that a root cau be determined to any degree of accuracy by a continuation of this process. In like manner we could find the other incommen- surable roots of (4), either the one between —8 and —4, or and —1. For, an incommensurable negative root can be found if the negative roots be transforvted into posi- tive ones before apply vig Horner' s Method. This can be done by Art. 758. We may obtain the approximate values of the other two roots from the quadratic resulting from removing the approximate root from (7). 1 +11.250 +30.187500 -.265625000 (.008 .008 .090064 .242220512 1 +11.258 +30.277564 j/2 + ii.258j/+30.277564=0 (8) y + 11. 258JI/+31. 685641 = 1.408077 whence, jj/=— 4.443 and —6.815. But, remembering that these roots are 3.75 less than those of (4), we really have, as the roots of (4), j/=-.693and -3.065. VI. Finally, collecting all the values of y found, and remembering that x=^\y, we obtain these results as the solution of j^=-4, or -1, or +3.758 + , or -.693, or -3.065 ^= -2, or -^, or +1.879 ... or -.345 ... or —1.533 ... 831. The Principle of Trial Divisors. The pro- cess given above for finding the successive figures of a root would be found in practice to be very tedious, since each successive figure is determined from among several NUMERICAL EQUATIONS. 665 digits by actual trial. But it will be found that after one or two figures of the root are obtained as above, that a suggestion of the next figure can be obtained in a very * simple way. To illustrate this, consider equation (7) in the last article: j/3 + 11.25y + 30.1875j/~.^65o25=0. We know that y is some number of thousandths plus something. We are determining the figures of the root one at a time, and at present merely desire the number of thousandths. By proper transformations in the equation it is evident that _ .265625 -^"j/' + 11.25j/+30.1875* Now, since jK is known to be a fraction, and less than one hundredth, the most valuable term in the denominator of the fraction is 30.1875; for, the higher the powers oi y the less account they are when _y is a fraction. Hence -^^sral ^ea^ly=-000S79+ (9) Whence, it is quite certain that 8 is the first figure of y, or the fourth figure of a root of (4). Since this same reasoning would apply whatever the given equation might happen to be, therefore, in any case, when two or three figures of a root have been obtained, a suggestion of the next figure can be had by dividing the absolute term of the depressed equation by the coefficient of the first power of the unknown number. This is known as the Principle of Trial Divisors. Of course, as we find more figures of the root and contin- ually depress the roots of the equation, the smaller y becomes and the more surely can we rely upon the sug- gested value. Thus, two figures of the approximate value of y, given by (9), are really correct, as will be seen. 666 UNIVERSITY ALGEBRA. 832. Arrangement of Work. We now give a more elaborate arrangement of the work of Horner's method as used in article 830. The lines that extend across the page are the divisions between the successive depres- sions. Follow each line accross and the numbers beneath it are the coefficients of the equation with roots depressed. The fifth figure of the root was obtained by the method of trial divisor; that is, by dividing .023404488 by 30.367756. 11.274 -12 _9 - 3 i8 15.00 6.97 - 8 - 9 (3.7589 -17.000 15-253 1.747000 1.481375 9.0 21.79 - .265625000 .7 7.28 .242220512 9.0 29.0700 - .02uoms .7 10.4 •7 .5575 29.6275 .5600 11.10 •05 30.187500 . 090064 II. 15 30.277564 •05 .090192 11.20 30.367756 • 05 11.250 .008 11.258 .008 » 11.266 .008 833. Summary. The equation just solved illustrates the method of procedure by which we may determine the real roots of any numerical equation. We briefly sum- marize the process in the following statement of advice for any case: NUMERICAL EQUATIONS. ^6j Any numerical equation being given : I. Transform the equation, if necessary, so that the coefficient of the highest power in f{pc) shall be unity and none of the other coefficients shall be fractions. II. Search among the positive and negative factors of the absolute term for integral roots by dividing f{x) by X minus each factor by synthetic division. Use the numeri- cally smallest values first. Depress the degree of the equation whenever a root is found. III. When all integral roots are found, test for equal roots, by noting whether the function and the first deriva- tive have a common divisor. IV. Tabulate the function and locate the incommen- surable roots by Art. 808, and approximate them as desired by Horner's method. V. As soon as a quadratic equation is obtained, solve it in the ordinary way. 834, The separation of the roots may be effected by Sturm's theorem, as explained in Art. 824. But the labor of computing Sturm's functions, renders the appli- cation of that theorem undesirable save in exceptional cases. Among these exceptional cases may be mentioned the one in which an equation has two or more roots lying between consecutive integers. For, if there be an even number of such roots, they would not be pointed out by Art. 808, but Sturm's theorem would indicate their presence. Thus, see the example in Art. 824. KXAMPI.KS. Solve the following equations : 1. j»;3— 9;r2+23;r— 15=0. 2. .;»;4_|.2;^3_2l;t:2_22;,;-f40=0. 66S UNIVERSITY ALGEBRA. 3. :tr^--5jir3 — 13:^2 +52jr 4- 60=0. 5. x^—x^—x^ + 19;i:— 42=0. 6. Jtr^— 3jtr4— 9:^3 -f21;i;2-10;r+ 24=0. 7. ;«;3--4;tr— 12=0. 8. ;t:3— 24;i;+44=0. 9. ^3^10;t:2+6ji;— 120=0. 10. ;r3+:r— 3=0. 11. ;i;3-3;i;+l=0. 12. 3;i;3 4-5;»r-40=0. 13. x3-17=0. 14. ;i;4-15=0. 15. x^—3.5x^--h2x-}-2=0. 16. .r3-15^-5=0. 17. ;t:4-8-r^ + 12.^2 _^3;t:--4=0. 18. x^-6x^ -\-8x'^ -17x + 10=0, 19. 2;i;5-7;v*-9;t3 + 33.;i;2 + 17;i;~30=0, CHAPTER XXXVI. DECOMPOSITION OF RATION AI. FRACTIONS. 835. A fraction in which both numerator and denom- inator are rational integral functions of x is called a Rational Fraction. If the numerator of the fraction is of a lower degree than the denominator, the fraction is called a Proper Rational Fraction. If the degree of the numerator is equal to or greater than that of the denominator, the fraction is an Improper Rational Fraction. An improper fraction may always be reduced to a mixed expression ; that is, to the sum of an integral expression and a proper fraction. 836. The problem of this chapter is to decompose any given rational fraction into a set of simpler fractions called Partial Fractions, whose sum is equal to the given rational fraction. If we wish to decompose an improper fraction, we first reduce it to a mixed expression, and then decompose the resulting proper fraction. Since any improper fraction may be treated in this way, we will confine our attention in the discussion which fol- lows to the case of proper rational fractions. 837. We will first decompose a rational fraction into the sum of two other fractions, then the second of the two resulting fractions into the sum of two more fractions, and so on. 670 UNIVERSITY ALGEBRA. Let 4^^ denote a proper rational fraction and suppose a root a occurs 71 times and no more in Fipc). Then we have Ax)^ Ax) F{x) {pc-aYF^ix) where, of course, F^{pc) denotes what remains of F{x) after removing the factor {x—cCy, and evidently F-^ipc) does not contain the factor {x — a). Now, whatever A stands for, we have fix-) ^ AF,{x-)+f{x)-AF^ix) {x — aJF^ {pc) (x — a) "F^ (x) AF,(x) _^/(x)-AF,(x) (x—a) 'F-^ ( Jf) {x — a) 'F^ (;t:) A ^ /(x)-AF,ix-) (x—ay (x—ayF^(_x) As the sum of the last two fractions equals the given fraction whatever A stands for, we may take A anything we please. Let us then, if possible, select A so- that the numerator of the last fraction shall contain the factor (x—a). Now, this numerator /(;«;)— ^/^i(;r) is some function of X, and hence, if it contains the factor x—a, a is a root of Ax)—AF^(x) by Art. 471. But if a is a root of /(x)—AF^(x) this expression must vanish when a is substituted for x by Art. 736. Hence, we must have /(a)-AF,(a)=0. Hence, ^ = -Wt\ F^{a) an expression which is evidently independent of x. With this value of A, the expression /(;»:) -~^i^j,(.:r) contains the factor (x—a), and hence the fraction f{x)-AF,{x) (x-ayF.ix-) can be reduced to lower terms. DECOMPOSITION OF RATIONAL FRACTIONS. 67 1 Now representing hy f^(^x) the quotient obtained by- dividing /(.r)—^/^i(ji;) by {x—d), we have where the numerator y^ is independent of x and the second fraction of the right-hand member is a proper rational fraction. In exactly the same manner as above, the fraction , /^ , -r mav be expressed as the sum of two fractions. Hence, we may write (;^;-a: ^^^i(^) {x-ay-^ {x-ay-'' F^{pc) Again, in the same way, we have (3) Evidently we can continue to obtain successive equations as long as there remains any power of {x~d) in the denominator of the fraction we last obtain. Hence, making successive substitutions in equations (1), (2), (3), etc., we obtain {x-ayF^{x) {x-ay^ {x-a)"-^'^'" '^ x-a'^ F^{x) where the numerators in the first n fractions in the second number are all independent of x, and the last fraction is a proper rational fraction. If now, a root b occurs s times and no more in F^ (x), then {x) ^ (x) F,ix) Cx^dyF,(xy in which F2 (x) represents the quotient obtained by divid- ing F,(x) by (x—dy. 672 UNIVERSITY ALGEBRA. In exactly the same way as above it may be shown that <^(^) _ B B^ B^, d>,(x} Evidently the process pursued thus far may be con- tinued until all the factors of the denominator are ex- hausted, when the decomposition comes to an end. /(x') Therefore, any proper rational fraction ~~-( can be decomposed into a set of partial fractions whose numera- tors are all independent of x atid whose denominators are the successive powers of the binomials x — a, x — b, etc., (a, b, etc., being roots of the given denominator); the highest power to which any binomial appears in any denominator being the power to which that same bi- nomial appears in the denominator of the given fraction. 838. We will next show that there cannot be two different sets of simple fractions whose sum equals the given rational fraction. Suppose, if possible, that the rational fraction v^y-t is F(x} equal to the sum of each of the two sets of partial fractions A A, A,^_, 6(x-) \ "T 7" _N»4-i "T •• • • -r— — - -r (x—aY {x—ay^^ x-r-a F^{x) and 7 ^+7 TTZT "^" ■ * • ^ ^"cv~~7^>' {x—ay {x—ay ^ x—a F ^{x) Each of these sums being equal to the given fraction, we have ^ ^1 , A,,_^ (x) (x—ay (x—ay-'^ x—a F^{x) DECOMPOSITION OF RATIONAL FRACTIONS. 673 If n and r are not equal to each other, suppose ri>r\ then if we transpose all the terms of the first member except the term _^ and represent the resulting sec- ond member by ^^_^|f.^,^^^y we have (x^ay (x—ay-^'ifCxy in which "^(x) does not contain the factor (x—a), and A is independent of x. If we put x= a in the last equation, we get ^ = 0. Hence, if w>r, we must have A — ^. Hence, if ^^0, n cannot be greater than r. In the same way it may be shown that if n i. ^., under the pres- ent supposition A=^~r-, — -r^- DECOMPOSITION OF RATIONAL FRACTIONS. 6/5 Evidently, however, we might have begun by finding the numerators corresponding to the various powers of x—b. To find B we would proceed exactly as above when A was found except that everywhere the sign of i would be changed. Therefore, the value of B is the same as A except in the sign of i. Hence, if ^ = #F^V If we represent the value of A by l+im, the value of B will be / — t7n. Therefore two of the partial fractions in the above series become in the case here considered l+im ^ l—im and {x—a—i^y (x—a+ipy Now it is readily seen that the sum of these two frac- tions is real. For, let Cx—a^tpy^L-ht'M, then (ix—a-\-zfiy=L—zM, l+nn l—im __2{/L + 7nM) L-ViM^ L^M" Z,2+^"2— Now, since {x—a—ipy'=L + iM, it is evident that L is a function of x of the nih degree, and M a function of X of the {n—l^st degree. Therefore, the sum of the two partial fractions is a fraction whose numerator is of the degree n and whose denominator is of the degree 27t. The denominator is in fact {x'^—2ax-\-a'^-\rP'^yy hence we may write A , B ^"{x) where '\(x) is of the degree n. (1) t^6 UNIVERSITY ALGEBRA. In precisely the same way the partial fractions ^ - and ^ when added together give a real fraction whose numera- tor is of the degree n—\, and whose denominator is of the degree 2(7Z — 1). The denominator in this case is (;t;2—2a;ir4-a2+/3 2 )'*-!, and hence we may write -^1 + ^ = ^^Z(fL_ .2) In a similar way we obtain ^. . Bj_ <^{^ ..^ (;t:_a-/iS)''-2"^(;t--a4-2W"-2 (;r2-2a;f -f a^ 4-/52)«-2 W Evidently this process of grouping partial fractions together as has just been done may be continued until all the corresponding partial fractions are grouped together into real sums. When this is done, instead of the origi- nal partial fractions corresponding to the roots a-fz^ and o.—i^ each occurring n times in the original denominator, we have the partial fractions If now these fractions be reduced to a common denom- inator and added we obtain a single fraction whose de- nominator is (.r2 — 2a.r + a2-f-/?-)" and whose numerator is a real function of x of the wth degree. xCf) (;»;2-2a;r + a2+y82)n Divide the numerator by (;r2 — 2a;f+a2-f ^2^^ ^^^ 1^^ Xi(^) denote the quotient and Cx-\-D the remainder, then X(;tr) = (;r2-2a;t: + a2+/32)^^(;^)4_C-_f.^. Hence, (;»;2~2a;t: + a2+/32)« Cx^-D ^ XiW (;«;2_2cu;-|-a2+/32)« ' (;^2_2a;t: + a2+^2)«- '-1 DECOMPOSITION OF RATIONAL FRACTIONS. ^TJ The same process may be repeated upon the second fraction in the right member and repeated again in the second fraction of the resulting right member, and so on until a fraction is obtained whose denominator is {x-—^a.x-\-aP'-\-^'^-''), We thus obtain ;z partial fractions whose numerators are of the first degree and whose denom- inators are the successive powers of {x'^—^o.x+aP'-\-P'^^. Hence ^— ^ Cx+D C,x-\-D^ C^X + D^ Cn-xX -\- Dn-^ ' / ^.9. O r -.2 I 0'2\H—'i "t" • • • "t" Reviewing now article 837 with the present article, we conclude that to each real root occurring n times in the denominator of a given proper rational fraction there cor- responds n partial fractions of the form given in the first n fractions of the second member of the equation in Art. 837, and to each pair of imaginary roots occurring r times in the denominator of the given fraction there corresponds r partial fractions of the form given in this article^ This may be otherwise expressed by saying that to a factor (:r— a), which occurs n times and no more in the given denominator, there corresponds n partial fractions of -the form given in Art. 837, and to an irreducible quadratic factor x'^ —Icxx-^oP- -\- ^'^ , which occurs r times in the given denominator, there corresponds r partial fractions of the form given in this article. DETERMINATION OF NUMERATORS. 840. Having found the form of the partial fractions into which a given fraction can be decomposed, it remains now to determine the values of the various numerators. 6/8 UNIVERSITY ALGEBRA. We will first take a particular example. After explain- ing this, the statement of the method to be pursued in any case will be readily understood. Let us decompose the Assume (^x+i)\x'-\-x^iy (^x-^iy^ x^i {^xf^x+iy x'-\-x+i Reducing the partial fractions to a common denomin- ator, adding, and clearing of fractions, we obtain x'^-'Zx'-'2=-A(^x'^ +x+\y +A ^{x+l)(x'^ +x+iy + {Bx-]-C)(,x+\y + (^B,x+C,')(x+l)\x''+x+r). Arranging the second member of this equation accord- ing to powers of x, we have x^'-^x—2={A^+B^)x'^-\-(^A+ZA^+ZB^ + C^')x^ + (2A+bA^+B-^AB, + '^C^)x^ + (2A+ZA^+B+2C^B^+ZC,)x Now as the sum of the assumed partial fractions equals the given fraction whatever value be given to x, the equation last written is an identical equation, and hence, by Art. 596, the coefficients of like powers of x on the two sides of the sign of equality are equal each to each. Hence, by equating coefficients, we have A,+B,=0. ^ + 3^,+3^, + Ci=0. 2^+5^1 4- i5+4i5i+3Ci==0. 3^+5^1 +2Z?+C+3i9i+4Ci = l. 2^ + 3^1 +i5+2C+i5i4-3Ci = -3. DECOMPOSITION OF RATIONAL FRACTIONS. 679 Solving these equations, we obtain the values ^=3. A,== 1. i?=2. i?, = -l. C=-2. Cx = -3. Introducing- these values in the assumed set of partial fractions, we have (x+iyix^+x-hiy C;^+l)2"^;c+l 2;c-2 -'X-S '^(x^-hx+iy'^x'^+x+l The method pursued in this example will render the following directions readily understood. First. If the fraction is improper, reduce to a mixed expression and consider then the resulting proper rational fractions. Second. If this fraction is not in its lowest terms, re- move all fractions common to numerator and denomin- ator. There will then remain a proper rational fraction in its lowest terms. TJiird. Equate this proper rational fraction to the sum of a set of partial fractions of the forms described above with undetermined numerators. Fourth. Clear the resulting equation of fractions by multiplying by the denominator of the fraction in the first member. Fifth. Equate coefficients of like powers of x in the two members of the resulting equations. Sixth. Solve the resulting equations and thus find the numerators of the assumed set of partial fractions. 841. The above method may be somewhat shortened, especially when the denominator of the given fraction has 680 UNIVERSITY ALGEBRA. no imaginary roots; that is, when it has no irreducible quadratic factors. This is done as follows : Multiplying by /^(:r), we obtain fCx) = Aoct>(x-)+A,(x-a')(ix)+ . . . +(;c-a)'',/.(;r). Since this equation is true for all values of x, let x=a. Then all the terms in the second member except the first term vanish, and we obtain Hence ^0 = 4-^ {z) z^'^ z'^ z^ z'^'^ z^^ z^'^ z'' z "^(^-1)10* Then, by the formula already explained, we readily obtain the following values, where the denominator being 1 is in each case omitted. ^^ = 1 + 5x10=51. ^2 = -2 -5x45 + 51x10=283. . ^3 = 1 + 5 X 120 - 51 X 45 + 283 x 10 =1136. i?4=0 - 5x210 + 51x120 - 283x45 - 1136x10 = 3695. i?g=0 + 5 X 252 - 51 X 210 +• 283 x 120 - 1136 x 45 + 3695x10=10340. ^g=0 - 5 X 210 + 51 X 252 - 283 x 210 + 1136 x 120 - 3695 X 45 + 10340 x 10 =25817. ^^ =0 + 5 X 120 - 51 X 210 + 283 x 252 — 1136 x 210 + 3695 X 120 -. 10340 x 45 + 25817 X 10 = 58916. DECOMPOSITION OF RATIONAL FRACTIONS. 685 If now we restore the value of z, viz.: x—^, we have all the partial fractions whose denominators are powers oi X — 2, and since the partial fractions whose denomina- tors are powers of x — 3 have already been found, we have completely decomposed the given fraction. Hence we have the following: 607 1678 4044 8790 17642^ {x-zy^{x-zy {x-zy'^ {x-zy {x-zy 82259 51631 . 5 . 51 . 283 {x—zy x-z ' (;r-2)8 ' {x-iy ' {x-iy 1136 8695 10340 25817 58916 ■^(;r-2)5'^(;tr-2)4"^(;r~2)«"^(;i;-2)2+^~2* We have purposely chosen an example from which, by ordinary methods, the most indefatigable calculator would shrink. Other rapid methods of decomposing rational fractions are given in the Cambridge and Dublin Mathematical Journal, vol. Ill, and in the Mathematician, vol. III. CHAPTER XXXVII. GRAPHIC RKPRKSKNTATION OF EQUATIONS. 843. The graphs of simple equations in two variables, of systems of two such equations, and of equations of the form y=ax^ + dx-\-Cy have already been considered: see pages 259-2G6. It now remains to consider the graphs of equations of higher degree and some of the general prop- erties of such graphs. EQUATIONS OF THE FORM^=/(;r). 844. Every equation of the form y=f{x) is the equa- tion of a curved line. For no matter what value be as- signed to X, y will have a single value, and the assigned value of X and this resulting value of y will together locate a point. But as ^ is a variable, the point {x, y) is not fixed in position, hntis a moving- poinL Also, since /(x) is continuous by Art. 748, if we make x change continuously, then y will change continuously, and the point must describe a contmuous path or airve, 845. As a particular instance, consider the equation jK=^^ — .6;t:2 — .88:r+.192. A table of values may be formed by assigning values to x and computing the corresponding values of y, as in Art. 387. The roots of/(;t:) have been inserted in their proper places in the table. The graph is shown in fig. 21. The distance from the origin to A, B, and C, respectively, are the three roots oif{pc). At E is represented a minimum value oi fix)^ and at E' is represented a maximum value. X y — 1 - .528 - .8 + .192 + .2 + 1 - .288 + 1.2 + 2 +4.032 GRAPHIC REPRESENTATION OF EQUATIONS. 68/ 'X' A/ .\^ Ic X Ir Fig. 21. 846. In drawing the graphs of equations it will be found frequently that the values ofy are very large com- pared to the corresponding values of ;r, thus the difficulty arises of conveniently representing the graph on a piece of paper of the usual size. This difficulty can be avoided, however, by using a smaller tmii of measure for the y's than for the x's ; for example : an inch as the unit of measure for x and one-sixteenth of an inch as the unit of measure for y. The proper adjustment in each instance must, of course, be determined by circumstances. Al- though the graph is condensed in one dimension by this process, yet many of the essential properties of the graph remain unchanged. 847. By the Center of a curve is meant a point such that all lines passing through that point and terminated by the curve are bisected by the point. Consequently if a curve has but one maximum and but one minimum point, the center, if there be one, must lie half way between these points. A curve is symmetrical with respect to its center, and if a curve be symmetrical with respect to a point, that point is the center of the curve. 688 UNIVERSITY ALGEBRA. KXAMPIyKS AND PROBI^KMS. Draw the graph of each of the following equations. In some cases, as in 4 and 5, fractional values must be substituted for x in order to determine the form of the graph. 1. y=x^—Sx'^—6x+8. 5. j^=x^ + Sx'^ + 2x, 2. y=x^—5x'^+4:. 6. y==x^ + 2x+S. 3. jy=x^—5x'^—4x+20, 7. jy=x^—Sx. /^, j;=x^—Qx^ + lix—6. S, jy=x^+Sx. g. y=2x^—7x^+4:x+4t. 10. In the graph of y=^f{pc), what represents the real roots of /(;»:) =0? 11. Changing the absolute term oif{x) has what eiOFect on the graph of y—f\xy> 12. If /(:i:) is a cubic, and a^, ag, and a^ represent its three roots, show that the center of y==f(x') is at |(a^ -f 0-2 +^3) units from the j^-axis and that the numerical distance from maximum or minimum point to vertical line through center is (ai^+a2^+^3^"~^i<*2~^i^3"~^2*3)^* 13. Determine the condition thsity=x^+ax^ + dx+6 has 710 maximum or minimum points. 14. If ^^ = 8<^, discuss the maximum and minimum points of y=x^+ax'^ + dx+Cy and the form of the curve. 15. Show the effect of Horner's transformation on the graph of jK=/W. In Horner's transformation, x-h^ is substituted for x in /{x). That is, all the x's in /{x) are increased if A is negative, and all tha x's in /(x) are decreased if /i is positive. In the first case the graph is moved bodily to the right and in the second case it is moved to the left. GRAPHIC REPRESENTATION OF EQUATIONS. 689 16. Show that jr=x^-^kx represents all /or?ns of curves represented by y=x^ +ax^ + dx+c. By Horner's method, Art. 7C2. the second term of x^-{-ax^ -\-dx+c may be removed. By example 15 this moves the curve to the right or left without changing its form. Then by dropping the absolute term, by example 11, the curve is raised or lowered, depending upon the sign of the absolute term. We then have an equation of the form 17. Show that the graph of any cubic equation of the form y=/(x) is symetrical with respect to one of its points, which is the point we have termed the center. Move the curve until its equation is of the form y=x^-{-/:x. Suppose x' ,y' is a point on the curve. Then we have y'=x'^ + /:x'. (1) Putting —x' for x and —y' for y in y=x^-{-kx, we get —y' = —x'^—kx', (2) which is the same as (1). Therefore, ilx' , y' is a point on the graph, the point —x', —y' is also a point. Therefore, the curve is symmet- rical with respect to the origin. Therefore, by Art. 847, the origin is the center of the curve. 18. Find the position with reference to the j/-axis, of the center and of the maximum and minimum points of y=:x^+ax'^-{-bx+c. Here/(-r)=x3 + ^x8 + ^-^+')=0. For example, consider the equation The following table of values may be easily obtained by assigning values to x and computing the corresponding values oiy. For each value of x we get two points of the graph, one below and one above the .r-axis. The points are seen to lie upon a circle. That all points of the graph must lie upon this circle may be seen from the equation itself, for since x'^-\-y'^=-2h and since x and y are always the legs of a right tri- angle, the hypothenuse of the same must always be 5. That, is all points of the graph must be 5 units from the origin, and consequently constitute a circle with radius equal to 5. X y -6 ±V- •11 -5 -4 ±8 -3 ±4 —2 ±4.6 -1 ±4.9 ±5 + 1 ±4.9 + 2 ±4.6 + 3 ±4 +4 ±3 + 5 +6 ±1/- -11 692 UNIVERSITY ALGEBRA. KXAMPI^KS. Draw the graphs of the following equations : I. x^+y-^iQ. 5. ;t;2+4y2 = 36. 2. x-'—j^=25. 6. 36j;2 + 100j/2 = 3- xy=12. 7. ;ir2— 4^2=36.1 4- y^=^9,x. 8. ^2 = 6;t:2+2j;* 9- (^ —2)2 + (;*r-3)2=25. 10. y' =6;f' '+ArS. II. x^ +J/3: =27. 849. Names are given to several of the graphs of the above set of equations. Thus : 1 and 9 are Circles ; 2 and 3 are Rectangular Hyperbolas ; 5 and 6 are Ellipses; 7 is a Hyperbola; 4 is a Parabloa. GRAPHS OF QUADRATIC SYSTKMS. 850. The theorems of Arts. 398-401 are elucidated in a striking manner when we apply the graphic method. Consider the system :r2+;/2=34 (1)) ;ry=15 (2)f • Drawing the graph we find a circle and rectangular hyperbola in- tersecting in four points. The co-ordinates of these points of intersec- tion constitute the solution of system, for since a point of intersection is found on both graphs, the co-ordinates of such point must satisfy both equations. The natural step in the solution of the system is to add twice (2) to (1), thence deriving the system ^2 4-2x;/+7^=64 (3)), 2^/-=:i0 (4)f''- The graph of this system is the hyperbola above and two parallel "northwesterly" straight lines, intersecting the hyperbola in the satne points in which it ivas cut by the ci cle. Thus the transformation of the equations of the system has changed the graph, but has preserved the points of intersection unchanged. It is the location of these points of intersection that we seek. GRAPHIC REPRESENTATION OF EQUATIONS. 693 The next natural step in the solution is to subtract four times (4) from (3), thence deriving the system x'^+2xy+y^=6i (5)) x^-2xy4.y^= 4 (6) f The graph of this gives us the same "northwesterly" lines as be- fore, but two parallel "northeasterly" lines in place of the hyperbola. The points of intersection of these lines are the same points noted above. The next step in the solution is to write x+y=±S {7) I J x-y=±2 (8)r- which is exactly the same as c and gives the same graph. From this we derive the four system j^=3r^' ^=5r*' ;'=~3r3- >^=-5r*- We say that the original system is equivalent (see Art. 396) to these four systems. Each of the systems ^j, ^g, e^, and e^ consists of two straight lines, parallel respectively to the ^-axis and jr-axis. The four points of intereection (5, 3) (3, 5) (-5, -3) (-3, -5) given by these systems constitute the four solutions of the original systems. Next consider the linear quadratic system x»-^y^=25 (1) X -{-y = 1 (2) Squaring (2), x«4-;''=25 (3) x* + 2yx-\-y^=i9 (4) Subtracting (3) from (4), x^+y*=2o (5) 2xy =24 (G) Adding and subtracting (5) and (6), x^-2xy-\-y^= 1 (7) x^-\-2xy-j-y^=49 (8) Taking square roots, x-y=±l (9) x-\-y=:±7 (10) Correct solutions: Introduced solutions: X=:zni r X=:^\ . X=-^\ X=-\\ . ^==4^1- y^^\J^- y=-\\J^' ^=_3fA- If we draw the graphs of (1) and (2), we shall have a straight line and circle intersecting at two points. The co-ordinates of these points of intersection constitute the solution K d. 694 UNIVERSITY ALGEBRA. If we draw the graph of system [d) we shall find a circle and twa parallel straight lines intersecting it. Of the four points of intersec- tion, two are introduced by squaring (2). System {c) gives a rectang- ular hyperbola and circle meeting in the same four points as before. System d (and e which is identical with it) give two sets of parallel straight lines, cutting the axes at an angle of 45°, and intersecting each other in the same points as above, The systems f-^.f^^f^^ 3-^^ f^ give us lines parallel to the respective axes, intersecting at the same four points noted above. Since the solution to any system of two equations con- sists of one or more systems of the form x=^a \ y=b] it follows that no matter what the graph of the original system may be, the solution results in a graph consisting of straight lines parallel to the x and y axes. KXAMPI.KS. In like manner to the above, the student may draw the graphs of the following systems and consider the solutions: I. 2. xy= 4. 3- x+j=6. xy—b. ;ir2+_j/2=20. X +j/= 6. 4- X -y = 2. 5- x-" xy = 5. =6. CHAPTER XXXVIII. DETERMINANTS. 851. I^et US take two simultaneous equations of the first degree and, from them, find the values of ;i; and y. a^x+b^y^c^ (1) a^x-^b^y=C2 (2) Multiply (1) by b^ and (2) by b^, whence a^b^x-^b^b^y^c^b^ (3) a^b^x-{'b^b^y=c^b^ (4) Subtracting (4) from (3), we have €•> b.j — c^b X hence ^=^ 4^' (5) Now multiply (1) hy a^ and (2) by «i, whence a^ao^x-\-cCob^y=a^c^ (6) a^a^x+a^b<^y=^a^c^ (7) Subtracting (6) from (7), we have hence ^^^iflZI^i. (3) It is to be noticed that the denominators in the values of X and y (equations (5) and (8)) are just alike and contain only the coefficients of x and y in the given equations (1) and (2). This expression a^b^—ac^b^ is called the Determinant of the four quantities^!, ^2 »^i» ^2- We agree to write this determinant in the form of a square, where the numbers a^, b^y a^, b^^ are arranged in the same order as they appear in the equations (1) and (2) with a vertical line before and after the square, thus : ^2 ^2 696 UNIVERSITY ALGEBRA. In order that this shall mean the same as a^h<^—a<^b^ it is evident that we must take the product of the diagonal terms running downward to the right, and from this product subtract the product of the diagonal terms run- ning upward to the right. Returning now to the value of ;t: in equation (5), it is seen that the numerator c^b^—c<^b^ is like the denom- inator except that ^j and c^ are written in place of a.^ and ^2* ^"^ ^^^ expression is called the determinant of the four quantities ^1, ^1, ^2> ^2> ^^^> ^^^^ ^^ other, may be written ^1 *i I ^2 ^2 I provided it is agreed to interpret this square array of numbers the same as the previous square array was in- terpreted. In the value of y in equation (8) it is seen that the numerator a^Ci—a^c^ is like the denominator except that c^ and c.^ take the place of b^ and b.^, and, just as before, the expression ^^^2— ^2^1 ^^Y be written in the form of a square, provided the square array oi numbers be interpreted as before. The values of x and y, written in the notation just explained, are x= C\ ^ ^^2 b^ a, b. «2 b. and y= a, f, «2 <^2 «1 «2 b, b. Each of these determinants in the numerators and denominators of the values oi x and y is expanded (J. e,^ written out in the ordinary form) by the same rule, viz. : Multiply together the numbers in the diagonal rmining dow7iward to the right and from this product subtract the product of the numbers in the diagonal rummig upward to the right. DETERMIMANTS. 697 The student should carefully notice these values of x and y. In each case the denominator is the determinant obtained by writing the coefficients of x and y in the same order as they appear in the given equatio7is, while in each case the numerator is obtained from the denominator by wr ti g the right hand members in place of the coefficients of the quantity whose value is sought. The results in this form are easy to remember and convenient to use. EXAMPLES. Find the values of x and y in the following equations, using the notation just explained: I. \2x-\-Zy=l2 Here jr= - 7 -1 1 12 8 _ 21^(-12) _33 " 9-(- 2)~11~^ 3 -1 I I 3 3| 3 7 I 2 12 I _ 36-14 ^- \ 3 -1 ~ 9-(-2) I 2 3| ^- \2x+by=lb r3;ir-7j/=10 3- Ibx-Zy^^ 2 852. Definitions. 22 4. 5. r3^-f2y=16 (7;r+q)/=38 r3;r+5>/=13 \lx+'6y^lZ In the determinant '^2 «2 ^2 the four quantities a^, b^, a^, b^y are called Elements.* The elements composing any horizoiital line are called a Row, and those composing any vertical line are called a Column. * These quantities are called Constituents in Salmon's Modern Hififher Algebra, in Chrysta.'s Algebra, in Adlis' Algebra, in Burn^ide aud Pautou'a Theory of Equations, aud in Todhuuter's Theory of Equations. 698 UNIVERSITY ALGEBRA. When the determinant of two rows and two columns is expanded, i, e., written out in the form a^b^—a^b^ each term is seen to contain two factors and hence the deter- minant is said to be of the Second Order. When ex- panded, each term is seen to contain one and only one element from each row and one and only one from each column. 853. I/Ct us now find the values of x, y, and z from three equations of the first degree containing three un- known quantities. a^x+b^y+c^z^d^. (1) a^x-^b.^y+c<2,z=d^. > (2) a^x-\-b^y-^c^z=d^, (3) Multiplying (1) by b^c^ — b^c^, (2) by — (^i^s — ^s^i) and (3) by b^c^ — b^Ci, we obtain + ^1(^2^3— ^3<^2>=^l(^2<^3— ^3*^2) (4) — ^2(^1^3— ^3^1>— ^2(^1<^3— ^3*^1):^ — ^2(^1^3— ^a^l>=— ^2(^1^3— ^3^1) (5) ^3(^1^2— <^2^l)-^+<^3(^1^2—<^2^l)j^ 4-^3(^1^2— ^2^1>=^3(<^1^2— ^2^1) (6) Adding (4), (5) and (6), we obtain [^l(^2'^3—^3<^2)— ^2(^1^3— ^3^1) +'^3(^1^2— <^2^l)]^ = ^l(^2^3--^3^2)-"^2(^J^3— ^3^l)+^3(<^l'^2— ^2^1) (7) ^^ ^1(^2^3— ^3^2) — ^2(<^1^8"-<^3^l)+^3(^l<^2— ^2^1) ^ON «l(^2^3 — ^3^2)— ^2(^1^3 = ^8^l)+^3(^1^2~^2^l) Similarly multiplying (1) by -— (<^2^3~"^3^2). (2) by ^1^3 ""^3^1 » ^^^ (^) ^y <^i^2~'^2^i> ^^^ adding the re- sulting equations together, we obtain — ^l(^2<^8-^8^2) + ^2(^1^3-^3^l)-^8(^1^2-^2^1 ) ^g-x — <^l(^2^3-^3^2) + ^2V;^1^3-«3^l)-^8(^i^2-^2^l) DETERMINANTS. 699 Again, multiplying (1) by <^2^3"~^3*2> and (2) by "•(<^i^3""^3^i)j ^^^ (^) ^y <^i^2"~^2^i) a^^ adding the resulting equations together, we obtain ^ ^l(^2<^3-^3^2)-^2 (^1^3 -^3^1)4-^3 (^1^2-^2<^l) qqn ^1(^2^3- ^3^2) -^2(^1^3 -^3^1) + <^3 (^1^2- «2^l) If we remove the parentheses from the denominators in equations 8, 9, and 10, each of these denominators is easily seen to be ^1<^2^3""^1^3^2+^2^3^1""<^2^1^3+^3^1^2""^3^2^1 (H) Now this expression (11) which contains only the coeffi- cients of X, y, z in equations (1) (2) (3), we wish to write in the form of a square array where the letters appear in the same order as in the three given equations, if it is possible to interpret such a square array so as to give the same result as the one here written. We must then, ii possible, interpret a^ ^1 ^1 ^2 ^2 ^2 *'3 ^3 *'3 (12) so as to give the same result as (11). The quantities composing the square array will be called Elements, any horizontal line will be called a Row, and any vertical line a Column, and the square array itself, or its equal (11), will be called a Determinant. As each term of (11) is the product of three quantities, the de- terminant is said to be of the Third Order. Now it is easy to see that the expression (11) can be obtained from (12) by taking the algebraic sum of all the products ob- tained by taking one and only one, element from each row, and one, and only one, from each column ; using the sign + or — before each product, according as the order of subscripts in that product is obtained from the natural 700 UNIVERSITY ALGEBRA. order 1, 2, 3, by an even or an odd number of interchanges of consecutive subscripts. Hence we may write a^ di Ci ab^c^-aj)c^^aj)^c^-ab^c^jraj)^c-aj)^c^ (13) Each of the denominators in (8), (9) and (10) is equal to the determinant (12); and, since in (8) the numerator is seen to differ from the denominator only in having a^^ a^, a^ replaced by d^, d.^^, d^, therefore the numerator in (8) is equal to the determinant a. l>i C\ d^ b, c. d^ b^ Cz As the numerator in (9) is seen to differ from the denom- inator only in having b^, b^, bo. replaced by d^, d^, d^, therefore this numerator equals the determinant d'l ^2 3 ^3 And similarly the numerator in the value of z in equa- tion (10) equals the determinant ^1 ^1 ^1 Cf"> ^2 ^2 b. d.. Hence we have x = d. b^ c^ d. b. Ci d. ^ <:% a, b. <^i «2 b. Ci ^3 bz <^3 a^ d. Cx «2 d.^ Cl «8 d^ c.i «1 b. c^ flj b. Ci «3 b. c» «i b. d. «2 bi di a, b. d. «1 b. -^i «2 bi ^2 «3 bz ^3 It is to be noticed that in these values of x, y and^a* the denominator in each case is the determinant formed by taking the coefficients of x, y and z in just the order in DETERMINANTS. 7OI which they appear in the three given equations, and that in each case the numerator is obtained from the denom- inator by substituting therein the right-hand members of the equations in place of the coefficients of the quantity whose value is sought. EXAMPLES. Find by determinants the values of x, y and 2 in the following equations : (\x-\- 6j/-3^=17. r ;ir+ jK+25'=83. 1. \ x+ 7j/4- 5'=35. 3. ] x+2y+ ^=82. (5ji:+13j/4 4^=82. (2x-\- y+ ^=31. r x+ 2y-hSz= 6. C x+Sy-i- 5^=4. 2. ] x-\- Sy+4z= 8. 4. ] 2x+5y+ ?,z=2. i2x+ 5>/+8^=15. iSx+dy+102=7. 854. There is no difficulty in extending the process above employed to a greater number of equations, but, after these illustrations, we prefer to treat determinants by themselves, independent of these application to a set of linear equations.* We will, however, return to the solution of a set of linear equations when we have obtained a number of properties of determinants. We have seen that a determinant of the second order, composed of four elements, is written ^1 ^1 a^ ^2 and is defined as being a^b.^-^a^^b /, also that a determinant of the third order, composed of nine elements, is written a^ b^ c ^2 ^2 ^2 i ^3 ^3 ^3 and is defined as being the expression, * "I,inear equations" are equations of first degree. 702 UNIVERSITY ALGEBRA. Similarly, a determinaut ot the wth order is composed of n^ quantities, called elements, and naturally written thus : «2 ^i <^2 d,. ■h «3 ^3 c» dl- ■It «4 ^ Ci d,. ■h The elements in a horizontal line are called a row and those in a vertical line, a column. The determinant is defined as being equal to the alge- braic sum of all products that can be formed by taking one and only one element from each row and one and only one from each column, the sign + or — being writ- ten before each product or term according as the order of the subscripts in that term is derived from the natural order by an even or an odd number of interchanges of successive subscripts, it being understood that the letters preserve the natural order. The collection of terms written out according to this definition is called the Expansion of the determinant. 855. It is to be noticed that the elements are here represented by letters with various subscripts. Obviously other symbols might have been chosen to represent the elements, but the advantage of this notation consists in the fact that the position of each element is indicated, the letter showing the column and the subscript the row to which any given element belongs. Take for example the element d^ ; since d is the fourth letter, the element helongs in the fourth column, and the subscript being 3, it belongs to the third row; thus its position in the deter- minant is completely indicated. Of course, the position of an element would not be thus indicated if there should DETERMINANTS. 703 be any disarrangement in the rows or columns of the above determinant, unless we knew exactly what disar- rangement had taken place. 856. Since from the above definition each term must contain one and only one element from the first column, therefore each term must contain a with some subscript; and since each term must contain one and only one element from the second column, therefore each term must contain b with some subscript. In the same way it follows that each term must contain ^ with some subscript, and so for the other letters. Hence all letters appear in each term of th^ expansion and no letter appears more than once in the same term. Again, from the definition, each term in th^ expansion must contain one and only one element from the first row; therefore each term must contain some letter with a subscript 1 ; and since each term contains on^ element from the second row, therefore each tei^i miisK: contain some letter with a subscript 2; and in the «ame way each term must contain some letter with the subscript 3, and so for the other subscripts. Hence all the subscripts ap- pear in each term of the expansion and no subscript appears more than once in the same term. From this it is seen that every term in the expansion of the determinant contains all the letters a, b, c, ■ .1 with all the subscripts 1. 2, 8, • • •;?, and if we please we may keep the letters in their natural order and the sub- scripts will be attached to these letters in eveTy possible orderi To illustrate > a determinant of the fourth order ^1 ^1 ^1 ^\ ^2 ^2 ^2 ^2 ^3 ^3 ^3 ^3 a^ b^ c^ d^ 704 UNIVERSITY ALGEBRA. expanded becomes —ab,i\d^ + ^//;^3 + ^2 Vx^4 — ^. ^4^ ~" ^«^4^i^3 + ^2 Vs^x + ^3^x^=<, - ^3^z^4< - ^3 Vt^4 + ^3^4^ ^ ^3 Vx^ ^ ^3 V/x In this expansion the signs are determined according to the definition by the order of subscripts. Take for ex- ample the term a^b^c^d^, where the subscripts appear in the order 42ol. This order can be determined from the natural order, as follows: Natural order 12 3 4 First, interchange 4 and 3 12 4 3 Second, interchange 4 and 2 14 2 3 Third, interchange 4 and 1 4 12 3 Fourth, interchange 2 and 1 4 2 13 Fifth, interchange 3 and 1 4 2 3 1 As 5 is an odd number, the sign before a^b^c^d^ must be -. 857. The rule for determining the sign of any term in the expansion of a determinant may be simplified by noting that the interchange of any two numbers, how- ever far removed, is the same as an odd number of inter- changes of successive numbers. For suppose any number of numbers, 12 34.../^ > >m- > ., Let there be r numbers between k and m. Then, if we wish to interchange fit and h, we have to pass m to the left successively over the r intervening numbers, and then over the h ; we have next to pass h to the right over the r numbers that originally separated h and 77i. In passing m to the left we have made r-f 1 interchanges of succes- sive numbers, and in passing h to the right we have made r interchanges, so in all there are 2r+l interchanges of DETERMINANTS. 705 successive numbers, and this is an odd number whatever be the value of r. We may then strike out the word suc- cessive in the rule of determining the sign of any term in the expansion of a determinant. The order 4231, derived by five successive interchanges, may be derived by a single interchange of 4 and 1. 858. By using subscripted letters for the elements of a determinant it can be expanded with considerable ease, but how can the terms be written out when other symbols .are used to denote the elements ? Suppose we wish the expansion of Now we know that the expansion of «i ^\ ^1 a b c d e f g h k (1) ''2 ^2 ^2 «3 ^3 ^3 (2) is «i^2^3""^1^3^2 — ^2^1^3 + ^2<^3^1 + ^3^1^2 — ^3^2^1 , and to obtain the expansion of (1) we must, of course, substitute for each of the elements in (2) that one which in (1) occupies the same position. The expansion of (1) then becomes aek—ahf—dbk + dhc-^-gbf—gec, A better method will be given further on. 859. Theorem I. 77ie expansion of a determinant of the nth order contains \n terms. From the definition the terms are obtained by taking the letters a, b, c,- - - in their natural order and their subscripts in every possible order. Whence, the number of the terms is the same as the number of ways of arrang- ing: ^ thinsrs taking all at a time, which is 1 • 2 • 3 • . «. 45— u. A. 706 UNIVERSITY ALGEBRA. 860. Theorem II. If in ajiy determinant the rows are changed iiito columiis aiid vice versa^ the value of the determinant is not changed. Let us take the case of a determinant of the fifth order. We are to prove d. ^1 b^ «3 bz «4 b. «5 b. d, d^ ^4 — d. d^ ^3 dz d. <^5 d. h <^5 d. e^ ei e-i ^th and rth subscripts interchanged, the sign being the same in each case. This term in the expansion of A' also appears in the expansion of A but with an op- posite sign, being determined from the previously selected term of the expansion of A by interchanging two sub- scripts, which, of course, changes the sign. In the same way every term in the expansion of A' is found in the ex- pansion of A with an opposite sign. Therefore, A= — A'. DETERMINANTS. 7^9 863. Corollary. If a determinant has tzvo rows or two columns ideiitical^ the determinant equals zero. For if we interchange the two identical rows or columns in the determinant represented by A, we get a determinant rep- resented by —A; but interchanging two identical rows or columns cannot change the determinant at all. There- fore, A=— A .-. 2A=0 .-. A=0. 864. Minor Determinants. **If in any determinant we erase any number of rows and the same number of columns, the determinant formed with the remaining rows and columns is called a Minor of the given deter- minant. The minors formed by erasing one row and one column are called First Minors ; those formed by erasing two rows and two columns are called Second Minors, and so on.** — Salmon's Modern Higher Alge- bra, If the given determinant is of the n\\i order, the first minors are of the («— l)st order, the second minors are of the (« — 2)d order, and so on. Thus, we may speak of rth minors or minors of the order n^-r indifferently. Minors of the first order are the elements themselves. The elements at the intersection of the rows and col- umns erased also form a minor of the given determinant called the Complement of the minor which is left. In any determinant the complement of an rth minor is a minor of the rth order. If the determinant is of the ^^th order the complement of a minor of the order r is a minor of the order n—r\ that is, the complement of an rth minor is an (;^— r)th minor. In any determinant there are as many first minors as there are elements and the first minors obtained by eras- ing the row and column intersecting in any given element 7IO UNIVERSITY ALGEBRA. is called the first minor corresponding to that elemerit. In the determinant cf"i b<^ C2 ^3 ^3 ^3 which we represent by A, if we erase the first row and h ^3 first column, there remains the determinant which is the first minor of A corresponding to a\, and be. ^ ^ is the first minor of A corresponding '1 «-! ^3 ^6 similarly to a^'y also to b^) and so on. is the first minor of A corresponding 865. Expression of a Determinant in Terms of the Elements in any Row or Column. Let the determinant «1 ^ <^i d. ^2 b. c^ d. «3 h H dz «4 b. ct d. be represented by A. Since, by the definition of a deter- minant every term in the expansion must contain some element from the first column, a certain number of terms in the expansion will contain a^, while other terms will contain a^, and so on. Collect together all those terms of A which contain a^ ; then, after taking out this common factor (2 J, there will remain an aggregate of terms which we will represent by A^, so that a^A^ will represent the algebraic sum of all those terms which contain a^. Referring to Art. 856, we see that A^ — b^c^d^ —b^e^d^ — ^3^2^4 + b^e^d^ + b^c^d^ — b^e^d^ . In the same manner we might collect together those terms in the expansion of A which contain the element DETERMINANTS. 7II ^2, and represent the algebraic sum of all these terms by ^2-^2^ then, from Art. 856, Similarly the algebraic sum of all the terms in the ex- pansion of A which contain a^ would be ^3:3^3 and the algebraic sum of those terms containing a^ would be a^A^. Now every term in the expansion of A must contain some element from the first column. Therefore, if we collect into one group those terms which contain ^1, and into another group those which contain a 2, and into an- other those which contain ^3, and into another those which contain ^4, then in these four groups we will be sure to have a/l the terms of A. Therefore, A=«i^i +^2^2 +^3^3 +^4^4- This is an expression for A in terms of elements of the first column. In a similar way we could obtain an ex- pression for A in terms of the elements of the second column or any other column or any row. If we select the terms in the expansion of A which contain any one of the sixteen elements and, after taking out this common factor, represent the remaining aggre- gate of terms by a capital letter of the same name and with the same subscript as the element we are consider- ing; then A may be expressed in any one of the following eight ways : A=^l^l-f«2-^2+^3^3+^4^4- (1) A=l^l^,-f^2^2+^3^3+^4^4. (2) A= C^ Ci + ^^2 ^2 + ^3 Q + ^4 Q- (3) A:=d,D,+d2n2 + d^D^+d^D^. (4) A=ai^i + ^ii5i+^iCi+^iZ>i. (5) A=a2^2 + ^2^2 + ^2 ^2 + ^2^2- (6) A=^3^3 4- ^3^3 + ^3^ + ^3 A' (7) A=a^A^+ d, B^ + ^4 C4 + d^D^. (8) 712 UNIVERSITY ALGEBRA, The explanation is here given for a determinant of the fourth order, but it is so evident that the process applies to a determinant of any order that we omit a separate ex- planation for the general case. 866. Expression of a Determinant in Terms of the First Minors Corresponding to any Row or Column. As we usually represent a determinant by A, so let us represent the first minor corresponding to ^^ by A^^ and similarly represent the first minor corresponding to any element by A with that element used as a subscript. We will prove that in the above eight equations the factor that multiplies any element is either + or — the first minor corresponding to that element. First, to prove ^i=A^^. The terms of A ^ are obtained from the terms of A that contain the element a^ by striking out this element; that is, they consist of the letters b, c, d in the natural order with the subscripts 2, 3, 4 attached to the letters in every order. Moreover, in those terms of A from which the terms of A j are derived, the element ^j stands at the head, and hence the sign is determined by the number of inter- changes of the last three subscripts. But the terms of Aflj also consist of the three letters b, c, d with the sub- scripts 2, 3, 4 attached to these in every possible order and the sign of each term is here also determined by the number of interchanges of the subscripts 2, 3, 4; hence, ^, = A,,. Second, to prove ^2 = "~^«,- The terms of ^2 ^^^ determined from the terms of A that contain the element a 2 by striking out this element; that is, they consist of the letters b^ c, d with the sub- scripts 1, 3, 4 attached to these letters in every possible DETERMINANTS. 7^3 order. The sign of any of these terms is the same as the sign of the corresponding term in the expansion of A and in this term of A it requires one interchange of subscripts to begin with to get the subscript 2 or the element a.^ at the head, and so the sign of any term oi A.^\s determined by a number* one greater than the number of subsequent interchanges in the subscripts 1, 3, 4. The terms of Aaj also consist of the letters 5, r, d with the subscripts 1, 3, 4 attached to the letters in every pos- sible order; but the sign of any term of Aag is determined by the number of interchanges of subscripts 1, B, 4; hence, the sign of any term in Aa^ is opposite to the sign of a corresponding termini 2 J ^^^ consequently -^2""= ^«,- In exactly the same way it may be shown that ^3= + A^, ^, = -A,,, ^3 = -^v Hence we see that the multiplier of any element in the expansion of a determinant is either + or — , the first minor corresponding to that element, the sign + or — being used according as the element is removed an even or odd number of steps from the element in the upper left hand corner; where, in counting the steps, we pass along the first row to the right until we are in the column in which the element is found, and then downward until we come to the element, but never pass along a diagonal line. Calling that diagonal running from the upper left hand corner to the lower right hand corner the Leading or Principal Diagonal, then the rule just given to determine the sign may be simplified. The sign + or -- is used according aa the element taken is an even or odd number of steps from any element in the principal diagonal. •The 8ig3i si j^us is this.number is even and minus if this number is odd. 7H UNIVERSITY ALGEBRA. All this is given for a determinant of the fourth order; but a careful examination of the discussion will show that it is equally applicable to a determinant of the «th order. 867. The above gives us a new way of expanding a determinant. Take, for instance, a determinant of the fourth order and express it in terms of the first minors corresponding to the elements of any row or column, say the first column : A= Now we can expand each of these determinants of the third order in terms of their first minors in the same way. «3^3^3^3 «4VA = «x v/4 -«. +''3 Kc,d, -«4 Kc.d, W3 a\yi. -K c^ d^ Cid^ + ^4 f 2 d^ Cz d^ -«2 I f>X c^ d^ -h c^ d^ c^ d^ + *4 c, d, Cz d^ +«» { ^ c-i d^ Ci d^ -b. c,d, c^d^ +^ ^2 d.^ — «4 1 *1 c, dj ^i d% -b. + -^3 C\ dx Ci d^ Expanding each of these twelve determinants of the second order we have the expansion of the determinant A as follows: )— ^2V3< + ^2V/3 + ^«Vx<-"^«V4<--^»Vx^3 + ^.Wx + «3V2^4-SV4<~«3Vx< + ^3V/x + ^3Vx<-^3Va^x (-^V2^3+^V3< + ^^M-^4V3<-^4Vx< + ^4Va^. The result agrees with the expansion given in Art. 856. By the process here given we can expand a determinant when the elements are represented by any symbols what- A= DETERMINANTS. 715 ever as easily as when the elements are represented by letters with subscripts : thus be I /-I /I -d = a e h b h + g c f ^aiek—hf^-dibk-hc) -\-g{hf-'ec) = aek^ahf — dbk-\- dhc+gbf—gec. EXAMPl^KS. I. Express the determinant a b c d e f g h k in terms of the minors corresponding to the elements in the second column and expand the resulting determinants of the second order and show that the result agrees with that in the last article. 2. Express the same determinant, in terms of minors corresponding to the elements in the second row and show that the final result is the same as before. 3. Find the value of each of the following determinants: 1 2 3 1 3 1 12 3! 13 2 3 3 3 2 3 4 14 4 14 4 14 4 3 3 4 3 3 3 3 3 3 868. Theorem IV. If all the elements of any row or column can be expressed as the sum. of two or more quanti- ties, the7i the determinant can be expressed as the sum of twa or more determinants. Take for example the determinant (^i+Z^i) ^1 ^1 (^2+^2) ^2 ^2 7i6 UNIVERSITY ALGEBRA. Expressing this in terms of minors corresponding to the elements of the first column, we get b. c, K+)3i) («2-f^2) V + -^^1 ^3 ''S 2 -a. by Cy ■\b. \-?^ by Cy bt c. 2 ^2 I I 1 ^1 I I J^2 I j ^ ^ But the expression in the first bracket is evidently equal to the determinant ttj b^ c^ 0-2 ^2 ^2 «3 ^3 ^3 and the expression in the second bracket equals the de- terminant Hence If the given determinant had been (a2+/^2+72) ^2 ^2 {^Z^Pz + yz) ^3 ^3 then, by what has just been given, this determinant is equal to (ai+/3i) b^ c^ (^2+^2) ^2 ^2 (a3+i^3) ^3 ^3 and supplying the value of the first of these from above, we have P , ^ ^1 Pi bi c^ Pz ^3 ^« (ai+^,)^^l a, by <^i iS. *1 <"l («2+/32) *2 Ci = 02 ^2 <^2 + /J2 ba C2 (as +^3) ^8 <^3 «3 ^ <^8 ^3 I'zC^ 71 ^1 ^1 + 72 ^2 ^2 73 ^3 ^3 («i+^i+7i) ^i fi (^2+^2+72) <^2 ^: (a8+^8+73) ^8 ^1 .1 «1 ^) ^1 Pi ^1 ^1 7i ^i ^ "2 = ^2 ^2 ^2 + /^2 ^2 ^2 + 72 ^2 ^^2 8 «3 ^8 ^3 /?3 ^8 ^8 73 ^8 ^3 DETERMINANTS. 717 Similarly if all the elements of the first column were the sum of any number of quantities, the determinant would equal the sum of the same number of determinants, the forms of which are evident from the example here given. Evidently this peculiarity might have presented itself in any other column as well as the first, or in any row. A precisely similar discussion would show that if all the elements of any row or column were expressed as the difference between two quantities, then the determinant could be expressed as the diiBference between two deter- minants. KXAMPI,^. Express the determinant 2 3 1 3 3 3 4 4 1 as the sum of two determinants in three different ways ; find the value of each of the resulting determinants and compare the sum with the value of the given determinant. Also express it as the sum of three determinants, find the value of each, and add. 869. Theroeni V. If all the elements of any row or column be micltiplied by a common factor the determinant is mnltiplied by that factor. Let us take the following determinant, which we rep- resent by A: a b c d e f g h k and multiply all the elements in the first column bj'' /», then we obtain ma b c VI d e f tng h k 7i8 UNIVERSITY ALGEBRA. Calling this A', expressing A and A' in terms of the minors corresponding to the first column, we get A=« e f hk ■ b c -•^ hk +sr b!=zma e f h k -'Vk +mg c f c I /I from which it is evident b!^=mt^. Corollary i . If all the elements in any row or column contain a common factor, that factor may be taken out oi each of the elements and placed as a factor of the remain- ing determinant. Corollary 2. Multipljdng any row or column by any number and dividing another row or column by the same number does not phange the value of a determinant. KXAMPL^. Verify each corollary in the determinant: 12 3 870. Theorem VI. If the elements of any row or col- umn, each multiplied by the same number, be added to or subtracted from the corresponding elements of another row or column, the value of the determinant is not changed. "^ *The wording ot the theorem should be carefully noted, for if the elements of any row or column be added to or subtracted from the corresponding elements of another row or column multiplied by the same number the determinant u changed. If from the determinant a b c de f g hk we make another by adding the elements of the second column to m times the ■elements of the first column we get I{ina-Vh\ b c \ {vtd^e )e f\ {mg-\-h) h k \ and this is just m times the first one. DETERMINANTS. 719 a b c rnh b c = d e f + me e f ghk mil h k Let us take tlie determinant ' a b c d ef ghk and add to the elements of the first column the correspond- ing elements of the second column each multiplied by m. Thus we get (^a-\-mb^ b c (^d-\-me) e f ig+m/i) h k Now, because each element in the first column is the sum of two quantities, therefore (<^ + ;;^^) b c (d-\-me) e f (^g-\-mIi) h k Taking out the factor m from the elements of the first column of the second determinant on the right side of the equation, we get (^a-\-mb') b c ( d+me) e f Ig+m/i) h k But the last determinant in the equation has two iden- tical columns and therefore vanishes, whence ( a + mb) b c (d+me) ef {g+mli) h k Similarly ( a — m,b) b c ( d—me) e f i^g—mli) h k Scholium. — When dealing with a numerical determi- nant in which the elements are large numbers, we may combine rows with rows and columns with columns ac- cording to this theorem so as to reduce the elements to a b c b b c = def +m e e f ghk hhk a b c = def ghk a b c b b c a b c C= def —m e e f = def ghk hhk ghk 720 UNIVERSITY ALGEBRA. Ismaller numbers and thus obtain a determinant easier to compute. 871. Theorem VII. If all the elements but one in a7iy row or cohcmn be zero, the detenninant may be reduced to one of the next lower order. Take, for example, the determinant «i b^ d^ an b.y do b. . . ^3 ^4 ^4 ^4 ^4 If this be expressed in terms of minors corresponding to the elements in the third column, it evidently equals 2 ^2 ^2 3 h ^3 4 <^4 ^4 a^ b^ d^ ^1 ^1 ^l ^1 ^j d^ -0 ^3 ^3 ^3 a^ b^ d^ +0 «2 ^2 ^2 ^4 ^4 ^4 -^4 ^2 <^2 ^2 ^3 ^3 ^3 whi ch equals -^4 a^ b^ d^ ac^bc^d.^ «3 ^3 ^3 872. To Compute the Value of a Numerical De- terminant. If we have to compute the value of a numerical determinant we first look to see if all the ele- ments of any row or column contain a common factor and remove as many common factors as possible in order to reduce the elements to smaller numbers ; then v/e seek, by some combination of rows with rows or columns with columns, to still further reduce the elements, especially aiming to transform the determinant so that in some row or some column all the elements but one shall be zero, when the determinant may be reduced to one of a lower order. We then treat the new determinant in a similar way, and thus by continual reductions we may find its value usually much more easily than by expanding. DETERMINANTS. 721 I^t US compute the value of the determinant. 12 6 3 6 3 1 9 4 1 10 3 2 12 24 6 4 9 2 2 5 12 Take the factor 3 from the first row and 2 from the last and we get 4 2 13 6 3 12 9 4 12 10 3 2 5 12 3 2 6 Subtract two times the first column from the second col- umn; also subtract the fourth column from the third; and subtract the fourth column from the fifth; 2 1 1 3 2 1 4 1 3 1 5 1 2 6 1 2 2 112 3 2 11 5 12 3 6 12 4 Subtract the fourth column from the first: -6 0112 2 2 11 2 12 3 = -12 2 12 4 112 12 11 112 3 112 4 -12 oil =-12 1 2 1 112 Subtract the third row from the fourth: 112 12 11 112 3 1 Subtract the first column from the second: -12 46 - U. A. 1 1 1 1 1 1 2 722 UNIVERSITY ALGEBRA. Subtract the first row from the second : -12 1 1 1 = 12 1 2 =24 which is the value of the determinant of the fifth order that we started with. 873. In article 865 eight different expressions were given for the determinant h h b. ^Z dz d^ each in terms of the elements of some row or some col- umn, and it was noticed that A^^^a^- ^2 d^ b^ c^ d-^ Ao = —^a=— ^3 ^3 d.^ b^ c^ d^ Keeping carefully in mind the meaning thus given to the capital letters with various subscripts it is evident that ^1^1 +^2-^2 +^3^3 +^4-^4 '•b. b^ c^ d.. ^i ^1 d^ ^3 ^3 ^3 -^2 b^ C^ <3 ^4 ^4 ^4 b^ c^ d^ + h. ^1 ^1 ^1 I I b^ C2 ^2 |~~^4 ^4 ^4 ^4 I I ^ Cl d. b^ Co, do b. C-i dl and this is evidently the expression of the determinant b^ by c^ d^ ^2 ^2 ^2 ^2 ^3 ^3 ^3 ^3 ^4 ^4 ^4 ^4 in terms of the minors corresponding to the elements of the first column. Now this determinant, having two identical columns, equals zero ; hence, /^l^l +^2^2 +<^3^3 +^4^4 = 0- DETERMINANTS. 723 In the same way we could obtain a relation connecting the elements of any row or column with minors corres- ponding to the elements of some other row or column. There would be in all twenty-four such relations given by the determinant <2j b^ Ci d^ a 2 ^2 C2 ^2 ^3 ^3 ^3 ^3 ^4 ^4 ^4 ^4 In the same way if we were given the determinant of the ni\i order: ^1 ^1 ^1 ' * ^1 ^2 ^2 ^2 • • -^2 3 *^3 ^i a^ ^4 ^4 • ^3 we could obtain %i different expressions for it, each one as multiples of the elements of a row or a column, and we could obtain %i{7i—\) other relations connecting the ele- ments of one row or column with the minors corresponding to the elements of another row or column. As samples, we write two equations of each kind and leave the student to write others. <3^2^2+^2-^2 +^2 Q+ • • a^A^ + b^B^ -f ^3(^2+ • • KXAMPI^KS. a,,B,=0. /3^2=0. I. Write the six different expressions 1 2 3 for and verify each expression. 724 UNIVERSITY ALGEBRA. 2. Write the twelve other equations expressing the relation between the elements of one row or column and the minors corresponding to the elements of another row or column of the determinant in example 1. 3. Express the value of a b d e g h c f k in six different ways. 4. Write the twelve other equations expressing the re- lation between the elements of one row or column and the minors corresponding to the elements of another row or column of the determinant in example 3. 874. Application to the Solution of Simultaneous Equations of the First Degree. Let us take n equations of the first degree containing n unknown quantities. an^x-\-boy-\-Co,2:+ • • • -\-l^v--m2, a^x+b^y-\-c^2+ - ' +l^v=m^. a,,x + b,,y -\-c„z-\- • • + l,,v = m„. Here we suppose that the determinant formed by the coefficients of the unknown quantities, viz.: a^ 0^ Ci • ^2 ^2 ^2 ' «3 ^3 ^3 • •/2 is not zero. a„ b„ c„ ' ' • l„ Multiplying the first equation by A^, the second by ^^2, and so on, we have a^A^x+b^A^y-\-c^A^2-\' . • . -\-l^A-^v—77i^A ^ a<2,A ^^^ so on, and add the resulting ^% equations, we get -^~ V' where A 2 is what A. becomes when the ^'s are replaced by the right members of the given equations. Again, if we multiply the first by Cj, the second by ^2, and so on, and add the resulting equations, we get ^3 2'==— 2. A where A3 is what A becomes when the r*s are replaced by the right members of the given equations. It is now evident that the value of any unknown quan- tity in the given set of equations is the ratio of two determinants, in which the denominator is the determi- nant of the coefficients of the given equations and the numerator is what the denominator becomes when the right-hand members are put in place of the coefficients of the quantity whose value is sought. 875. Another Method. Form the determinant (a2^+^2JV+ • • • +hv—m^ b<^ c^' ' 'I2 (a„x+d„y+ • • • l„v—m„) d„ c„- - ■ /„ 726 UNIVERSITY ALGEBRA. Each element in the first column is formed by trans- posing the right-hand members of the given equations, and hence each of these elements equals zero, therefore the determinant itself equals zero. Now each element in the first column is expressed as the algebraic sum of ;^ + 1 quantities, hence the determinant can be expressed as the sum of 72 + 1 determinants. Whence ax b c I I KCn' • •/. Ky K^. +...- bc^- ninbnCn* • • /. =0. b„y b,, r« . . . 4 All these determinants, except the first and last, vanish; for after taking out the common factor from the first column we have left a determinant with two identical columns. Moreover, after taking out the common factor x from the first column of the determinant, we have left the deter- minant A, and the last determinant is evidently what we have called A^; hence A;c— A^=0, or A;i:=Ai, or _Ai ■^^ A the same as before. Similarly, the other unknown quan- tities may be found. 876. If the determinant called A should equal zero, we cannot obtain a definite, finite set of values of the un- known quantities. If all the numerators are also zero, the unknown quantities are i7tdeterminate diVidi the equations given are not independent. If, however, some of the numerators are not zero, then the equatiois cannot be satisfied by any finite set of values, in which case the equations are said to be incompatible. Thus, the equations 2;t:+ y+ 5^=19 7jir-f4y+ 12^=51 DETERMINANTS. ^2^ are independent and compatible and therefore form a solvable set of equations. In this example A=— 2, Ai = — 2, A2 = — 4, A3 = — 6, whence :r=l, ^^=2, ^r=3. The equations 2x-\- y+ 5^=19 Sx+2y+ 4^=19 7;r+4j/4- 14^=57 are compatible but are not independent, the third being de- rived from the other two by adding the second to twice the first. In this example A=0, Ai = 0, A2=0, A3=0, so that the value of each unknown quantity assumes the inde- terminate form TT- The equations 2x+ y+ 52'= 19 Zx-\-2y+ 4^=19 7;c+ 47+ 14^=50 are incompatible with one another. If we add the second to twice the first, we get 7;i:+4j/4-14i?=57, but this contradicts the third equation. In this example A=0, Ai = 42, A2 = — 49, A3 = — 7, so that no finite values of X, y, z satisfy the equations. 877. Let us now take n equations of the first degree containing n—\ unknown quantities: a^x-\-b^y-\-c^z-\- • • • -f /i=0 «2-^+^2j^+^2'2'+ ••• 4-/2=0 a^x-\-b^y-\-c^z-\r • • • +/3=0 dnX -f bny-\- C„Z+ - ' ' + 4 = The absolute terms are written on the left-hand side of the equations because it is better for the method here pursued. It is to be noticed that the number of equations 728 UNIVERSITY ALGEBRA. is one more than enough to enable us to find the values of the unknown quantities. Represent the determinant of the known quantities by A, whence A= ^1 ^1 ^1 ' ' 'h ^2 ^2 ^2 * * * ^2 ^n "n ^n ' * ' *' n Now, if we add to the last column x times the first, y times the second, z times the third, and so on, the deter- minant is not changed in value; therefore A= K Cn' * ' {a„x-\-b^y-\- c,,2-\- . . . -f 4) But from the given equations it is evident that every element in the last column equals zero. Therefore, A=0. We have tacitly assumed that the given equations were compatible with one another and have shown that the determinant equals zero; whence the following theorem: If n equations of the first degree containing 7i — 1 un- known quantities are compatible zvith one another the deter- minant of the known numbers equals zero. But this determinant may be zero when the equations are incom- patible, as in the set '1x-\- y-\- 4^—16=0 Sx-\- y+ 2^—11=0 Sx+Sy-h 8<^— 38=0 7^+3)/+10^-40=0 2 14 -16 3 12 -11 8 3 8 -38 7 3 10 -40 Subtracting twice the second row from the third we have two identical rows; hence, A=0. But if two Here A= DETERMINANTS. 729 times the fisrt equation be added to the second, we get 7^+3y+10<3'— 43=0, which contradicts the fourth equa- tion, whence the system is incompatible. Let us see if we can tell when several equations are compatible. For the sake of definiteness in language let us take four equations, ^2-^+^2 J^ + ^2^ + ^2=0 «4^+<^4J^ + ^4'S' + ^4 = and suppose three of these, as for instance the last three, are independent and compatible. Then from these three we find the values of x, y, z to be x=^ Changing numerators so that the column of ^'s shall be the last column in each case and taking out the factor — 1 if it occurs, we get jr=- These determinants are easily seen to be minors of -^2 b^ ^2 -dz ^3 ^z —d^ b^ c^ ^2 -^2 ^2 ^3 ~^8 ^Z ^4 -^4 ^4 , ^ = ^2 bc^—d^ ^3 ^3-'^3 ^4 ^4-^4 «2 ^2 ^2 ^3 <^3 ^3 ^4 ^4 ^4 y y — a^ ^2 ^2 ^3 ^3 ^^ ^4 ^4 ^4 ^2 ^2 ^2 ^3 ^3 ^3 ^4 ^4 ^4 ^2 ^2 dn b^ c^ d^ ^4 ^4 ^4 , J^ = (^2 ^2 ^2 ^3 ^H ^3 ^4 ^4 ^4 , '2'= — ^2 ^2 ^2 ^3 ^3 ^3 ^4 b^ ^4 ^2 ^2 <^2 «3 ^3 ^3 ^4 ^4 ^4 «2 ^2 ^2 ^3 <^3 ^3 ^4 /^4 ^4 ^2 ^2 ^2 ^3 ^3 ^3 ^4 ^4 ^4 Representing this by previous notation. x=^ — b\ ^1 ^1 2 2 2 ** ^4 (^4 ^4 c/4 A, we may write according to Vx J/; ^+^/-) (1) We shall now show that a rationalizing factor may be found for (1). The notation will be simplified if we give a particular value to p. For example: let ^=4. It will be easily seen afterwards that the same method may be applied whatever the value of p. I^et ^0^^+^l^^ + ^2^"^ + ^3=^ JJ 2 1 1 then ^ia^+-^2^^+^3^^+^o = ^^^^^> also, ^2^^ + -^3^^ + ^0^^ + ^l=^^^^> 3 2 1 3 and A^a'i-V A ^a'^-\- A ^a'^ -^^ A<2^^ma'^. I 2 3 Transposing vi^ ma'^, ma'^, and ma'^ to the left mem- bers, and forming the determinant of the left members, we have, by Art. 878, Aq A^ A2 (A^—m) Ay A2 A^ (A^—ma^ ^^ A2 A.^ Aq {Ay—PtaT) A^ Aq Ay {A^—ma"^ 732 UNIVERSITY ALGEBRA. Whence, b}^ breaking the determinant up into the differ- ence of two determinants, we have Aq A^ A2 1 Aq Ai A2 A^ A^ A 2 A^ aJ A I A2^A^ Aq A^ A^ Aq aJ ^■2 A.^ Aq A^ A^ Aq A^ aJ ^3 Aq A^ A^ The determinant on the right side is rational as far as aJ is concerned. Therefore, the determinant on the left side is the rationalizing factor for niy as it is the multiplier that renders m rational. lyikewise, a rationalizing factor may be found for a rational integral function of any other surd. If we have a rational integral function of several surds, we may rationalize the expression as far as one of those surds is concerned, then rationalize with reference to one of the remaining surds, and so on until the expression is entirely rational. An important result of this discussion is the fact that every algebraic equation can be rationalized. Thus, the solution of rational integral equations includes the solu- tion of all algebraic equations of whatever kind. 14 DAV i'^j, uatRowBD *- ^ t niped below, or Kenewedbooksaresub^ec^ \§o«oT^''"' p P306O71 THE UNIVERSITY OF CALIFORNIA LIBRARY