COLLEGE ALG-EBEA-. BY LEONARD EUGENE DICKSON, Ph.D., ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO. SECOND EDITION, CORRECTED. NEW YORK: . JOHN WILEY & SONS. London : CHAPMAN & HALL, Limited. X^iS Copyright, 1902, BY L. E. DICKSON. PRESS OP BRAUN WORTH h CO. BOOK MAMUFACTURERB BflOOKLVN, N. V. PREFACE. This text is intended primarily for the college and the tech- nical school. By treating only the subjects usually given in the college course in algebra, space has been gained for a more detailed exposition of the more difficult topics. As the extent and the character of the review at the beginning of the course upon topics prescribed for entrance to college varies so widely, and as the review is usually conducted in an informal manner, it seemed best to the author to leave to the instructor the review of the ele- mentary principles, but to give in the text review exercises. x\s to the order of the subjects, the aim has been to present first those topics which are readily mastered by the student, and to reserve for the latter part of the text the questions grouped around the subjects involving infinite series. In reviewing the subject of simultaneous equations, the student is led naturally, almost unavoidably, to the determinant notation. Determinants are there- fore treated early in the text, an order of presentation shown by actual experience to give very satisfactory results, especially in arousing the interest of the student. In the chapter on graphic algebra, the first principles of coordinate geometry are introduced and applied to the study of simultaneous equations and inequalities. In this connection is presented an elementary account of the solution of numerical equa- tions, chiefly from the graphic standpoint. The arrangement is such as to admit of a very brief course or of a fuller course involving Horner's method of synthetic division. The practice of 474!l^bo IV PREFACE, emphasizing graphic algebra in courses for technical students com- mends itself also to the general student. • Attention is invited to the proofs of the fundamental theorems on logarithms, the treatment of mathematical induction and the illustrations showing the necessity of the different steps in the process, the examples from the physical sciences of the topic varia- tion, the complete proof of the general binomial theorem independ- ent of the principle '^permanence of form," the establishment of the relations between the roots and the coefficients of the quad- ratic, cubic, and quartic equations prior to the proof of the general theorem, the solution of those equations before introducing the assumption that every equation has a root (here proved in the Appendix). The attempt has been made to present the subjects limits and infinite series in as simple form as is consistent with rigor. Forty-five sets of exercises, averaging over fifteen to a set, are given at very short intervals in the text. Some historical data have been introduced, with no attempt to give the source, the subject-matter being classical. The author is under great obligation to Dr. Moulton, of the Department of Astronomy of the University of Chicago, who read with care the entire manuscript and offered numerous suggestions as to the form of presentation, most of which have been adopted. Likewise the thanks of the author are due Professor J. W. A. Young of the same University, who examined the more critical chapters and offered valuable suggestions. Proof-sheets of the entire book were carefully read by Professor L. L. Conant of the Worcester Polytechnic Institute, whose correc- tions and suggestions have led to numerous improvements. Finally, the author is indebted to Professors Moore and Young of the Uni- versity of Chicago, who examined critically the proof-sheets of certain portions of the text. CuiCAGO, Januarv, 1902. CONTENTS. [See the Index at the end of the book.] PAGE Kumber in Algebra; Surds and Imaginaries 1 Exponents; Logarithms 10, 17 Table of Four- place Logarithms 24, 25 -^ ilL Factor Theorem; Quadratic Equations 27, 29 IV. Simultaneous Equations; Determinants 35 V. Ratio; Proportion; Variation 58-61 VI. Arithmetical, Geometrical, and Harmonical Progressions.. . : .. 64-71 VII. Compound Interest and Annuities. 73 VIII. UndeterminM Coefficients; Partial Fractions 76, 80 IX, Permutations and Combinations 85 Binomial Theorem for Positive Integral Index 90 Multinomial Theorem 92 X. Probability (Chance) 94 *** XI. Mathematical Inductitm 99 ■*■ XII. Limits; Indeterminate Forms 104, 106 XIII. Convergency and Divergency of Series 113 XIV. Power Series; Undetermined Coefficients 125, 126 Expansion into Series; Reversion of Series 126, 128 XV. Binomial Theorem for Any Index 130 XVI. Exponential and Logarithmic Series 136, 140 Natural Logarithms; Interpolation 141, 142 XVII Summation of Series; Recurring Series; Generating Relation; Generating Fraction 145, 148 The Method of Differences 148 XVIII. Graphic Algebra. Coordinates; Graph; the Straight Line 152-160 Simultaneous Equations 164 Simultaneous Inequalities 166 Solution of Numerical Equations 168 Horner's Method of Synthetic Division 169 Descartes' Rule of Signs; Location of Roots 176 V vi CONTENTS. CHAPTSB PAGE XIX. Theory of Equations. Solution of Cubic and Quartic Equations 180, 185 Solution of Certain Equations of the Fifth Degree 189 Reciprocal Equations 190 Fundamental Theorem of Algebra 193 Relations between the Roots and the Coefficients 194 Fractional, Surd, and Imaginary Roots 198, 199 Symmetric Functions of the Roots 200 Miscellaneous Exercises 205 APPENDIX. Argand*s Diagram 207 De Moivre's Theorem 208 Solution of Cubic in the Irreducible Case « . .• 210 Proof that Every Equation has a Root 211 Index ••• 213 SYMBOLS AND ABBREVIATIONS. = , equal. JJr^ nPn S6. ^, not equal. A. P., G.R, H.P., 64. iz , approaches, 106. sin, cos, 208. 00 , infinity, 69, 105. 1^1 — absolute value of t, 114. 6-2.71828.,.., 137. w! = 1 X 2 x 3 X . . .X ^., 80. 1 — |/ — 1. I COLLEGE ALGEBRA. CHAPTER I. NUMBER IN ALGEBRA; SURDS AND IMAGINARIES. 1. The natural numbers or positive integers (1, 2, 3, ...) make it possible to enumerate the objects of a group considered for the time as equivalent entities. It has been established that primitive counting was done on the fingers and that in many languages the numeral "> is merely the word for hand, 10 for both hands, and 20 for the whole man (hands and feet).* AVhile 5, 10, 20, and 60 have been used as bases, 10 is the usual. base. If a group of objects can be partitioned into equivalent smaller groups, each smaller group or a combination of them is a fraction (I, f, 1^, ...) of the original group. Abstractly, a fraction is the quotient of two positive integers. Fractional results may or may not admit of an interpretation in a particular problem. A shepherd would declare it to be impossible to separate his flock of 50 sheep into three equal flocks ; but would find no theoretical difficulty in dividing a 50-foot rope into three equal pieces. The algebraic statement for each problem is the same : to find x such that *Thus in the language of the Tamanacs, the word for 6 is "one on the other hand "; the word for 12 is ''two to the foot "; for 15, **a whole foot "; for 16, ''one to the other foot "; the word for 20 is "one Indian"; for 40, "two Indians "; etc. 2 NUMBHt^' fN '^bpiBkA: ^-SJURD^- AND IM AGIN ARIES, [Ch. I 3a; = 50. The formal solution is :?: — ^- ; the possibility of its interpretation depends upon the character of the special problem. The Egyptians used fractious before 1700 B.C.* and resolved them into sums of unit fractions. Thus f was written 3 15, which meant \ + -^^. By the Babylonians, f was written 37 30, which 37 30 meant -^ -f ^^^ Instead of fractions, the Greek geometers used the ratio of two numbers or two magnitudes. For the introduction of negative numbers (as well as the decimal positional system and the symbol 0), algebra is indebted to the early mathematicians of India (between 500 and 600 a.d.). We now' find it convenient to write — 15° for 15° below zero ; — 100 ft. for 100 feet below sea-level, thereby abbreviating our map notations. The determination of a number x such that h -\- x = c leads to the algebraic solution x = c — h. li h exceeds c, the result c — h \^ a negative number. Such a result would be excluded if the problem were to find how many feet of rope must be added to a rope h feet long to make a rope c feet long. But the negative result leads us to restate the problem so that the required c-ioot rope is seen to be obtained by cutting o^h — c feet of rope from the Z>-foot rope. In this connection, we note the Eoman notations IV for 4, VI for 6, IX for 9, XI for 11, which seem to have been of Etruscan origin. The term rational number includes positive and negative in- tegers and fractions. All other numbers are called irrational. The solution of equations of the form x''' — A^ where A is a rational number and n a positive integer, introduces two classes of irrational numbers. Thus, for ^^ = 2, and A a positive integer not the square of a rational number, the square root of A^ denoted by the symbol X^A, is an irrational number called a quadratic surd. Similarly, 4'^2, V — ^, VA, A not the cube of a rational number, *Rhind papyrus, *' Directions for Attaining to the Knowledge of All Dark Things." Sec. 2] COLLEGE ALGEBRA. 3 are surds of the third order. In general, V A, where A is positive if n is even, is a surd of order n. The second class of irrational n . ■ numbers are defined by the symbols V A, where A is negative and n even (§ 4). While the equation x^ = 80 possesses the formal solutions x=± 1^80 (the positive root is designated VSO, the negative — VSO), the possibility of the interpretation of one or both results depends upon the character of the particular problem. It is possible to form a square of area 80 square feet, but impossible to arrange 80 square blocks of equal size in the form of a square and yet preserve the form of each block. 2. Tlie fact that surds really exist as such may be illustrated by showing that V2 is not expressible as a rational number. If we take the side of a square as the unit of length, the diagonal is of length V2, But it is proved in Geometry that the side and diago- nal are incommensurable (see § 55). Hence 1 and l/2 have no common measure. It follows that V2 is not equal to a rational number. For, if (1) ^-^ = 1^ then — would be contained p times in V2 and q times in 1 and hence be a common measure of 1 and V2, To give a purely algebraic proof, suppose that equation (1) holds p . . . true, the fraction — being in its lowest terms, so that p and q are integers having no common divisor. By squaring and multiplying by q'^, we get 2q^ = p^, so that 7/ and therefore p is divisible by 2. Setting p = 2r, we get q^ — 2r^, so that q is divisible by 2. Then p and q have a common divisor 2, contrary to hypothesis. 3. But with the introduction of rational numbers and surds, we do not meet all the demands which are made upon a number system. rWe are led in Geometry to consider the number n which expresses 4 NUMBER IN ALGEBRA; SURDS AND IM AGIN ARIES, ^ [Ch. I the ratio of the circumference to the diameter of a circle and to approximate its vahie by considering the perimeters p^ and P^ of an inscribed and a circumscribed regular hexagon, the perimeters jOj, and Pj2 ^^ ^^ inscribed and a circumscribed regular polygon of 12 sides, etc. From the results, true to four decimal places, for a circle of unit diameter: A = 3 ^ Pu = 3.1058, . . . , p,,, = 3.1415, . . . Pg = 3.4641, F,, = 3.2153, . . . , F,,, = 3.1416, . . . we obtain a succession of numbers between each pair of which the value of 7t must lie. By proving that P,< Pu< P2,< ' - < P^M < . . . < ^, P, > P,, > P,, > . . . > P33, > . ,,>7t, and that the diiference P„ — Pn can be made to differ from zero a? little as we please by sufficiently increasing the number of sides w, we have pointed out to us, with as great a degree of precision as we may desire, a certain limit, which we take as the value of n. In an analogous manner, we can define the number 1^2 by means of two sequences of rational numbers. 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 3, 1.5, 1.42, 1.415, 1.4143, 1.41422, By the arithmetical process for the extraction of a square root, wo find that the value of 1^2 lies between each pair of corresponding numbers in the sequences. In general, two such sequences of rational numbers proceeding by a given law are said to define, by a limiting process, a number.* The value of the number may be determined to as great a degree of approximation as may be desired. All such numbers as well as all rational numbers are called real numbers. ♦The above sequences which defined the number n can evidently be replaced by sequences of rational numbers related to the Pn and P„ . Sec. 4] COLLEGE ALGEBRA, 5 4. An even root of a negative number is called an imaginary 4 . 6 quantity. Thus y — 1, r — 2, r — 1 are imaginaries. Unlike surds and other real numbers, an imaginary can not be expressed approximately in terms of rational numbers and hence has no inter pretation in strictly arithmetical problems. By the introduction of imaginaries, we may give a formal solution of the equations a:* = ~ 1, a;'^ := — 2, and, indeed, of every quadratic equation. By extending the system of all real numbers by the introduction of the quantity 1^ — 1, we obtain the quantities a -{-I V — I, where a and h are arbitrary real numbers. We shall see that the system of these complex quantities a -{- h V — 1 forms a number system within which may be performed all algebraic operations including the solution of all algebraic equations, so that a further extension is unnecessary. The employment in algebra of imaginaries has therefore a great practical value in that the operations may be effected without the limitations otherwise necessary. To further justify this extension, we recall that negative, fractional, and surd numbers were introduced to enable us to give a formal solution of many simple problems which would otherwise have remained insolvable, and that the possibility of the interpretation of nega- tive, fractional, or irrational results depends upon the nature of the particular problem.* 5. If a -\- Vb = c -{- Vd, where a, h, c, d are rational numbers and Vb is irrational^ then a = c, b = d. From a — c -\- Vb = Vd, we derive, after squaring, 2(a - c) Vb = d-b-{a- cy. Unless the coefficient a — c is zero, we could express Vb as a rational number, contrary to assumption. Hence a = c, so that b = d. * A possible interpretation of complex qaantities is given in the Appendix. The instructor may prefer the illustration by means of operations which com- bine a rotation with a magnification. Thus — 1 rotates through 180°, V — 1 through 90°, 4 -}- 3 V — i magnifies five-fold and rotates. See Chrystal's Algebra, I, p. 239. 6 NUMBER IN ALGEBRA; SURDS AND IMAClNARIBS, [Ch. I 6. Let us attempt to extract the square root of a -{- Vb^ where Vb is a true surd and a is rational. We seek a result of the form Vet -j- V fd in whieli a and /3 are rational. Setting i/a + Vb := Va + V/3, and squaring, we find that By the above theorem, we may equate the rational parts and also the irrational (surd) parts. Hence a = a -\- ft^ b — 4:a/3. Then {a - f3f = (tY + I3f - 4:a/3 = a^^ - b, so that az=^{a + Va^ - b), 13 = ^{a - Vd^ - b). By assumption a and jS are rational. Hence the square root of a -{- Vb is expressible as a sum of tivo quadratic surds Voc -\- V ft if, and only if, c? — b is the square of a rational number. For example, if aj == 6, Z> = 20, c? — b is the square of 4. Hence a + |/^ = 6 + l/20 is the square of V'oc -\- V~ft = Vh -\- \. When the problem is solvable, it may usually be done by inspec- tion as follows. Put the expression a-\- Vb into the form m -\- 2 Vn by taking m = a, n = ^/4. The required root is Vol -f V ft, where a -\- ft ^=^ m, aft = /^. Thus 6 + ^^20 = 6 + 2 |/5 = (1 + Vlf, since 1 -}- -5 = 6, 1 • 5 — 5. Tims 16 + 6 V7 = 16 + 2 V'^ =^{Vl + V'^y\ sine:) 7 + 9 = 16, 7- 9 =r 63. 7. Denote by i the symbol V —1. Then + i and — i are the roota of :c^ r= — 1. By the symbol V — c^ where c is positive, we shall mean V — I Vc= i Vc, where Vc denotes the positive square root of c. Thus i2=~l, i^=-i, i^ = + l, 22»= (-!)«, ^2^ + ^ = ^(-l)^ Sec. 8] COLLEGE ALGEBRA, 7 If c and d are any two positive real quantities, V~^ V — d — i Vc 'iVd=: — V'cVd-= -- V7d. In introducing the equation x^ =^ -- c as an equation to be solved by algebra, we are tacitly assuming that x may be combined with itself and with real numbers according to the laws of algebra for the com- bination of real numbers, so that d V — c— V — c d^ d -\- V — c = V — c -\- d^ etc. In addition to these assumptions, we assume that complex quantities may be combined according to the laws holding for real numbers. Then {a + bi) ± (^ + fti) ^{a±a)-^{h± ft)U {a + hi) {a + I3i) = {aa - 1(5) + (aft + la)i, a + [51 _ {a -\-l3i){a — hi) _aa -\-h/3 aft — ha . a + hi ~~ {a + hi) {a - hi) ~ d^ + U^ "*" d^ + W ** Hence the sum, differe?ice, product, or quotient of any ttoo comj^lex quantities is itself a comi)lex quantity.'^ In freeing the denominator of the above fraction from imagi- naries, we used the multiplier a — hi, called, the conjugate of the denominator a -f hi. The sum and the product of two conjugate complex quantities are both real. 8. TJie three cuhe roots of unity are 1, G9 = - i + i V"^=~3, G?'^ = - I ~ 1 l/^Ts, so that G3^ — \, 1 + &? -|- G9^ = 0. The roots of x^ = 1 are :?: = 1 and the' two roots of '^ -, = ^2 _[_ ^ _j_ 1 ^ 0. X — 1 Completing the square in the quadratic equation, we get Hence x=— ^±V — ^. Setting - ^ + ^ V ~ 3 - oa, the second root is - 1 - 1 V'^^3 = ( _ ^ + 1 i/Tr3)2 ^ ^, 9. I7i an equation hetiueen two comjjlex quantities, the real parts are equal and also the imaginary parts. * The complex quantity a + hi is real if & = 0. 8 NUMBER IN ALGEBRA; SURDS AND IMAGINARIES. [Ch. I Let a-^- bi = a-}- ^i, where a, b, a, /3 are real numbers. Then a — a z= (/3 —.b)i. Upon squaring, we find that the number {a — a)^, which is positive or zero, must equal the number — (/3 — by^ which is negative or zero, so that each must be zero. Hence a = or, b ^ ft. In particular, \i a -\- bi — 0, then a = Q, b = 0, 10. The square root of any complex quantity may ahvays be ex- pressed as a complex quantity.^ Let the given complex quantity be « + bi, where a and b are real and i = V — l» We seek real numbers o^ and y which will make Va -j- bi = X -\- yi. Squaring, a -\- bi = x^ — y^ -{- 2xyi, Equating the real parts and also the imaginary parts, ^2 _ y2 — f^^ 2xy = b. Then {x^ - y^ + ^.x^ = {x^ + y^f = a'^ + P. Since x and y are to be real, x^ + y^ must be positive. Hence a;2 -|- ^2 _ |/^2 _j_ ^2 (positive square root). Having the sum and difference of x^ and y^, we derive „ Va^ + b^^ + a , Va^~+~^ - a x^ = 2 > f = -2 • Since Va^ + b^ is positive and greater than a, the expressions for x^ and y^ are positive, so that real values of x and y may be determined by extracting the square roots of positive quantities. Since 2xy = Z>, the sign of y is determined as soon as the sign of x is chosen. Hence there are always two and only two square roots of a -\- bi. For example, to find the square roots of 5 — 12i, we have x^ — y^ = 5, 2xy = — 12, whence x^ -}- y^ = 13, x^ = 9, y^ = 4. The square roots are ± (3 - 2i), * Contrast with the theorem of § 6. The extraction of higher roots of complex quantities is done very simply in terms of trigonometric ratios fsee Append ixl. Sec. 10] COLLEGE ALGEBRA. 9 The inspection method of § 6 may be extended to find the square roots of certain complex numbers a + ^ 1/ — 1. Thus for -3+4 V^^, set r — 3 + 2 V — 4:= Vx-{- V — y {x and y positive). . •. -3 + 2 |/"^^ :=^x - y + 2 V — xy. x — y— — Z, xy — ^ (by §9). Hence :?: = 1, ?/ = 4, so that one square root of — 3 + 4 V— 1 is 1 + 2 s/~^\. EXERCISES. Express with rational denominators ^ 2- ya ^ i^5 + 3 i/3 + i/5 3. 2+ |/3* * |/5 -2* ' 1 - |/15 * i/3 + |/5 +^i/iO ^ 1 +y6 3 + i/~^r5 6. 7. V3 - 1^5 + VlO* * i/2 - 1^3 + |/5 * * 2 + 4/ - 1* 3 + 5 V^^ 1 - 4/2. CHAPTER 11. EXPONENTS; LOGARITHMS. 11. If m is a positive integer, d^ denotes the product of m factors each of which is a. Similarly, if 71 is a positive integer, a"^ z=z a . a .,, a^ to n factors. Hence d^ , a"" = a . a , . , a to m -\- n factors = «"* + ''. We may therefore state, for the case of positive integral indices m, n, The First Law of Indices. The index of the product of tzvo poivers of the same quantity is the sum of the indices of the factors : (1) a"^ . fl^'^ = a"* + ". As a corollary, we derive the formula a^.a^. ^^. . . . a«= «"* + "+^ + --- + «. 12. For the division of two positive integral powers of «, a.^ 0, a"^ _a . a, a. . , a (to m factors) a" a , a . a . . , a (io n factors) ^= a . a , . , a {to m --' n factors) if m > n. We may state, for m and ?^ positive integers, m > n, The Second Law of Indices. The index of the quotient of two powers of the same quantity is the excess of the index of the numer- ator over the index of the defiominator : (2) «V«^ = a"^ - " {a ^0, m>n), 13. If m and n are positive integers, we have, by definition, [a'^Y ^za"" .(f,,. a"^ (to n factors) ^ ^m + m + ... + m -. ^mn (by §11). Hence, for positive integral indices, we may state Sec. 14] COLLEGE ALGEBRA, ii The Third Law of Indices. The index of the nth power of a^ is the product of the indices m and n: - - (3) (pry = a^. 14, We next extend the use of the symbol a'' to cases in which n is negative or fractional^ assuming that such new symbols d^ will satisfy certain cases of the above first law of indices, and proceed to determine what meaning, if any, may be attached to the gen- eralized symbols. It is later shown (§ 15) that the symbols, with the meanings thus obtained, satisfy the three laws of indices for m and n any rational numbers. For this reason the interpretations of the symbols are justified and the desired permanence of the algebraic laws is attained. Consider the symbol a-^, where q is any positive integer. Since the symbol shall satisfy the first law of indices, a^ .a^.a^...{toq factors) = a^ + ^^-^ + "-^^o,terms) ^ ^i ^ ^^ Hence a « must be such that its ^th power is a, that is,* a'i = ya. Similarly, the symbol a^^*^, where ;; and q are positive integers, must be such that p p p p aJ . aJ. . . (to g f actors). = ^i^-^ + y ^•••(*°«^®^^'^> _ ^p^ so that a^^^ must be a ^th root of a^. Hence 1 Since the symbol a'l obeys the first law of indices, we find that a « . a 5 . . . (to jt? factors) = ci'^ z=z (^ Va ) . * The radical sign is used to denote a particular root. Thus |/4 = + 2, - 4/4 = - 2; hence 4^ = + 2. 12 EXPONENTS; LOGARITHMS, [Ch. II Hence the two resulting values for a^^'' must be equal, a true theorem on radicals. Thus VS'^ =: \ yS ) =4. Consider the symbol a^. By the first law of indices, so that a^ = a!'/oJ' ==1. Hence, if a be any number different from zero, a^ — \, Consider the symbol a ~% where r is a positive integer or frac- tion, and assume that it may be combined with the symbols already defined in such a way that the first law of indices holds. Then a-^'-dr z=.ar-'' ^ a'= 1, or Hence a'"" denotes the reciprocal of oT, As examples, 9^^ r= VO^ = I/36 =.33 = 27, 4 " * = 1 -^ 43/2 ^ 1 ^ ^743-^ 1/g^ ( _ 27)" ^ = 1 ^ ( - 27)* = 1 -^ ( ^ - 27)' = 1 -^ (- 3)2= 1/9. 15. We have been led to the following definitions: (Def.) a^= V~a^, aP = 1, a-^^^ = 1 -^ a^^S so that we hav6 a definition for a"" for every rational number n. In order that these generalized symbols a^, a^ shall prove to have practical value, it is essential that they shall obey the three formal laws of indices which have been proved to hold for positive integral exponents (§§ 11, 12, 13). In proving that this is the case, we make use of certain properties of radicals, namely: (A) vT= 7T'. (B) i/J- Vt = Vt^i, (c) \^i^ ^r= VUi' Sec. 15] COLLEGE ALGEBRA, 13 I. To prove that dJ^dP- = 0^ + "" whe7i m and n are rational numbers. (1) Let m and n each be a positive fraction, including the case in which the denominator is unity. Then «««» = Va''j/a'- (Def.) = 1/a'« y^"^ (by A) = Ya"' ■ a"! (byB) ~ -"^a^' + rq (by first law, § 11) ps+ rq =z a ^^ (Def.) P.+ Z = a^ «/ (2) Let m be a positive fraction — and n a negative fraction ^ T p r I*— — , and let — > — . Then ps — rq is positive. a'ia « — \/a^ -^ \/a'' = ?^ ps — rq ,,so that a'^a'' = «'^ + '' in this case. (3) Let m = ^, 7^ = - -, with ^ < -. q s q s (Def.) (by A) (byC) (by second law, § 12) (Def.) By case (2) a^a~^ = a^~ ^ (since - > -j. \ s qj Taking the reciprocal of each member, we have by (Def.), 14. EXPONENTS; LOGARITHMS. [Ch. II P T P T (4) Letm = — — , 7^ = , where — , - are positive fractions. ^ ^ q s q s ^ Taking the reciprocal of the two equal expressions in case (1) and applying the definition of a negative exponent, we get a 9a ^ = a ^« sy z= a « «. (5) Let finally m be rational and 7i be zero. By (Def.) II. To prove that a^ -^ d^ = aJ^"^ ivlien m mid n are rational, a'»-^a™ = fl^^". — - a"*. a-« (Def.) since m and (— n) are rational numbers, so that Theorem I applies. III. To 'prove that {a'^'Y = ^"''' when m and n are rational, (1) Let m be a positive fraction and n a positive integer. Then Va«/ = f a^.f aP. . . (to n factors) (Def.) = \/a^'a^ ... (to n factors) = \/a^ (by B) (§ 11, corollary) (2) Let w and n be positive fractions. Then (Def.) (Def.) = \'ai [by case (1)] pr = ««* (Def.) (3) Let w be a positive and n a negative number, integral or fractional. Then, using the definition of a negative exponent twice, ^""^^'^ " 7^-" ^^ ^ ^~"" fby case (2)1. (4) Let m be negative and n positive. Then («-")" =(i^r=(;^»' Sec. 15] COLLEGE ALGEBRA. 15 also for n a positive fraction in view of since [— = ~n'^^ ^^"^^ ^^^" positive integral n by inspection, and \WI ^ \l0l to'' ^^f^r ?^^/^ Next, {a^y^ = a^'\ for /i and n positive, by case (2). Hence (5) Let, finally, m and n be both negative. Then ia-^V*" — z r — — — [by case (4)1 = a^\ (Def.). e laws of indices therefore hold when the exponents m and n are any rational numbers. As an exercise, the student may prove that, for any rational number 7^, By way of illustration of the laws of indices, we note that 3 1 3 1 5 3 1 13 ^H Express with positive indices 7. 3a- I 8. 2-^0^-2. 9. S^Ba?"*. 10. 2~^G~^ 11. V x^ -^Vx~^' 1 2 i 1 8 .1 1 a" a 'a' = a' ~''^ ' z=z a^ = 1. EXERCISES. Find the numerical value of S\ 2. 16 '. 3. 9 \ 4. \g) . 6. 25 '. 6. lOO"" '. l6 EXPONENTS; LOGARITHMS, [Ch. I] Express by radical signs with positive indices 12. al 13. J. 14. a. 16. a~^ 16..-5«l n.5^\ 18. /.-V.-^ 19.§^\ 2 • a x' 20. ST. 21 Simplify 22 . a^a~%a)~K(^''y\ 23. aH'-a-^^l 24. (a^ l^h^ " V 4/5 - 4 |/^F^. 25. a ^6 a*5' ' 5-^* 26. 2^(2^-T-2"^^2"~^ 27. ( 2")""^ h- (s^-O^""'. 28. Multiply a;~^ +y~Hyaj~ ^- y'^l 29. Multiply a*" + a"^ + 1 by a" "^ + a" 2 ^ 1. 30. Divide 32aj " ^ + 12aj" * + lOaj" ^ - 12 by 2iB- ^ ~ 1. 31. Factor a^ - &^ :? - y, x~ ^ - 27y~ \ 32. Find the square root of a '* — Qa^ -j- 6a^-\- 12a" + 4. 33. Find the square root of 4a — 4a® -f- a + 2a -\-a^ — 4a'. 34. Solve aj - a;^ - 30 = 0; aj^ - 3aj* -f 2 = 0. 35. Solveaj^ = 8«~^ + 2; a;"*+ 16 = 8aJ~^ 16. What meaning, if any, may be attached to the new symbol 3^^^ , in which the index 4/2 is not rational ? Since 4/2 can not be expressed as a fraction (Chapter I), the symbol 3*"^ is of a charactei not previously considered. But 4/2 may be approximated by a deci- mal fraction to any desired degree of approximation. For example, 4/2 lies between 1.414 and 1.415; it lies between 1.41421 and 1.41422; etc. We have obtained a definite meaning for the expres- i4J_4. 1415 sions 31^00 and 3 ioott. It is natural to say that the value of 3*'^ lie$ between the values of these two expressions. As a closer approximation, the value attached to 3^^ lies between 141421 , 141422 3Ti7iyirTrir and 31^^^^^. Sec. 17] COLLEGE ALGEBRA. 17 Continuing the process of approximation, we obtain two fractions — and — ^^, which differ by an amount — which can be made as n n n small as we please, and such that 4/2 lies between the two fractions. By this limiting process, tliere is pointed out, with as great a degree of precision as we may desire, a so-called limit, which we define to be the value of 3*"^ . Similar definitions apply to the symbols 2*'^ , a^^ and, generally, to d^ where n is any real number. A series for a^ in ascending powers of n will be given in Chapter XVI; so that a more practical method is derived for the calculation of a^ If decimals correct to five places be used for 3 ^^ and 3 ^ , the product of the decimals will equal 3 ^^ + '^^^ to an approximation correct to five places. * It follows that in numerical work we may employ the laws of indices when dealing with symbols like 3*"^ by means of their approximate values. We make the assumption that the laws hold for the symbols 3^^ , . . . , themselves. LOGARITHMS. 17. By the first two laws of indices, two powers aP" and aP^ of the isame quantity a may be multiplied or divided by merely adding or subtracting the indices m, n. Now addition and subtraction are [simpler operations to perform numerically than multiplication and jdi vision. It would thus appear desirable to be able to express all real numbers as powers of a single number a, called the base. Only ipositive numbers will be employed as bases. For example, taking a = 2, we have i=:2'^ 8=:2.^ 1 = 2^ i = 2-^ i=2-3, |/^=2^ 4/8 = 2"'. *In employing only five decimals, the fifth place in the product may vary 3y unity from the correct result for the fifth place in the most unfavorable lases. Thus, if only five places be retained in .972385, the square would be ncorrect by more than .000009, which would be replaced by .00001 when only ive places are retained. l8 EXPONENTS; LOGARITHMS. [Ch. Il'i But we do not find by inspection the power of 2 which will give certain numbers, for example, 3. By trial, we find that 9 3 8 3 9 10 2 < 26" < 22 4 < 3 < 22 4 < 2~6- < 22. 38 Now 22 4 — 2.996 to three decimal places. The required exponent is slightly greater than f f , but is less than ||. Just as the symbol /j^3 was introduced to denote the real positive root of the equation x^ = 3, so now we employ the symbol 'logg 3, read logarithm 3 base 2, to denote the exponent x which makes 2"^ = 3. Definition. — The exponent of the poiver of the base a luhich gives rise to a oiumher N is called the logarithm of N to the base a; symholically , (4) iV^i=:a^«^a^. Thus, for a = 2^ the above relations give 2 = log, 4, 3= log, 8, O = log,l, -l=--log,i, -3 = log,i j i = logy2, f-log,4/8, IK log, 3 < If. ' Similarly, 1000 = 10^ gives 3 = log,, 1000; .01 = 10- 2 gives - 2 = log,, .01. 18. For JV =■ a, equation (4) gives log^« = 1. Also a^ — 1 gives loga 1 = 0. Hence, whatever the base may de, the logarithm of 1 is zero and the logarithm of the hase is unity. Since logarithms are exponents, the three laws of indices (ex- ponents) give rise to three corresponding properties of logarithms. I. The logarithm of a product equals the sum of the logarithms of its factors. Let the factors be N and M. By the definition (4) we have (5) iV=a^«^a^, M = a'^'^a^. Hence, by the first law of indices, and the definition (4), . •. logaNM = log^JV + logJL II. The logarithm of a quotient equals the logarithm of the divide7id minus the logarithm of the divisor. Sec. 19] COLLEGE ALGEBRA. 19 Let the dividend be N and the divisor M, Applying the second law of indices to the identities (5), we get M N By the definition (4) applied to the number -^, we get N M N \0gaM. III. The logarithm of the ptli -poiuer of a number equals p times 'he logarithm of the numler. Let iV^ be the number. Raising the two members of (4) to the 30wer p and applying the third law of indices, we get 3ut, by definition (4) applied to the numiber iV^, .-. ^log,iV^^=7^1og,iV. 19. Compare the logarithms to the base 2 which are given in 17 with the logarithms of the same numbers to the base 8: = log84, l^loggS, O^loggl, -i^logsi, -1 =:=log8|-, .. . ¥e observe, for each number iV, that logg N = \ logg N, But = logg 2. Hence there is a constant multiplier logg 2 in passing rom logarithms to the base 2 to logarithms to the base 8. The student may readily verify the results in the table Number N . Log^io^ Log.oo^ Log^.^iV 10,000 4 2 12 1,000 •3 3/2 9 100 2 1 6 10 1 1/2 3 1 0.1 - 1 -1/2 - 3 0.01 - 2 - 1 - 6 0.001 -3 - 3/2 - 9 20 EXPONENTS; LOGARITHMS, [Ch. II For the numbers iV^of the table, logiooJ^^= i l^gio^j t^G multiplier which converts logarithms to the base 10 into logarithms to the base 100 is ^ = logjoolO. Similarly, a logarithm to the base 10 must be multiplied by 3 = logio^ 10 to give the logarithm of the t same number to the base 10^. In general, suppose that the first base is a and that the second base is b. By the definition (4), . •. h l«g/ z=z a ^^^a^ = {b ^°^6") ^^^a^. Applying the third law of indices, we get logt,]Sr=]ogaN. logf,a. To pass fro7n a table of logarithms to the base a> to a table of logarithms to the base b, we employ the constant multiplier log^ a, th6\ logarithm of the old base with resyect to the neio base, EXERCISES. 1. Find log2 32, log^ 32, logi 2 |/8, logi ^64, logs 5 i/5. 2. logc h . logb N = logc a . loga J^. 3. loga J^ -^ logo ^ = l0g6 J^ -^ logb M. 4. Express logio |, logio 60, logio 450 in terms of logio 2 and logio 3. 6. Solve 52-» - 5^ + 1 + 6 = in terms of logio 3 and logio 3. 6. Simplify log ( |/144 |/i08 -f- 1/1728" \/m). 20. Logarithms to the base 10 are known as Common Loga- rithms* and are used in numerical calculations. The chief advan- tage of the base 10 lies in the fact that from a known logarithm of a number ^may be derived by inspection the logarithms of all numbers differing from iV^ only in the position of the decimal point. Thus, if we know that logio 2-23 = .3483, then logio 22.3 = log^olO 4- logio 2.23 == 1.3483, and logio 2230 ^ 3.3483. The decimal part .3483 common to the three logarithms is called tlie mantissao The integral part of a logarithm is called the characteristic; it * Introduced in 1615 by Briggs (1556-1630), a contemporary of Napier (1550-1617), the inventor of logarithms. For a history of logarithms see the article by Glaisher on Logarithms in the Encyclopaedia Britannica. Sec. 21] COLLEGE ALGEBRA. 21 depends upon the position of the decimal point in the given number. Thus logio 2.23, logio 22.3, logio 2230 have respectively the charac- teristics 0, 1, 3. 21. That a shifting of the decimal point in any number iV^does not alter the mantissa of logio ^ follows from the fact that the shifting of the decimal point p places to the right multiplies iV^by 10^, the shifting q places to the left divides iYby 10^. But loglo(i^^x 10^) == \og,,N + p\og^^ 10 = i? + logio iV, j logio (^-^ 10^) = logio N-q logio 10 = - ^ + logio ^ In determining the mantissa we may tlierefore disregard the decimal point in the number. Thus, for the number 22.3, we use the sequence of digits 223 in looking for the mantissa in the Table of Logarithms given below. In taking the mantissa from the Table, the decimal point is to be placed before the digits found. Hence for logio 22.3 the mantissa is .3483; by inspection, 22.3 lies between 10^ and 10^ so that the characteristic is 1. Hence logio 22.3 = 1.3483. 22. We may determine the characteristic of the logarithm of my number iV^by inspection. The place immediately to the left of the decimal point is known as the units place. If there be digits to ,Lhe left of units place, let their number be p. Then N =^ M X 10^, vvhere if is a number whose first digit is in units place, so that M .ies between 10 and 1 = 10^ The characteristic of logio if is evi- lently zero and therefore that of logio ^ is p. But if the first signif- cant digit in iV^is in the nih place to the right of units place, then N= M -- 10% if being defined as before. Thus, if JSf = 0.0024, ^ = 3 and M — 2.4. Then the characteristic of logio iV^ is — ^^. iVe may combine the results to give the theorem : The char act eristic of the logarithm of a number is -\- n if the first significant digit lies n places to the left of units place, but is — nif the first significant digit lies n places to the right of units ilace. Example. Find the logarithms base 10 of 2400, .24, and .0024. From the Table, we get log 2.4 = .3802. Hence log 2400 = 3.3802, log 0.24 =T.3802, log 0.0024 = 3.3802. 22 EXPONENTS; LOGARITHMS, [Cn. II As in this example, the negative sign belonging to the charac- teristic is written above the latter. Otherwise, — 1.3802 would bj taken to mean that both 1 and the decimal .3802 were negative; whereas the mantissa is understood to be always positive. Hence 1.3802 denotes - 1 + 0.3802. If we desired to find log ^/.'M, m should divide log .24 == 1.3802 by 2. To do this, we write i i(T.3802) =r: |( - 2 + 1.3802) = - 1 + .6901 = T.6901. 23. There follows on pages 24, 25 a table of Logarithms to the bas< 10 correct to four decimal places. For its origin see Chapter XVI 24. To find hy the TaUe the logaritlim of a given numter.' Suppose the given number to consist of three digits, as 2.23. Th« first two digits 22 are to be found in the left-hand column headed i\^ On a line with these and in the column headed by the third digit 3i we find 3483. With the decimal point prefixed, the result .3483 i the mantissa of log 2.23. Since there are no places to the left o; units place, we get log 2.23 = 0.3483. Similarly, log 7 = log 7.0 == 0.8451. If the given number consists of four or more digit^Has 2.2325 we determine the logarithms of the numbers of three digits whicl are respectively less than and greater than the given number. Thu log 2.2300 = 0.3483 log 2.2400 z= 0.3502 The logarithm of 2.2325 lies between these two logarithms, whost difference is .0019. But 2.2325 exceeds the smaller number 2.230( by y%'V of the difference of the two numbers. Hence log 2.232i must exceed log 2.2300 by approximately -f-^-^ of .0019, so tha log 2.2325 = 0.3483 + .0005 = 0.3488.t * Henceforth the base is supposed to be 10. f The value .0005 is used for .000475, since the former is the decimal of fou places nearest to the latter. It would be deceptive to retain more than fou places, since the results from the Table are only true to four places, and th retention of six places would indicate our belief in the accuracy of the fifti and sixth places. Sec. 25] COLLEGE ALGEBRA, 23 In Chapter XVI, we establisli the principle here involved, namely, /or a relatively small increase in a numler^ its logaritlijn increases proportionally. For example, from the Table : log 9.34 = .9703, log 9.35 = .9708, log 9.36 = .9713, log 9.37 = .9717, log 9.38 = .9722, log 9.39 = .9727, the difference between adjacent logarithms being 5 in four cases and 4 in one case. The difference 1 in the numbers 934, 935 , . . . , 939 is relatively small. On the contrary, the principle is not valid if applied to the numbers 100, 200, 300, whose differences are not relatively small. Thus, from the Table, log 100 = 2.0000, log 200 = 2.3010, log 300 = 2.4771. 25. To find hy the Table the number corresponding to a given logarithm. Let the given logarithm be 3.9779 and the corresponding number be xY. Since the mantissa .9779 does not occur in the Table, we take those numbers in the body of the Table, viz., 9777 and 9782, which are just larger and just smaller than 9779, respectively. Then log 9500 = 3.9777 1 j. .0002 = difference log N = 3.9779 y \0005 = difference log 9510=: 3.9782 Hence the difference between 9500 and iV^must equal f .of 10, the difference between 9500 and 9510. Hence J^ = 9500 + 4 = 9504. If the given logarithm is 3.1795, the work is as follows: log .00151 = 3.1790 log JV ' = 3.1795 log .00152 = 3.1818 |. .0005 = difference * .0028 = difference Hence the difference betw^een .00151 and iVmust equal 2\ot .00001, the difference between .00151 and .00152. Hence J^ = .00151 + .000002 = .001512 to four significant places. /For 2^ x .00001 = .00000178 +, we take .000002 as the additive part to obtain iV, being a decimal all of whose figures are reliable. Had we used a six- 24 EXPONENTS; LOGARITHMS. TABLE OF FOUR-PLACE LOGARITHMS. [Ch. II N O 1 2 3 4 5 6 7 8 9 0000 0000 3010 4771 6021 6990 7782 8451 9031 9542 1 0000 0414 0792 1139 1461 1761 ,2041 2304 2553 2788 2 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 3 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 4 6021 6128 6232 6335 6435 6532 6628 6721 6812 6902 5 6990 7076 7160 7243 7324 7404 7482 7559 7634 7709 6 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 7 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 8 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2830 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 ■ 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 •4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 88 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 COLLEGE ALGEBRA. TABLE OF FOUR-PLACE LOGARITHMS. 25 N 1 2 3 4 5 6 7 8 9 50 6990 6998 7097 7016 7024 7033 7042 7050 7059 .7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 • 7185 7193 7202 7^10 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 "8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 950.4 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9052 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9V82 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 26 EXPONENTS; LOGARITHMS. [Ch! H place table of logarithms, the additive part would have been retained in the form .00000178. 26. An equation in which the unknown quantity appears in an exponent is called an exponential equation. Such equations may usually be solved with the aid of logarithms. Example. Find the value of a? in 3^^ + ^ = 3^^~*. Taking the logarithm of each member of the equation, we get (2 a? + 1) log 3 = (bx - 4) lo^'g 2. .-. a; (2 log 3 - 5 log 2) = - 4 log 2 - log 3. . •. X (log 3^ - log 2"^) = - log (2* X 3). , ^ ^ - log 4_^ ^ - 16812 ^ L6812 ^ ^^^^^^ log 9 - log 32 1.4491 .5509 to four decimal places. The denominator may be calculated otherwise, since it equals log -5 = log ?l^ = log (.28125) = 1.4491. 2 10 " EXEKCISES. 1. Find the common logarithms of 6752, 5.4321, 0.04682. 2. What numbers have the logarithms 2.1516, T.2222, 43333 ? 3. Solve 3^-' - 450, lO'"'^ := 4'"'^, 3^ + 1 s^^-^ ^ 1000. Using logarithms, calculate to four significant places : 8124 X .00345 *• .-00069^8-7X2- 5.t/l2345.- 6.(84.625).. 100,— 10,- 7. V45.24 X 1^.0004 X |/12345. 8. 2T 9. 1/5 X |/8765. Solve by logarithms the simultaneous equations: 10. 3'^ + '^== 10, 5^-' = 125^'^-'. 11. 3^ + ^ = 6'', 2^ = 9 X 2^-\ 12. Find the number of digits in 25^^, in 2*T CHAPTEE III. FACTOR THEOREM; QUADRATIC EQUATIONS. 1 27. It is observed that x — a is a factor of x^ — d\ x^ ■— ^ax-{- 2a^, x^ — ax^ — x -f «, and tliat eacli expression vanishes when a is written in place of x. In general, if an expression E^ involving x has a factor x — a, and if Q^ is the quotient arising from the division of B^ by x — a, then E^ = {x- a)Q^, so that E^ vanishes when a is written in place of x. The inverse of this theoi^em is called The Factor Theorem. If an integral expressio^i involving x vanishes tulien a is iuritte7i in place of x^ it lias ill e factor x — a. Divide the given expression E^hj x — a until a remainder R is obtained which does not involve x. If Q^ be the quotient, then E^,= {x-a)Q,+ R. Substituting a for x in this identity, we get where E^ is the value of E^ when a is w^ritten for x. By hypotliesis, E^ vanishes, so that the remainder R is zero. Hence E^ is exactly divisible by x — a. Incidentally, we have also established The Remainder Theorem. Upon dividiyig an integral expression E-^ iy X — a, we oUain as the remainder Ea-, the value of E^ when \j^ki^ written for x. ^f Example 1. To verify tliat x - a is a factor of a:r> — a^, we put a for x in tlie expression and get ap — aP — 0. Example 2. a? — 1 divides :i? --2x^ -\-\, since 1 — 2-1-1 = 0. 27 28 FACTOR THEOREM; QUADRATIC EQUATIONS, [Ch III 28. We next apply the factor theorem to prove tliat, if n be a positive whole number, x"^ — y'^ is always divisible by x — y^ but that x"" + y"" is never divisible hj x — y. Putting y in place of x, the expressions become, respectively, To prove that x"" + V"" has the factor x + y when n is odd, but not when n is even, we note that x-\-y — x— { — y). Upon writing — ^ in place of x^ x"^ + 2/'' becomes ( — l)*y + y"^, which vanishes if n is odd, but not if 7i is even. Similarly, x"' — y"^ has the factor x -\- y ii n \^ even, but not if n is odd. In particular, r'' — 1 is always divisible by r — 1. We have r^ -\ = (r - l){r + 1), ^3- 1 = (r- l)('r2 + r+ 1), r^ _ 1 = (r - l)(r3 + r^ + r + 1). To establish the general formula (1) r''-l = {r - l)(r^-' + r"-2 + . .. +r + l), we perform the multiplication on the right and obtain (^n _|_ ^n-1 _|_ _ , _|_ ^2 _^ ^) _ (^.n-1 _|_ _ ^ _|_ ;. _|_ 1) :=^ ;.n _ 1^ Substituting - for r in (1) and multiplying by y'\ we get (2) x""- y'^={x-y){x''-^ ^x""-^ y-^x^-^ y'^^ . . .+ ^^"~'+^"'*')» Eeplacing y by — y we obtain, //* -^^ 15 odd, Eeplacing ?/ by — i/ in (2), we obtain, if n is even, (4) x""- y''^ {x + y){x''-'- x^'-hj+x'^-^y'^ - . . .+ xy^'-^-y''-'). 89. The preceding formulae may be applied to find a factor which will rationalize any binomial surd. For example, ^3- 1 ~ (^3"- OCt'B + f/3 + 1) ' 3-1 Sec. 30] COLLEGE ALGEBRA, 29 If the given surd \^ x — y, where x = \/a and y = ^h, then a;" and y"^ are rational if n be a multiple of both p and q. Applying formula (2), we see that the rationalizing factor is the rational product being x''' — ^^ If the given surd is x + y, where x, y, n are defined as above, then, for n odd, formula (3) gives the rationalizing factor and the rational product x^ -\- y'^] wliile for n even, the rational product is x"^ — ?/" and, by formula (4), the rationalizing factor is x^-y-^x^-hj -\-x^-hf — ... -\-xif-'^ — y""'^* For example, to rationalize |/7 + y 5, the rationalizing factor is (/7 )'-G/ry i/b + (4/7 )' ^25"- (/f)' . 5+4/7". 54/^- 5^25" and the rational product is (4/? )^ — (t^S )^ = 7^ - 5^ = 318. RXEHCISES. Decompose into three or more real factors 1. ^* - y\ 2. x^ - y. 3. x + /. 4. oj^ - /. Without actual division show that 5. 18a?^^ + 19^'^ f 1 is divisible by ic + 1. 6. 2«* - i^^ - 6aj^ + 4a; - 8 is divisible by ^ -L ' 7. x^ - ?>x -f- 3ic^ - 3a? + 2 is divisible by a;^ - 3ic + 2. Without actual division find the remainder when 8. x^ -^x ^-x-^ is divided by a; + 3. 9. x' -f %xy^ -f- y^ is divided by cc — 2y. Express with rational denominator 1 v/ 1 1 + |/r 4^2" i/9" 4/3 - V5 V'2 -1-1 1 - 1/4 4/2 + 4/3 14. {x -\- yf — a;''' — y^ and (a; -f y^ — a?^ — ?/^ are divisible by a; -\- xy-\-y^ 30. A quadratic equation is an equation wliich involves the square, but no higher power, of the unknown quantity. When all I 3© FACTOR THEOREM; QUADRATIC EQUATIONS. [Cu. Ill the terms have been transposed to the left member of the equation, the quadratic equation has the form (5) ax^ + hx+c = 0, where a, h, c are known constants. If upon substituting for x a particular number x^, the left member of (5) reduces to zero, so that tlie equation is satisfied, x^ is called a root of the equation. A quadratic equation (5) may often be solved by inspection by factoring its left member. Thus x^ -{- x — 12 = may be written {x — ^){x -[- ^) =^ 0, so that the roots are + 3 and — 4. When the factors are not evident, we complete the square of the terms involving x. For example, x^ — ^x—^l = may be written, after adding and subtracting 9, (:^; _ 3)'^ _ 100 = 0, {{x - ^) + 10\\{x - d) - 10] = 0. Corresponding to the factors x-{-7, :c — 13, we obtain the roots -7,13. 31. To solve the general equation ax^ -{- bx -\- c = 0, where a 9^: 0, we divide by the coefficient a of x"^ and obtain the equation (6) xij^^x+- = 0. Adding and subtracting the square of .- — , we get Z CI Kir-£-3=»- Since stny quantity is the square of its square root, the left member may be regarded as the difference of two squares., Factoring, we get (^+2« + ^4a2 aSr^^a ^ia' a f " ^- . Each factor furnishes one of the two roots (7) -=-f±^-=^'. ^ ' 2a 2a «EC. 32] COLLEGE ALGEBRA. 31 In particular, we derive the factorization The 7'oots of the quadratic eq^iation ax^ -{- hx -\- c = are — Z> + \/U^ — 4:ac ^ — b — i/¥ — 4ac (9) a =: ^ , /3 = '-——^ . ^ ^ 2a ^ 2a According as P — 4ac is positive, negative, or zero, the roots are real and unequal, imaginaj^y and unequal, or real and equal. For the last case, W — iac, formula (8) becomes ax^ -\- l)x -\- c = aix + , . 32. Taking the sum and the product of the roots, we get (10) a + ^=-K a^ = tzJ^l^:^) = t. ^ ^ a 4:a^ a Comparing these results with the quadratic (6), we conclude that, in a quadratic equation tuhere the coefficient of x^ is unity ^ the sum of the roots equals the negative of tlje coefficient of x^ the product of the roots equals the constant term (the term not involving x). For example, in the above quadratic x'^ — Qtx — 91 = 0, having the roots — -7 and 13, the sum of the roots equals + 6, the product of the roots equals — 91. To give a second proof by a method applicable also to equations of higher degrees (Chapter XIX), consider the equation* (x - a){x - y^) = x'^ — {oc + /3)x + a/3 = 0, Since x — a vanishes for x = a, the equation has a root a. Simi- larly, it has a root /3. By inspection, the sum of the roots is the negative of the coefficient x- and their product is the constant term. *By using the symbol =, read identically equal, we emphasize the fact that we have here two different forms of the left member of the equation, the two members of which are connected by the sign =, read equal. I 32 FACTOR THEOREM; QUADRATIC EQUATIONS. [Ch. Ill The quadratic equation v/hose roots are given may therefore be constructed as follows: x^ — (sum of roots):c -|- (product of roots) = 0. Thus, the quadratic equation whose roots are — 2 and 4 is ^2 _ ( _ 2 -f 4).T - 8 = a.'2 - 2.^ - 8 =: 0. Example 1. li a and ft denote the roots oi x^ — px ~{- q r= 0^ find th( values of ex' + fS\ a^ + fi\ a:^ + ft\ Since a -\- (5 = jp and af5 — q, we have ^2 + /52 =: (or + ftY - 2«:yS = f- - 2g, a:3 _^ /53 ^ (tr 4- fif - 3aVi - 3a/?2 ::^ ^3 _ 3^^^ a* -f y^* = (a2 4- ^2)2__ 2aVi^ = (i?^ -"Iqf - 2^^ == p* _ 4^2^ 4. 2?^ Example 2. Determine the quadratic equation whose roots are the cube< of the roots oi x' — px -^ q — ^. If a and (i are the roots of the latter, we have (Ex. 1) a^ + /5^ r= _p3 - 3i?g, d'ff' ^ q^. The quadratic equation with the roots a^, fS^ is therefore y'^ - {p^ - ^pq)y 4-^ = 0. Example 3. Determine the condition upon the coefficients of the equatioi ax^ -\- hx -\- c = Q in order that one root may be four times the other. h c By formulae (10), a -{- /3 = > aft = ~. Since ft is, by the condition 0/ Ob ^ imposed, equal to 4a, we must have 5a: = , 4^2 = — . a a Equating the resulting values for d', we get the condition ^H' ^ 256^^-. EXERCISES. Form the equations whose roots are 1. 4, - 5. 2. c 4- (Z, c - d. 3. 0, 4. 4. i - i. 5. 2 4- 1^3; 2 - l/a 6. - 3 4- 1/7; - 3 - |/7 7. V'^T, - \/ '^^. 8. 4 ± \f -Z. 9. Find the condition that one root of ax^ -\- hx -\- c — ^ shall be (1) th( double of the other; (2) the negative of the other. 10. If a and ft are the roots of x' — j?aj -f 5- — 0, find the equation whose root« are (1) -^- and' -— ; (2) a^ft and aft'-, ^3) p and q; (4) a 4- /5 and a" 4- , (5) a 4- I and /i+ ^; (6) « + ^ and i 4- ^. Sec. 33] COLLEGE ALGEBRA. 33 Factor the expressions and derive the roots of the corresponding equations: 11. x' - 53a; + 150 12. x^ - ^^x - 200. 13. ^x^- 18 -j- 21a;. 14. x^ -\- 2ax — 52 _ 2ab. 15. x'^ — iax + Sa\ 16. x^ — ^x'^ + 4. 17. ^2 _ ^ _ ^2 4. i 18. -^ + ^^- 2. 19. ^2 + ^-29 x^ ■ a^ x -[- \ x^ ' ic^ 33. Symmetry. An expression is said to be symmetrical with respect to two or more letters if they enter the expression in such a manner that it is unaltered when any two of the letters are inter- changed. Thus 2a + 2Z> and a^ ~\- h^ are symmetrical with respect to a and b\ a-\-d -\- c and ah -\- ac -\- he are symmetrical with respect to a, h, c. An expression like ah^ + ^^ + ^^^ possesses only a partial sym- metry; it is unaltered when a is replaced by h, h by c, and c by a, but is altered if two letters are interchanged. It is said to possess cycle-symmetry. Example 1 . Factor the cyclo-symmetrical expression ;S' = a\b - c) + h\c — a) + c\a - b). Writing b in place of a, the expression becomes b\b - c) + b\c - 5) = 0. Hence, by the factor theorem, a — 6 is a factor of 8. Similarly, b — c and c — a are factors of S, results following also from the cyclo-symmetry of S. Since 8 is of the fourth degree in the letters a, b, c, there remains a fourth factor la-\-mb-^ nc of the first degree in a, b, c, so that o (11) ; — T 77 ^ = la -\- mb -\~ nc, {a — b){b — c){c — a) The numerator and denominator of the fraction both possess cyclo-symmetry and therefore also the fraction does. Hence la -f- mb -\- nc^lb -\- mc -f- na. From this identity we derive I = m = n. To prove that ^ = — 1, we observe that the term — a'^(5 — c) of the denominator of (11) must divide into the term a\b — c) of 8 to give the term la of the quotient. Another method is to employ particular values, say a = 0, b = 1, c = 2, when the identity (11) be- comes - 6/2 = m + 2/i = 3^. Hence 8 = — {a — b){b — c){c — a){a -\-b -\- c). Example 2. Factor the cyclo-symmetrical expression E~{a-bf^{b- cf + (c - af. I 34 FACTOR THEOREM; QUADRATIC EQUATIONS, [Ch. Ill As in Example 1, we see that E is divisible by the product P~ [a - h){h - c){c - a). Since E and P possess cyclo-symmetry, the quotient E/P is a cyclo-symn metrical expression of the second degree. Set E/P = l{a^ -f b'' + c2) -f m{ab -[- 5c -f- ca). For a = Oy b = 1, c = — 1, we find that 15 =. 21 — m. For a = 0, b—l, c~ 3^, we get 15 = 5^ -j- 2m. Hence I = 6, m — — ^. Hence E= 5{a— b){b — c){c — a){a^ + b'^ + c- — ab — be — ca). EXERCISES. Factor the following cyclo-symmetrical expressions: 1. ab{a — b)-{- bc{b — c) + ^^^(c — a)- 2. a\b - c) + 6^{c - «) + c'(« ~ *). 3. ^25^(61 - ^>) H- b''c\b - c) + c'-^a2(c - a). 4. ab{d^ - b^) -f bc{b^ — c^) + ca'c^ ~ a% 6. a*(62 - c') + 5i(c2 - a^) + c\(i' - b''). 6. aW - c'f -{- b\c' - a'^f + c'^a^ _ 52^3^ CHAPTER IV. SIMULTANEOUS EQUATIONS; DETERMINANTS. 34. If we assume that the values of the unknown quantities X, y, . . . are the same in each of several equations, the equations are called simultaneous equations. Consider the pair of simultaneous equations ^ ^ ^ 6x-2tj = 2. To solve for x, we multiply the first equation by — 2 and the second equation by -f- 3 and add the resulting equations. Then 14:X =14, X = 1, In a similar manner, we find that y = 2. 35. Consider the general pair of simultaneous equations a^x + % = c^, + hy = ^2- To solve for x, we eliminate y between the equations. This may be done by multiplying the first equation by b^ and the second by— b^ and adding the resulting equations. Then {a,b, - a,b,)x = {c^b, ~ c^b,). Employing the respective multipliers — cf^ and «p we get The quantities in the parentheses are all of the same form. We therefore employ as an abbreviation the symbol (3) ^'!;!-"xJ.-«A. 35 (2) \'^' 36 SIMULTANEOUS EQUATIONS; DETERMINANTS, [Ch. IV called a determinant of the second order. It equals the difference j of the two cross-products. The above relations may now be written j (4) X = y «o c. By division, we obtain the values x, y satisfying equations (2).,; a, b. In formulae (4), x and y are each multiplied by the symbol «o b. which is formed of the coefficients in the left members of equations: c. b: (2). The determinant c,b. is derived from the preceding deter- y = 2-4 6 2 minant by replacing the a's by c's. We may express our results by the following theorem : For the simultaneous equations {2)^x times the determinant D of the coefficients of the left members equals the determinant obtained bij\ substituting the constants c^^ c^ in place of a^, a^ in the first column, of D ; y times D equals the determinant obtained by substituting the constants c^ , c^ in place of b^ , b^ in the second column of D, ' Applying the theorem to example (1), we get 2-3 _|- 4 - 31 2-3 6 - 2 "^ ~| 2 - 2| 6-2 Evaluating these determinants by the definition (3), we get 14:2; = 14, Uy = 28. EXERCISES. Solve by determinaiits the pairs of simultaneous equations : I, Sx — y = 34, 2, 3aj + 4y = 10, 3. x — y — 5, x-\- Sy = 5d. ^x-\- y = 9. 5^ - 4y == 40. 4. ax — by = c, 5. x -}- y = 1, 6. ax -\- by = a\ ex — ay = b. ax -\- by = c. bx — ay — ab. 36. Consider the three simultaneous equations r c^i^ + b,y + c^z = \ , (5) I ^\^ + b.;y + c^z = \ , I %^ + b^y + c^z = \. Sec. 36] COLLEGE ALGEBRA, 37 ir we can combine these equations in such a way that y and z shall be eliminated, the resulting relation will, if it contains x, de- termine the value of x. Multiplying the first equation by M^ , the second equation by M^ , and the third equation by M^ , and adding the resulting equations, we get (0) {a,M, + a,M, + a,M^)x + {\M, + b,M^ + b,M,)y We desire tliat tlie coefficients of y and z shall vanish. It will be shown that this will indeed happen if we take (7) M, M,^^ K c. ^1 ^1 1 1 , if3 = 1 1 h Cs *2 ^'2 Substituting for these determinants their values as given by the definition (3), we find that (8) h,M^ + \M^ + d,M, = ^,(^2^3 - ^3^2) - ^2(^1^3 - ^3^1) -r ^3(^1^2 - K^i) ^ 0- In a similar manner, c^3f^ + ^2^2 + ^3 ^^^3 — ^- Hence (6) be- (!omes (9) {a, M, + a,M\ + a,M.:)x = ^,if, + h.M, + /^3.¥3. For the coefficient of x, with the values of M^ , J/^ , M^ inserted from (7), we shall employ the following symbol: a, J, c. (10) + ^3 M a, Z^2 ^2 «3 ^3 ^3 It is noticed that the symbol is formed of the coefficients of the left members of equations (5), the coefficients retaining the same relative positions in the symbol as in the equations. We may de- rive the right member of equation (5) from the coefficient of x by replacing a^, a^, a^ by Jc^ , Ic^^h^y respectively. Hence equation (9) may be written in the form (11) «1 h Ci K K c, a. K '-, X = K h ^5 'h K <\ K K Cr ■38 SIMULTANEOUS EQUATIONS ; DETERMINANTS. [Cir. IV Such symbols are called determinants of the third order. If the determinant on the left is not zero, we obtain by division the value oi X. 37. The preceding method may be seen to be a direct general- ization of the method employed in §35 for the solution of simul- taneous equations in two unknown quantities. To solve equations (2) for X, we employed the multipliers l^ and — h^ [notice the alternation in sign^ corresponding to the signs -{-, — , + in the ^1 w multipliers (7)]. We observe that, in the determinant V ^2 stands diagonally opposite to «j and that ^^ lies opposite to r/./, so that the multipliers used in solving for x lie opposite to the co- efficients of X in the equations (2). For the three equations (5), the nuiltipliers (7) used in solving for x lie opposite to the coelli- cients of x. Expressed more exactly, the first multiplier M^ niay ^j be derived from the array of coefficients in the left members of (5)*| by erasing the row a^, h^, c^ and the column a^, a^, a^ containing h c of the remaininer coefficients. ftj and taking the determinant ^3^3 The second multiplier M^, apart from its sign^ may be derived by erasing the row and the column containing a^ and taking the deter- of the remaining coefficients. Similarly, J/gis gotten minant ^3^3 1 by erasing the row and the column containing a^. The resulting determinants are called the minors of «i , a^^ a^, respectively. In solving equations (2) for y, we employed the multipliers — «2 ^^^ + ^r Apart from their signs, they are, respectively, the minors of b^ and h^ (the coefficients of y in the given equations) in the determinant {h\ for y, we should employ as multipliers where This suggests that, in solving equations ^1? +^2' — ^3» (12) -B.^ «o ^, 0^ C^ «i C,l 3 2 . ^.= I 1 . ~^3 = I ' «3 Cs ^3 ^3 fl, «^J Sec. 38] COLLEGE ALGEBRA. 39 which are, respectively, the minors of h^, 1)^, h^ in the determinant (10) of the coefficients of the left members of (5). Then (13) - a,B, + a,B, - a^B, = 0, -c,B, + c,B^ ~ c,B, = 0, so that, for M^= — B^, M^= + B^, M^- — B^, equation (6) gives (14) {~\B, + \B, - \B,)y = - k,B, + \B-\B,. The coefficient of y, with the values of ^^ , B^^ B^ inserted from (12), may be expanded as follows: (15) - \a/.^ + \a^c^ + d.a^c^ - \%c^ - l^a^c^ + \a^c^. It is therefore equal to the determinant (10). The right member of (14) is derived from the coefficient of y by replacing Z>j, l^, h^ by k^, /bj, ^3, respectively. Hence (14) becomes (16) I Incidentally, it was seen that the determinant on the left can be expanded according to the elements of the second column by multiplying h^ by the negative of its minor B^^ h^ by its minor B^, ^3 by the negative of its minor B^^ and taking the sum of the re- sulting products. Proceeding in a similar manner, we find that «1 ^ Ci a, K <■' «. ^ c. y = a. K c.^ «, K f. », K c. (17) «1 * fl, T^: a. I. Hence to solve equations (5) for any one of the unhnoion quan^ titles Xy y, z, loe equate the product of the unhnoion and the deter- minant D of the coefficients of the left memlers of (5) to the deter- minant oMained from D hy substituting the constants k^, k^^ k^ in place of the coefficients of that unknoion. 38. It follows also from the preceding developments that, if A^, A^, Jg, B^, B^, B^, (7i, Cg, C^ denote the minors of «,, a^, a^. 40 SIMULTANEOUS EQUATIONS; DETERMINANTS. [Ch. IV ^1? ^2^ ^3^ ^1? ^2' ^3' respectively, in the determinant D, these rela- tions hold: /^i^i-&2^2+M3 = 0, -b^B^-i-b^B^-b^B^ = D b^C^-b^Cr^-^b^C^ -■ 0, Ci^l-C2^2 + C3^3 ==0, -Ci^i+Cg^g-^sA = 0> CjCi-CgCa+c^Ca /> The three expressions for D give three methods of expanding the determinant D, To expand D according to the ^elements of any column, we multiply each element by its minor, with the proper sign prefixed, and form the sum of the products. The proper sign depends upon the position of the element, as exhibited by the following scheme: + — + — + — + — + Compare the black and white spacing on a checker-board. 39. The second equation in the above set may be written ^1 K ^1 = b,A, - \A^ + b^A^ = ^2 K ^2 ^-^3 h ^3 Hence a determinant vanishes if its first and second columns are alike. Similarly, the third equation gives c,A, ^2^2 \ ^3^3 ^3 "^3 Hence a determinant vanishes if its first and third columns are alike. Similarly, — c^B^ -\- c^B^ — c^B^ — shows that a determinant vanishes if its second and third columns are alike. The last result also follows directly from the definition (10) of the determinant. Combining these results, we see that a cleterniinant of the third order vanishes if any two of its cohimns are alike. Sec. 4i)] COLLEGE ALGEBRA, 41 40. Upon expanding the three determinants I K a. <^i a.^ c. h c. h a K a. e. > a. c. h ? c^ K «= K a. c» A c. h c. K a, according to tlie elements l\, h^^ b^, we obtain in each case But these determinants were obtained from D by interchanging two of its columns. Bi/ the interchange of any two columns of a determinant of the third order, its value is changed in sign. The theorem of § 39 may be derived as a corollary to the last theorem. Indeed, if D has two columns alike, it is evidently un- altered upon interchanging them, whereas it must change in sign. From D ~ — D, we derive D = 0. 41. As an example, consider the simultaneous equations ^+ y + ^ = 11, 2x ~ 6g — z — 0, 3:^; + 4^ + 2z = 0. Employing the definition (10), we find that 1 1 1 2 - 6 - 1 = 11. 3 4 2 Hence formulae (11), (16), and (17) give 11 1 1 1 11 1 1 1 11 tWx^ 0-6—1, lly = 2 0-1, 11^=2-6 0. I 042 302 340 ach determinant has two zeros in one of its columns. It is therefore best to expand the determinant according to the elements of that column. The results are, respectively, 1 -6 -1| -11 3 -1 3 2 I( mce X = —8, y = -7, z = 26. -11.7, 11 = 11.26. 1 42 SIMULTANEOUS EQUATIONS; DETERMINANTS, [Ch. IV 1 .1 1 1 1 1 a h c X — h h c a^ ¥ c^ Jc" P ^2 We may factor D without If we substitute I? for a in D, 42. Consider the simultaneous equations ^ + y + ^ =h (18) ax -\- bi/ -{- cz = Jc, c?x -f- y^y -f- c^z = P. The value of x is given by the relation (19) Denote the coefficient of x by D. making use of its expanded form, we obtain a determinant having its first and second columns alike, which therefore vanishes by § 39. Hence, by the factor theorem, a — b is Si factor of D. Similarly, a — c ernd b — c are factors of D, Since D is of the third degree in the letters a, b, c, we may write (20) D = m{a - b){a - c){b - c), where m is a numerical factor not involving a, b, or c. For the particular values a — 0^ b — 1, c = — 1, the identity (20) becomes 2 = — 2m, whence m = — 1. The same result follows from a comparison of the coefficients of a term, like bc'^, in the expanded forms of the two members of (20). Changing the sign of the factor a — c, we may state the result: 1 1 1 {a — b){b — c){c — a). (21) a b c d^ W 0^ Changing a into k, we obtain (h — b)[b — c){c — k) as the value of the determinant in the right member of (19). Hence {a — b){c — a)' In a similar manner, it is fouiid lliat {k- c){a - Jc) _ {Ic -Jt){bj-Jc) [b - c)[a - by ^ ~ {a"- a){b - r)' y I \ Sec. 43] COLLEGE ALGEBRA^ 43 43. If, in place of the third equation (18), we take a^x + l^y + c^z — h\ the values of x^ y, z depend upon determinants of the form 111 A ^ a 1) c r/4 ^>4 ^4 It follows as above that A is divisible by the product P={a-h){b- c){G-a), Since the degree of A in its expanded form is five, the quotient A/P is of the second degree in a, h, c. By the interchange of a and Z>, P is changed into ~ P and A into — z/ {§ 40), so that A/P remains unaltered. Likewise z//P is unaltered by the inter- change of a and c or by the interchange of h and c. Hence A/P is symmetrical in the three letters a, J), c, so that we may set (22) A/P = m{a? + ^^ + c^) + 7i{ah + ac-\- he), where m and n are numerical values independent of «, Z>, c. We may evaluate m and n by substituting in the identity (22) special values for a, Z>, c, such, however, that the denominator P does not vanish. For ^ = 0, b = I, c= — Iwe get f = 27)i — 7i. For a ~ 0, h — 1, c = 2, we get -^4 = 6m + 2n. Solving these two simultaneous equations, we find that m- =1, n = 1. Hence the above determinant may be factored thus : ^ r={a- b){I) - c){c - a){a^ + V^ -\- c^ -{- ah -\- ac + he), EXERCISES. Solve by determinants the following sets of simultaneous equations, can- celling the extraneous factors in 3, 4, 5, 6 : I. X -{- y -\- z = 0, 2. X - 2y -i- z = 12, oj + 2^ -f 3s = - 1, ir + 2y 4- 32! =: 48, aj + 3y + 6s = 0. 6ic + 4^ + 32 = 84. . x+ y-]- z :=!, ^. X -^ y -\- z = 1, ax -\- by -\- cz = k, a^x + h'^y + c^z — W, a^x + Wy + c^z = k^. a^x -f Iry -\- cH - k^, 44 SIMULTANEOUS EQUATIONS; DETERMINANTS, [Ch. IV b. ax -\- y -\- 2 — a — 8, X Ar ay -\- z = — 2, X -{- y -\- az — — 2. 6. ax -\- by -\- cz = 2a -[-h -{• c^ hx -\- cy -{- az — a + 2& + c, ex -\- ay -\- hz = a + 6 -}- 2c. 44. If k^ = 0, h^ — 0, h^ = 0, equations (5) become (23) a^x+b^y+c^z = 0, a^x-^-b^y-^c^z = 0, a^x+b^y+c^z = 0. According to the general method of solution, we have «j b^ c, (24) Dx = 0, Dy = 0, Dz = 0, Z> = It I) ^ 0, we obtain only the evident set of solutions x = y = z=:0. But, if Z> = 0, the relations (24) impose no conditions upon x, y^ %, In this case, equations (23) might have solutions other than the evident set re = ^ = 2; = 0. Let us seek the solutions for which 2; #: 0, for example. The first and second equations (23) may then be written ^1 7.2/ •^17-2/ ^z ^z C.' X u For the unknown quantities — and - we obtain the relations z z- (25) «1 h X - <^1 \ «1 \ l- a. -^1 «2 h z - c. \ «. h z «2 -c. It remains to inquire whether the resulting values of — and — z z satisfy the third relation (23), which may be written (26) «37+^37 + '^, = 0. z z ^1 ^1 To avoid fractions, we multiply (26) by X tJ the values of — , — . The result is clearly a,b. — Cy b. \a. — c, a, b. 1 1 7 + *, 1 + c. 1 1 -c, b. >, -c. «2 K before substitutinpr = 0. Sec. 44] COLLEGE ALGEBRA. 4i, The left member is seen to be equal to D, Hence, if i) = 0, the third condition (26) is satisfied by the values of — , ~ given by z z (25). We must examine the special case when 0^ = «, X 1/ vanishes, since equations (25) then fail to determine — , ~. If z z A= ^ ^2 ^2 does not vanish, we divide the first and second equations (23) by x and obtain solutions for -, — , which satisfy the third equation if. ( X X and only if, D = 0. Similarly, if a, i?,= ^2 X z does not vanish, we obtain solutions — , — , if Z) = 0. y y There remains the case A^ =0, ^3 = 0, C^ = 0. Since a. , Z>i, Cj are tacitly assumed to be not all zero, we suppose that a^ i^ 0, for definiteness. We may then set a^ = pa^, whence a^h^ — ap^ =: gives 2>2 — 7^^!, while a^c^ — af^ = gives c^ = pc^. Hence the second equation (23) may be derived by multiplying the first equa- tion by p, AVe need therefore only consider the first and third equations (23). As shown by the above method, these two equa- tions determine the ratios of x, y, z except in the case = 0. In the latter case, the third equation is a consequence of the first. Since the equations then reduce to a single one, arbitrary values may be assigned to y and z; and the equation then determines x. Wr may therefore state the following theorem : «1 h = 0, h «i = 0, «i ^1 «» h ^3 ^z «s Cz 46 SIMULTANEOUS EQUATIONS; DETERMINANTS. [Ch. IV The 7iecessary and sufficient conditions that equations (23) have solutions x^ y, z^ not all zero^ is that the determinant D shall van- ish. If D — 0, the equations reduce to two equations or to one equation, according as the minors of the nine coefficients are not all zero, or are all zero. In the first case, the ratios x : y : z are deter- mined., so that one unhnoiim is arbitrary j i7i the second case., two of the unhnowns may he chosen arbitrarily. Given, for example, the simultaneous equations 2^ + 3^ - 4^ = 0, ^x + by-z = (), Ix + \ly — 9^ = 0, the determinant D is zero, and relations (25) become = 17, z - 10. «1 ^ c, «2 K c. a. K c. 45. Theorem. A determinant of the third order is not altered in value if tve multijjly the elements of any column by a constant and add the results to another cohimn. Consider a determinant D and its expansion: =: a^A^ - a,A, + a,A^. Multiplying the ^'s by 7n and adding the products to the a's, we get a^ + ^^^1 ^1 ^1 a^ + 7nb, b, c, = {a^ + 7nb,)A^ - {a, + 7nb,)A^ + {a^ + 7nb,)A^ ^3 + ^^^3 h ^3 irr (a^A^ - a^A^ + a,A,) + m{b^A^ - b,A^ + b,A^) = Z>, since the expression multiplied by m is the expansion of b, b, c. which is zero by § 39. The theorem is therefore true for this case. To extend the proof to the general case when the elements of the ith column of D are multiplied by m and added to the /tli column, Sec. 46] COLLEGE ALGEBRA. 47 the resulting determinant being called Z>„, , we observe that the /th column of D may be interchanged with the first column and the ith column interchanged with the second column of the new deter- minant, giving the final determinant D^ = ± D. Doing likewise with D,n , we get the determinant Z>,', = ± D^. By the proof given above, Z)'„ = D\ Hence i>,„ = D, As an example, consider the determinant 2 11 . D^ 6 2 1 12 3 1 Multiplying the second column by - Oil D= 2 2 1 G 3 1 Multiplying tha tliird column by — 1 and adding to the second, 2 and adding to the first. (27) 1 i) = 2 1 1 = - 2. 6 2 1 The aim is to obtain two zeros in one row or one column. 46. Theorem. If the first, second, and third colnmns of a de- terminant of the third order he taken as the first, second^ and third roius, resijedively^ of a new determinant, the resulting determinant equals the original determinant. Starting with the determinant i), given by formula (10), we form the determinant i)' = The terms of D and D' which involve a^ are seen to be equal by inspection. The terms of D* which involve a^ are — h^a^c^ -|- c^afi^^ the same as in B. The terms of D' which involve a^ are b^c^a^ — ^A^3 J ^1^6 same as in D, Hence D — D\ «1 «2 «3 1^', K a^ a^ a^ a. \ K ^3 ^"l-: 3 ^3 -^ 2 ^2 3 + c. h c, c. C, 48 SIMULTANEOUS EQUATIONS; DETERMINANTS, [Ch. IV It follows also that the determinant D maybe expanded accord- ing to the elements a^, Z>j , 6*, n^ its first row, viz., Since any row may be interchanged with the first row if we change the sign of D, we obtain the following expansions according to the elements of the second or of the third row : D= - a^A^ + \B^ - c^C^ , B = a,A^ - b,B^ + c,C^. For example, the determinant (27) may be expanded thus: I) = + 1 -2. Applying the present theorem that the rows and columns of a determinant may be interchanged to the results of §§ 39, 40, wej may conclude that the value of a determinant of the third order u changed in sign tvhen any tiuo of its roios are interchanged, and also that a determinani of the third order vanishes if any two of its\ rows are alike. Similarly, from the last section we derive thd theorem that a determinant of the third order is not altered in\ value if lue multiply the elements of any row hy a consta7it and add the results to the elements of another row. ■ 47. Theorem. A factor common to the elements of any column or any row of a determinant of the third order may he removed and placed as a factor in front of the resulting determinaiit. The proof follows from the relations : ma^ ma^ mao ma ^2 mc^ ^ ma^A^ — nia^A^ + ^^^^3^3 = ^^ ma^A^ — mh^B^ + mc^C^ = m «. \ c, "2 K c. 9 «3 h Cz a, h Ci «2 !>. C2 «s h Cz and similar relations derived by interchanging rows or columns. Sec. 47] COLLEGE ALGEBRA. 49 As an application, consider the cyclic determinant: \a h c A^ ^3 _|_ J3 _|_ ^3 _ 3^^^.^ Upon adding the last two columns to the first column, the resulting determinant, equal in value to J by § 45, has all the elements of its first column equal to a -\- 1) -\- c, which is tlierefore a factor of the determinant. Since aj\-h-^c and the expansion of A are botli symmetrical in the letters a, h, c, the remaining factor is a sym- metrical expression of the second degree. Hence A = {a-\-l + c) \m{a? + V^ + c') + n{ab + ac + Ic)']. Hence the right member must have unity as tlie coefficient of a^ ind zero as the coefficient of a%. Hence m = 1, m -^n = 0. We lave therefore tlie important factorization 28) A = a^+b^+c^-^abc={a+I) + c){a^+b'^+c'^-ab-ac-hc). The factor of the second degree may be decomposed into two maginary linear factors. Let go be the quantity introduced in § 8' ,0 that GD^ = I, Multiplying the elements of the second column of d by 00 and those of the third column by oo^ and adding the prod- icts to the elements of the first column, the sums are a -{- bao -\- CGO^, c + aao + boo^ ~ Go{a -{- bao -{- coo^), b + coo -\- aoo^ = Go'^{a + boo ~\- cro^). lence a + bGD-\- coo^ is a factor of A. Employing the multipliers »2 and G9, we obtain the factor a -{-boo'^ -[- coo. Hence 28') A = {a-[-b-\- c){a + boo -\- coo^){a + boo^ + coo). EXERCISES. Solve the following sets of simultaneous equations : 1. a; + y + 3z = 0, 2. a^ + 3y + 4^ := 0, Z. 2x - ^ + 42 = 0, aj-f2y+2s=:0, 4a; -I- 12y 4- 162 = 0, x^ 3y - 22 := 0, a.-f5y- 2=0. 3a! + 9^ + 122 = 0. a? - lly + 142 = 0. Factor the following determinants without expanding them: 4. a b c 5. 1 a be 6. d d d a" 52 c' 1 b ca c b a a^ b^ c^ 1 c ab ab ac be 5© SIMULTANEOUS EQUATIONS; DETERMINANTS. ^ 8. [Cn. n . a2 a2 _ (J _ cf he I C2 C2 - (a - Z>)2 ,^6 I 48. As definition of a determinant of the fourth order, we take d' h^ 6' he ca ah (29) ^ />. a^A^ - a.^A^ + a^A^ - a^A^, l^U ^\ ^4 ^4 where .Ij denotes the minor of a^ obtained by erasing tlie row an^ the cohimn containing a^, A^ the minor of a^, etc. ; that is, A^ h e-i d. h Cz d. h c. d. \ ^1 ^h h c^ dz b, 6\ d. A.= K Ci d. h c^ d. h c. d. K c. d. h ^^2 d. K ^3 d. The terms of (29) which involve 5j are seen to be -«A c^ d^ c^ d^ c, d. c, d. + (^A c, d. ~ ''*^' c d = -b. 1/3 t,lg c, d. c^ d^ c, d. c, d, «i Ci d, ^3 3 c, d. -aA c, d. + a A C3 d. = K a, c, d, «« c, d. which equals — b^B^, where B^ denotes the minor of h^ in the deter minant D defined by (29). Similarly, the terms of D which in volve ?>., are (^A which equals ly^B^, where B^ denotes the minor of h^ in D. Simi larly, the terms of D which involve h^ may be combined int( — ^3^3, the terms which involve h^ may be combined into h^B^ Hence we have (30) D=- \B, + \B, - \B, + ^4^4- ' But by interchanging the first and second columns of D, we get K a. Ci d, K ". ^ d, h «» Cz d, K a. c^ d, = \B, - \B, + \B, - h,B, D. *The definitions of determinants of orders 5. 6, etc., are quite similar. They have properties analogous to those of determinants of orders 2, 3, 4. [3ec. 49] COLLEGE ALGEBRA, 51 Hence a determinant of the fourtli order is changed in sigji by 'ntercha7iging its first and second^ columns. Upon interchanging the ^'s and c's in the identity (29), we note hat the signs of ^4^, A^, A^, A^ are changed (§40). Hence the lew determinant of the fourth order is the negative of D, Simi- arly, upon interchanging the ^'s and ^'s, or the ch and ^Z's, the ign of D is changed. In order to interchange the a's with the c's, we may first inter- ihange the a's with the Z>'s, then interchange the a'^ with the c's, md finally interchange the Z?*s with the c's. The sign of the de- -erminant is changed at each of the three steps. The preceding results may be combined into the theorem: A determinant of the fourtli order is changed in sign ly inter - 'hanging any tiuo of its columns. As a corollary to this theorem, we have the result: A determinant of the fourth order vanishes if a7iy two of its olumns are alike. 49. By the definition of the determinant symbol, we have d, d^ = {a^ + 7nb;)A^ - {a^ + 7nh;)A^ ^3 + («3 + ^'^^3)^3 - K + 'inh,)A^ = {a^A^ - a,A^ + a,A^ - a,A^ + m{\A^ - \A^ + h^A^ - h,A^). The first quantity in parenthesis equals the determinant (29); he second equals m times a similar determinant with the «^'s re- )laced by &'s, a determinant having its first and second columns like and therefore equal to zero (corollary of §48). We have herefore proved that a determinant of the fourth order is not Itered in value if the elements of the second column be multiplied •y a constant 7n and the results added to the first column. As in §45, we extend the theorem to the case in which the olumns in question are arbitrary, say the ii\\ and ji\\ columns, ndeed, by an even number ofv interchanges of columns, we may «j + mh^ \ Cl flj + mh^ \ «2 «j + mb^ \ C, a^ + mb^ h C4 52 SIMULTANEOUS EQUATIONS; DETERMINANTS, [Ch. IV, bring the iih. column into the position of the first column, and the yth into the second column. Each interchange of columns onlj changes the sign of the determinant (§48). We may therefore state the theorem: A determinant of the fourth order is not altered in value if thi elements of any column are multijMed hy a constant and the ]prod\ ucts added to the corresponding elements of any other column. As an application, we prove that the determinant J = a^ nc. \ a^ nc^ c^ a^ nc^ c^ a^ nc^ c^ vanishes identically. Multiplying the third column by -— n and adding to the second column, we obtain an equal determinant hav< ing only zeros in the second column. Interchanging the first and second columns, we obtain a determinant which vanishes, as shown by the' definition (29), whereas its value is — z/. 60. Theorem. A determinant of the fourth order is U7ialtered in value if its corresponding roics and columns are interchanged. In fact, if the given determinant is (29), the resulting deter- minant is i «1 «2 ^3 «4 \ \ h K Cj c^ c^ c^ d^ d^ d^ d^ where A^ is the minor of a^, B^ the minor of Z>j, etc., in D. In view of the corresponding theorem for determinants of the third order (§46), these determinants Jj, B^, C^, D^ are equal to the minors of a^, \, Cj, d^^ respectively, in the original determinant (29). Hence in the two determinants D and (29), the terms in- volving a^ are the same. The terms of (29) which involve b^ were seen in § 48 to be given by — l)^B^, the same as in D, The terms of (29) which involve c^ are evidently D^ = a^A^ - \B^ + c^C^ dj),. F Bbc. 51] COLLEGE ALGEBRA, {-a,){-c,) + «3(- ^l) -^4(-0 53 = 6', a. Similarly, the terms of (29) which involve d^ are given by — d^D^ It follows that a determinant (29) of the fourth order can be expanded according to the elements of the first row by taking the ;um of the products of each element and its minor, with the proper ;ign prefixed, viz., a^A^ — h^B^ + ^i^\ — dj)^, I 61. Combining the theorem of § 50 with the theorems of §§ 48, p, we derive at once the following results: I A determinant of the fourth order is changed in sign hj inter- 'hanging any two of its roivs. It vanishes if a7iy two of its roivs •re alike. It is not altered in value if the elemeiits of any row are mltipUed ly a constant and the products added to the correspond- ng elements of any other roiu. 52. By the definition (29) and the proof in § 50, a determinant 9 of the fourth order can be expanded according to the elements f tlie first column or of the first row as follows: ) = a^A^ - a,A, + a^A^ - a^A^, D = a,A^ - l^B^ + c^C^ - dj)^, iiiice any row can be brought into the first row by an interchange f rows, and likewise any column into the first column, the deter- linant being changed in sign by each interchange, we derive the xpansions: ) ^ -\B^-\-\B-\B,-^lfi„ D ^ -aJ^+\B,-c,C,+d,D, O = + c,C-c,C,+c,C-c,C\, D - -^a,A,-b,B,+c,C-d,D„ ) = -d,D,^dJ)-c\D,-^dJ)„ D=- aJ,+b,B-c,C,+d,D^. 'he sigus are determined by the positions of the elements. The olio wing scheme, analogous to a checker-'board, is useful: + — + — — + — + + — + — — + — + 54 SIMULTANEOUS EQUATIONS; DETERMINANTS. [Ch. IV 53. As an example, we evaluate the determinant D = 3 - 1 - - 5 ^ 2 ' ~ I We seek an equal determinant having three zeros in one row or one column. To avoid fractions, we select a row or column con- taining ± 1 as an element, say the last column.* Adding it to the first and third columns, and, after multiplication by 7, to the sec- ond column, we get - 1 6 2 3-10 1-1 D = - 15 1 -2 - 15 1 — 1 — (> 26 - 2 () 26 -2 3 3 - 10 1 upon expansion according to the second row. Multiplying the last row of the determinant of the third order by — 2 and adding to the second row, we get D= - - 15 1 46 -4 z=: -3 - 10 1 - 15 46 = - 42. 54. Consider the general set of simultaneous equations a^x -\- h^y + c^z -\- d^w =p k^ , (^2^ + \y + <^2^ + d,io = h, , a^x + \y + c^z + d^io = \ , To solve for x^ multiply the first equation by the minor A^, i\H second equation hj — A^^ the third by A^, and the fourth by — A^ and add the resulting equations. By formula (29), the coefficienl * If none were present, we drst work to that end, unless all the element! of one row or column are multiples of one element fas is the case with th( first column of the above determinant of the third order). Sec. 54] COLLEGE ALGEBRA, 55 of X in the sum is the determinant D of the coefficients of the left members. The resulting equation becomes + {d^A^-d^A^-\-d^A^-d^A^)w = k^A^-h^A^+h^A^-Tc^A^, The coefficients of y, z^ w all vanish since they are the expan- sions of determinants having two columns alike (§48). The right member is the expansion of a determinant derived from D upon replacing the «'s by ^'s. Hence the relation becomes Dx^ To solve for y^ we use the multipliers — ^i , -\- B^, — B^, -\- B^, The coefficients of x, z^ 2V are seen to vanish, while that of y is, by (30), - h,B^ + b,B, - \B, + \B, = D. To solve for z, we use the multipliers + (7^ , — C,, + ^3> — ^r Similarly for iv. We obtain the following results: «1 K C^ d. K \ C^ (h a, «3 C, d. X — a. K c. ch K h' c. d. Dy. «i K Cj d. a, k. c. d. «3 ^3 ^ d. a. k. c. d. Dz "l *i h < a. K h d. "3 h K d. «* h h d. Dtv «1 ^ Cx k, «2 K <\ k «3 h Cz h a. I. tv k, Hence the product of any one of the unhnown quantities hy the determinant D equals the determinant obtained from D upon re- placing the coefficients of that unhnoivn hy the constants k^, , , . k^. As an example we take the simultaneous equations ^-\- y + ^+ w = 0, ^+^ + 32; + 4W =: 0, .^ + 3i/ + 62; + lOiu - 0, a; + 4y + IO2; + 20w = — 1. 56 SIMULTANEOUS EQUATIONS; DETERMINANTS. [Ch. IV The determinant i> of the coefficients of the left members is found to equal + 1. Hence X = -1 1 2 3 4 10 20 1 1 1 1 1 1 1 2 = + 1 2 3 4 zzz 12 z:z: 3 6 10 3 7 3 7 1, y = 1 oil 1 3 4 1 6 10 1 -1 10 20 - 1 111 1 3 4 111 2 3 _ 2 3 1 6 10 5 9 ~ 5 9 Similarly, z = ^^ to = — 1, These values of x, y, z, w satisfy the given equations. EXERCISES. Solve by determinants the sets of simultaneous equations : 2. ax -h by -\- cz -\- dw ■- k, a^x -f b'^y 4- c^z 4- d'^w = k\ a^x + b^y -\- c^z + d^w — J. By the ratio of any two quan- tities of the same kind is meant the ratio of the two numbers ex- pressing the number of units contained in the quantities. Thus fin the ratio of 2 months to 10 days is expressed by the number — = 6; the ratio of 12 to 20 cents is expressed by the number 10. Evidently the ratio a : h equals the ratio ma : mh. The ratio of c , r , c r cs -, to —equals — -7- - = -^. as a s dr If the ratio of two quantities can be expressed as a rational number, the quantities are said to be commensurable ; otherwise, they are said to be incommensurable. Thus the diagonal and side i/2 of a square are incommensurable, since their ratio = J__ = |/2 is not a rational number (§2). The ratio aa : hft is said to be the ratio compounded of the ratios a : l and a : /?. Its value 7-^ equals the product of - and (X -, When the ratio a : & is compounded with. itself the resulting r' 58 Sec. 56] COLLEGE ALGEBRA. 59 ratio a? : IP' is called the duplicate ratio of a : b. Likewise a^ : b^ is called the triplicate ratio oi a \b. 66. Problems involving several equal ratios /-.x ace w i=d = j = --- = ' are usually solved most simply by eliminating the numerators a, Cy e, ... of the fractions (1), by substituting in the proposed identity their values (2) a — rb, c = rd, e = rf, .... Example 1. Given — = — = — , prove that b d f a^ — 2c^ -f 4^^ _ ace b^ - 2d'' + 4/3 ~ b^' Making the substitution (2), the two fractions become b' - 2(^ -^ 4/3 ~ '^' ~bdf ~ Example 2. If the ratios r» 3-' 7* • • • are equal, each equals a -[- c -\- e -\- . , . & + t? +/+... • Making the substitution (2), the fraction reduces to r X y z Example 3. Prove that -=-=—, when it is ariven that a b c ^ bz — cy _^cx — az _ ay — bx a ~~ b ~ c ' Applying the result of Ex. 2, each of the latter fractions equals a{bz - cy) -\- b(cx — az) -f c{ay — bx) _ Hence bz — cy — 0, ex — az = 0, ay — bx = 0, and therefore z _^ y X _z y _x c ~ b* a ~ c' b ~ a' EXERCISES. 1. Arrange the ratios 5:6, 14 : 16, 41 : 48, 31 : 36 in descending order of magnitude. Find the ratio compounded of them. 2. What value of x makes 3-|-a;:4 + « = 5:6? 3. Find the duplicate and triplicate ratios of 3 : 5, also of 6 : 7. I 6o RATIO; PROPORTION ; VARIATION. [Ch. V 4. Find two numbers such that their difference, their sum, and the sum of their squares are as 1 : 15 : 113. 6. Find two numbers whose sum is 77 and whose ratio is 3 : 8. It a : b := c : d = e :f, prove that ;^| 6. a^b'' + Sa'e^ - 66^f : b^ + 36^^ _ 5;^5 ^ ^4 . ^4^ . ' 7. a'c + Sc^e + le^c : b^d + Sd'e -f If'd = e^'.p. 8. V{ac - Zc' -f ^e') : \/{bd - M' -h 4/*^ = c : c?. 9. a2 -f a6 : c^ + c equals the ratio c : d, the four quantities «, h, c, d are said to be proportionals or in propor- tion. The proportion is written a : b :: c : d, or also a : h — c : d, and is read « is to ^ as c is to d. The terms a and d are called the extremes, h and c the means. In any proportion the product of the extremes equals the product of the means. Thus, from a : b = c : d we get - — -- ad = be, b d Inversely, \i ad — be, then a, b^ c, d are in proportion. If three quantities a, b, c are such that a \ b =■ b \ c^ so that J2 — ac, then b is said to be a mean proportional between a and c, while c is said to be a third proportional to a and b. If a \ b = c \ d, then a ± b : b = c ± d : d. In proof, we add ± 1 to each term ot v = -i- Then — 7— = — - — . b d b d Dividing the terms of the equation in which the plus sign holds by the terms of the equation in which the minus sign holds, we get a'\- b _c -\- d a — b^ c — d' Sec. 58] COLLEGE ALGEBRA, 6i Hence, ii a : b = c : d, then a -{- b : a — b — c~{-d:c — d,a, result said to be derived by composition and division from a : b = c : d. EXERCISES. 1. It a: b = b : c, then a : c = a^ : b'K 2. If a : b = c : d, then b : a = d : c, and a : c = b : d. 3. It a : b = c : d, and e -.f = g : h, then ae : bf = eg : dh. 4. Find the mean and third proportionals to 4 and 16; to a* and a^U*. It a '. b ■= c : d, prove that 5. a'c - ac' : bH - bd'' = {a - cf : (b - df. 6. pa^ -f- qh'^ : pd^ — qb'^ = pc^ + 9'^^ : pc^ — qd^, 1. b - d:a - c= \/l}' 4- d' : i^a'-^ + cK It a : b = b : c = c : d, prove that 8. a:b-\- d^ c^ -.c'd-i- d\ 9. a + 5c? : 2(^ - 3(^ = a^ _[_ 5^3 . 2a^ - Sb\ 10. id' + 6^ + c2)(62 -f c^ + (?2^ := (a6 -{- bc-h edf. 11. Find four proportionals the sum of whose squares is 530, the sum of the extremes being 23 and the sum of the means 13. 58. Variation. If two variable quantities A and B depend upon each other in such a way that when A is changed in a certain ratio B is changed in the same ratio, B is said to vary directly as A . For example^ a body moving at the uniform rate of 5 miles an hour will move 10 miles in 2 hours, 30 miles in 6 hours, etc., so that the distance varies as the time. Between the distance B reckoned in miles and the time A reckoned in hours, the following relation holds: B — 6 A, In the general case, if B varies directly as ^, then B = niA, where m is some constant number. If B varies directly as the reciprocal (or inverse) of A, then B Q is said to vary inversely as A^ and B = -j, where c is some con- stant number. 69. The Law of Boyle states that the volume of a given mass of any gas at a constant temperature varies inversely as the pressure. Thus if V is the volume when the pressure is F, the volume be- comes ^Fwhen the pressure is 3P, the volume becomes ^^Fwhen the pressure is —P. I 62 RATIO; PROPORTION; VARIATION. [Ch. V The Law of Gay-Lussac and Charles states that, if the pressure be constant, the volume of a mass of gas varies directly as the tem- perature measured in degrees Centigrade* above — 273° C. A gas therefore expands by -^^ of its volume at 0° for every 'increase in temperature of l"" Centigrade. Thus 273 volumes of air at 0° become 273 + ^ volumes at f Centigrade, and the latter become 273 + T volumes at 1 °. From the preceding law and the Law of Boyle, we would expect that, when the pressure P and the temperature 7'(in degrees above T — 273° C.) both vary, the volume Fof the gas would vary as ^, By actual experiment this statement is found to be true for mod- erate values of Tand P. It may however be derived from the two laws stated by means of the following algebraic theorem : 60. If A depends only on B and C, and if A varies as B iclien C is constant, and if A varies as G tvJien B is constant^ then A will vary as the product BC when B and G change simtiltaneously . Let the quantities A, B, Chave initially the values a, h, c and suppose that a change in B from Z> to ^ and a simtdtaneotis cliange in G from c to y together cause a change in A from a to a. We may, however, consider that the changes in B and G take place successively. First, if G remains at the constant value c while B . changes from h to /?, then A will change from a to some value «' intermediate to a and a. Applying the hypothesis for this case, we get a _ b a'^p' Next, let B remain at the constant value /?, which it just attained, while G now changes from c to y. Then A must complete its * The freezing-point of pure water is 33° Fahrenheit or 0° Centigrade; the boiling-point of water is 212° Fahrenheit or 100° Centigrade. Hence 180 de- grees Fahr. = 100 degrees Cent. Sec. 60] COLLEGE ALGEBRA. 63 change from the intermediate value a' to the final value a. Ap- plying the hypothesis for such a change, we get a y Multiplying together the two equations, we get a _ be a ~ py' Hence, if B and (7 change simultaneously, A varies as BC. Example. — Triangles with the same base are proportional to their alti- tudes, and triangles witli the same altitiide are proportional to their bases. Hence the areas of triangles vary as the products of their bases and altituiles. EXERCISES. 1. If X varies as y and if a; = 6 when y — 15, find x when y = 10. 2. If X and y each vary as z, then x -{- y and \/xy each vary as z. 3. If y varies as a -f- b, and a varies directly as x, and h varies inverse 1}^ as x"^, and if ?/ = 19 when ic = 2 or 3, find y in terms of x. 4. If X varies directly as y and inversely as z, and \i x — 2 when y — o and z — 4:, find y when x — 12 and 2 = 6. 5. When a body falls from rest its distance from the initial point varies as the square of the time it has been falling, and its velocity varies as the time. If a body falls 400 feet from rest in 5 seconds, how far does it fall in 10 seconds? How far in the tenth second? If the velocity at the end of two seconds is 64, what is the velocity at the end of 5 seconds ? At the end of 10 seconds ? 6. An amount of gas measures 100 cubic feet at 0°; find its volume at 10° Centigrade, the pressure remaining constant. 7. Given 500 cubic feet of air at 10° Centigrade, find its volume at — 10° C, the pressure being unchanged. 8. If the temperature of a gas is raised from 0° to 30° Centigrade, and the pressure increased .tenfold, what becomes of 500 cu. ft. of air ? 9. What is the increase in pressure of the air in an air-tight room, when the temperature is raised from 0° to 40° Centigrade? 10. Given that the area of a circle varies as the square of its radius, show that a circle of 5 inches radius equals the sum of two circles of radii 3 and 4 inches. CHAPTER VI. ARITHMETICAL, GEOMETRICAL, AND HARMONICAL PROGRESSIONS. 61. A series of quantities is said to be in arithmetical progres- sion when the difference between any term (after the first) and the preceding term is the same throughout the series. The following series are examples of arithmetical progressions:; 2, 4, 6, 8, . . ., 271, . . . -1, -3, ~5, -7, . . ., - (2;i-l), . . . a, a-{- d, a + 2d, a -\- 3d, . , ., a + {n — l)d, . . . In the first series the common difference is 2, in the second series it is — 2, in the third series it is d. The nth term in the first series is 2n, in the second series — {2n — 1), in the third series a-{- {n — l)d. In particular, for n = 4, the fourth terms in the respective series are 2-4 = 8, — (2-4 — 1) = — 7, « + (4 — 1)^^ z=z a -\- 3d. We may state the theorem : In an arithmetical progression tvith the first term a and the common difference d, the nth term is a -|- {n — \)d, 62. When three quantities are in arithmetical progression, the middle one is called the arithmetical mean of the other two. But, if a, Z>, c are in A. P.*, we have h — a = c — b = common difference. *A. P. is an abbreviation for arithmetical progression. Similarly, G. P. will be used as an abbreviation for geometrical progression, H. P. for har^ monical progression. 64 Sec. 63] COLLEGE ALGEBRA, 65 Heuce h = |(« + c), so that tlie arithmetical inean of any two quan- tities is half their sum. It is their '' average value." When any number of quantities are in arithmetical progression, the terms between the first and last are called arithmetical means between the first and last quantities. Thus 7, 10, 13, 16, 19 are in A. P. and 10, 13, 16 are the three arithmetical means between 7 and 19. 63. Sum of a7iy 7iumher of terms in arithjnetical progression. Let S denote the sum of n terms in A. P. of which a is the first term, I the last (viz., the ni\\) term, and d the common difference. By § 61, Z = rt + {n - l)d. We have S=a+{a + d) + {a+ 2d) -\- . . , J^ {I - 2d) + {I - d) + I Writing the series in reverse order, we get S=l+{l-d) + {l-2d) + ,.. + {a + 2d) + {a + d) + a. Adding the corresponding terms of the two series, we get 2S={a + l) + {a + l) + {a + l) + ,.. + {a + l)=n{a + l), [there being n terms each {a -\- 1). Hence '(1) 8=M<^ + 1). Here ^{a + I) is the average of the first and last terms of the series, also the average of the second term and the term preceding the last, etc. If 71 is odd, there is a middle term, the ^{n + l)st term of the series, whose value is seen to be a + {^n + 1) - 1 \d = !{« + a + {n- l)d\ = ^a + 0- Hence ^{a + I) is the average throughout the series. We derive a I 7 second proof that the sum of n terms is n 7" . 64. Example 1. Find the sum of 19 terms of the A. P. 1, 5, 9, . . . The 19th term Hs 1 + (19 - 1)4 = 73. The required sum is 19 . ^-~ = 19 . 37 - 703. 66 ARITHMETICAL PROGRESSION. [Ch. VI Example 2. The 11th term of an A. P. is 12, and the 19th term is 36; find the 40th term. The conditions give a-{-10d = 12, a+lSd= 36. Hence 8^^ =24, d = 3, whence a = — IS. Then the 40th term is a -f 396? = - 18 + 39 . 3 = 99. Example 3. Insert six arithmetical means between 8 and 29. We are to construct an A. P. of 8 terms such that the first term a is 8 and the eighth term a -{- Idis 29. Hence Id = 21, d = d. The required means \ are therefore a + d = 11, a ^ 2d = U, 17, 20, 23, 26. ' Example 4. How many terms of the A. P. 48, 40, 32, . . . must be taken i so that the sum may be 144? Let the number of terms be n. Since the common difference is — 8, the i nib. term is ? E 48 — S{n — 1). Hence 144 = /S = ^n{a -{- I) = ln{9^ - S{n - 1)]. Hence 144 = n{62 — 4/1), so that 7i2 — ISn -\-2Q~in - 4)(/i - 9) = 0. For 71 = 9, we get the A. P. 48, 40, 32, 24, 16, 8, 0, - 8, - 16, whose sum is 144. Since the last five terms have the sum zero, 7i = 4 is a suitable value. EXERCISES. 1. Find the 13th and 41st terms of the series 6, 12, 18, . . . 2. Find the 20th and 40th terms of the series — 5, — 3, — 1, . . . 3. Find the 10th and 60th terms of the series 1, 6, 11, . . . Find the sum of the following series : 4. 5, 11, 17, . . . to 30 terms. 5. 12, 9, 6, ... to 21 terms. 6. 2^, 4, 5|, ... to 37 terms. 7. a, Sa, oa, ... to a terms. 8. a, 0, — a, . . . to 4:a terms. 9. 13, 9, 5, ... to 100 terms. Find the common difference and the number of terms in the A. P. ; 10. The first term is 6, the last term 180, the sum 2790. 11. The sum is 72, the first term 27, the last term — 18. 12. The last term is — 32, the sum — 266, the first term - 6. 13. The first term is a, the last term ISa, the sum 49a. Find the A. P. in which 14. The 7th term is 1 and the 31st term is — 77. 15. The 12th term is 214 and the 41st term is 739. 16. The 54th term is — 125 and the 4th term is zero. How many terms must be taken of 17. The series 15, 12, 9, ... to make the sum 45? 18. The series 42, 39, 36, . . . to make the . um 315? Sec. 65] COLLEGE ALGEBRA. 67 19. The series 16, 15, 14, ... to make the sum 100? 20. The series - 10|-, - 9, - 7i to make the sum — 42? 21. Insert 7 arithmetical means between 269 and 295. 22. Insert 15 arithmetical means between 23 and 71. 23. Insert 8 arithmetical means between — 80 and — 50. 24. Insert 10 arithmetical means between 6^ — 5^ and Qy — 5aj. 25. Find 5 numbers in A. P. whose sum is 80 and the sum of whose I squares is 1640. I 26. Find three numbers in A.P. whose sum is 39 and product 2184. 27. The sum of the first n odd numbers is n\ 28. Find the sum of all the odd numbers between 200 and 400. 65. A series of quantities is said to be in geometrical progres- sion when the ratio of any term (after the first) to the preceding term is the same throughout the series. The following series are examples of geometrical progressions: 1, 2 , 4 , 8 , . . ., 2«-i , 2- . , ... q Q -1 1 (l\n-i (\\n-2 (a, ar, ar^, ar^, . . ,, ar''~'^, ar"" , ... In the first series the commoii ratio is 2, in the second series it \, in the third series it is r. The ^th term in the first series is -\ in the second series ^{\Y~^, in the third series ar""-^. In particular, for n — 4, the fourth terms in the respective series are h2^ = 8, 9{|-)^ = i, ar^. We may state the theorem: m ://?' a geometrical jjrogression with the first term a and the com- IB^ ratio r, the nth term is ar'^~'^. ^ 66. When three quantities are in geometi'ical progression, the middle one is called the geometrical mean of the other two. But, if «, h, c are in G.P., we have he ~ = Y = common ratio. a b Hence h'^ = ac, so that the geometrical mean of any two qxiantities is tlie square root of their product. It follows from §57 that the •geometrical mean of a and c is the mean proportional between a and c. 6S GEOMETRICAL PROGRESSION. [Ch. VI When any number of quantities are in geometrical progression, the terms between the first and last are called geometrical means between the first and last quantities. Thus 2, 10, 50, 250, 1250 are in G. P., and 10, 50, 250 are the three geometrical means between 2 and 1250. 67. Theorem. The sum of n terms of a geometrical progression whose first term is a and common ratio r is ^ Since the ^th term is ar'^~'^, we have 4?^ = a + ar + «r2 + . . . + ar""-^ = a(l + r + r^ + . . . + r''-^)A But formula (1) of § 28 gives r^ — 1 1 — r'* 14-r + r2 + . . . +r~- r — 1 1 — r' 68. As an example, the sum of 20 terms of the geometrical progression 1, |^, ^, . . . is S. 1 — /iASO The sum of 50 terms is 2 - (|)*^ the sum of 100 terms is 2 —{^y\ It appears that the sum of n terms 8^ is less than, 2, but differs from 2 by an amount which decreases as n increases. In fact, S^^ , S^ , 8,^ differ from 2 by (^y\ (i)^ (|)^ respectively. By taking n sufficiently large, 8^ will differ from 2 by as little as we please. In fact, for each increase of n by unity, the difference in question is divided by 2. The statement is made more evident by observing that {\Y = yV < TTT' so that /1\*"* 1 I \2 ) < 10^ ^ • ^^ • • • ^M^ - 1 zeros before 1). 1 Hence (|)*"* may be, made as small as we please by taking m large enough. We have established the following theorem: Sec. 69] COLLEGE ALGEBRA. 69 By taking a sufficiently large numier of terms, the sum (3) l + i + i + i + .-. can he made to differ from 2 hy as little as we please. The theorem may be iUustrated by the diagram : I i 1 ^ 1^ 1^ 1 ■ A line two inches long is divided into two equal parts; the second part is subdivided into two parts each ^ inch long; the second sub- part is divided into two parts each \ inch long, etc. Taking the first of each pair of parts and forming the sum, we get the expres- sion (3). After any number of steps, the sum is less than the length 2 inches of the whole line; but the difference from 2 inches, being the last piece of the line, tends towards zero as the operation is continued. 69. For the general series a, ar'^, ar^, . . ., the sum S,^ differs a ar^ from - — '■ — by the quantity . If r is a proper fraction, whether positive or negative, the numerical value of r^ decreases as n increases and may be made to approach zero as near as we please by sufficiently increasing the value of 71. We will denote by S^ the limit of the sum Sn of n terms when n increases without bound, and speak of the result as the sum to infinity. Hence, in a geo- metrical progression with the first term a and common ratio r nU' merically < 1, the sum to infinity is (4) S^ 1 - r Notice that, in a geometrical progression with r > 1, the result (4) no longer holds. For example, if a = 1, r = 2, the sum 1 + 2 + 4 + 8 + 16 + .. . + 2^-^ increases without bound when the number of terms n increases, whereas has the value — 1. 70 GEOMETRICAL PROGRESSION [Ch. VI Example 1. Find the geometrical progression whvose second term is — 6 and whose sum to infinity is 8. Let a be tiie fir^t term and r the common ratio. Then ar = - 6, ---^ =z 8. 1 - r By division, r(l — r) = — ~ , Solving this quadratic equation, r = - | or + |. The value r = | is impossible, since 8^ would then exceed 8. Hence r = — |, so that the series is 12, — 6, 3, . . . Example 2. Find the value of the recurring decimal .6 23 E. 6 23 23 23 . . . Aside from .6, it may be regarded as a G. P. with the ratio (x\j)'s 6 2^ . _^ , ^^ . ""lo'^wv ^102 ^ 10* "^ • • 7 _ 6 23 / 1 \ _ ^ , ^3 100 " To "^ lO'i 1 _ _L ) ~ 10 "^ W ' 99 \ lOV _ 6^ 23 _ 617 "■ 10 "^ 990 " 990* EXERCISES. 1 Sum 12 -[- 18 + 27 + . . . to 8 terms, also to n terms. 2 Sum 36 — 12 + 4 — ... to 7 terms, also to n terms. 3 Sum f^ 2 4- I — • • • to 71 terms also to infinity. 4 i uiu .5 -f 15 -f -045 -{-... ton terms, also to infinity. 6 Evaluate .023; 1.466; .142857; .052. G. Find the G. P. \utli 2d term — 4 and sum to infinity 9. 7. Find three numbers in G. P. such that their sum is 62, and the sum of their squares is 2604 8. If an odd number of quantities are in G. P., show that the first, the middle, and the last of them are also in G. P. 9. Insert between 4 and 18 two numbers such that the first three shall be in A. P. and the last three in G. P. Sec. 70] COLLEGE ALGEBRA. 71 10. Find three numbers in G. P. whose product is 1728 and the sum of whose products by twos is 624. 11. Insert four geometrical means between 5 and IGO ; between \ and 27. 12. Find a G. P. with 81 as the 5th term and 21 as the 2d term. 13. Find a G. P. the sum of whose first and second terms is 5 and such that every term is 3 times the sum to infinity of all the terms that follow it. 14. Sum to n terms x -\- a, x^ -\- 2a, x^ -j- 3a, . . . 15. Sum to 2n terras (t -|- 3, 'Sa — 6, 5a -{- 12, . . . 16. Find the ratio of two numbers whose arithmetical mean is double the geometrical mean. 17. Find a, b, c, given that their sum h 70, that a, b, c are in G. P., and that Aa, 5\ 4c are in A. P. 18. Explain the paradox of the race between the hare and the tortoise. The latter travels at the rate of 1 mile an hour and the former travels 2 miles an hour. The tortoise has a start of one hour. While the hare is covering the distance the tortoise travelled during the preceding period, the tortoise moves ahead a new distance, etc. Does the hare overtake the tortoise ? 70. Several quantities are said to be in harmonical progres- sion when their reciprocals are in arithmetical progression. Of three quantities in II. P., the middle one is called the harmonical mean of the other two. If a, b, c are in II. P., then -» t. - are in A. P., whence ci c d a~ c h ' 1) '~ a c* Hence the harmonical mean b ol a and c is 2ac I a -\- To insert 71 harmonical means between a and c, we construct t'.ie 11. P. of 71 -f- 2 terms of which a is the first and c the last term. For example, to insert 4 harmonical means between 1 an.d •I-, we observe that 1, 2, 3, 4, 5, 6 are in A. P. and therefore that 1^ h h h Ty i are in H. P. 72 HARMONICAL PROGRESSION. [Cii. VI EXERCISES. 1. The geometrical mean of any two quantities is also the geometrical mean between their arithmetical and harmonical means. 2. The conditions that a, b, c may be in A. P., G. P., or H. P. are i a — h '.!) — c '.'. a : a, a — b : b — c :: a : b, a — b : b — c :: a : c, respectively. 3. If a, b, c are in G. P., then a -{- b, 2b, b -{- c are in H. P. 4. If a, b, c are in H. P., then a, a — c, a — b are in H. P. 6. If x^, y\ 2' are in A. P., then y + 2, z -\- x, x -{- y are in H. P. 6. If a, b, c are in H. P., then 2a — b, b, 2c — b are in G P. 7. If a, b, c are in A. P. and b, c, d are in H. P., then a : b = c : d 8. If three distinct numbers a, b, c are in A. P., while a\ b^, c^ are in H P.^ then — —, 6, c are in G. P. 9. Insert four harmonical means between | and f^ ; between 2 and Y CHAPTER VII. COMPOUND INTEREST AND ANNUITIES. 71. Problem : To find the amount A of a given sum or principal P at compound interest for n years at the rate of R per cent a year. To simplify the formulae, set r = R/100. The amount of both principal and interest at the end of the first year will be P + Pr — P{1 + r). At the end of the second year, the amount will be {P(l + r) } (1 + r) = P(l + r)\ At the end of the third year, the amount will be P(l + r)^, etc. Hence at the end of n years, the amount will be A = P{1 + r)\ If the interest be compounded semi-annually, the amount at the end of n years is found, by a similar argument, to be If the interest be compounded quarterly, tKe amount will be Example. Find the amount of $1000 in 20 years at compound interest at 6 per cent a year, payable semi-annually. Since P = 1000, n = 20, r = .06, the amount is A^ = 1000(1. 03)«. By the four-place logarithmic table, pp. 24, 25, we have log 1.03 = 0.0128. .-. log A^ = log 1000 + 40 log 1.03 = 3.5120. .-.^2 = 3251, 73 74 COMPOUND INTEREST AND ANNUITIES. [Ch. YII the last figure being wholly unreliable. With a seven-place table, log 1.03 = 0.0128372, and log A^ = 3.5134880, whence A^ = 3262.03. 72. Problem: To find the present value P of a sum A which is to be paid at the end of n years, allowing compound interest at R per cent. Set r = R/IOO. The amount of P at the end of n years must equal A. Hence P-A{1 + r)-^ 73. A fixed sum of money paid annually is called an annuity. The subject finds immediate application in Insurance. The sim- plest problem in annuities is the following : To find the amount A of an annuity a allowed to accumulate for n years at compound interest. The first annual installment a is due at the end of the first year and hence draws interest for n — 1 years; its amount will be a(l + ^)"~^. The second annual installment a will amount to a{l + rY~^, etc. The last installment a draws no interest. Hence, the amount of the n installments will be ^ = •« + « (1 + r) + «(1 + rf + . . . -f a(l + r)"-2+ a{l + rY'^ by the formula for the sum of geometrical progression of ratio 1 + r. 74. Problem : To find the present value of an annuity a to continue n years, allowing compound interest. By § 72, the present value of a due one year hence is a{l-\-r)~'^] the present value of a due two years hence is ^^(1 + ^*) ~^? • • • j ^^^^ present value of a due n years hence is a(l + r)T"". The total present value is a{l + r)-'-{-a{\+r)-^-\- . . . +«(l + r)-« Sec. 74] COLLEGE ALGEBRA. 75 Since the ratio (1 + r)~^ is less than unity, the geometrical progression may be summed to infinity, giving - . Hence the present value of a perpetuity (perpetual annuity) a is —, the rate of comDOund interest as above. Thus a perpetuity of $100 has the 100 .04 presen+^ value — r^ = 2500 dollars, if compound interest is reckoned at 4 per cent. EXERCISES. [The calculation of (1 + ^)" for r small and n large by means of our four- place table of logarithms is not very accurate. By a six-place table, log 1.02 = .008600, log 1.03 = .012837, log 1.04 = .017033, log 1.05 = .021189, log 1.06 = .025306, log 1.025 = .010724.] 1. Find the amount of $1000 in 30 years at 6 per cent compound interest, the interest being paid annually. 2. What sum of money will amount to $900 in 1^ years at 4 per cent com- pound interest, interest being compounded semi-annually ? 3. What is the present worth of $5000 due in 10 years, allowing compound interest at 5 per cent paid semi-annually ? 4. In how many years will a sum of money double itself at 5 per cent compound interest, due annually ? 5. Find the amount of an annuity of $1000 in 20 years, allowing com- pound interest at 3 per cent per annum. 6. What is the present value of an annuity of $500 for 30 years, allowing interest at 5 per cent compounded semi-annually ? 7. A person borrows $10000 at 5 per cent compound interest. How much must he pay in annual installments in order that the whole debt may be paid in 15 years? 8. What should I pay for a perpetuity of $500 to begin 10 years hence, if compound interest is reckoned at 3 per cent ? CHAPTER VIII. UNDETERMINED COEFFICIENTS; PARTIAL FRACTIONS. 76. A variable quantity is one which may assume different values (usually an unlimited number of values) in the same investi- gation. A constant quantity is one which retains the same value throughout the discussion. Thus in a given geometrical progression whose first term is a and common ratio is r, the Tith term 4 is rtr"~^ We consider a and r to be constants, while n is a variable ; by giving to n the values 1, 2, 3 4, . . . , in succession, we obtain for t^ the values a, ar, ar^, ar^, . . . , respectively. The area of a circle of radius E is ttEK To obtain different circles we let E vary; but tt is a constant number. A variable y is called a function of a variable x if to every value that may be assigned to x there corresponds a definite value of i/. It sometimes happens that certain isolated values may not be as- signed to X, since y then ceases to have a definite value (see § 100). In the preceding examples, t^ = ar"~Ms a function of 7i, the area ttE^ of the circle is a function of the radius E. Similarly a:*, 3x -\- 1, ^x^ log x are functions of x. It is often convenient to employ symbols which emphasize the functional dependence. Thus, for asmd r constant and 7i variable, we used /"* to denote the function «r" ~ ^ of w. Likewise, ^x and log X are functional symbols. 76 Sec. 76] COLLEGE ALGEBRA. 77 76. By formulae (8) and (9) of § 31, we have the identity (1) ax^ -{- hx -\- c~ a{x — a)(x — /?), if a and /3 denote the roots of the equation ax'^ -\- bx -\- c = 0. To generalize this result, consider the expression U^ ~ ax"" + bx""-' + cx""-^ + . . . +kx + l, called a rational integral function of x of degree 7i. If the ex- pression E^ vanishes when a is substituted for x, then, by the factor theorem, x — a is 'd factor of B^, The first term of the quotient Q obtained upon dividing B^hj x — a is clearly ax^~^. Hence F,= {x-a)Q, Q = ax^-'+... Similarly, if Q vanishes when ^ is written for x, then Q = {x-/3)Q\ Q'^ax^-'+,.. Hence F^= {x- a){x - /3)Q\ so that E^ vanishes when ^ is substituted for x. 7/"* there be n distinct valued a^ ^, , , , , v of x for which E^ vanishes, a continua- tion of the preceding argument shows that E^ has the 7i factors X — a, x — /3, . , , y X — V and a final numerical factor a, so that (2) E, = a{x - a){x - (3) . . . {x - v). Then a, /3, . , , , v are roots of the equation E^ = 0. 77. Theorem. A ratmial integral fu7icti07i of the nth degree in X cannot vanish for more than n values of x^ unless the coefficients of all the poivers of x are zero. Suppose that the function E^ vanishes ior n -\-l distinct values X and a, ft , . . , V oi x. Then E^ must have the form (2), and must moreover vanish for x = \. Hence a{X - a){X ^ /3) . , . {X - r) = 0. * The question of the existence of such values a, /jf, . . . , is considered in Chapter XIX. 78 UNDETERMINED COEFFICIENTS; PARTIAL FRACTIONS. [Cii. VIII Since the differences X — a, . . , , X — r all differ from zero, a must be zero. Hence B^ reduces to hx"" ~'^ -\- cx^"^-]- . . . Since this function of degree 7i — 1 vanishes for more than n — 1 values, b must be zero. Similarly, c = Oy . , . , k — 0, I — 0. Another statement of the theor-em is the following: A7i equation of degree n cannot have more than 7i roots unless all the coefficients are zero. 78. Suppose that the two functions of degree n are equal in value for more than n values of x. Then vanishes for more than n values of x. By the previous theorem, the coefficients of all the powers of x must be zero, whence Po = ^0 ' ]\^ 9l^ • • • ^ Pn-l = qn-l, Vn = 9^ If ttuo rational i)degral functions of the nth degree in x he eqnal in value for more than n values of x, the coefficients of like potuers of x are equal. In particular, if two rational integral functions* of -x each of iinite degree are equal in value for all vahies of x, the coefficients of like powers of x may be equated to each other. 79. To illustrate the application of the preceding theorems to problems on integral functions, we consider certain examples. Example 1. Find the conditions on a, h, c, d in order that a^y^ + ^^^ -\- ex -\- d may be a perfect cube for all values of x. It must be the cube of an expression Ix -\- m oi the first degree, in which I and m are, as yet, undetermined coefficients. Set * aa? -\- hx^ -\- ex ■{- d~{lx -\- mf ~ Px^ + Sl'hnx'^ + dlm^x -\- m\ Since the first and third expressions are equal for all vahies of x, ' a = l^, b — SPm, c = Slm\ d = m^. Hence I — a^, m — d^. The second and third conditions then give h = ^a^dK c = 3a^c?l * The symbol E is employed in identities. Here the relation is an identity in x. Sec. 79] COLLEGE ALGEBRA. 79 Inversely, if h and c have these values, then ax^ -[- bx^ -\- ex -\- d = {a^x -j- d ^f. Example 2. Determine the value of k so that x^ 4- ^x^ ^ ^x -]- k shall be divisible hy x^ -{■ x -{- 1. The quotient must have the form x"^ -}- 7ix -f- k. Set x^ + ^x^ f 3.C + ^ = («2 j^ x-\- l){x^ -\-hx^ k). Expanding the right member and equating coefficients of like powers of x in the two expressions, we have the conditions -r 7i -}- 1, 4 = ^^ -f /i -I- 1, 'd --^h^ k, which are all satisfied if ^ — — 1, ^ = 4. » Example 3. Find the sum of the squares of the first n integers. Since the sum depends upon n, we assume that -I- 22 -{- . . . + (7^ - 1)2 + ?i2 = ^ + hn -f en' + dii^ + en^ -{- fn^ + . . ., wnere a, b, c, d, e, f, . . . are undetermined coefficients, each independent of n. Changing n into n -\- i, we get 124-224- . . .-^n'-j-{ni-lf = a+b{n+l)-i-c{n-\-iy-]-d{n-^lf^e{ni-iy+. . . Subtracting the previous identity, we get n^-\-2n^l~b + 2cn + c-\- Mn^ + Mn -^ d ^ Aen^ + 6en^ -{- ien + e -i- . . , Since this equation holds for every value of n, the coefficients of like powers of n may be equated. Hence e = and /, and all succeeding coefficients, must be zero. Also, dd ^ 1, 3^ + 2c = 2, (Z + c + 5 = 1, wheuce d = I, c — I, b = I. Hence 12 + 22 + . . . + (?l - 1)2 -f ?,2 = ^ 4_ 1^^ 4_ J^2 _|_ J^3^ If 71 — 1, there is a single term 1 in the series, so that 1 = ^ + J + i + *, or a = 0. Hence I2 + 22 + . . . + 712 z= ^n{n + l){2n + 1). EXERCISES. 1. When is a'^x^ + bx^ -\- ex' -\- dx ~\- p a perfect square? 2. Find the condition that x^ — Spx -f- 2q may have a factor of the form {x - c)2. 3. Find k so that there are solutions x, y, z not all zero of X -{- y — z = 0, 2x — y -{- dz — 0, x -\- ky -\- z = 0. 4. If ax^ + bx^ -}- ex -\- d is divisible by x'' + h, then rfd — be. Using undetermined coefficients, establish the sums 5. 1 + 2-^3 + ...+^==^ i^'in + 1). 6. 12 4- 32 4- 52 + ... -f {2n - If' = ln{^v? - 1). 8o UNDETERMINED COEFFICIENTS; PARTIAL FRACTIONS, [Ch. VII 7. 1^ + 2^ + 33 + . . . -f- 71^ = ln\n + 1)2. 8. 13 + 33 + 53 + . . . + (271 - \f = n\2n'' - 1). 9. 1* -h 2* + 3* + . . . + 71* = ^\n{n + l)(27i + l)(37i2 -^ 37i - 1). 10. 1-2 + 2-3 + 3.4 + . . . + n{n + 1) = M^i + 1)(^ + 2). 11. 1.2-3 4- 2.3.4 + 3.4.5 + . . . + n(n + l)(n +2) r=: ^n(7i+l)(7i+2)(;2+3)r 12. The sum of the cubes of the first n integers equals the square of their sum. PARTIAL FEACTIONS. I 80. It is desirable to be able to express a complex fraction as the sum of simpler fractions, called partial fractions. Thus /3^ 4-5:. _ 1 « \*^I 1 _ Q^ I 0V2 ~~ 1 „ ^ "I 1 — 3a; + 2a;2 1 — a; ' 1 ~ 2a;* To expand the complex fraction in the left member into a series of ascending powers of x, we may expand the partial fractions on the right and add the resulting series. Indeed, by the theory of geo- metrical progressions, if r be numerically less than unity, the fol- lowing expansion holds: = 1 + r + r2 + r^ + . . . 1 - r Hence, if 2x is numerically less than unity, i — X 3(1 + 2a; + 4a;2 + . . . + 2*^a;" + ...), l-2a; 4. /it* 81. To explain the general method employed to decompose into partial fractions a given complex fraction whose denominator is of higher degree in x than the numerator, consider the fraction in the left member of (3). We observe that the fractions and ^ ^ 1 — X I Sec. 82] COLLEGE ALGEBRA. 8l r — , when reduced to the denominator (1 — x){l — 2x)^ will 1 — tlx contribute to the formation of a complex fraction with the given denominator and with a numerator of the first degree, while no new fraction witli a denominator of the first degree will have this prop- i erty. Hence if our aim is to be attained, the decomposition must be of the form 4: — 5x ^1 ^ 1 -'dx + 'Zx^ 1 - a; ^ 1 - 2a;' in which the undetermined coefficients a and b are to be determined. If this be possible, we shall have 4t-bx = a{l - 2x) + b{i -x)^a + h - x{2a + b). In view of its origin, this relation is to be true for all values of x except 1 and ^. Then, by § 78, the relation must be true for all values of x, and the constant terms as well as the coefficients of x may be equated. Hence rt -f ^ = 4, 2a + b = 5, Hence « = 1, ^ ~ 3, giving the true result (3). 82. As a second example, we decompose into partial fractions 4:X^ +3X-1 {X- iy(x + 2y It is clear that fractions with the denominators a; -\-2,x — l,(x — 1)^ and with constants for numerators may contribute to the formation of a fraction with the denominator (x + 2){x — 1)^ and with a luinierator of degree at most two. There is nothing gained by employing also fractions of the forms mx -\- n rx -\- s {x^2){^^' (^^^Tp' since they are themselves expressible as sums of partial fractions, \{rn + n) \{2m - n) r s -^ r x-l "^ x-\-2 ' x-l'^{x-iy^' 82 UNDETERMINED COEFFICIENTS ; PARTIAL FRACTIONS, [Ch. YIHI respectively. Moreover, -, — — r-^ cannot be expressed as the sum* of fractions with denominators of the first degree. The decomposi- tion will therefore give partial fractions of simpler form than the given fraction only when 4a;2 -)- 3^ - 1 + ■ (x - 1)2(^+2) x + 2^ x-l^ {x - If for suitable values of the undetermined coefficients a, t, c. Then (4) 4a;2 + 3a; - 1 = a{x - If + h(x - l){x + 2) + c[x + 2). Equating the coefficients of x^, of x, and the constant terms, we get a + Z> = 4, - 2a + /^ + c = 3, a — 2h+2c= - 1. I Solving by determinants or otherwise, we get « = 1, Z> = 3, c = 2, A simpler method of determining a, Z>, c is to substitute special values for x in (4). We observe that, in view of § 78, relation (4) is an identity and is there fore, true for the values x=\, — 2, etc. For X — \, we get G = 3c ; for a; = — 2, 9 = Ort ; for x — 0, — l = a — U + 2c. Hence 6? = 2, a = l, b = 3. 83. The presence of imaginary numbers offers no difficulty. To decompose the following fraction, we set 42 — 19a; ^ J- ^ • ^ {x-4:){x^ + l) x-4: x^\/'-\ x-\/'-\ .-. 42-19.^ = a{x^-^\)-^l{x-^){x-\/~^)-^c{x-^)(pc^\^~^). Equating coefficients, we obtain the values of a, h, c given below. With the explanation made at the end of § 82, we may simplify the work. For a; = 4, we get — 34 = 11a, whence « = — 2. For x -/ — 1, we get 42 - 194/ - 1 n= c(4/ - 1 - 4) (2|/ - 1), c=l-\- VV - 1. Sec. 84] COLLEGE ALGEBRA. 83 For X = — \/ — I, we get h — 1 — ^^-\/ — 1. Hence 42-192: -2,1- -VV - 1 , 1 + VV - 1 (^ _ 4)(:^^ + 1) x-4. ' x + \/ -l x-\/ -I The second and third partial fractions are conjngates, since one is derived from the other by replacing 4/ — 1 by — |/ — 1. Hence 2x — 11 their sum must be real (§ 4). In fact, their sum is \ . We might have avoided the introduction of imaginaries by setting 42 - 19.T a , ex-^f 42 - 192: = a{x^ + 1) + {ex+f){x - 4). Observing that this relation is an identity by § 78, we may set 4, whence a = — 2. Equating the coefficients of x^ and of x, we get 6 = 2, - 19 == - 4^ +/, whence /= - 11. 84. If the degree of the numerator of the given fraction equals ar exceeds the degree of the denominator, w^e first divide the nu- merator by the denominator and obtain a remainder of degree less than the degree of the denominator. For example, 122:^ + 102:2 _ 14 iQ:c - 8 = 42: + 6-' 3^C2 _ 9;^ _ 1 ' ' 3^2 _ 22: — 1 ' Proceeding as usual, the second fraction may be decomposed into IO2: 2 'dx -\- 1 2: — 1 The division mentioned may be avoided by assuming that 122:3 + 102:'^- 14 , , , ^^^ , ^ o2r - 2x — 1 ' ' 32: + 1 2: — 1 ind determining «, h, c by the usual method. 84 UNDETERMINED COEFFICIENTS; PARTIAL FRACTIONS, [Ch.VIII EXERCISES. Resolve into partial fractions: 7aj- 1 a? -V ar' -f 15aj + 8 (X- x" - - ^){x + 1) - 3aj 4- 6 aj*+ aj2-f- 2 * a;(aj - l)^ °- (05 -f b){x' + 1)* 10 - ■ - ' - 11 ^^' - ^ + 1 12 aj* + 1 ' • (a; - If • ■'^- (ar* - l){x + If 13. Expand into series the fractions of Exs. 1, 3, 5, 7, 9. 1 - 5a; 4- 6a;2* ^x" -x-{-% . a^4-« • 2a^-3 x'-^l' 5aj3 4- 6a;2 + hx i CHAPTER IX. PERMUTATIONS AND COMBINATIONS; BINOMIAL AND MULTI- NOMIAL THEOREMS FOR POSITIVE INTEGRAL INDEX. 85. Three letters a, l^ c may be arranged in six ways ahc^ acl, lac^ tea, cab^ cba, Jhile there are only three groups (or selections) each of two letters isen from three letters a, Z>, Cy viz., ab^ ac, be, we obtain six arrangements of three letters two at a time, namely, ab, ba, ac, ca, be, cb. Definitions. Each of the arrangements which can be made with r things chosen from n things is called a permutation of the n things r at a time. Each of the groups (selections) which can be made by selecting r things from 7i things is called a combination of the n things r at a time. Thus there are six permutations of a, b, c two at a time : ab, ba, ac, ca, be, cb. There are three combinations of a, b, c two at a time : ab, ae, be. In a combination, the order in which the letters are written is indifferent; in a permutation, the order is essential. Thus the pairs ab and ba give the same combination, but give distinct per- mutations. - _ P^ S6 PERMUTATIONS AND COMBINATIONS, [Cn. IX 86. The 7mml)er of jjermtitalioiis of n different tilings r at a time is n{n — l){n — 2) . . . (/^ — r + 1).* AVe are to find the number of ways in which we can iill r places when we have n different tilings at our disposal. For the first place we may take any one of the n things; for the second place any one of the remaining n — 1 things; for the third place any one of the now remaining n — 2 things, etc. Hence the r places may be filled in n{7i — l){?i — 2) . . . {71 — r -j- 1) ways, since the rth factor is 71 — (r — 1). Corollary. The number of permutations of 71 different things taken all at a time is 71(71 — l)(?z — 2) . . . 3 •2-1. This product of all the natural numbers up to and including 71 will be denoted by the symbol 7i\ which is read ''71 factorial." f For example, there are 3! = G permutations of a, I), c three all a time (given in §85) and 3-2 = permutations of a^ Z>, c two alj a time (given in § 85). 87. We notice that each of the three combinations al), ac, be o\ the letters a^ b, c taken two at a time furnishes exactly two permid tations of a, b, c two at a time; thus the combination V/Z> furnishes the two permutations ab and ba. Moreover, the six resulting per- mutations give alt the permutations of «, b, c two at a time. Simi- lar remarks hold true in the general case next considered. Let „6r denote the number of combinations of 71 different thing} r at a time. Let ^P^ denote the number of permutations of 7 different things r at a time. AVe have shown that ^F^=n{n-l){n-2). . . (m - r + 1). Each one of the ^C^ combinations consists of a set of r differen things, which may therefore be arranged or permuted in exactly rPr = t\ distinct ways. Every such arrangement is a permutatioi of the 71 things taken r at a time. By starting with a suitabh * The number is zero if r > n. The product then has a factor 71 — n = f The symbol I n is also used. Sec. 88] COLLEGE ALGEBRA. 87 combiiuitioii of r of the things and arranging- tliem in a suitable way, we may rer.ch any given pei'mutation of the n things r at a time. Hence „(7, X r\ = „P, = n{n - 1) , , , (n - r + 1). . r - n{n-l){n- 2). ^jjn^^ + 1) • • "^ 1-2-3 . . . r Upon multiplying the numerator and the denominator by {7i — r){n — r — 1) . . . 3 • 2 • 1 = (m — r) ! , we get * C - '''' 88. The nnmher of comMnatioiis of n different things r at a time eqiials the nnmher of comhinations of n tilings n — r at a time. When a set of r things is selected from 01 things, there is left a set of n — r things. Moreover, any given set of n — r things may be left (negative selection) by making a suitable selection of r th ings. Hen ce „ C,. ~ ^ C,, _ ,. . The proposition also follows from the formula for ,,(7^.. Thus ri '_:_: : _ "• _ p " -^' - (;, _ r) ! [;/ - {n - r)] ! {n - r) ! r\ ~ "" '" n^n 89. Theorem. For positive integers n mid- r, The combinations of n + 1 different letters a^^ a^, • • • ? «w> ^^i + i ^' ^t a time may be separated into two sets, according as the combination contains the last letter <7,, + i or does not. If the last letter be taken, there remain only r — 1 letters to be selected from *For r = 71, the fir.it result gives nCn — — = 1, w^liile the second gives n\ ,iCa — — ~. The second formula therefore holds for r = n only when we adopt the notation 0! = 1. Similarly, the result in §88 holds for r = n only when we put uCq ~ 1. 0^0 >c„ A. A A. A. A A. ,0, A, Ao fi. A A, 88 PERMUTATIONS AND COMBINATIONS. [Ch. IX the n letters «j, a^, , , ., a^^ which can be done in nCr-i ways. If the last letter be not taken, we must select r letters from the n letters «j , rt^, . . ., a„ , which can be done in ^0^ ways. The sum of the two numbers n^r-i ^^^ n^r must equal n + i^^r- The formula remains true for r = 1, if we take ^G^ = 1, there being one way of selecting no objects from n objects. An interesting application is the construction of Pascal's Triangle: 11 --^ 1 2^ 1 1 3 3 1 ,C, 14 6 4 1 The above formula now shows that any number n + i^^r equals the sum of the number ^0^ just above it ancj the number „(7,._i to the left of „(7^. For example, ^C^ = ^C, + ,C\, or 6 == 3 + 3. The numbers in the next row of the table on the right would there- fore be 1 5 10 10 5 1. It appears that the numbers in the oith row of the table are the binomial coefficients in the expansion of (1 + ^Y [§ 91]. 90. We proceed to determine the number of permutations of n things taken all at a time, when the things are not all different. There are only three permutations of a, a, h taken three at a time, namely, aah, aba, laa. If we replace the two a's by two distinct letters a^ and a^ , the first permutation aal) will furnish two permu- tations a^ajb and a^afi. In this way, we reach the 3-2 = 6 permu- tations ot a^y a^, h. In general, let there be n letters, p of which are a^s, q of which are <5>'s, and r of which are c's, so that n =p -^ q -\-r. Let P denote the required number of permutations of these n letters n at a time. Consider any one of the permutations as aa . , , abb . . . bcc . . . c. Sec. 90] COLLEGE ALGEBRA. 89 If we replace the p letters a by 2^ distinct letters «i , ^2 , . . . , a^, we may derive from the given permutation jr?! permutations a^a^ . . . apb , . .he , . , Cy a^a^ . . . aj) . . , he , , . c^ etc. Hence from the P permutations of the «'s, ^'s, c's, we derive P ,p\ permutations oi a^, a^., . . . ^ a^, h, . , , , h, c^ . . , , c. Similarly, if in one of the new permutations we replace the q letters bhj q dis- tinct letters h^^ h^, . . . ^ h^, we may obtain exactly q\ permutations a^a^ . . . aphj)^ , . .h^c . . . c, a^a^ . . . a^h^h^ . . .h^c . . .c, etc. Hence the P . p\ permutations of a^, a^, , . . , ap, h, . , , ^h^ c^ , , , ,c give rise to exactly P , p\ q\ permutations of a^, a^, , , , , ap, J^, . . . , hq, c, . . . , c. Finally, if the r letters c be replaced by r distinct letters Cj, c^, . . . , c^, the resulting number of permutations on the p -\- q -\- r^n distinct letters will be P,p\q\r\ But the num- ber of permutations of n distinct letters n at a, time is nl Hence P.p\qlrl = n\ P= .^; . plqlrl By a similar proof, the number of permutations of 71 things n at a time, p of which are alike, q alike, r alike, , , , , t alike, is seen to be p\q\r\...t\ ('»=i' + ?+'- + ---+0- Eor example, the number of permutations of all the letters of the 11! word Mississippi is ^^ ^ ^ — 34650. EXERCISES. 1. In how many different ways can five boys stand in a row ? 2. How many numbers of six digits can be formed by using the numbers 1, 2, 3, 4, 5, 6 ? How many numbers of four digits ? 3. How many numbers of three digits (the first not zero) can be formed with 0, 1, 2, 8, 4, 5 ? How many of five digits ? 90 BINOMIAL THEOREM, [Ch. IX 4. Find the number of permutations of all the letters of the word animal ; of the word relative. In how many permutations do the vowels and conso- nants alternate ? How many of the latter end with a vowel ? 5. How many numbers less than 1000 can be made with the digits 0, 1, 2, 3, 4, 5, 6 ? How many with the digits 1, 2, 8, 4, 5, 6, 7 ? 6. In how many ways can seven boys form a ring ? a row ? 7. How many sums of money can be made with 4 pennies, 3 dimes, 2 quar- ters, and 1 dollar ? 8. Prove that n + ^Gr-^l = nCr + 1 + ^nGr-{-nGr-l. 9. If„C5 = „(7io, find^Cg. 10. In how many ways can 4 men and 2 boys be chosen from 12 men and 5 boys ? - '• ^ C>C } 11. In how many ways can six gentlemen and six ladies be arranged at a round table so that no two gentlemen are adjacent ? 12. How many triangles can be formed by joining three of the vertices of a general octagon ? 13. Prove that nCr + \ = n-lGr + n-^Gr + n~^Gr-\- . . . -\-rGr. 14. By considering the combinations of s letters some of which are chosen from rtj, «2J • • • > ^i ^ _|_ X > Hence „(7,, = ,, (7^ _i according as = 1, namely, according as r —^(pi-\-l), since the inequality sign must be reversed when the signs of the two members are changed. Hence the number ,,(7^ increases as r increases, so long as r is less than ^{n -\- 1), but de- creases as r increases when r > ^{71 -\- 1). The case r — ^{71 + 1) occurs only when n is odd; then ?• — 1 = ^ — r, so that ^0^ = n^r-u in agreement with the result of § 88. Hence, if 71 is odd, the coef- ficients n(^hoi + i) ^^^^ n^h(n-i) ^^'^ cqual and are greater than the remaining coefficients; if 71 is even, the coefficient ^C^^ is the great- est coefficient. 92 MULTINOMIAL THEOREM, [Ch. IX Observe that the number of terms in the expansion of {a + ly is n-^-l, and that, for n even, „Q„ is the coefficient of the ^7i + 1st term, which is then the middle term ; while for 7i odd, there is no middle term, but there are two terms, the ^{n + l)st and the \{n + 3)rd, at the middle of the expansion, their coefficients being n^i(»-i) 9,nd n^i(n + i)' ^c may state the result: In the expansion of {a + Vf, the greatest coefficient is the ..licldle term if n he even ; while for n odd the middle pair of terms have the greatest coefficients, EXERCISES. 1. Find the greatest coefficient in the expansion of (a -f Vf, 2. Find the greatest coefficients in the expansion of {a -j- Hf. 3. Two coefficients „Cr and nCs are equal only when r-\-8 = n or r = 8. 4. If the 10th and 12th coefficients are equal, find the 4th. 6. If 2«C3 = 24nC4, find 71. 6. The sum of the coefficients in the expansion of {a -f &)^ is 2". 7. The sum of the coefficients of the odd terms of a binomial expansion equals the sum of the coefficients of the even terms. 8. If there be a middle term, its coefficient will be emn. 9. nCi+2nC2 + 3nCs+. • • -^ r nCr +. . . + 7i n Gt = /i 2"-l. 10. nCx -2n(7-, + 3n6^3- . . . + ( — l)"-l^i nO. = 0. 11. nCo + inO« + Jn(74 + i„a+... - ^. {2n) ! 12. uOq nOi -j" '^^i nOi -f- n^a n^3 -}-... -{-nOn—1 nOn — (w 4- 1) ! (71 - 1) !* 13. nCx - i„(7u + inC3 - . . . + ( - 1)^-^1 n(7n = i + 4+ J + . . . + ^- 14. nCo« + nO.' + n^.' + . . . + nCn' = (2^) ! - (t^ ! )K MULTINOMIAL THEOEEM. 93. Let 7^ be a positive integer and consider the product (a+Z> '\-c-{-d){a-{-h-\-c-\-d) . . . {a+I)-\-c+d)^ [n factors]. Every term of the product is of the form each exponent a^ /3, y, S being a positive integer or zero. But various terms will be equal and may be added. For a, p, y, 6 Sec. 93] COLLEGE ALGEBRA. 93 given, the terms equal to a'b^c^d may all be obtained by selecting the letter a from a parentheses, h from (3 parentheses, c from y parentheses, and d from the remaining 6 parentheses, the selection to be made in every possible way. By § 90, this selection may be dene in exactly n\ a\ fi\y\d\ ways. Hence the general term in the expansion of (rt+J+c-j-^)"^ is n ! a \ fi \ y \ d -^a%^c''d\ Example 1. Find the coefficient of a^h'^c'^ in (^ i X f = 2"f6'' since the probability of throwing an ace at one throw is ^, and the probability of not throwing an ace is f . Example (Poincare). From two urns of like exterior, the first contain- \ ing 4 white and 6 black balls, the second containing 3 white and 2 black balls^ one ball is drawn at random. Find the probability P that a white ball is I drawn. Since the total number of balls in the two urns is 15 and the number of; white balls is 7, one might be led to say that the probability P of drawing a white ball is 7/15. But this solution is incorrect as it assumes that the events are equally probable. Indeed, the x)robability of drawing a particular ball I *We may state the condition that ^ and B are independent events in the following form. The probability that A will happen, the probability that A \ will happen if P happens, and the probability that A will happen if B does not i happen shall all three be equal. See Poincare, Calcul des Probabilites, Paris, 1896. Sec. 96] COLLEGE ALGEBRA, 97 from the first urn is 1/10, while the probability of drawing a given ball from ;the second urn is 1/5. The correct solution is as follows. As each urn is equally likely to be selected, the probability that the first urn will be selected is \, the probability that the second urn will be selected is |. If a ball is taken from the first urn, ithe probability that it will be a white ball is -^q. Applying the theorem which relates to two independent events, we find that the probability that ithe first urn will be selected and that a white ball will be taken from it is = Y%. Analogously, the probability that the second urn will be selected "and that a white ball will be taken from it is J . | = }^. The required prob- ability P is therefore f^ -j- ^^^ = ^. EXERCISES. 1. From a bag containing 4 white and 8 black balls, 3 balls are drawn. •Find the chance (1) that all are white ; (2) that 2 are white and 1 is black. 2. A has 3 shares in a lottery in which there are 3 prizes and 15 blanks ; B has 2 shares in a lottery in which there are 3 prizes and 9 blanks. Com- pare their chances of drawing at least one prize. 3. If 6 coins are tossed, find the chance for 3 heads and 3 tails. 4. If 3 Latin, 5 Greek, and 6 French books are placed on a shelf at random, find the chance that the books of each language will be together. 5. The letters of the word probability are placed at random in a line. Find the chance that two vowels will come together. 6. Find the chance for a sum 4 at one throw of two dice. 7. Find the chance for a sum 15 at one throw of three dice. 8. Find the chance for a sum of at least 6 at one throw of two dice. 9. In a lottery of 100 tickets, there are 5 prizes of |100, 10 of |50, and 10 of $20. Find the value of a ticket. 10. From a bag containing 5 twenty-dollar bills and 15 ten-dollar bills, a person is entitled to draw 2 bills. What is the value of his expectation ? 11. Find the probability of throwing an ace at least once in three succes- sive throws of a single die. 12. From a bag containing 4 white, 5 black, and 6 red balls, 3 balls are drawn in succession, each being replaced prior to the next draw. Find the probability that (1) the balls are of different colors ; (2) the first is white, the second black, the third red ? 13. If two events are dependent (the occurrence of one being contingent upon the occurrence of the other), the probability that both will happen equals the probability of the first multiplied by the probability that when the first has happened the second will follow. Prove a similar theorem for any num- ber of events. 14. Solve Ex. 12 when the balls are not replaced after being drawn. 98 PROBABILITY {CHANCE). [Cii. : 15. If Pis the probability of -the happening of an event and therefor 1 — P the probability of its not happening, the probability that the event wil liappen exactly r times in n trials is nCr P'Xl-P)^-^ 16. If 6 coins are tossed, find the chance for a single head. 17. Find the chance of throwing exactly 3 aces in 6 throws of a die. 18. Find the chance of throwing at least 3 aces in 6 throws of a die. 19. The probability that A can solve a given problem is ~ and B's proba bility is f . Find the probability that both will solve it. 20. A number of 3 digits is formed at random from the 10 figures 0, 1 2, . . . , 9. Find the chance that the sum of the 3 digits is 25. 21. A's probability of being alive 20 years hence is J, B's probability is | Find the probability that both will be alive. 22. Two balls are drawn from a bag containing 3 red, 4 white, and 5 blac balls. What is the chance that both are of the same color ? 23. A has twice the skill that B has. What is the chance that A wins ' games before B wins 3 ? 24. What is the chance that in 5 numbers taken at random exactly tw^ begin with the digit 9 ? That at least two begin with 9 ? CHAPTER XL MATHEMATICAL INDUCTION. 97. By the method of snraming an arithmetical progression we find that the sum of the first ^z- odd numbers is '?^^. This result may also be proved by tlie following method. We observe that 1 + 3 = 2'^ 1 + 3 + 5 r= 32, 1 + 3 + 5 + 7 = 42. Let us suppose that we have continued this numerical verification of the theorem as far as the first m odd numbers, so that 1 4- 3 _j_ 5 4- . . . -f (2m - 1) = ?7^l Consider next the sum of the first m + 1 odd numbers. This sum may be derived from the preceding sum by adding the m -\- 1st odd number, which is 2m + 1. Adding it also to the second member m^, we get 1_}_34_5 + . . . + (2w-l) + (2m+l)=rm2+(2m+l) e (ni+lf. In the left member of this equation we have the sum of the first m-\-l odd numbers ; in the right member we have the square of m -\- 1. Hence the theorem holds also for n = ??i -f- 1 when it has 'been established for n = m. But we observed from the identity 1 -|- 3 = 2^ that the theorem holds for ?i = 2 ; hence b} our proof it holds for 7^ =: 2 + 1 = 3. Being, true for n = 3, the theorem holds for 7^ = 3 + 1 = 4. Being true for n = 4,' it holds for 71 = 4 -[- 1 = 5, «tc. It therefore follows that the theorem holds for any positive integer n. 99 loo MATHEMATICAL INDUCTION, [Ch. XI 98. This method of proof is called mathematical induction. There are two distinct parts in the proof as applied to establish a theorem true for every positive integral value of n. On the one hand, we have to show that, if the theorem holds for any particular value m of 7i, it will then hold for the next value m -\- 1 oi n. On the other hand, we have to show that the theorem is true for the initial values of n, say ^ = 1, or ^^ = 2, so that there may be a starting point (free from '^ifs^') for the induction process from m to m -\-\ (from '^ if true f or m " to '^ then true for m + 1 "). We must have a ladder by which to climb from any round (the mth) to the next round (the m + 1st) ; but the ladder must rest on a solid basis so that we 'can get on to the ladder (the ?^ = 1 or ^ = 2 rounds). To illustrate the necessity of both parts in the proof, consider the two following examples. Whatever be the value of the number c^ the equation 1 + 2 + 22 + 23 + ... + 2" = 2" + ^ + c will be true for ?i = m + 1 if true for n = m. Indeed, we have (l_^2 + 2-^ + ...+2"^) + 2"^ + i= (2^ + ^+c) + 2"* + izz:2^ + 2_^c, which is the original formula for ^ = m + 1. The ^^ ladder" is therefore perfect. But to be able to climb up the ladder to the general round (the ^th round), we must be able to get on the first round ; the formula must be true for n = 1, which requires 1 + 2 = 2^ + c, whence c = — 1. * Hence the only true formula is 1 + 2 + 2'-^ + 2'^+ . . . + 2^ = 2" + ^ - 1, I a result also proved by the formula for the sum of a geometrical progression. As the second example, consider the theorem due to Gauss that, for prime* numbers n, the circumference of a circle can * A positive integer n is called prime when it is divisible by no positive integers other than unity and n itself. Thus 2, 3, 5, 7, 11, ... , are primes. Sec. 99] COLLEGE ALGEBRAl ' loi be divided into ii equal parts by means of ruler and compass if, and only if, 71 has the form 7^ = 2^"" + 1. For m = 0, 1, 2, 3, 4, we find that n — 3, 5, 17, 257, 65537, respectively, and that each of these values of 7^ is a prime number. It was consequently supposed by Fermat that the expression 22*^+1 was a prime number for all positive integral values of m. Later, however, Euler proved that this supposition was false by showing that, for m = 5, the resulting number 2'^' + 1 = 2^2 + 1 == 4,294,967,297 is divisible by 641. 99. Example 1. Prove by mathematical induction that the sum of the cubes of the first n natural numbers equals \ln{n -\- 1) p. By trial, 1^ + 2^ == { ^2(3) } ^ Suppose, then, that the theorem holds for any particular value m of n, so that 1^: lH-2'+33-f-...+m3 = ||w(m + l)}2. Adding {m-\-lf to both members, we get 13 _|_ 23 4- 33 -f . . . -I- m^ + (w + If =. (7/1 -f 1)^ I (T^j + 1) + ^' [ = (^ + l)^{^^^^[={i(m4-l){- + 2)l^ \ Hence the theorem holds also for the value m -\-lof n. Being true for 71 = 2, the theorem therefore holds for n — Z, etc. By the two parts, the theorem is proved to hold for every n. Example 2. Show that x^ — y^ is divisible by a? — y if 71 is a positive I integer. I The theorem is evidently true for n — i and /i = 2. Since ajj.t — ym^x'^-^{x — y)-\- y{x^ ~ ^ — 2/"* ~ ^)» the theorem will be true for n =: m \i true for n — m—\. In fact, if a;"* - 1 — 2/"^ - 1 is divisible by oj — y, the identity shows that a;"* — y"^ will also ibe divisible by a? — y. Hence the theorem follows by induction. Example 3. Prove by induction the Binomial Theorem for all positive integral exponents n. By § 91, the theorem in question is given by the formula {a + hf' = c^a^ + c^a^-^b -f- c^a^-'^b"^ + .;. . -f Cra^-^br + . . . + <^r= ~ 1'^ _r ' . . . , tJn = t. lo? MATHEMATICAL INDUCTION. [Cii. X] The theorem is true for n = 1, since {a -\- h)^ — a -{- b ; also for tz r= 2, since {a + bf =: d-^ + 2ab + 6'^ r^ Co^^ -f c^ah -f Cg^'^. J Suppose that the theorem has I)een established for 7i — m, so that Multiplying both members by a -f h, we get (a + bf^ + 1 = ro^^" ^ ^ + (^0 + c^)a'^b + (c^ + c^'ia'^-^b^ + . . . But ^0 = 1, Cm= 1 , and Co + ^1 = ^ 4- 1, m{m — 1) _ 'm,{m + ^) Ci + ^2 = ^ + ■ 1-2 " 1-2 _ w(77i — 1) . . . (w — r --f- 2) ??^ (m — 1) . , . (7?^ — r -f 1 ) c^_i + Cr - 1^2^77(7^1) ^ 1-2. ..r m[m — 1) . . . (m — r + 2)[r + (?7J — r + 1)| ~ ~ 1 -2. ..(r - l)r ' (?y^ + l)7/^(m — 1) . .. (m — r -f 2) "^ 1 • 2 . . . (r^^ 1)7- .., (a-^^>)»n + l :=«ni + l + (m + 1)^^^ + '^^ ^ ^^^^ ^"^ " ''^' + ' ' * But this formula is the same as the initial formula for {a-\-by^ ior n = m-\-l. Hence the theorem is true for n = m-\ 1 if true for n = m. Also the theoreii is true for 7i = 1. Hence it is always true. In fact, being true for 7i = 1, ii is true for n — 2, and therefore for n =d, etc. Example 4. If n is a prime number, iV^"— iV^is divisible by n. Denote iV" — iV^by the functional symbol /(iV^), Then /(Jf+l) -f{M) = \(M-{- 1)« - (1/+ 1;} - {.¥« - M] 1 /w upon expanding {M-\- 1^ by the Binomial Theorem. The first and last terms are evidently divisible by n. Also is an integer, bcinLr a binomia" coefficient, and is divisible by n, since 2 does not divide tlie i rime n {n > 2, otherwise the term M'^-- does not occur). In geucral, the coefficienl ^'^~ 1 .V ' — ^^ -¥«-»* occurs only when n > r and is then Sp:c. 99J COLLEGE ALGEBRA. 103 mteger. Moreover, it is divisible by n since there is no factor in coninion witli n and thc3 denominator r ! In fact n is greater than r and hence cannot divide any factor of r ! ; while, inversely, no factor of r ! can divide the prime number n, .'. /(i/+ 1) ^f{M) -f a multiple of n. Hence, if / {M , is divisible by n, so is also/(J^f + 1). But/(1) = 0. Hence f{2) is divisible by n ; therefore also/(3), etc. This theorem U known as Fermat's Theorem. Incidentally it was shown that in the expansion of {a -f &)^ for n a prime number, all the binomial coef- ficients except the first and last are divisible by n. As illustrations of the theorem, we observe that 2^ - 2 =z 5 X 6, 8^ - 3 = 5 X 48, 4^^ - 4 = 5 X 12 X 17. EXERCISES. Prove by mathematical induction that 1. P 4- 2-' + 32 h . . . + ^' ^ ln{n + l)(2/i + 1). 1-2 "^2 3"^ 3.4 ■"^* • • '^ n{n i-l)~ n -\-l' 3. {a -^ b -\- c -{- dy^ — a^^ — b^ — c^ — d" is divisible by the prime n. 4. x^ — y^^ is divisible by ic -|- ^ when n is even 6. 2.4 -f 4.6 + 6-8 + ... +27^(272, + 2) =rz^-(2ri +2X2/1 + 4). ^ ^ , m ' m{m + 1) , m{7n + l)(m + 2) , ^ , . ^ . 7. 1 + J + ""i7^ + — 1 : 2 . 3 + . . . to 7i + 1 terms equal** 6. 2 7'^ + 3-o'» — 5 is divisible by 24. 1) m(?7^ + l)(m "~ "^ 1 •2-3 (m 4-l)(m + 2). . . (m + n) r^2 . . . w 8. l-2 + 2-3 + 3-4 + ...+ 7i(7i + l) =ri7^(7l + l)(^l + 2). 9. l-2-3 + 2-3-4 + ...+7i(/i + l)(?i + 2) = ln{n + 1 K^ + 2)(7i + 3)^ 10. 1=^+2=^+ . . . + ^^' = (1 + 2 + . . . + 71)'^. 11. (1^ + 25 + . . . + 7l5) + (17 4- 2^ + . . . + 71^) :=r 2(1 + 2 + . . . f '/^^ 12. cl' + 2^ + . . . + ?i.^) + 3(1^ + 2^ + . . . + 71"^) = 4(1 + 2 + . . . -f- /i)3. CHAPTER XIJ LIMITS ; TNDETEKMWATE FORMS. 100. Preliminary Remarks on Limits. — In testing the penetra- ting power of a bullet fired at a given distance with a particular rifle and uniform loads, we determine the limit to the thickness of the pine board through which the bullet will pass. In Geometry, the length of the circumference of a circle is obtained as the limit of the perimeters of the inscribed (also of the circumscribed) regular polygons as the number of sides increases without bound. Of all triangles having a given perimeter and a given base, the isosceles has greatest area, so that there is a limit to the area of such a tri- angle. The last two illustrations offer a contrast in one respect, the variable inscribed regular polygon does not reach its limit, the circumference; but in the last example the variable triangle reaches the limiting form of greatest area, the isosceles triangle. In the algebraic formulation, we consider a variable x and a lunction/(^) to which there is a definite value for each value that X may take (see definition of function, § 75). A Necessary Condition for a Limit. — In order that the func- i'ow f{x) shall have the limit I as x approaches a, where / and a are fixed quantities, it must be possible to make the difference between /(cc) and I as small as we please by taking x sufficiently near to a, [Complete definition of limit in §§ 103-104.] This condition is satisfied by the function 2^ -f 4 if a = ^, ^ = 10, since the difference between 2a; -j- 4 and 10 is proportional T04 Sec. 101] COLLEGE ALGEBRA, 105 to the difference between x and 3. In this case we may let x become 3, whence 22: + ^ becomes 10. The question is not so simple with the function ^ ^ ^ X ^ a which is not defined for x = a. Indeed, upon setting a; = «, we are led to the symbol ^ , which may be regarded to be equal to any number whatever if, indeed, it have a value at all (§ 102). At any rate, f{x) does not have a definite value for x = a and hence, by the definition of a function, is not defined for x = a. We may, however, show that f{x) approaches a limit as x approaches a. As long as X differs from a, f{x) equals x -{- a. Hence f{a + .01) = 2a+ .01, f{a + .001) = 2a 4- .001, f{a + .0001) = 2a + .0001, . . . By making the difference between x and a sufficiently small, but distinct from zero, the difference between f{x) and 2a may be made as small as we please. That this necessary condition is also a sufficient condition for the limit 2a is shown in § 105. 101. The Term Infinity.— The value of the function — for any X value of X except zero is readily computed. The operation division ceases to have a meaning when the divisor is zero ; there is no number q such that , q = 10, since, for every number q, . q is always ; 10 cannot be separated into parts each equal zero. But as X decreases through positive values, — increases and may X be made to exceed any assigned value (say 10,000) by taking x sufficiently small (viz., less than one- thousandth). Thus — • in- creases without bound; for brevity, it is said to ^^ become infinite," or to '^approach infinity^' (symbol, -[- 00 , or merely 00), terms ► lo6 LIMITS; INDETERMINATE FORMS. [Ch. XII which mean only *' increases without bound. '^ Similarly, as x increases through negative values, — decreases, and 2i^x approaches X zero, — decreases without bound ; it is said to become negative X infinity (symbol, — co ). A variable which does not increase with- out bound nor decrease without bound is said to remain finite. \0x 102. While tlie function — is not defined for x—0, division x by zero being excluded (§ 101), it has the limit 10 as x approaches 10a; zero. In fact, — =10 for every Xy^ 0. But if we consider the numerator and denominator separately, we have functions which approach as a; ^0^ and the quotient of their limits takes the form ^. But the same symbol would be obtained from the % ^x \()x functions — , --, , whereas their limits as a; li: are 1, 3, 10, XXX respectively. Hence the symbol ^, considered apart from its origin, does not possess a definite value and so is called an indeterminate form. lOx For every value of a; > the function — has the value 10, so that its limit as * i?; = oc is 10. If we consider the numer- ator and denominator separately, we have functions which increase without bound as x does, and the quotient of their limits has the form ^. But, as x r:^ co , the functions -, — , — - have the ^ XX X limits 1, 3, 10, respectively, while each may lead, in the sense ex- f)Uiined, to the symbol ^-. Hence ^o ^s called an indeterminate form, not possessing a definite value. * Eead "x increases without bound." x ^ — cc is read '\v dctreases without bound."' But a' = 1 is read **as x approaches 1." Sec. 102J COLLEGE ALGEBRA, 107 Writing the above functions in the forms x , —^ x , -, x , — , vC/ %4y JU and considering the limits of the separate factors as x^^, we are led to the symbol . go in each case. Similarly, as x = qo , we are X S^ 1 0^ led to the symbol 00 . in each case. But — , — , — have the re- spective limits 1, 3^ 10 as :i^ — or as a; == go . Hence tlie symbols . CO and co . 0, when considered apart from their origin, are Mide- terminate forms. As x~l, the two members of the identity 2 !_ __ 1 - x lead to the respective symbols go — oo and ^. Thus go — 00 is an indeterminate form, when considered apart from its origin. If, for X ~ a, di function of x assumes an indeterminate form, we determine its limit as ^ = «. If the given function \^f(x)/g{x), where fix) = and qix) = as a; = «, we seek -\A- •^ ^ x — a g{x) Example 1. As x ~ 1, (1 — ^)/{\ — ^^) leads to the symbol 0/0. But 1-x^ _ {i-x){l -\-x-\-x^ - \ -x^-\-ai^) l-a^~ [\ - x){\ -{- X ^Id^) ' •. limit 1 — x^ _^ Example 2. Find the limit as a? £= 00 of f.x) = ^x + \' which may be said to lead to the symbol 00 /oo . Dividing numerator and denominator by x^, we see that limit fix) ^ limit 4 - d/x 4- l/x'' a; zz 00 X :b 00 1 _ 5/aj _^ i/x' ^=4. io8 LIMITS; INDETERMINATE FORMS, [Ch. XII Example 3. The limit, as x approaches 3, of ^3^- 3 _ v/3 ( V^- l/3") «-3 ( |/a.+ |/3)( \/x- |/3) is |/3"/2 1^3 H i For a; = 3, the fraction leads to the symbol 0/0. EXERCISES. State which indeterminate form arises for the value of x given in each of the following problems. Find the limit of the function as x approaches the value given. . Vx-1. for x = l. 6. ^ "—^ for x V^-^ ' ^ - ^ 2 - V^x „ (x — d){7x + 1) . x^ -h^x 11. g-^^^ + l for ^ = 2. 12. i^I+5^_^J^El for .^. = - 1. X — 2 1-i-x 103. In giviog a complete definition of limit, which shall give not only a necessary condition for a limit, but the necessary and the sufficient conditions, we give in this section the definition of a limit as 7t = 00 , and in § 104 the definition of a limit as :c = «, where a is a given constant. The former case is the one used in the later chapters of the text. In giving a necessary condition for a Umit (§ 100), nothing was said of the effect arising by Jetting x approach a by different methods, for example, by passing through positive values only, negative values only, or rational values only, or eveii integral values only, etc. Thus the sum of an even number of terms of the geometrical progression 1, — 1, + I^ — 1> + l^--- is zero, the sum of an odd number of terms is + 1- As the number of Sec. 103] COLLEGE ALGEBRA, 109 terms is increased without bound, there is no limit approached. If n grows large through even values only, the necessary condition of § 100 for the limit zero is satisfied, whereas we do not admit that there exists a limit of the progression. In § 68 was discussed the sum and it was shown that the difference between S^ and 2 may be made as small as we please by choosing n sufficiently large. Thus to make ¥^^ ^ Too ' ^* ^"^^^^ ^^ *^^^ n>7] to make — -- < j^, it suf- fices to take n > 10. Then Sn is said to have the limit 2 as n in- creases indefinitely. Definition.* If the function /(w) differs from a constant I by an amount less than an assigned positive number e, however small, for all values of n which exceed an assignable number (whose value depends upon the value of e), then/(?^) is said to have the limit I as n increases indefinitely and we write limit f{?i) = L 71 ± CO If, in the preceding example, e be y^^, then 2 — Sn < e for n > 7. If e be yoVo' then 2 — Sn < e torn > 10. Consider the repeating decimal .3 = .333 . . . and set Sn^ .3333 ... (to n decimal places). For er^.OOl, ^-S.-^^-,^ < e. For e^.OOOOl, ^-6;= ^^oVtto <^. In general, for e = 10~% | — aS'^ < e f or every 71 ^ e. Hence limit Sn = i- 71 iz CO *For an infinite limit I = od , the definition would read : If/(«) exceeds an assigned positive number E, however great, for all values of n which exceed an assignable number, then limit f{n) = 00 . n ^ 00 no LIMITS; INDETERMINATE FORMS. [Ch. XII The limit of .6 e .666 ... is |, being twice the limit of .3, a result following from the theorem : T/', as 71 increases indefinitely^ f{n) lias the limit I, and if cis amj constant, then c » f{n) has the limit c . I. In fact, for an assigned positive number e', however small, /(/z) differs from I by an amount < e' for every value of n whicli exceeds an assignable number. Hence c . f\n) differs from c ,1 by an amount < ce' for those values of 7i. Taking e' = — , we conclude that, for an assigned positive number e, c .f{n) differs from c . Z by an amount < e for every value of n which exceeds aa assignable nuQiber, so that c ,f{n) has the limit c . I. 104. By way of introduction to the general definition below, a function having a limit and a function not having a limit as x approaches 3 are first given. The limit ot 2x -\- 4: as x approaches 3 is 10. In fact, 10 and 2x + 4 differ by an amount < e for every x between 3 — — and3 + -|. Let a function E{x) be defined for positive values as follows : E{x) = X, for X an integer; B{x) equals the integer just greater than X, foYX fractional or irrational. Hence as x increases from 2.1 up to 3, including 3, £J{x) remains equal to 3; but as x decreases from 4 toward 3, excluding 3, B(x) remains equal to 4. Thus E{x) does not have a limit as x approaches 3.* General definition of limit. If the function /(.t) differs from a constant I by an amount less than an assigned positive number e, *In a more restricted sense of limit, as x approaches 3 through vahies immediately less than d,.E{x) has the limit 3. Similarly, by §101, as x approaches through positive values, - has the limit + oo ; as ^c approaches 10 through negative values, — has tlie limit — oo. 3ec. 105] COLLEGE ALGEBRA. IM however small, for all values of x between a — 6 and a-\- 6 (except, perhaps, x — a), where c^ is a constant and d an assignable positive number depending upon e, then f(x) is said to have the limit Z as a; approach es a, and we write limit f{x) = I, x — a For example, tlie limit of the above function 2.^• + 4 as a; ap- proaches is 4, since the difference 2x is numerically < e for every r between — — and -f i^- The definition is to be modified if either a or / is infinite or if both are infinite. Thus, if/(:^") exceeds an assigned positive number E, however great, for all values of x between a — d and a-\-6, where «5 is a constant and d is an assignable positive number, then f{x) has the limit -j- go as :?; approaches a, and we write limit f{x) = 4" ^ - X i= a Similarly, the limit — co may occur. The case « = oo is treated in § 103. For example, limit 1 _ limit 4x — 3 _ limit 4 — 3/x _ X i=: '^2-'^^ n- =. CO 2x+ 1 ~ ^ = 00 2 -f 1/x ~ * 105. For all values of x different from a, the functions f{^)= ^.^,Zl ^ F{x)^x+a are equal. But /(a:) is not defined for x = a, since it then iassumes the indeterminate form ^. B\\tf{x) differs from 2a by less than e for all values of x between a — e and a -\- c. Hence limit /(2:) = 2« — limit F{x). X ± a X ± a In a similar manner, we may prove the general theorem : If tioo functions of x are always equal ivhenever each is defined and if each function has a limit as x^a^ the tivo limits are equal. 112 LIMITS; INDETERMINATE FORMS. [Ch. XII Suppose that two fLinctions f{x) and ^ (re) have the respective limits F and G 'd^ x approaches a. Then for an assigned positive number e/2, however small, the difference between /(:c) and t anc the difference between g{x) and G are ea^h < e/2 for all values of iij between a — S and a-\- 6, where d is an assignable positive number,^ Hence the difference between /(a:) -\- g{x) and i^+ 6^ is < e for al values of x between a — 6 and a -\- d, which proves the first of th( following theorems: The limit of the su7)i of two functio7is equals the sum of theii limits. Ihe limit of the difference equals the diff'erence of the limits. Denote by | 7i | the numerical value of 7i, whether n is positive oij negative. We may write I f{x)g{x) - FG\^\ f{x)g{x) -f{x)G\^\ f{x) G-FG\ ^mcQ. f[x) has the limit F, and g{x) the limit G, then I 1/(^)1-1^1 1, the absolute value of x"" increases without bound as n does, so that the series is divergent. If 1 2: | < 1, x"" has the limit C as n increases without bound (§ 69). We may show tliat limit a _ ^ n^ 00 1 — x To take a numerical case, let a: = |, so that 8n-= ^ — 3(f)^ To make 3 — S^ < e, it is necessary and sufficient to take log 3 — log € ^ ^ log 3 - log 2* There remain the cases x = ± 1. For x = -{- 1, S,^ = n and limit Sn = limit n = co , n :^ CO n = 00 * As explained before, | t I denotes the mimencal value of the real number t ; thus 1 + 2 I = 2, I — 2 I =^ 2. The symbol U | is read absolute value of t. Sec. 108] COLLEGE ALGEBRA. 115 For a; = — 1, Sn is 1 or according as n is odd or even, and hence the series 1 — l + l— l+---is divergent. Note that is now \. We have therefore proved the theorem : The infinite series \ -\- x -\- x^ -\- . . .is cojivergent if\x\ <1, and is divergent if \x\ ~ 1. An infinite arithmetical progression is always a divergent series. [f a be the first term and d the common difference, the ni\\ term ^„ is « + {n — l)d and the snm of the first n terms is (§ 63) 'Sn = ^n{a + tn) = in[2a + {n -l)d]. .', limit Syt =^ 00 . n = CO Hence the series is divergent. Since limit ^^ = qo , the preceding result may also be derived from the tlieorem : In a convergent series, the limit of the nth term, as n increases wit ho 21 1 hound, is zero. In proof,, we note that S^ and S^^^ both liave the limit S^ so .hat their difference 4 l^^s the.limit zero (§ 105). 108. Comparison Test. Of ttvo series witli all terms 'positive, ^'■i + «2 + «3 + • • • ^ ^'Z + «2' + ^'Z + • • • :» \et the first be convergent. If a\^ ~, a,^ ^ from an assigned value of n, omvards^ then the second series is coywergent. Let m be the assigned value of n, so that ^4 < ^« (for any n ~Z m), [f S\ denote the sum of the first n terms of a-^' -\- ci^' -{- • • • > ind S,, the sum of the first n terms of a^-[- a ^ -{-,.. , with the .imit 8, then ^'n ~ ^m ^ Sn — Sm < S — S^* Il6 CONVERGENCY AND DIVERGENCY OF SERIES, [Ch. XIH Hence S^ remains finite for every value of n. But 8' is a posi- tive variable which increases as n increases. Hence there exists number, say 8', toward which 8' approaches arbitrarily near bu does not pass. Thus 8'^ has the limit ;S''. Comparison Test. Of two series loitli all terms positive, | ^1 + ^2 + ^3 + • • • > ^i' + ^2' + «/ + • • • ^ let the first he divergent. If a^ y an, from an assigned value of n omvards, the second series is divergent. With the preceding notations, we have ' ' K" ^^m ^ ^n - ^m (any n = m).\ Since 8^ is always positive but does not approach a finite limit, it must increase without bound as n does. Hence limit >S^^ Tz limit 8^ -\- 8!^ — 8^ ~ ^ - n-^co n-:^ (JO 109. The simplest harmonical progression is 1, ^, ^, ^, , Since the wth term - approaches zero as 71 increases without bound, so that the individual terms are ultimately very small, it might be supposed that the series would have a finite sum and hence be con- vergent. However, by comparing the two series l+i+l + i + i + i + i + l+TV-f.-- +tV + . . i + i + i + } + i + f + l + i+ !+••• + iV + .- we observe that each term of the second is equal to or greater than the corresponding term of the first. If the first is divergent, the second is divergent (§ 108). But the first equals and is* divergent. To show that there is a boundless number of terms ^ in the last series, we observe that a term | was derived fromj each of the parenthesized groups of 1 + (i) + a+ i) + a+ i + ^ + 1) + (i+ • • • + iV) + • • • » 3ec. 110] COLLEGE ALGEBRA, II 7 1111 he final terms of the groups being ■^, k^^ ^^ ^y ' • ' y so ^^^.t the lumber of groups is boundless. Hence the series s a divergent series, 110. As a generalization of the preceding theorem, the series 2) 1+1+1 + 1+ +1 + ^/ ^p ~^ 2^ d^ 4:^ n^ s divergent if p 'Z 1 and convergent if p > 1. Let first ^ < 1 (including the case/> negative). Then — > — (for any positive integer n>l), 'ndeed, if both terms of the inequality be multiplied by n^ it )ecomes n^~^ > 1 and hence is true, since 1 — p is positive. Com- )aring the two series (1) and (2), we have shown that each term of 2), after the first, is greater than the corresponding term of (1), vhich is divergent. By § 108, series (2) is divergent. Let next jt? > 1. Comparing (2) with the series r) 1+1 + 1 + 1+1+1 + 1 + 1 + ..., 2^ 2^ 4^ 4^ 4^ 4^ 8^ lach term of (2) is equal to or less than tlie corresponding term of 2'), thus 3^ < ^. 5? < 4?' • • • I^^^^® ^^ (^') ^^ convergent, 2) will be convergent (§ 108). But (2') may be written 1 + 1+1 + 1+ -i^fi_l) 2 >eing an infinite geometrical series with the common ratio — , which 3 less than unity since jt? > 1. li8 COM/ERGENCr AND DiyERGENCY OF SERIES. [Ch. XII: 111. The series (2), including (1) as a special case, are stand ard series, by a comparison with which a given series may ol'tei be proved convergent or divergent, as the case may be. Example 1. Each term of the series 1 ^ 4 ^ 9 ^ 10 ^ ^ TiV ^ is greater than the corresponding term of series (1), since 71 + 1 1 , 1 1 n^ n n^ n Since (1) is divergent, the given series is divergent. Example 2. The series whose ;/th term is _ w + 3 n^ + 1 , ^ f , 71 + 3 _ 7i + 3/2 4 „ IS convergent, m tact, Un < — —t. — ., — — — . Hence each term c 7L'^ < n-^ < n^ the given series is less than the corresponding term of 4 4 4 j^2 + 2"2 "^~ • • • "^ ;^ + • • • ' which is convergent, being 4 times series (2) for j9 = 2. Example 3. For positive values of x and a, the series X X -}- a X -\- 2a x -\- 7ia X is divergent. Multiplying 2£ by a and setting y = -, we get Let the integer just greater than y be m. Then a:S -4- ^ 1 , , 1 m m -\- i m + 2 m -\- m the latter being only a part of the divergent series (1). EXERCISES. Test the series for convergency or divergency: 1 - -I- - + ... -I = f- . . . 2. - -4- — 4- . . . + -i- -f- . . . 1.2'^2.3"^ ^//(n-fl)"^*" ' 1.2.3~^2-3 4'^""^n(/i-f-l)(7i-f 2) ' Sec. 112] COLLEGE ALGEBRA, 1 19 fi -La- J-4--l-,_^ e ^ I ^ I ^ I 3-42 "^4.5*''^ 5-62 '^* * * 3'-'-42'^ 4^-52'^ 5^-6"^ "^ ** * 1^3^ 5^ •lP^2P^3i>^ ^ nP ^ 9. 1 7—r. 7T + / — T-TTT — i— k; -[-••• for a; and y positive. 10. 1 + 2 - 3 + 1 + 2 - 3 + . . . 112. An infiyiite series is convergent if the terms are alternately positive and negative, if each term is numerically less than the pre- ceding, and if the nth term has the limit zero as n± cc, 1 In the series u^ — u^ + 21^ — ii^ + . . . , let each tc be positive and let * ^ ^ . . limit ri 2l,> u,> zf,> u^> ,., , ^^^u^ = 0. By considering the sum of an even number of terms, I S,n = {^h - ^2) + (^s - '^4) + • • • + (^2n ^1 - W2J, we observe that ASgn is positive. Hence ^2n + 1 = ^2n "T ^2» + 1 is also positive. Writing S2n + 1 in the form we see that Szn + i < w^. Hence /S^2n < '^^- Since the quantities S^, S^, . , . 82^, ' ' ' ^1*6 increasing positive numbers, each < n^, Sc^ must have a finite limit I as n increases without bound. But limit (a q \ — li^^i^ ./ — o Hence /S^2n + i ^^so approaches the limit /. The series ti^ — 1/2 + • • • is therefore convergent. Example. The series 1— i + i — i + .-. is convergent. But 12^3 4^ * The student should make the proof for the example 1— ^ + i — i + ..» I20 COhlVERGENCY AND DIVERGENCY OF SERIES. [Ch. XIII is divergent (see end of § 107), since limit n-\-l = 1. For the latter series the third condition of the theorem is not satisfied, while the first and second conditions are satisfied. 113. In comparing a given series with a standard series, it is often convenient to remove a Ji?iite number of terms of the given series, the removal of which subtracts only a finite quantity and hence does not alter the convergency or the divergency of the given series. Thus in it is convenient to remove tlie first five terms, whose sum is finite. There remains the infinite series 5!i^ + 6 ' 5'r, .5 <5! [^ + ^©^(I)V-} The series in brackets is an infinite geometrical progression whose sum is g = 6. Hence the series (3) is convergent (§ 108). 114. Theorem. An infinite series all of loliose terms are positive is convergent if, after any particular tenn, the ratio of each term to the preceding is alioays less than some fixed quantity lohich is itself less than unity. The terms preceding the particular term in question may be removed without altering the convergency of the series, since their number is supposed to be finite. Let the remaining series be de- noted by u^ + n^ + Uz + tt,-T . . ., and let -^ < r, -' < r, -* < n ^ . ~ f r < 1). u, u, u^ Sec. 115] COLLEGE ALGEBRA. 121 Multiplying together the first two inequalities, the first three, etc., we find that — < r^, — * < r^, etc. Hence All the quantities being positive, we find by addition that ^^ + W'2 H- ^8 + «^4 + • • • < ^1(1 + r + r^ + r^.+ . , .), The infinite geometrical progression in parenthesis has the sum , since r < 1. Hence v^ -\- u^ -{-..» -{- itn is a positive 1 — r quantity which increases as 11 increases, but remains < 2_ ' -> ^^^ hence has a finite limit. But nothing follows in the case r = 1, since 1+^'+^^+ • • • H-r*"""' increases without bound as n does. Although u^-\- u^-\- , , . -\- ti^ is less than the former sum, it may have a finite limit or increase without bound. 115. Theorem. A series luitli positive and negative terms is convergent if the series derived from it ly making all the terms positive is convergent. Since the series with all terms positive has a finite sum ^, the sum Sr, of the first n terms of the given series lies between — ^ and + 2. Hence, if we show that S,^ approaches a limit as n increases indefinitely, this limit will be finite and the series conver- gent. Let Pp be the sum of the p positive terms in 8n, and Nq the sum of the q negative terms in Sn after their signs are made posi- tive. Then S, = Pp-Nq {p + q = n). Also Pp and N^ are each positive and less than ^. Hence Pp ap- proaches a fixed value P, and N^ a fixed value A^ as ^ increases without bound, so that either p or q increases or both increase without bound. Hence limit Sn = P — N— fixed finite number. W£r CO 122 CONVERGENCY AND DIVERGENCY OF SERIES, [Ch. XIII 116. In view of the preceding theorem, the result of § 114 leads to the more general theorem : Eatio Test. A series is convergent if, after any particular term^ the absolute value of the ratio of each term to the 'preceding is always less than some fixed quantity ivhich is less than unity. Thus, in the series (3) of § 113, the 7ith and n + 1st terms are 5n-l 5^ {n-l)V n\ The ratio of the latter to the former is 5/w. Hence, if 7^ > 5, the ratio is at most | and hence is always < \\, for example. Hence the series is convergent. Note that here also 5 terms were removed. For the series (1), the ratio of the ni\i term to the preceding term is — \ = = 1 . While this ratio is less than 71 n — 1 n n unity for every value of n, it is impossible to find a proper fraction /"such that 1 < fior every n. Indeed, as^ = Go, 1 il. To put the matter in another light, if one assigns to/ the value .9999, the ratio 1 will exceed / = .9999 as soon as n > 10000. ' n ^ Hence the theorem fails to prove the convergency of 1 -)- i + i + • . . By § 109, the series is divergent. Series (3) is a special case of the series The ratio of the {n + l)st term to the nth. term is x/71 and is, in absolute value, less than a quantity less than unity for all values of n greater than x. Hence, if X be the least positive integer equal to or greater than | a: | , we need only remove the first JT terms of the series, so that in the remaining series the ratio of each term to the Sec. 117] COLLEGE ALGEBRA. 123 preceding shall be in absolute value always less than a fixed quan- tity less than unity. Hence series (4) is convergent for every x. 117. An infinite series all of ivliose terms are of like sign is divergent if, after any particular term, the ratio of each term to the preceding term is alivays greater than or equal to unity. Remove the terms preceding the particular term in question and denote the remaining infinite series by «^i + ^^2 + ^^3+ • • • . ^2 ^^^^ ^^3 ^ «^2^ • • • . •. i^i + t<2 + t^3 + • • • + ^'n ^ ^^ + ^1 + '^1 + • • • ^ nu^. Since nu^ = cc as /i = 00, u^ + ^2 + ^^3 + • • • is divergent. For example, the ratio of the nth term to the {n — l)st term in 172 "I" 2^3" + ** •■^(^- \)n'^ n{n^l) "*" * * * \^ Z{n — 1) -T- {n -\- 1) and hence is greater than 1 for all values of n greater than 2. Hence the series is divergent. Ratio Test. An infi^iite series of^^ositive and negative terms is divergent if, after any particular term, the ratio of each term to the preceding term is numerically equal to or greater than a fixed quan- tity r > 1. Beginning with the term in question, let the series be ^1 + ^a + ^'3 + • • • > where, for every n^ U„ > Hence | '^^n + 1 1 c ^"^ t '^^i I ' ^^ ^^^^ limit 1 1^^ ^ ^ | = 00 . But a series is ^ n±co divergent unless the nth. term approaches zero as7uncreases (§ 107). 118. A more convenient form of the ratio tests of §§ 116, 117 is given in the equivalent theorem: 124 CONyERGENCY AND DIVERGENCY OF SERIES, [Ch. XIII ffin any series ii^ + ?^2 + ^s + • • • there exists * a limit y __ limit the series is convergent if I < 1, divergent if I > 1. Ifl = lya further test is necessary. As an example, consider the logarithmic series (Ch. XVI), /yt /yi /yiO /y«4 /ytfl — 1 /y»H 1 2 + 3 , 4 + • • • ^ ^^ n-1 ^ ^^ n '" The ratio of the ^^th term to the {71 — l)st term is n \ nJ Hence limit | r | = | ^ | , so that the series is convergent ii\x\ < 1 , but divergent if |a;| > 1. For x = -\- 1, the series is converg- ent by § 112. For x = ^ 1^ the series is the divergent series --1-i-i-... (see §109). EXERCISES. 1. —r + —7-^ ::: — h . • • IS converffent if a and b are positive. a a-{-b a-{-2b a-f-36 »• r2 - 2^ + 3^4 - A + • • ■ '^ convergent. For what values of x are the following series convergent ? ••|+i+? + ---+^ + -- 6.1 + 3. + 3a. + 4x3+... '•2+8+4+--- + f+3 + --- *• ^+4+---+-^ + --- 9. *+3+5 + y + .-- 10. l + 3+3, + 45 + --- /ys j>A 3.7 2* 3^ 4* "•*-|] + il-f!+--- ^2.1 + ^, + 3- + j-, + ... * This limit does not exist for the series 1 -{- 2x-{-x'^ -\-2a^ -^x* + 2x^ + . .. CHAPTER XIV. POWER SERIES; EXPANSIONS INTO SERIES. 119. An infinite series of the form (1) a^ + a^x + a^x^ + . . . + an-ix""-^ +«»^" + • • . 5 in which each a is independent of x, is called a power series in x^ since each term involves x merely in the form of a power. It may be employed in an investigation only when it is convergent. To apply the ratio test of §118, we investigate the limit of the xct ratio -^ — ^ as n increases without bound. If the limit exists, its absolute value equals \x\ -r- I, where limit 71 — 00 Noting when \x\ -r- Z is less than 1 or greater than 1, we find that The series (1) is convergent if \'^\ < I, divergent if \^\ > L For the case \x\ = /, a special investigation is necessary. Corollary. If a power series converges for x = a, it converges for every value of x numerically less than a, 120. Suppose that series (1) converges iov x = a^ a 4^ 0. Then iox X — OL the first of the following limits exists and is finite: limit / , 9 , , n\ limit , . , , „_|x ^ 3^ 00 iV+^2^^+ • • • -h«^n^") = ^ ^ ^ 00 («l+ V + • • • + ^n^"" ')> the equality being true by § 105. Hence the series a^ -\- a^x + . . . is convergent forx = a and therefore, by the preceding corollary, for every x numerically less than a. In particular, it has a finite 125 126 POIVER SERIES; EXPANSIONS INTO SERIES. [Ch. XIV value for a; = 0, so that the product x{a^ -\- a^x -\- , . .) vanishes for a; = 0. We may therefore state the theorem : If the series a^ + a^x + ci^x^ -\- , . . is convergent for a value of X different from zero, the series approaches the liniit a^ as x approaches zero. 121. Theorem. If the ttvo infinite series % + «l^ + «2^^ + • • • + "n^"" + . . . , b, + \x + \x^ + . . . + h^x^ + . . . are convergent for a; = o', where a is some quantity different from zero, and if the series are equal for every value of x such that r\^ «' L then a^ — h^, a^ = b^, , . . , a^ = bn, - » » , and the ttvo series are identical. By the preceding theorem, the series approach the respective limits a^ and b^ as x approaches 0. Hence a^ = b^. Then a^x + «2^^ + . . . , b^x -{- b^x^ + . . . are equal and are convergent for x ^a. Hence a^ + a^x + . . ., ^^j^h.^x + .. are equal and are convergent for < a: p a- . As before, these series approach the respective limits «5j and Z>i as x approaches 0. Hence a^ = b^ Proceeding similarly, a^ = b^, . . ..> a^ = b^, . . . The theorem is a generalization of that on rational integral functions, § 78. Hence, under certain conditions, we may equate the coefficients of like powers of x in two power series. This principle may be applied to the solution of problems involving infinite series with undetermined coefficients. 122. The expansion of a function into a power series is valid only when the resulting series is convergent. For example, 1 l-x' \ '\- x -\- x^ -\- x^ -\- , , . ad in^mfum, Sec. 132] COLLEGE ALGEBRA, 127 are equal ^i \x\ < 1, but not if | ^ | > 1. Thus, for x = 2, the' fraction equals — 1, while tlie series has an infinite sum. Example 1. Expand (1 — ic)/(l + ^^) ii^^o a power series. We seek a power series convergent for values. of x such that p I ^ a , and equal to the given fraction for those values of x. Employing undeter- mined coefficients, we set 1 — X — -— 2 = <^ + ^^ -h ^^'' + d^^ -\- ex* -{- . . , Multiplying both members by 1 + ^^, we have I - X = a -{- bx-i- {a-i- c)x^ + (5 + d)x^ + (c + e)x* + . • • for all values of x for which the series is convergent. Then a = 1, b - - 1, a + c = 0, b -\- d = 0, c -^ e = 0, . . . 1 — X .^ z:z 1 — X — Q.^ -\- x^ -^ X* -}- . , . (when convergent). I ...„....„..„,.,.. .„„._„„ the more direct derivation of the series by means of the expansion I - ;-^ = (i--)(r3^,-.)) = (i--)a--^ + ^^--«+---). holding for a;^ < 1, namely, for — 1 < x < -\- 1. Example 2. Expand \^1 — x into a power series. Assume 4/I — :c = a + bx -\- cx^ + dx^ -\- ex* -{- . . . For values of x for which the series is convergent after its terms are made positive, the square of the series may be found by the elementary rule for squaring multinomials. Hence, for such values of x, »1 -x = 0" + 2abx -{- (62 + 2ac)x' + {2ad + 2bc)x^ 4- • • • . •. a"^ = 1, 2ab = - 1, b"^ + 2ac = 0, 2ad -f 2^>c = 0, . . . For the positive square root, a — ^ 1. Then b = — ^, c= — I, d = — -^^^ I In the next chapter this series is shown to be convergent if j^l < 1. 5a;2 ^ 4^^ _|_ 3 1 -\- X -\- x^ -\- x^ By the method of partial fractions (Chapter VIII), we get Example 3. Expand ^ — r^;ri~:^\ — 3 ^^^^ ^ power series. 5^2 -f 4.^ -p 3 _ 2 3.^ 4- 1 ^ - (1 + x)(l -{-x^)~ l-\-x~^ l + x^' T28 POIVER SERIES; EXPANSIONS INTO SERIES. [Cn. XH Then, if x^ < 1, the following expansions are valid: ^ = 2|1 ^ a^ + a:^ - a!3+ ... + (- l)na;n + .. .}, j^ = (3a; + 1){1 -cc^ + a^ -... + (- 1)^3^2- 4- ... }. .•./=3 + aj + ic2_5a,3_^...^j2_f_(_i)n|a.2n_|_|_2 + 3(_i)n}a,2n + i_^... 123. The reversion of a series y = a^ + a^x + a^x^ + . . . consists in expressing a; as a series in ascending powers of y, X — 'b^-\-'h^y ^ l^y^ + • • • "The method is applied only to conver- gent series and the solution is valid only when the resulting series is convergent. If a^ = 0, then ?/ = if a; = (§120), so that h^ — 0. If a^ ^ ^^%Qi y — a^ — %\ after reversion, X ^ c^z Ar c^z^ + c^z^ + . . . = c^{y - a,) + c^{y - a^Y + . . . Example 1. Effect the reversion of y = x — ^x^ -{- M^ - 4a^ + . . . Assume x = ay -{- by"^ -\- cy^ -\- dy^ + • • • SnLsMtuting the value of y, we get x = a{x- 2a2 + Sa;^ -. ,.) + h{x-2x'' +...)' + c(a;- 2a;^ + . ,f -\-d{x-.. .)*+... =zax-\-{h- 2a)x'' + (c - 4& + da)x^ + (tZ - 6c + 106 •- Vi^ + . . . We find Id succession that a = 1, b = 2, c = 6, d - 11 Uence i x = y + 2y'-{-5y'-}-Uy' + ... Example 2. Find a series in y that will give a solution ol x"^ — 2x -{- S = y. Setting y — 3 = 2, we have z = x^ — 2x. Hence z change? in sign when x does. Hence the required series for x contains no even powers n* z- Ket X =^ az -\- b^ -\- c^ -\- dz' -\- , . . .'. X = a{-2x-\-x^) -f b{-2x + a^f + c{- 2x -\- x^f -^ d{-2x + x") -J- = - 2aa; + (a - Sb)x' + {12b - S2c)x^ + ( - 6& + 80c - 128d)x'^ + . 1 , -1 -3 ^ -3 ••• ^=-T' ^^16"' ^ = T28' ^ = ^'-- .-. x = - i{y - 3) - j\{y - Sf - jUv - ^f - ^U(y - 3)' + . . . The solution x is valid only for values of y for which the series is convergent The fact that the coefficients of the even powers of z are zero may also be verified by computing them directly, as was done for the odd powers. Sec. 123] COLLEGE ALGEBRA. 129 EXERCISES. Expand to five terms in ascending powers of x : «• . .'^"^."^ . » 6. ,_}^Z^ 2 • 7. (1 - x)\ 8. (1 + ic)- '. Find the coefficient of the general term in the series for ' {\-x){l-yx^)' ' l + 3uj + 2.c2' ' x^x^' ' x' -\- X Obtain the reversion of eacli of the series IZ. y = X - X' -^ x^ - x^ -^ . , . 14. 3/ = ^ + Y+ -^ -f y -h . . . Solve by series the equations lb. y = x - xK 16. y = 1 -h "ix' -f ^x^ -h 6a*. Vl. y ^\'\ x-\- a» CHAPTER XV. BINOMIAL THEOREM FOR ANY INDEX.* 124. Consider the power series in x /.x . . . nin—l) ., , n(7i—l){n—%),..(n—r-\-l) ^, (1) i+nx+ -^--^x^-\., . .+ -^ \,2\.,.r ^^^ + ' If n is a positive integer, the series is a finite series terminating | with the n + 1st term, since the {n + 2)nd and all subsequent terms ; contain the factor n — n. In this case the series is the expansion : of the binomial (1 -\-xY [see the proofs in §§ 91 and 99]. When n is not a positive integer, the series is an infinite series, since no one of the factors ?^ — 1, ^ — 2, . . . can now be zero. The | first step in the study of the series is consequently to determine for what values of x^ if any, the infinite series is convergent. We employ the ratio tests of §§ 116 and 117. The (r + l)st term u^^i is exhibited in the series (1). The rth term is consequently _ n{n — \)(n — 2) , . , (n - r + 2) _, Ur — ^^ 1-2-3... (r- 1) Ur \ r J \ r Hence this ratio approaches — a: as r increases without bound. Its absolute value will ultimately become greater than unity if |a;| > 1, so that the series is divergent in that case (§117). But for | a: | < 1, the absolute value of the ratio will be less than some fixed quantity which is itself less than unity' for all values of r which exceed *The theorem was discovered by Sir Isaac Newton. The proof given is a modification of that due to Euler. 130 ; Sec. 125] COLLEGE ALGEBRA, 13I \ a certain number and therefore the series is convergent (§116). \ For example, if n = 6|, a; = — -i-, tlie ratio will be a positive frac- tion for r ^ 3. [It may be shown* tliat the series is convergent for re = 1 provided u < — 1, and for x = ■— 1 provided 71 > 0.] The series (1) is convergent if x is numerically less than unity ^ hut is divergent if x is numerically greater than unity, 125. Suppose that x has a fixed value which is numerically less than unity. - The series (1) has a finite sum whose value depends upon 7^; we denote the sum by/(yi). Then the series (2) l+m:r+ ^^^ V + . . . + -^ 1.2.., r "" + • ' • will have the finite sum f{ni). If the series (1) and (2) be multi- plied together and the product be arranged according to ascending powers of x, the coefficient of x will evidently be n + ni and the coefficient of x^ will be n(n — 1) , mim — 1) (?^ + m) (71 -\- m — 1) A__ + nm + —3-^- = ^^ . Likewise, the coefficient of x^ is seen to equal {n + m){7i -\- m — l){n -{- m — 2) 1.2.3' • As far as the first four terms, the product-series is of the form (1) when n is replaced by 7i -\- m, giving series (4). We proceed to prove that this result holds true for all the terms of the product- series. This is readily proved for the case in which 7i and m are any positive integers. For, in this case, the series (1) is the expan- sion of (1 + xY and series (2) is the expansion of (1 + x)"", so that the product is (1 -f :r)'' + "', whose expansion is the series (4), since ^ + m is a positive integer. Hence, if 71 and m arc positive integers, (3) f{7i) xf{77i)=f{n + ni). * Charles Smith, Treatise on Algebra, Art. 338. 132 BINOMIAL THEOREM FOR ANY INDEX, [Ch. XV Since n and m are arbitrary positive integers, they must be rep- resented by general letters, the restriction to positive integers being supplementary and not expressed in the notation for 7iand m. On account of the generality of the notation for n and m, the way in which the coefficients of the series (1) and (2) combine to give the coefficients of the product-series must have been 2^ formal process independent of the supplementary restriction that m and n are posi- tive integers. The argument is based upon a principle known as "the permanence of form.'^ Its validity in the present case is established in § 126. Hence the form of the coefficients of the product-series remains the same when that restriction is removed, so that the product-series is />«\ ^/ I \ 1 I / I N 1 {n -\- m)(n -\r m —\) „ , (^) /{^ + ?«)== 1 + (^ + m^ + ^—^ — '-\^ ^-x^ + . . . whatever the values of n and m may be. Since |a; j < 1, the latter series is convergent. Formula (3) therefore expresses the relation between the finite sums of the three series (1), (2), and (4). 126. To give an explicit proof of formula (3), consider the coefficient of x"^ in the product-series, r being a fixed positive integer. It is clearly the sum ^ of r + 1 terms, one of which is the coefficient of x"" in series (1), and another the coefficient of x^ in series (2). We wish to prove that this sum '^ equals (n + m){n -{-m — \) . . . {n-\- m, — r -\-\) ^^ 1.2. . .r • Multiplying both ^ and /by r!, the equation to be proved is* n(n — 1) . . . (7i — r + 1) -f rmn(n — 1) . . . (^^ — r + 2) + . . . + m{m — 1) ... (m — r +1) =: (n -f m){n -^m — \) . , . {n-\- m. — r •\- 1), * Denoting for brevity the product p(p — 1)(7? — 2). . . (;? — «-f-l) byp» the relation is ^r + rCi nr-\m[ + rCg Tlr -2 ^2 + • • • + r^r - i W, W,, _ , '-|- W^ = {n + m)r and is known as Vandermonde's Theorem. Compare Ex. 14, p. 99. Sec. 1271 COLLEGE ALGEBRA 133 \ in each term of which n and 771 enter to the degree r. Now the re- ' lation has been established for the case in which n and m are both positive integers. Let 7i be an arbitrary quantity and let the rela* tion be expanded and the terms arranged according to descending I powers of 71, We obtain an equation of the rth degree in 71, Let \ m be any particular positive integer. Then the equation is satisfied ' by an infinite number of values of 71, namely, the positive integers. Hence (§ 78) the equation is an identity in 71 and is satisfied by every value of 71. Hence the relation is established for 71 arbitrary and ?/i any positive integer. Consider the original relation for any particular value of 72, whether or not a positive integer. We expand the members of the relation and arrange the terms according to the powers of m, obtaining an equation of the rth degree in m» By the previous case, this equation is satisfied by all positive integers m, whose number exceeds r, and is consequently satisfied by every value of m. Hence the relation is true for 71 and m both arbitrary quantities. It follows that -2" =/and hence that the product-series is iden- tical with the series (4). 127. By repeated application of (3), we find that f{n) XM) Xf{p)X... Xf{q) =f{7i + 77i+p+, . . + q). Let there be 5 terms n, ?/?, p, . , , , q and take each equal to the fraction r/s, where r and ^ are positive integers. Then w:-)r=<-+^+--+r)=/ex')=/«- Since r is a positive integer, f{r) = (1 -j- xy. Hence {/(:t)P=(i + -)- /(It) = (!+-)'• Hence^ ifn is a positive fraction - , series (1) equals (1 -f- x)s. 134 BINOMIAL THEOREM FOR ANY INDEX. [Ch. XV In the identity (3) put n = — m. Then /(»Ox/(-«0=/(o) = i. •■••^^-"^=/lk = (rT-^' = ^' + ^"^"""' if m is a positive fraction. Hence if n is a negative fraction, series (1) equals (1 -|- xy. We may now state the final theorem: If X is in absolute value less tlia7i ^mity, and if n is any rational 7Uimher^ 7ve have the expansion ...«., , n(?i — 1) ^ , 7ihi — l)(n — 2) „ ' (5) {l-\-xr=^l + 71X + -A___Z^2 ^ _A _|V_ _^^3_j_ ^ ^ ^ This result is known as the Binomial Theorem. Since any real number may be approximated to any desired degree of approximation by a rational number, we can establish, by a limiting process (§ 16), the Binomial Theorem for any real index 7i, 128 Example 1. Expand (1 -f ir)-iby the binomial theorem. If I oj I < 1, formula (5) gives, for n = — 1, the result = 1 -X + X'' - x^ -i- . . . -i- (- lYxr + . . Tlie resulting series is an infinite geometrical progression with the ratio -X. Hence (§69) its sum is - — ■ — when Icri < 1. vo / 1 -^ X II Example 2. Expand 4/I — ^ by the binomial theorem. Setting — y rrr X, n = {, formula (5) gives, if |^| < 1, (i-^)^ = i + i(-^) + M^\-^)^+...+iiti^^ ,. r. -i ^ 12 l-3-5...(2r-3) ^ •. |/1 - 2^ =: 1 - ty - 4/ - . . . ^^^ y^ - . . . This result agrees with that obtained by undetermined coefficients (§ 122^. Example 3. Find ^99 to six decimal places. upon setting y ^ .01 in the formula of Ex. 2. .-. i/99 =: 9.949874 -f. The terras beyond the fourth do not affect the first eight decimal places. Sec. 128] COLLEGE ALGEBRA, 135 Example 4. Find 1^66 to six decimal places. = 4(1 + .0104166 - .0001085 ^ 3 32 1/iy 5/n«_ n 9i^32j "^81\32y • • ■ .0000019 - . .) = 4.0412400. EXERCISES. X Expand to four terms and give the (r 1 {l-\-x)-"\ 2. (1 - x)-K 3. (1 5. |/1~^^2^^. 6. 7/-r-f^- 7. x\l - x'i l)st term: 3a;)^. 4. (2 - ^x)K '=^'^ 8. \/^ - x^. Find to five decimal places the value of 9. \/ll. 10. ^"12(5. 11. ^1002. 12. f 3r2a 13. Find the coefficients of x^ and x^ in (1 + 2.?; -|- %x^)-'^. 2 — x^ 15. Find the coefficients of x^»\ a-^'" - 1, a;^"* + 1 in (1 + a; + x^) - *. 16. The coefficient of x^ in (1 — a; -f x^ — x^ -^ x^)-^ is zero. 17. The coefficients of x^ and x' in (1 + a;-V ^^ -\-x^)~^ are zero. 18. Prove that (1 - ic)-2 = 1 + 2icH-3aj2 -f . . . + [r-^l)x^-\- . » o 14. Find the coefficient of x'^ in the expansion of CHAPTER XVL EXPONENTIAL AND LOGARITHMIC SERIES. 129. If ?^ > 1, we have, by the Binomial Theorem, / , 1 V'"^. -I I ^^ I '^^^{^^^ ~ -^) I ^^^f^^^' ~ 1)(^^^ "" ^) I -1 + ^+ 2! "^ 3! +• For the case a: = 1, we have V + n) -1 + 1^ 2! + 3! +••■ . 1 _L ^ J. ^^^ ~ ^/^) _L a--(a^ - l/w)(a; - 2/w) • • 1 + a; H ^1 1 — 3I H • • • {> + ' + <^^ + -(' a result true for any value of /j > 1. Taking the limit as 71 in- creases without bound, we get l + =r + 2-i + 3T+-..=(l + l + 2T + 3!+---)- The general term of the series on the left is x"- ,. .^ x(x - l/n){x - 2/n) . , . \x - {r - l)/n} —r = limit . ,^ o • r! ^ . ^ 1-2-3 . . . r n — CO 136 Sec. 129] COLLEGE ALGEBR/I, I37 For small values of r this evaluation of the limit is evident. To prove that the result holds true when r becomes large, we denote by Ur the fraction whose limit we seek. Then u. I '^ * r \r n ^ ml ,\ limit Ur — - limit «^r-i« r ' n^co n=co Oj2 /yi3 But the limit of w„ was seen to be —r; hence the limit of ^„ is ttt. ^ 2 ! ^ 3 ! By induction, it follows that the limit of u^. is x^'/rl Set (1) ,Hl + H-^ + ^ + ^+...+^+-.. The above result may now be written (2) ^^i + :r. + lj + l^ + ...+^+... To evaluate e, notice that jj is one-fourth of ^, etc. Hence e = 2+ .5 + .1666 + .0417 + .0083 + .0014 + .0002 + . .. Hence e = 2.7182 to four decimal places. To ten places, we get (3) e = 2.7182818284. We may now expand a^ into a power series in y. Since a = e^^'^e^, ay = e^ ^«^A by the third law of indices. Putting x = y log^a, series (2) gives (4) a^ = 1 + y log,a + l^y^log^af + ~y\\ogsf + • • • This result is known as the Exponential Theorem. The relation (2) is valid for any a:, and (4) for any y. 138 EXPONENTIAL AND LOGARITHMIC SERIES, [Ch. XVI 130. We may give a second proof of the result (2), following the method employed for the Binomial Theorem (Chapter XV). Set i^(^) = 1 + o: + |_r + -gj + . . . +^ + . . ., For every value of x and z^ these series are convergent (§ 116). Form the product of the series F{pc) and F{z), The coefficient of x^'z^ in F{x) • F(z) is 1/ (r!^!). But x^'z^ occurs in the series F{x -f- 2;) only in the expansion of the term {x -^ z)^^^ -~ {r -\- s) ! and occurs there with the coefficient (as shown by the Binomial Theorem for positive integral index) 1 (r+5)! __ 1 (r+ 5)1 r\s\ r\ sV Hence the series obtained as the product of the series F(x) and F{z) is identical with the series F(x -\- z)» v .-. F(x)- F{z) = F{x + z), for all values of x and z. It follows that F{x,) . F{x^) . F{x,) . . . F{x:) =: F{x^ + x, + x^+...+ x,), for any positive integer 7i and any values oi x^,x^, , , , ^ x^- Taking the particular values x^ = x^=^ , . . = 2:^ = 1, we get \F{l)Y = F{n), But F{1) = 1 + 1 + ^ + . . . = e. Hence F{n) = e^ so that (2) is proved for tlie case x = n, ^ positive integer. Taking next Sec. IBIJ COLLEGE ALGEBRA. I39 the particular values x^ = x^= ... = ir^ = — , where m also is a positive integer, we get By the case just established, F{m) — e'"'. Hence Hence formula (2) is proved for the case x — ^ — , a positive frac- tion. To extend the proof to the case x — —f, f being a positive fraction, we note that F{-f)'F{f) = F{0) = 1, whence Hence formula (2) holds for all rational values of x. By the method of limits (§§ 16, 105), the result may be extended to any real \alues of x. Ill 131. Example. Find the sum of 1 -f -^-: + -j^ + -^ + • • • «^ infinitum. For X =2 -\- 1 and x — — 1, formula (2) gives ... . + .-. = 2 + 1 + 1, + ! + ... The required sum is therefore ^{e -\- e- i). EXERCISES. 1. Show that ^,-1 = 1 + I + 1 + A + . . , 2. Evaluate a"" - b'' + ^(a' - b') + -{a' — 5^) + . . . 3. Expand (e-^ + e^^) -^- e^^ into a power series in x, P 2=^ ^! I i' 1 ! ' 2! 31 4 ! ' 4. Prove that 5^ = t^ + -^ , + oi + x, + • • MO EXPONENTIAL AND LOGARITHMIC SERIES, [Ch. XVI 6. Prove that 1 = — 4- — -}-Ai_t_L,.. 2! 3! 4! ' 5! "^ 12 o'-^ 8^ 6. Prove that ^ =z 1 + ^^+ ^ + |j + . . . . 7. Findthesumofl + ^^+^' + ^V.. 132. Theorem. If \x\ <1^ log^ (1 + x) equals the sum of the series /y»2 /y.3 /y.4 ^r (5) ^_| + |-.^-+...._(_l)^?-+... Substituting 1 + ^ for a in the equation (4), we get (1 + ^)i/ = 1 + 2^ log,(l +:,) + ^^2]log, (1 + :r) [2 + . . . Also by the Binomial Theorem, we have, when | a; | < 1, (i+.)«=i+,.+fcil.»+. . .+^fcil^__(i^i±l),.+ . . . In the second member the coefficient of y is .+(ri.>..+...+(-')(-^^ -(-+'v +..., which, upon simplification, is series (5). The coefficient of y in the series l-{-y log^ {I ~\- x) -\- . . . is log^ (1 + x). Since the two series are equal for all values of y^ we may equate the coeffi- cients of like powers of y (§ 121), so that the theorem follows. By equating the coefficients of ^^, we find, similarly, that illog. (1 + x)\^^ ix^ - i(l + ^)x^ + HI + i + \)x* -... 133. Keplacing a: by — ic, we obtain from (5), for | a; | < 1, /y2 /y«3 />«4 /y,r (6) \og^{\-x) = -x---~--- ...---... » Sec. 134] COLLEGE ALGEBRA. 141 Subtracting from (5) and using log -7- == log a — log h, we get (7) log4-±| = 2(0. + I' + I' + y + . . .). Substituting in (7), 'in — n ^1^1 + .'^ in X = ; , so that -— ' — — --, m -^ n\ 1 — X n we get, for any positive values of in and n, 134. 7'able of logarithms to the base e = 2.71828 +; natural logarithms. For 771 = 2, n = 1, formula (8) gives log. 3 =2{i+i.(i)^+ia)^+i(i)^ +...}, from which we get log^ 2 = .693147 to six decimal places. For 771 z=: 3, 71 = 2, formula (8) gives log, 3 - log, 2 = 2 li + mr + my + my + . . . i = 2{.2 + .002667 + .000064 + .000002} = .405466 (to six decimal places). By carrying the work to more places, the sixth place is seen to be 5 instead of 6. Adding log, 2, we deduce log« 3 = 1.098612. For 7n = 5, 71 = 3, formula (8) gives log, 5 = 1.609438. We may now deduce log, 4 = 2 log, 2, log, 6 = log, 2 + log, 3, log, 8 = 3 log, 2, log, 9 = 2 log, 3, log, 10 = log, 2 + log, 5. We get log, 7 from (8). We thus get the numbers in the second column of the following table : I 142 EXPONENTIAL AND LOGARITHMIC SERIES, [Ch. XVI N log, N logio ^ 1 2 0.693147 .301030 3 1.098612 .477121 4 1.386294 .602060 5 1.609438 .698970 6 1.791759 .778151 7 1.945910 .845098 8 2.079442 .903090 9 2.197225 .954243 10 2.302585 1.000000 ' To obtain log^^ iV^, we multiply log^ JVhy [see § 19] 1 1 logic .4342945. loge 10 2.302585 Hence the numbers in the third column of the table are derived from the corresponding numbers in the second column by mul- tiplication by the constant .4342945, which is called the modulus of common logarithms (namely, logarithms to the base 10). Loga- rithms to the base e are called natural logarithms or Napierian logarithms (in honor of their discoverer, Lord Napier). 135. Interpolation. For m =n -\- a and m = n -\- b, formula (8) gives ^^^^ (^^+")-^^^^ " = M 2^ + M2..+. log, (,,+^)_log, n = 2 i -1- ^ 1 (-—Y \2n+b'^ 3 \2n+b/ "^ 5 \27i+b) If a is relatively small compared with 71, the 3rd, 5th, and higher powers of a -^ {27i + «) may be neglected for approximate results. Similarly for b. Then, approximately, (9) |log {n + a) —\ogn\ } :{log {7i-\-b) — log 71}= a : b, a ^ b _ a 271 -\- b since 2n + a * 27i + b b ' 2)1 + a and the last factor differ from 1 by a — b 2n + a hold for common logarithms base 10, If (9) holds for logarithms base e, it will also Sec. 136] COLLEGE ALGEBRA, 143 To consider a numerical example^ let the base be 10 and let n = 4550, a = 2, Z' = 10. Then —-^^ — =. -^ z= .00022 to five 2}i 4- a 91u2 decimal places: 2^ — -^ = — —- = .00110 to five decimal places. ^ 2)1 + 911 ^ Their cubes may be safely neglected. By a Table of Logarithms, log 4550 = 3.65801, log 4552 = 3.65820, log 4560 = 3.65896. . •. log 4552 — log 4550 = .00019, log 4560 — log 4550 = .00095. Hence by the formulae or by the Table, the diiferences of the loga- rithms are approximately proportional to a : ^, viz., 1:5, so that (9) is true. To pass from the results given by the formulae (base e) to those given by the Table (base 10), we must employ as multiplier the modulus, practically .4343 in value. Multiplying log, 4552 - log, 4550, of value 2 (.00022), by .4343, we get logio4552 - logi,450, of value .00019. For relatively small differences in tlie niimhers^ the logarithm's increase is approximately in proportion to the number^ s increase, 136. Example 1. Expand loge(l — x -{- x^) into a power series. ' log (1 + ^ + ^=^) = log ^-^1 - log (1 + ^) - log (1 + 0^) r i^\n 2 The coefficient of x^ is if n is not divisible by 3, but is ( — 1)« if n n n is divisible by 3. Example 2. If ex, (5 are the roots of ax^ -\- hx ■■{- c — 0, then log {a -hx^ c^') = log ^ H ^^x i^^ '^ ^^ ~ • h c By § 32, we have a -f /? = , a6 = — . Hence a — hx -\- cx^ — «(1 -f ax>j{X -\- ftx) log {a--bx + ex") = log ^ + {ax - Id'x' + • • ) \ {fix - |/5V + ...). 144 EXPONENTIAL AND LOGARITHMIC SERIES, [Ch. XVI Example 3. Prove that e is incommensurable. Suppose that e — —, m and n being positive integers. Then n ^ "^ ^ "^ 21 "^ 8i "^ • • * "^ wl "^ (7i + 1)1 "^ Multiply both members by n ! •■• '''^''-'^'■ = '^''^'' + ^l + (n + ln+%) ^ \n+l),nl^)in+Z) +-- The series beginning with l/(/i -|- 1) is less than 71 + 1 ' (^ + 1)2 ' (7i 4- 1)3 ' • • - n Hence the sum of the series lies between — ■ — - and - and is therefore a n -\- I n proper fraction. Hence would the integer m{n ~ 1)! equal an integer plus a fraction. Since this is impossible, e cannot equal a fraction. [EXERCISES. 1. Find the general term of the expansion of loge (1 -f « -f- a?^ + a?), 2. Find the general term of the expansion of loge (1 — %x -\- 2x^). /7>2 /7>3 nA n,1 nfi Z.liy = .-^+^-^+. ,.then. = 2, + |, + |- + ... 4. loge (. + .)- log. (n-c) = 3 (^+ ^+ ^+ ^^ + . . .). 6. log.2 = -^+^ + ^+^ + .,,. 6. logeS = 4 + r^+ Fr^+ 5^^ + • • • r + s 2rs 1 / 3r« \' 1 f 2rs \* '• ^"Se-y^^ - :;^r^^, + -^[^_f^->j + -5\^^rf-^) + • • • g-l, 1 a!''-! 1 x> - 1 '• '"S' * - i"+l "*" 3 (« + 1/ + "3" (a; + 1)» + • CHAPTER XVII. SUMMATION OF SERIES. 137. The method of undetermined coefficients (Chapter VIII) and the method of mathematical induction (Chapter XI) have been employed to effect the summation of certain classes of series. Like- wise, the sum of an arithmetical or a geometrical progression has been obtained (Chapter VI). We here discuss methods of greater generality for finding the sum of a series ^1 + «^2 + ^'3 + • • • + ^n + • . . In case the series is an infinite series, the result is valid only when the series is convergent. For an arithmetical progression of common difference d, we ; have u^_i — ti.^^ — d=iu^— n^_i , whence ii,, — 2?^„_i + w„_2 = 0. I The latter relation is called the generating relation, since the series may be constructed by means of it. Thus, taking n = 3 and 4, we get W3 = 2ii^ — tc^ , v^ = 22/3 — i(^ = 3if^ — 2u^, The series is there- fore u^ , u^ , 2u^ — u^ , 3ii^ — 2Wj , . . . , of common difference u^ — n^. For a geometrical progression, the generating relation is w„ — rUn-i = 0, r being the common ratio. Inversely, every two- term generating relation leads to a geometrical progression. 138. In general, a series ic^ -\- ii^-\- u^~{- . . , is called a recur- ring series of order Jc, if it has a generating relation of order ^, tin + Oi'^n-l+ ^2^n-2 -f- • • • + OjcVn-k =0 {c^, . , . , Cj, COUStauts), I4<^ SUMMATION OF SERIES. [Ch. XVII I which holds iTue for n — Jc -\-l, h ^2^ , . . , on to the end of the series when the latter is finite, but holds ad infinitum when the series is infinite. • In addition to the above examples, consider the series Since it is not a geometrical progression, the generating relation, if one exists, must be of order at least two. Trying the value Ic — 2, we have u^ + c^Un-i + c^^(^n-2 = for ^^ = 3, 4, . . . Hence nx^-^ + c^{n — l)x''-^ + c^{n — 2)x''-^ = 0. .-. n{x^ + c^x + cj — c^x — 2c^ = (for 7i = 3, 4, 5, . . .). .-. x^ + c^x + c^=0, - c^x -2c^=0 (§ 78). • . C| — /ClXj Cn — X » Hence there is a generating relation u^ — 2xu^_^ + ^^^n-2 = of order two. But this generating relation determines the above special series only when it is also given that u^ = 1, ii^ = 2x. In fact, the relation only expresses ^^g , ^^ , . . . in terms of u^ and v^, 139. In general, a generating relation of order Tc determines a particular series only when the first Jc terms of the series are given. On the other hand, if it is known that a series has a generating relation of order ^, but the constants c^^ c^, . , . ^ c^^ are not known, we must have, in addition to the first h terms of the series, the next h terms in order to determine c^, . . . , 6V If the general term is given, the preceding conditions are evidently satisfied and the generating relation is found as in the above example. Example. Find the generating relation of the recurring series 1 + ic + aj2 + 2a.'3 -4- 3^^ -f 2x^ + 2^« + 7ic' + . If the generating relation were of order two, Un-\- CiU^ -i-j-c^Un-z — 0, we would have x^-{- CiX-{- c^ — 0, 2x^ -f CiX^ + ^2^ = 0> whence x = 0. Assmn- ing, therefore, that the generating relation is of order three, Un-i-CiUn-l-{~ Cu1tn-2-\- C^Un-S = 0, I Sec. 140] COLLEGE ALGEBRA, I47 we have, for n = 4, 5, 6, the conditions CiX^ -\- CoX -Y Cz — — 2a^, 2ci a^ -\- c-iX^ -\- CzX — — Zx^, Hence Cx — — x, c^ = Scc^ Cs = — Zx^. The conditions for n = 1 and 8 are seen to be satisfied. Hence Un — xiin - 1 -j- 2x^Un - 2 — Z^Un _ 3 — 0. 140. Problem. To find the sum of n terms of a recurring series of order ttuo, Let the generating relation be n^ -\- pxu.^_i -^ qxhi,^_o — 0. Then pxSn = pa^x +pa^x^ -f- . . . 4- pa,,_^x''-^ + pa,,_^x'', qX^Sn = q%X^ + • . . + 5'^n -3^''"^ + qCln~'i^''+ qC^n-l ^""^ *• Since a^x^-\-2M^x^+q%x^^0, , . . , a,,_^x''--''-\-pa,,_>iX'''-^-^qa^_^x"'-^ — 0, Sn+px8n-\-qx^Sn=a^-^a^x-^pa^x-]rpa^^^x''-\-qa^_^Qf+qa,,_^x''+^ . ^ ^ ^0 + {(^i'\-P%)x + {pcin-i+ qctn-2)x'' + qa,,_iX''+^ ' * " 1 -{-px -\- qx^ t 2: be a value for which the series is convergent, so that (end of §107) limit aj^_iX''~'^ = 0. Then a.^^^nf' and a„_ia;'*+^ will also approach the limit zero. Hence _ ^0 + (^1 + ya^x "^ \^px-\-qx^ ' For a recurring series of order one or tliree we find, in a similar manner, that the sum to infinity is, respectively, ^0 ^0 + K + ;^^o)^ + (^2 + P^x +IfO^^ \-\- px"* \-\- yx-\- qx^ + rx^ 148 SUMMATION OF SERIES, [Ch. XVIL the generating relation being, respectively, Ur, + pxu^_^ = 0, u^ -\-pxnn-i + qxhi^_2 + rxH^_^ = 0. For the series of § 138, we have ^o = 1, «i = 2, p = — 2, 5' = 1. For the series of § 139, ao = 1, ai = 1, a^ = 1, p = — 1, q = 2, r = — S, 1 + 2a;2 •• ^OD 1 - aj + 2a;^ - Sx^' 141. The fraction obtained as the sum to infinity of a recurring series is called the generating fraction, since the series may be ob- tained by the expansion of the fraction (§ 122). When the generating fraction can be expressed in terms of partial fractions, whose denominators are powers of binomial ex- pressions, the general term of the recurring series may be readily obtained by the binomial theorem. For example, the series 1 + 9:r — Wx"^ -\- 57a^ — 159aj^ + • • • has the gen- erating relation Vn-\- 2xtin-i — Sx'^nn-2 = 0, so that (§§ 140, 80) 1 + llaJ 3 2 l-\-2x-3x:' 1 - X 1 4- 3« = 3(1 + ic + aj2+ . . . + aj^+ . . .)- 2(1 - dx -}- 9x^- \-(-^ Sxy J^ . . ,), in which the coefficient of aj'' is 3 — 2(— 3)**. EXERCISES. Find the generating relation, the generating fraction, and the coefficient of aj**, of the recurring series: 1. 1 + 4a? - 2aj' + lOa;^ - 14a^ + . . . 2. 1 _ 5a; + 34a;2 - 260a;3 _|_ 2056a;* - . . . 3. 5 - 8a; -h 66x' - 17Qx^ + 800a;* + . . . 4. 1 4. 6a; - 4a;*^ - 40a;3 - 112a;* - 32a;5 + 704a!« 4- . . . 5. 1 - 5a; + 9a;- - 13ar^ + . . . 6. 2 -f 3a; + 5a''^ + 9ar» + . . . 7. 4 - 5a; + 7a;2 _ lla;3 _^ _ . 8. 1 + 7a; - a;^ -f 43a;» + . . . 9. la + 2« -f 32 + . . . + 7i2, 10. 18 + 2« + . . . -f n\ Sec. 142] COLLEGE ALGEBRA. 149 THE METHOD OF DIFFERENCES. 142. If the first term of any series is subtracted from the second term, the second from the third, etc., a series is obtained which is called the series of the first order of differences. Proceeding simi- larly with the latter series we obtain the series of the second order of differences of the original series. Similarly, we obtain the series of any order of differences. Series: w^ u^, u^, w^, . . . 1st order differences: u.^ — u^, u^ — ii^, u^ — u^, , , , 2nd order differences: ii^ — 2u^-{-n^, ^^ — ^^^+^^2' • • • 3rd order differences: u^ — 3n^-{-^if,^ — i{^, ... Denote hj D^, D^, D^, . . . , the first terms of the series of the 1st, 2nd, 3rd, . . . , orders of differences, respectively: A = ^^ - '^^p A = ^'3 - -^'2 + ^^' A = ^^4 - ^^^3 + 3^/2 - ^^. . . . We observe that the expressions for 71^^ ti^, n^ have the same coef- ficients as the binomial coefficients in (a + by, {a + by, (a -\- b)'^. We proceed to prove by mathematical induction that (1) «„ + ! = Ml + «A + — i:2~A +---+nCrD,+ .-.+D„. Suppose that the theorem has been verified for any series as far as 'the (m + l)st term (as has been done for m = 1, 2, 3). Hence (2) u^^,= n,+mD,+^C,D,+^C,D,+ . . . +„,aA + . • - + I^m- Writing the analogous formula for the series of the first order of differences ii^—u^ = D^, u^—u^, . . , , u^j^i—u^, ^^ + 2— '^m + i? • • • , the first terms of whose 1st, 2nd, 3rd, . . . , orders of differences are i>2^ />3, Z>^, . . . , respectively, we get 150 SUMMATION OF SERIES. [Oh. XVII Adding (2) and (3), and applying ^(7^ + ^(7. _i = ^ + i(7^ (§89), we get But this is formula (1) for the case n = in -{-1. Hence, if the law holds for n = m, it holds for n — m -{- 1. But the law holds for n = 1/2, 3. It therefore holds for any positive integral value of n. Example. For the series 5, 8, 11, 14, ... , the series of the first order of differences is 3, 3, 3, ... ; that of the second order, 0, 0, . . . Hence D^ — 3, i>2 = 0, i>3 = 0, . . . , i>r — 0. Hence Un + i = 5 -|- 3;^; this result may also be derived as the {ti 4- l)st term of an A. P. of common difference 3. 143. Problem : To find the sum S,, of the first 71 terms of the seines (4) v^, v^, t'g, ?;,,..., v^, . . . Evidently S^ is the {n + l)st term of the series (5) 0, V^, V^ + l\, %\ + 1^2 + ^3' • • • ' ^i + ^'2 + • . . + ^^n, . . . • The series of the first order of differences of (5) is clearly (4). Hence if rZ^, d.^, d^, , . . ^ are the first terms of the series of the 1st, 2nd, 3rd, . . . , orders of differences of (4), then v^, d^^ d^, d^, . . . , are the first terms of the series of the 1st, 2nd, 3rd, 4th, . . . , orders of differences of (5). Hence the {n + l)st term of series (5) is -[by formula (1), with 21^ ^=0, D^ = v^, D^ = d^, D^ = J^, . . .] 2) s -nv I ^ ^^ - ^^ d -4- !^(!!_:zi_)(!^ -d,+ . + ?i ^'r <^^r - 1 + • • • + ^4 - !• 144. Example 1. Find the Tith term and the sum of the first n terms of 1, 15, 40, 79, • 135, 211, . . . 1st order differences: 14, 25, 39, 56, 76,... 2nd order differences: 11, 14, 17 20, . . . 3rd order differences: 3, 3, 3, . . . 4th order differences: 0, 0, . . . Hence the ??th term and the sum Sn are respectively 1 -w 1 /O * O Sec. 144] COLLEGE ALGEBRA. 151 , n{n - 1) , , , n{n - l){n - 2 ) ,, , n{n - l){n - 2)(n - 3) ^ = iin{?.ri' + 26/^2 _l 69 ;i - 74). Example 2. Find the sum of the squares of the first n integers. Series: 1, 4, 9, 16, ... , r\{r + 1)\ {r -f 2)'^ (r + 8)2, . . . 1st order differences: 3, 5, 7, , 2r + 1, 2r + 3, 2r + 5, . . . 2nd order differences: 2, 2, 2, 2, . . 3rd order differences: 0, , 0, . . . , v(n— 1) ^ ■ nin - l)(?i — 2) _ 7? , , .,,_ , ,^ •■• '^'^ "" '' + \ 2 ^ "^ — r^"^3 — T ^'' + ^^^^'' + ^^* Example 3. Find the number of shot arranged in a complete pyramid ^Yhose base is an equilateral triangle. The respective layers contain the following number of shot: 1st order: 2, 3, 4, ,>•+!, r + 2, r -f 3, . . . Srul order: 1, 1, . . . . ,1, 1, . , . 3rd order: 0, , 0, . . . EXERCISES. 1. Find the sum of the first 7i natural numbers. 2. Find the ninth term and the sum of the first nine terms of 2, 9, 28, 65, 126, . . . Find the 7ith term and the sum of the first n terms of 3. 1, 8, 25, 52, 89, . . . 4. 2, 12, 28, 50, 78, 112, . . . 5. 1, 11, 35, 79, 149, 251, ... 6. 2, 20, 48, 86, 134, 192, .. . 7. 1, 4, 10, 20, 35, 56, . . . 8. 4, 8, 0, - 32, - 100, - 216, . . . 9. 1^ 2^ 3=\ 4^ . . . 10. 1*, 2\ 3*, 4S . . . 11. 1-3, 4-7, 9-13, 16-21, .. . 12. 1-2, 2*4, 4-7, 8-11, 16-16,... Find the number of shot in 13. A pyramid of triangular base each of whose sides contains 20 shot. 14. A pyramid of square base, 15 shot in a side. 15. A truncated pyramid of 30 layers, with 50 shot in a side of the square base. 16. A truncated pyramid, whose base is an equilateral triangle with 100 shot in a side, and whose top has 56 shot in a side. 17. A complete rectangular pile with base 9 by 15 shot. 18. A complete rectangular pile with 15 layers and 20 shot in the longer side of the base. 19. A complete rectangular pile of 71 layers with ?7i shot in the top layer contains ln{n -f l)(3m -j- 2n — 2) shot. I CHAPTEE XVIII. GRAPHIC ALGEBRA; SIMULTANEOUS EQUATIONS; SIMULTANEOUS INEQUALITIES; SOLUTION OF NUMERICAL EQUATIONS. 146. The numerical results of an investigation are often ex- hibited in a table of statistics accompanied by a diagram. For a comparison of the results, the diagram shows at a glance what would be gotten from the table only after a minute study. Suppose that the number of inches of rainfall in the 1st, . . . , 12th months of a given year was 2, 1, 0.5, 3, 4, 3.5, 1, 0, 2, 1.5, 0.5, 1 respectively. We may represent these figures graphically by marking off the proper number of vertical divisions above the base line at the point indicating the corresponding month (Fig. 1). n I f v V p . ) \ \ \ / \ \ <^ oL 2 : — — —. J \ n ) 11 Suppose that a quantity y varies as the quantity a; in such a way that y — 2x. Then for a: = 0, ?/ = 0; for a; = 1, z/ = 2; for :c = 2, ^ = 4 ; for a: =: — 1, «/ =: — 2 ; for a; = — 2, ^ = -- 4 ; etc. We may represent this relation graphically as in Fig. 2, in which the values of x are measured along the horizontal line OX (the positive values to the right of and the negative values to the left of 0) and the corresponding values of y are measured in the vertical 152 Sec. 146] COLLEGE ALGEBRA, 153 direction (upwards or downwards, according as the value is positive or negative). It is easily shown that the points so located all lie on a straight line AOB [see § 151]. Y -f— 1 t_ 1 L / L 1 t 1 t Y' Fig. 2. 146. Definitions.* The two perpendicular lines X'OX and F' OF are called the a;- axis and the ^-axis respectively; together they are the axes of coordinates ; their intersection is called the origin of coordinates. To plot the point whose coordinates are x and y^ we mark off x units on the ic-axis from the origin 0, and on the vertical line (parallel to the y-axis) through the point thus reached we mark off y units; the final point reached is designated (-2,4-3) (-2,^3) (-i-2»4-6) ( + 2,-3) Fig. 3. as the point (x, y). In Fig. 3, we have plotted four points whose pairs of coordinates differ only in their signs. * The system was introduced by Descartes in 1637. 154 GRAPHIC ALGEBRA; SIMULTANEOUS EQUATIONS. [Ch. XVIll Since there are innumerable pairs of values x., y which satisfy the equation y = 2x, there will be a boundless number of corre- sponding points {x, y) which lie on a line, called the graph of the equation y — 2x, It would appear from Fig. 2 that the graph of y — 2x is a straight line A OB [compare § 151]. 147. The graph of the equation x^ -f ^^ _ 95 will be a circle of radius 5 and center at the origin 0. Indeed, if w^e join with the plot P of any point (x, y) such that x^ -\- y'^ — 25, OP is the hypothenuse of a right-angled triangle whose base is x and vertical side is y, so that {OPf — x^-\-y^ — 26, OP = 5. Hence every such point Plies at the distance 5 from 0. The circle is given in Fig. 4. \ Y R \ \ \ \ ^ ^- P / ^ / \ \ / \\ M Q \ X V j \ \ / \ / V -^ ^^ ^ \, A The graph of the equation 2x-\-y •=-\^ is a straight line (§ 151 ). Hence two points will determine it; for example, (0, 10) and (5, 0), whose plots are R and Q in Fig. 4. It appears from the figure that the circle and the straight line RQ intersect in the points P and ^, the ])lots of (3, 4) and (5, 0). The corresponding algebraic problem consists in determining all sets of values a;, y which satisfy both of the equations x^^y'^ = 2h, 2a; + ?/ = 10. Sec. 148] COLLEGE ALGEBRA, 1 5 5 AVo must therefore consider the two equations to be simultaneous (§ 34). Setting ^ = 10 — "^x in the first equation, we get »x^-^x+lb = 0, x = ^ or 5. Hence the only sets of solutions are ^ = 3, y = 4; x — b,y = 0, 148. In general, the problem to solve two simultaneous equa- tions in x^ y is equivalent to the problem to find the coordinates of the points of intersection of the graphs of the two equations. In particular, two simultaneous equations of the first degree in X and y have at most one set of solutions in common. This result '* follow^s algebraically from the solution by determinants (§ 35), and geometrically from the fact that the two graphs are straight lines (§ 151). For example, the simultaneous equations 2x-\-y =: 10, Ax—?,y = 0, liave only the common set of solutions x =^ ^, y = 4; their graphs, the straight lines EQand OF (Fig. 4), have only the common point P, viz., (3, 4). 149. If we attempt to solve by determinants the equations 2x + y z= 10, 2x + y = 4=, we get O-o; = G, O-y — — 12, so that there are no common sets of finite solutions. Plotting the graph of 2x -\- y = 4, we obtain, a straight line wljich apparently does not intersect the graph of 2x -{- y = 10. The fact that the graphs are parallel lines may be proved by observing that the difference between the ^-coordinates corresponding to the same ^-coordinate is always 6. Hence the graphs have no point of intersection in the finite part of the plane. EXERCISES. 1. Prove that the graphs of x^ -f ^2 _ 25 ^j^^^ 2x -}- y — 1^125 are tangent. 2. Do the graphs of x^' -}- y'^ — 25 and 2x -\- y = 15 intersect ? 3. Find the intersections of the graphs of x^ -f y^ = 25 and x - 5y = 0. 4. Find the point common to the graphs of the three equations 2x - Sy = 7, Sx - iy = 13, 8^^ - lly = 33. 6. Find the interseciions of the graphs otx^ -\- y"^ — 9 and x -{- y = 2. I 156 GRAPHIC ALGEBRA; SIMULTANEOUS EQUATIONS. [Ch. XVIII 6. Show that a parallelogram is formed by the graphs of 2. a'^ b ' h^ a 1, -+^ = 2, a b ^4_ ^ 150. Theorem. The area of the triangle ivhose vertices are the points (x^, y^, (x^, y^, and [x.^^ y^) equals the determinant X, Vi i X, Vt i a;. Vz i Let the three vertices be P, , P^^ P^ , respectively. Evidently, triangle P^P^P^ equals trapezium Pj^j^gPg — trapezium P^Q^Q^P^ — trapezium P^Q^Q^P^ But, by geometry, the area of a trapezium equals one half the sum of the two parallel sides multiplied by the perpendicular distance between them. Since x^ = OQ^,x^ = OQ^,we have x^ — x^ ~ Q^Q^; similarly, x^ — x^^ Q^Q^ , x^ — x^= QiQr Hence A = K^s + yi)(^s - ^'i) - ^{y, + ^i)(^. - ^1) - \{y, + y,){^z - ^,) Sec. 151J COLLEGE ALGEBRA. 157 Usiug the determinant notation (§ 36), we get (1) A =1 X, yx 1 x^ y. 1 X, y> 1 I For example, the area of the triangle with the vertices (0, 2), (3, 0), (4, 1) is 2 3 4 1 151. Theorem. The graph of an equation of the first degree (2) Ax -\- By +0=^0 is ahuays a straight line. Consider three points {x^, y^), {x^, yj, (x^, y^ of the graph. Then (3) Ax,-^By,+ C=^0, Ax, + By,-^ C =Q, Ax, + By,+ C =^0. The area of the triangle whose vertices are the three points is given by formula (1). li A 4^ Q, we multiply the elements of the first column of the determinant by Ay the elements of the second column by B^ the elements of the third column by (7, and add the last two sets of products to the first set. We have (§§ 45, 47) Ax,^By,JrG y, 1 y, 1 ^ A =i Ax,^By,+ C y, 1 Ax, + By, + C y, 1 = 1 y, 1 y, 1 = 0, upon applying the hypotheses (3). E ^ence A = 0, S< ) that the three points lie in a straight line. Let the third point {x^ , y^ move to any new position on the graph. As the area is again zero, the new point lies on the straight line determined hy {x^, y^) and (^2 ' ^2)- Hence every point of the graph lies on that straight line. If ^ =: and B ^ Oy we multiply the elements of the second [column by B and add to them the products of the elements of the 15S GRAPHIC ALGEBRA; SIMULTANEOUS EQUATIONS. [Ch. XVIII third column by C, The resulting determinant has three zeros in the second column, so that ^ A = 0, whence A = 0. In this case the line (2) becomes By -\- C = and is parallel to the a;-axis, since the ^-coordinate remains constant. It remains to prove that, inversely, every point on the straight line joining Qx^^ y^) and {x^, y^) lies on the graph of (2). This result is shown in § 153 below. 152. To find the coordinates of the point which divides in a given ratio the straight line joining tivo given poiiits. Let the first point P^ be {x^ , y^ and the second point P^ be p. %-y< p,/ ' Fig. 6. {x^, y.^). Let the given ratio be m^-.m^. Let P,- be the point which divides P^P^ internally in this ratio, and P^ be the point which divides PiP^ externally in this ratio. Let their coordinates be {Xi, y,) and (cr,, y^, respectively. Since P^Pi\ P^Pi -■ r)i^'- ^^2 ^^^ ^i^e ' P^Pe — '^^I'l^h^ we have - Vx _ ^i — ^1 _ ^^^ - Vi ^2 - ^i ^^'2' ^ 1 / ' z 1 ih -Ve _^i-X, _ »«-. Hence (4) (5) y% - Ve •-»-, - X, ■'' «/, + m, ' «, «»i + m, ' ^^ m, — m, ■ Sec. 153] COLLEGE ALGEBRA. 159 Corollary. The middle point of the straight line wining {x^ , ?/J and {x^ , y,) is (6) (^% H^). Note. The dista^ice hetween tivo p)oints {x^, y^ and (x^, y^ is (7) V{^, - x^f + (y, - yJK P. R y^-y. Pe X ^i-^« 1 1 1 1 i Fig. 153. We may now prove that if the graph of Ax -\- By -\- C =0 contains two distinct points (2-^, y^ and {x^, y^)^ it contains every point on the straight line joining them. By hypothesis, the first and second relations (3) are satisfied. In view of the last section, it remains to be proved that the points P^and P^ with the coordinates (4) and (5), respectively, lie on the graph of Ax + By +(7=0, whatever be the valnes of m^ and m^. The condition for F^ is that Ax,+ By,+ C= -^{Ax^+ By,) + -^ {Ax,+ By,) + shall vanish. In view of the first and second relations (3), it equals -(- 0) + Wj + 7n,^ m, + m, Similarly, Ax^ + By^ + O vanishes. {-G) + l6o GRAPHIC ALGEBRA; SIMULTANEOUS EQUATIONS, [Ch. XVIII I 164. From what precedes, every straight line is the graph of some equation of the first degree. If two points (x^, y^ and {x^, y^ on the line are given, the required equation may be written X y \ (8) x^ y, A =0. Indeed, upon expansion according to the elements of the first row (§46), there results an equation of the first degree in a;. and y, which must therefore represent some straight line (§ 151). But the line coincides with the given straight line. In fact, it contains the point {x^y ^j), since the replacement of x by x^^ and y by y^ in the determinant in equation (8) gives a determinant having the first and second rows alike, so that the equation is satisfied (§ 46). Similarly, it contains the point {x^, y^), A second proof follows from § 150, since the area of the triangle formed by {x, y), {x^, y^), {x^, i/J, is one-half the determinant in (8); but the area must be zero if the point {x, y) lies on the line joining {x^, y,) and {x^, y^). 155. Example 1. Find the equation of the straight line making an inter- cept a on the a^-axis and an intercept b on the ly-axis. The line through the points {a, 0) and (0, b) has the equation = ab — ay — bx = 0, = 1. X y 1 a 1 b 1 It may be written in the convenient form — \- ^ '' a b Example 2. The medians of any triangle meet in a point and each median is trisected by that point. Let the vertices Pi, Pg* ^3 1^^ (^i» 2^1 )> (^2» ^2)* (^3» ^s^ respectively. By the I -h52 Vi 4- y2\ 2 ' 2 / Let Q be the point which divides the sect P^M^^ internally in the ratio 2 : 1. By formula (4), the coordinates of Q are corollary of § 152, the middle point M^^ of the side P^P^ isf — which divides coordinates of 3!i + m^ + Xt i + 1 Sec. 155] COLLEGE ALGEBRA, i6l ^*- 2 + 1 ~ 3 Similarly, the middle point My^ of the side P^P^ is ^^-i^^ ^^^ r ^^^ the coordinates of the point which divides the sect Pg^is internally in the ratio 2 : 1 are seen to be the values aj/, yt just given. In a similar manner, or by the symmetry of the result, the point [xi, yi) divides the sect P^M^^ inter- nally in the ratio 2 : 1, ifjs being the middle point of the side PjA- Hence the medians meet in a point and trisect each other. Example 3. Find the equations to the straight lines which pass through the point (1, — 3) and make equal intercepts on the two axes. By Ex. 1, the required lines have the equations a — a —b — b since a line through (1, — 3) will either have a positive intercept on the aj-axis and a negative intercept on the y-axis or else have negative intercepts on both axes (as a diagram will show). Substituting 1 for x and — 3 for y, we have for the respective lines the conditions \ a^--a'-b^-b whence a = 4, b = 2. The lines are x — y = 4: and x -{■ y = — 2. Example 4. To find the coordinates of the center of the circle inscribed in a given triangle. Let the vertices P^, Pg, Pg be (x^, 2/i)» (^2» ,V2^> (^3* ^s) ^^d the opposite sides «!, s,^, S3. The bisector of the interior angle at P^ meets the side P2P3 at a point §1 which divides it internally in the ratio s^ : 8.2. Hence the co5rdinates of Q^ are (§ 152) «3 -{- ^2 *'3 "4" ^2 But P2Q1 : P3C1 = «3 : «2 gives P^Q^ + P.^Q^ : P^Q^ = s^ + 8^ : s^, whence The bisector of the interior angle at P3 meets PiQi at the required center (7/ Hence CMivides P^Q^ internally in the ratio s^ : P^Qi, viz., Sj • o"- Hence the coordinates of G are yc — / ^3^3 ~r ^2^2 \ 8^X^ + S^X^ + 8^X^ ^ gl.Vl + ^2^2 + ^3^3 «1 + «2 + «3 "" «1 H- «2 4- «3 ' i62 GRAPHIC ALGEBRA ; SIMULTANEOUS EQUATIONS. [Cii. XVIII Thus, if the vertices are (0, 12), (5, 0), (35, 0), we get (Note, § 152) s^ = 30^ «2 — 37, Sg = 13. The coordinates of (7 are therefore Xc = 8, Pc = 4^. EXERCISES. Find the equation of the straight line 1. Cutting off 3 on the ic-axis and 4 on the y-axis. 2. Cutting off — 2 on the a;-axis and — 5 on the ^-axis. 3. Passing through the points (0, 0) and (3, — 3). 4. Passing through the points (5, 6) and (3, 4). 5. Passing tlirongh the point (1, 4) and cutting off — 3 on the ?/-axis. 6. Find the equations of the diagonals of the rectangle formed by the lines X = 2, a; — — 3, 2/ - 4, ?/ = — 5. 7. Find the equations of the lines through the origin and the points trisect- ing the segment of the line i^x -\- ^ = 12 intercepted between the axes. 8. Show that the coordinates of the centers (Jj, O2, C\ of the circles escribed to the triangle with the vertices {x-^y y^), {x^, y^, (ojg, 2/3) and the opposite sides *1> *2» *3> ^^® ~ ^1^1 + ^2^2 + ^3^3^ - ^l.Vl + ^2^2 -V hVz (f(^i. Q^ oppOSitC S^\ ■^ «! + Sj -j- % ~ *1 + ^2 ~i~ **3 s^x^-8.,X2-^s^x^ ^ s^y^ - ^2^2 + ^3.^3 (for G, opposite s.), *i *2 r *3 ^1 — ^2 + *3 with similar expressions for Cg, the sign of s^ being changed. 9. Find the centers of the inscribed and escribed circles for the triaui^b whose vertices are (9, 0), (0, 12) and lO, 40). 10. Find the centers of the inscribed and escribed circles for the triangle w^hose sides have the equations 4a? — 3y = 0, ^x — Ay 4- 12 = 0, dx-\-4:y-^2 -- U. 11. Find the equations of the sides of the triangle with the vertices (-1, -2), (2, -3), (1, 4). Find the coordinates of the points which divide internally and externally 12. the line joining (1, 2) and (4, 5) in the ratio 2:3. 13. the line joining (— 2, — 3) and (- 4, 7) in the ratio 5 : 6. 14. the line joining (0, 0) and (6, 8) in the ratio 4 : 5. 15. the line joining {c -}- a, c — d) and (c — d, c -]- d) in the ratio c : d. 16. The lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another. 17. If Pi , F-i, ..., Pn denote the points (a?, , yj), (X2 , ^2), . . , , (x„ , y„), and P1P2 is bisected at the point M\; M^P-i is divided at M-^ in the ratio i : 2 ; M^Pa is divided at Mz in the ratio 1:3; /I/3A is divided at M^ in the ratio 1 : 4; etc.; prove that the final point Mn-i has the coordinates 1 1 -{xx + aJa + aJa + . . . + Xn\ ~{yx + 2^2 + 2/3 + • • • + Vn). Sec. 156] COLLEGE ALGEBRA, 163 18. Show that the area of the polygon P1P2 . . . Pn of Ex. 17 is 19. Find the area of the polygon whose vertices are (4, — 7), (5, — 2), (3,4), (1,1). 20. Find the area of the triangle with the vertices {a, b), {— a, b -\- c), {— a, c ~ b), 21. Find the coordinates of the point of intersection of the two lines join- ing the middle points of the opposite sides of the quadrilateral whose vertices, taken in order, are (0, 3), (0, 0), (4, 0), and (6, 7). 156. To find the graph of tlie equation y^ = 4:X, we observe that a negative value of x gives an imaginary value of y, so that the graph lies to the right of the y-axis. For each positive value of x til ere are two values of y, namely 2\/x and — 2\/x, so that the graph is symmetrical with respect to the i?;-axis. Plotting the pairs of points (1, + 2), (1, - 2), (2, 24/2), (2, - 24/2), (4, 4), (4, - 4), (16, 8), (16, - 8), ... , and the point (0, 0), we may trace the graph AOB in Fig. 8. The graph is called a parabola. Y A ^ ■^ ^ y^ ,^ ^-• ^ / ^ ^ \ / ^ \ (/ \ ( X X \ 1 \ X / \ \ N / ^ ^ ^ k. ^ y ■ "^ ^ ^ Fig. 8. The figure also gives the graph of the equation x^ -\~ y^ = 10:r, which may be written {x — 5)^ -\-y^ ~ 25. Since the latter is derived from X^ -(- ^2 __ 25^ the circle of Fig. 4, by writing a; — 5 in place of X, it is likewise a circle of radius 5, but referred to a set of axes 1 64 GRAPHIC. ALGEBRA; SIMULTANEOUS EQUATIONS. [Ch. XVIII of coordinates obtained by moving the previous ?/-axis 5 units to the left. Hence the graph of ^c^ + ^^ _ ^^q^ |g ^ circle of ladius 5 and center (5, 0). Its intersections with the parabola y'^ = 4:X are seen to be the three points (0, 0), (6, |/24), (6, — 4/24), found by solving the equations considered to be simultaneous. Similarly, y^ = 4:X intersects the circle x^ -{- y^ = 25 in just two points; it intersects the circle {x — 6)* -\- y^ = 25, with radius 5 and center (6, 0), in four points; it intersects the circles {x — c)'^ -{-y'^ = 25, of radius 5 and center (c, 0), in no point if c > 7^. In fact, if we eliminate y^ from the latter equation by means of y^ = 4:X, we get a quadratic for x, with the roots 6' - 2 ± |/29 - 4c. (x + 5)2 +^2 = 25 and y^ — 4x intersect only at the origin (0, 0). Two simultaneous equations of the second degree ifi x and y may have 4, e3, 2, 1, or no sets of real solutions x, y. 157. In order that the three simultaneous equations (9) a^x + h,y -\-c^ = 0, a^x + h^y -\- c^ = 0, a^x + % -\-c^ = shall have a common set of solutions x, y, the determinant n- ^1 \ ^1 Cf, \ c, (7, Z>, C^ must vanish. This follows from § 44 when we set 2; = 1. To give another proof, multiply the elements of the first column by x and the elements of the second column by y and add the results to the corresponding elements of the third column; the new elements of the third column will then be zero, if x and y be the common solu- tions. Hence must D = 0. Inversely, if B = 0, equations (9) have at least one set of solu- tions X, y in common (§ 44). In fact, the third equation (9) then follows from the first and second equations. Their graphs are Sec. 158] COLLEGE ALGEBRA, 165 straight lines which, in general, intersect in a point (ic, y). But the second and third equations both follow from the first if all the minors of D vanish (§ 44), in which case there are innumerable sets of solutions x, y in common. The three graphs are then coincident lines. For example, the lines (Fig. 9) RP OP SP 2x + y - 10 = 0, 4:X — 3y =z 0, 2x^y - 2 = The \ Y n ■\ / / R \, / / / \ / ' / \ / / \ y \ y \ n y \^ y \ / \ / \ X / / \ 's / \ / • k / \ Fig. 9. intersect in the point P, viz., (3, 4). determinant D is seen to be zero. 158. Theorem. Two points {x^y^ and (^2^/2) ^^^ ^^ ^^^^ same side or on o^)posite sides of the graph of Ax -\- By -\- O =: cxcording as the quantities Q.^Ax, + By, + C, Q,= Ax, + By^+ _ are of the same sign or of opposite signs. Through (x^ , y^) draw a parallel to the ^-axis and let it meet the graph of the given line in the point {x^ , ?//) so that, Ax, + By^'+ C = 0, y,'=-{Ax,+ C)^B. '-' y,-y^^U^^.+ By,+ C)^B^QJB. , Through {x^. y^ draw a parallel to the «/-axis and let it meet : the given line in the point {x^^ y^). As before, y^ — yl — QJB, Then (x^, y^ and {x^, y^ are on the same side or on opposite sides of the given line according as y^ — y^ and y^ — y^ are of the same sign or of opposite signs, i.e., according as Q^ and Q^ are of the same sign or of opposite signs. Corollary. The point [x^, y^ and the origin ^0, 0) are on the 1^6 GRAPHIC ALG.; SIMULTAhlBOUS INEQUALITIES. [Cn. XVITI same side of the graph of Ax -\- By -\- C —0 if, and only if, Ax^-^- By^ + C ^^^ ^ \\^YQi the same sign. 159. A point {x^, y^) lies inside, on, or outside a circle of radius r and center at the origin (0, 0) according as x^^ + ^1^ — ^'''^ < 0, = 0, or > 0. In fact, the equation of the circle is x^ -\- y^ — r^ (compare § 147). A point {x^, y^) lies inside, on, or outside the graph of fl-i- ^ - 1 a' ~^ d-' ~ X ^ -9/2 (called an ellipse) according as ~ -\- ~ — 1 < 0, =0, or > 0, €1/ In proof, draw through the point {x^, y^ ar parallel to the ^-axis and let it meet the graph in a point {x^^ y/), so that £l f li^ - 1 • For the parabola ^^ = \x whose graph AOB is given in Fig. 8, we will say that a point lies within the parabola when it lies on the same side of the graph as the line OX, By a proof similar to that just employed, we find that the point (x^^ y^ lies within, on, or without the graph of y'^ = 4:X according as y^^ — 4x^ is less than, equal to, or greater than zero. 160. The preceding results enable us to treat .very simply prob- lems in simultaneous inequalities. Example 1. Resolve the simultaneous inequalities fi~x-y+l> 0, f, = 2x-{-y-6>0, /3 = - 5^ + 2y + 10 > 0. The graphs of /^ = 0, f ^ = 0, f^ = are given in the accompanying figure. By the corollary of § 158, f^ is positive for all points on that side of r 8ec. 160] COLLEGE ALGEBRA. the graph of /^ = on which the origin lies, tive only for the points on the opposite side three inequalities are therefore satisfied simultaneously by all the points in the shaded triangle and by no other points. The vertices of this triangle are (f, |), (4, 5), p/, Jg"). Hence the solutions X, y are such that Similarly for/g; but /a is of /2 = from the origin. 107 posi- The < ic < 4, 0, x^ - 10a' 4- y2 ^ 0. The sets of values x, y must be such that the corresponding points {x, y) lie outside of the parabola y^ — ix and inside of the circle {x — 5)'^ -|- y'^ = 25. These points therefore lie within the two crescent-shaped areas in the figure of § 156. Hence the solutions are V2i Ax, , 0, -3x -\- 2y i- 6 > 0, ic + 2^ - 4 > 0. 10. X - y -]- 4: > 0, X - 2y -\- 5 < 0, a- + 5y - 4 < 0. 11. X + y - d > 0, 2a; - 3y + 1 > 0, 2a' - 3^ - 5 < 0. 12. x^ + 2^2 - 25 < 0, 2a H- 2/ - 10 > 0. i;;. a2 H- 2/' - 25 < 0, 2a + 2^ - 10 < 0, a + 2^ - 2 > 0. 14. y^ < 4a, {x - 6f -f y' < 25, y > 0. 15. The equation to any straight line through the intersection of \ax -{- by -\- c = and Ax -{- By -\- C = may be written (ax -\- by + c) i- m{Ax -\- By -{- C) = 0. 1 68 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS. [Ch. XVIII 16. Find the eqaation of the straight line th»")ugh (0, 1) and the inter- section oi y -\- X = with x — 'dy -[- 1 = 0. SOLUTION OF NUMERICAL EQUATIONS. 161. Consider first the quadratic equation i?;^ — 6a; — 3 = 0. The graph of y = o;'^ — 6a; — 3 is given in the figure. It is roughly deter- mined by the points ( — 1, 4), (0, — 3), (1,-8), (2, -11), (3, -12), (4,-11), (5, - 8), (6, - 8), (7, 4). The sym- metry with respeot to the line x — ^ fol- lows from the form y -\- 12 = (x — 2y in which the equation may be written. The dotted lineF "have the equations l_Y i r J 1X4 \ ! X 4 it 1 4^ m ± T t r ^ L V J - X ^ 2 4 iL i ^It t - X-^4 4^4 ) ' / IV|/ y'~y + y^ 0, X' ^x • 3 = 0. Keferred to them as new coordinate axes, the equation is y' = x''^. To this form any equation of the form y = x^-{- px + q may be reduced The graph is called Fi<*- "• a parabola.* To solve the equation ^2_g^_3 _ Q consists in finding the values of x which makey = 0. In the graph, it consists in finding the a:-coordinates of the points in which the graph crosses the a;-axis. The one lies between and — 1, and the other between 6 and 7; the exact values are evidently 3 ± |/l2. It is clear that all quadratic equations in which tlie coefficient of x^ has been made unity, lead to the same graph, the variation being the relative position of the axes of coordinates. 162. Consider the cubic equation x^ + ^x^ — 7=0. The graph of y =z x^ -\- ^x^ — 7 is given in the figure, as sketclied through the points (- 4, - 7), (- 3^, - -J), (- 3, 2), (- 2^, 2|), (-2,1), (-1|, -1|), (-1,-4), (-^,-61), (0,-7), * The parabola y'^ = 4a;, whose graph is given in § 156, may be reduced to the above form by taking x' — y, y' — 4x. This transformation turns the picture through 90" and magnifies the resulting figure in the vertical direction. Sec. 163] COLLEGE ALGEBRA, 169 Y r~\ ' \ i / X / / / / / / / {h - H)> (1. - 2). (1-1. - -829), (1.2, .488), (1.25, 1.203), (1.5, 5|). The roots of the cubic lie between 1 and 2, between — 1 and — 2, and between — 3 and — 4. More closely, the positive root lies be- tween 1.1 and 1.2. To determine the second decimal place, the process of direct trial of 1.11, ..., 1.19 becomes laborious. To shorten the work, set x = 1.1 -{- z, whence the relation becomes (10) y =^ z^ -\~ 7.3^2 _|_ 12.43^ - .829. In determining z to make y = 0, the terms z^ and 7.3z^ are of little conse- quence, since z is of the denomination hundredths. We should expect, there- fore, that z lies between .06 and .07, being given approximately by 12 A3z — .829 ~ 0. In fact, y is negative when z = .06 and is positive when z = .07. Setting z = .06 -|- w, the relation becomes (11) y = tv^+ 7ASw^ + 13.3168W - .056704. Using the last two terms only, we find that ?/ = if w = .0042 +, to four decimal places. In verification, to = .0042 makes y nega- tive and w = .0043 makes y positive. Hence the positive root of the cubic is 1.1642 +. 163. The calculations may be shortened by Horner's method,* which is based upon the process called Synthetic Division. The process of the division of x^ -f 4:x'^ — 7 by a; + 3 may be exhibited as follows: Fig. 12. x^ + 3x^ X^ — '7 X -{- 3 = divisor x^ -\- X — 3 = quotient X^ + Zx - 3x - 3x - 9 + 2 = remainder. * Published in 1819 by W. Gt. Horner. I70 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS, [Ch. XVIII Since the literal terms x^, x^^ x may be supplied from their relative positions, we may indicate the work by detached coeflicients : 14 1 3 - 7 1 1 3 -3 ~ 3 - 9 1 3 divisor LI — 3 quotient 2 remainder. We agree to omit the first term of each partial product (1, 1, — 3, respectively) since it is merely a repetition of the number above it. Next, if the second term of the divisor be changed in sign, the new divisor being 1 — 3, the sign of the second term of each partial product will be changed, so that the subtraction previously per- formed is replaced by addition. The work for the division by a; -}- 3 thus becomes the following : 1-3 + 9 2 Arranging the work in a horizontal row and omitting the coefficient 1 in the divisor 1 — 3, the work stands thus: Dividend =1 4 0-7 Partial products = —3—3+9 Quotient =1 1—3", 2^ - 3 remainder. In this final form the process is called synthetic division. We next divide 3a;' +• x^ — ^Ix^ +21^ _ 5 by a; — 2. Dividend ==3 1 0-31 21-5 Products ^ 6 14 28 - G - 12 - 24 - 6 + Quotient = 3 7 14 - 3 - 6 12 - 3, II: ; rem. Sec. 164] COLLEGE ALGEBRA. 171 # The quotient is dx^ + Ix^ + 14.t* _ 3^^ ~ 6^2 _ i^x - 3, and the remainder is — 11. 164. Ee turning to the example and graph of § 162, we see that one root is a; == 1 + ^^, n being a positive decimal. Then y={\ + uf + 4(1 + uf _ 7 = ?^3 + 72^2 + 11?^ - 2. Without the expansion just made, the final form ii^ -{- , , .—2 may be obtained by synthetic division. If the degree of the given equation is large, say 6 or 7, tlie second process is much shorter than the first. Since ii = x — \^ we have ^ = 2;3 + 4a;2 - 7; y= (x - 1)^ + 7{x - If + ll(a^ _ 1) _ 2. Hence if we divide the first expression by x — 1, the remainder must be — 2, as is the case with the second expression. The respective quotients are x^ -{- dx -{-5 and {x — 1)^ -|- 7{x — 1) -f- H? and they must be equal. Dividing them by iz: — 1, the quotients are x -}- 6 and {x — 1) + ^y ^ii^ the remainders are each 11. It is thus clear that in the division of x^ + 4^^ — 7 by :c — 1 the suc- cessive remainders — 2, + H^ '^» 1 must be coefficients in reverse order of the transformed equation tt^ + '^^^^ + H^^ — 2- ^Y syn- thetic division this work may be done thus: 1 4 0-7 I 1 15 5 5 5, — 2 = first remainder 1 6 1 6, 11 = second remainder !_ 1 , 7 = third remainder. The general theorem involved is stated in the next section. 165. Giveny = p^x"" -{-p^x''-'^-^- p^x''-^^ + • . • +Pn-i^+Pn «'^^ X =^ u -\- Ji, to oMain the expression for y in terms of u, loe divide p^x'' -\- '-'-{- Pn ^y ^ — h and call the quotient Q^ and the remainder 172 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS, [Ch. XVIII rp then divide Q^hy x — h and call the quotient Q^ and the re- mainder r^, etc. The reqm^^d expression for y is Po"^"" + ^n^f^''~' + . . . + r,u^ + r,u + r^. In proof, we observe that the final expression equals F^ plx - hf + rr,{x - 70— ^ + . . . + r,{x - hf + r,{x--Ji) + r,. To prove that E is another form of the initial function y z=z p^x"^ -\- , , , -\- p^^ we divide both E and y hy x — h. By inspection, the quotient ol E is p,{x -« hf-' + r^[x - hy-^ +. . ,+ r,{x - h) + r, and the remainder r^ By the notations of the theorem, the divi- sion of y = PqX^ -{-... -\- pnhj X ~ h yields the quotient Q^ and the remainder r^ Hence will E = y if Qi =Po{^ - ^'T-' + rn{x - hy-' + . . . + r,{x - A) + r,. Dividing each member hjx — h^ the quotients are respectively (12) (?„ Po{^-'^Y~' + rn{x-hf-'+,.. + r, and the remainder is r^ in each case. Hence will E — y if the ex- pressions (12) are equal. After n — \ such divisions, the quotients are Qn-\ and pj^x — A) + ^^^ and E will equal y if these quotients are equal. Dividing them by a; — A, we obtain in both cases the quotient j»j) and the remainder r„. Hence E-^y, A more direct proof follows from the remainder theorem (§ 27). Assume that y ^ E, where the coefficients r„, . . . , r^, r^ of E are as yet undetermined. Setting x = h, vfQ get A^" +i^i ^""' + . . . +i^n-i/^ + ;?n = ^. But the left member is the value of y when h is written for x and is therefore the remainder left on dividing ^ by ^ — h (§27). By inspection, the right member r^ is the remainder left on dividing E hy X — h. Since the two remainders are equal and the two divi- dends y and ^are assumed to be equal, the quotients must be equal: Qi - Poi^ -hr-' + rn{x -hy -' + ... + r,{x ^ h) + r,. Sec. 166] COLLEGE ALGEBRA, 173 Proceeding as before, the division of Q^h-^ x — h leaves a remainder equal to r^ ; etc. Hence the undetermined coefficients Vn, - * - , r^, r^ in U turn out to have the values given in the theorem. 166. Kesuming the equation y = u^ -{- 7u^ -\- llu -- 2 of § 164, where ti is a decimal, we seek the value of u which makes y — 0. The powers u^ and v? are small relative to llu — 2. Hence u is approximately .1. In verification, we observe that y is negative for u = ,1 and positive for u = .2 and hence is zero for some intermediate value (by the graph or by the formal proof in § 170). Setting 7t = ,1 -\- z, the equation for z is found by syn- thetic division (§ 165) as follows: 1 7 11 - 2 I J^ .1 .71 1.171 1 7.1 11.71 , — .829 = first remainder r^ .1 .72 1 7.2 , 12.43 =: second remainder r, .1 1 , 7.3 = third remainder r^ .'. y = z^ + "T.dz^+UASz - .829, a result identical with (10). Hence, as in § 162, z= ,0Q -\- w, where w is of the denomination thousandths. By synthetic division, we form the equation (11) for 10 : 1 7.3 12.43 - .829 .06 V-. . .06 .4416 .772296 1 7.36 12.8716 , - .056704 .06 .4452 \ 1 7.42 , .06 13.3168 V\ 1 , 7.48 y=w^ + 7.48?(;2 _|_ 13.3168^; - .056704. Then ^ = for «^ = .0042 -{-, so that the positive root is 1.1642 +. 174 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS. [Ch. XVIII 167. For a negative root — x oi x^ -\- Ax^ — 7 = 0, we may solve the corresponding equation x^ — 4cX^ -[-7 — with the root -f- ^) or we may proceed with the given equation by the usual method. Thus for the root lying between — 3 and ~ 4, we set X == — 4 + ^, where t is positive and of the denomination tenths and proceed as usual. The work is exhibited compactly as follows : 4 ~7|-4 -- 4 - 7 |_j6 - 4 16 -4 ~ 4 16 •^ 9 - 8 .6 ~ 4.44 6.936 - 7.4 11.56 — .064 .008 .6 - 4.08 - 6.8 , .6 7.48 •*■ y - 6.2 .008 ~ .049536 .0594437 - 6.1.92 7.430464, - .0045563 .000617 .008 - .049472 - 6.184, ,008 7.380992 ^ 9 - 6.176 Hence, to six decimal places, the root is— 4+. 608617=— 3.391383. It is seen that the final terms 16^ — 7 of the first transformed equation do not suffice to determine t very accurately; whereas in the later transformed equations the last two terms determine the figure of the root sought. Similarly, the third root is — 1.7728, to four decimal places. \ I Sec. 168] COLLEGE ALGEBRA. 175 168. Consider as anew example the equation y=x^-\-&x'^-\-8x-\-S, whose graph is given in Fig. 13, liaving been traced through the points (1, 23), (.5, 13|), (0, 8), (- .5, 5f), (- 1, 5), (- 1.5, 6i), (- 2, 8), (- 3, 11), (-4, 8), (-4.5, 2f), (-5, -7). By accurate plotting in the vicinity of the two bends in the curve, we find that they occur at x = — .845 and x = — 3.155, approximately.* The corresponding y- coordinates are 4.9209 and 11.079, respec- tively. The fact that the lowest point of the bend which points downwards 'is above the X'Uxi^ follows more simply from the form y z=z (x -\- "^Y — 4:X in which the equation may be written. Hence, for — 2 < X < 0, X -{- 2 and — ix are posi- tive and therefore also y. On account of Fig. 13. the double bend, the graph crosses the :?;-axis but once; for, if the branch below the :r-axis should bend and cross the axis again (necessarily at the left ot x = — 5), or if the upper branch should bend and cross the ic-axis (necessarily at the right of X = + 1), it would be possible to draw a straight line which would meet the graph in at least four points. To show that this is impossible, we note that the equation of the straight line has the form y = 7nx -{- n. The ^-coordinates of its intersections with the curve are given by mx -{- 71 = x^ + 6x^ + 8:r + 8. Hy § 77, this equation of the third degree has at most three roots. To give an algebraic proof, we observe that, for :?; < — 6, :^ -f- 2 < — 4, so that the condition (x + ^Y < 4^ will be satis- fied if - {x + 2Y < X, viz., if - (2; + f)2 < - (3)2^ which is Y "o T ^ By the Cfalculus, the values of x are the roots — 2 ± J 4/I2 of the equa- tion 3^2 4- 12aj + 8 =r 0. 1 176 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS. [Ch. XVIIl true, li X > 2, then i?; -|- 2 > 4, and the condition {x -f 2)^ > 4a? will be satisfied if {x + 2)^ > x, viz., if [x + If > J, which is true. I 169. Descartes' Rule of Signs.* An equation f{x) = zvith real coefficients cannot have more positive roots than there are changes of sigji inf{x), and cannot have more negative roots than there are changes of sign in f{— x). For example, the terms of f{x) ~ -\- x^ -{- 6x^ + 8^ + 8 = all have + signs, so that there are no positive roots; there are three changes of sign in /( — x) = — x^ -^ 6x^ — 8x ~{- 8 =z 0, so that there are not more than three negative roots. The equation f{x) ^x^ — 6x^ — r^-s _|- 7:^; _ 2 = has at most three positive roots, and at most two negative roots since f{-x) = -x^-5x^ + x^-'7x-2=zO has two changes of sign. Hence the equation must have at least 9 — 5 ^4 imaginary roots [see § 181]. Suppose, in general, that the positive roots of f{x) = are ^1, fljg, . . . , «p. Then /(a;) is divisible (§ 27) by the product {x — a^{x — a^ , . , {x — ap). If the quotient is Q(x), the roots of Q{x) = give all the negative and imaginary roots of f{x) = 0. Let the signs of the terms of Q{x) be, for example, + +^_- + + -+. To find the signs of the product {x — a^)Q{x), we employ as multi- plier a binomial with the signs -| . The signs of the products are therefore Sum: +±-q:+_qi:p + -4-- * Descartes' La Geometrie, Ley den, 1637. For an account of his life and work, see Ball, History of Mathematics, pp. 270-279. f Sec. 170] COLLEGE ALGEBRA, 177 We observe that in the product an ambiguity of sign replaces each continuation of sign in Q{x), the signs before and after an ambiguity or set of ambiguities are different, and a change of sign is intro- duced at the end. Hence for every isolated ambiguity and the sign just preceding and just following it, the number of changes of sign (whether we take the upper or lower sign in the ambiguity) is the same as for the three corresponding signs in Q{x). An analogous result holds for a set of ambiguities if we take all the ambiguities in the set to be alike. But if the latter are not taken alike, there is one or more additional changes of sign in the product than in Q{x), There is always an additional change of sign at the end. Hence the product contains at least one more change of sign than Q{x), After multiplying in the p factors x — a^, x — a^ , , . . , x — a^, we obtain /(x), which therefore has at least ^ more changes of sign than Q{x), Hence /(a;) has at least jy changes of sign. The number of positive roots is thus '^ c, c being the actual number of changes of sign in f(x). The negative roots of f(x) = are the positive roots of f(^—x) = 0, so that the number of the former is less than or equal to the number of changes of sign in/(— x). Thus x^ — f^x^ -^ x-\-l = has at most two positive roots and at most two negative roots and consequently has at least two imaginary roots. . 170. Consider the graph of y =f(x), where f{x) is a rational integral function of x (§ 76). H there be two points of the graph, one of which is above the x-^xis and the other below, it may be proved that the graph must cross the x-^x\s at one or more intermediate points. Thus if f{a) is positive and/(Z>) is negative, there is some value of x intermediate to a and h for which f(x) is zero. The proof consists in showing that, as x changes gradually from a to ^, f{x) also changes gradually and assumes all interme- diate values between /(a) and/(5), so that in passing from a posi- 1 178 GRAPHIC ALGEBRA; NUMERICAL EQUATIONS, [Ch. XVIII tive to a negative value it passes through the value zero. While this is readily proved (see example below) when /(a;) is of the form x"" +pX-'^+ . . . +i?n, the case under consideration, it is not true for all functions /(^). For example, it fails for the functions y — 2\/x and y — \/li)x — x^ whose graphs were constructed in § 156, since there are two values of y corresponding to each value of x. Consider the example f(x) = x^ -\- 4:X^ — 7. As x changes from a to a + /^^ f{^) changes from f(a) to /{a + ^0' ^^^ increase being { (^ + /O' + 4(a + 7^)2 _ 7 [ - ] a^ + 4«2 _ 7 [ | = (3a2 -f. Sa)h + {3a + 4)¥ + h\ ' If h is small, this difference is small, and by taking h sufficiently small, this difference can be made as small as we please. Hence if x obliges by small amounts from a to b, f{x) will change by small amounts from/(^) to f{b). Moreover, a curve joining two points which lie on opposite sides of the :r-axis must cross that line an odd number of times; but m!ret cross it an even number of times, or not at all, when the points lie on the same side of the ^-axis. We may state the theorem: If f{x) he a rational integral function with real coefficients, and the values f{a), f{h) given ly siihstituting real numbers a, b for x are of opposite signs, the equation f{x) — ?ias an odd number of real roots lying betweeyi a and b; but has an even number of real roots or no real root lying betiueen a and b, if f{a) and f{b) are of like sign. Example 1. Every equation of odd degree has at least one real root of sign opposite to that of the constant term. Thus x^ -f ^^^ — 7 = was seen to have a root between + 1 and + 2. Let the equation be /(a;) = x^ -\- aa?'^- ^ -\- , . . -\- c = 0, and let x approach the values — 00 , 0, and -f go . The sign of /(O) is the same as that of c. To find the sign of/(± oo), divide /(ic) by a!"-i, which is positive since 7i — 1 is h c even. The quotient x -\- a -\- + • • • ^ — ^rri differs from ic by a function which approaches zero as x approaches ± 00 . Hence f{-\- 00 ) is positive and I Sec. 170] COLLEGE ALGEBRA, 179 f^— oo)is negative. Hence, if c is positive, /(ir) changes its sign between a? = — 00 and x — and therefore /(a;) = has a negative root. If c is nega- : tive, f{x) changes its sign between x = and x = -\- a:^ , ^o that/^a?) = has a. : positive root. Example *2. Every equation of even degree, whose constant term is nega- tive, has at least one positive and at least one negative root. Upon substituting — 00, 0, + 00 for x, the results are -f cjo , c, -j- 00 , re- spectively, w^here c is negative. Hence there is a real root between — 00 and 0, and a real root between and + ^ • (EXERCISES. 1. Plot the graph of y - x^ -{- 2x — \ and solve iu^ + 2« — 1 = 0. 2. Plot y = a^ -\- 2x -\- 20 and find to five decimals the root of V-f-- 2x + 20 == such that - 8 < a? < - 2. 3. Plot y = x^ -{- X — d and find to five decimals the root 1 < oj < 2. k4. Plot y = x^ — 4lX, What are the roots of x^ — Ax = '^ 6. Show that the graph of y — x^ -]- Sx^ -\- 2x -^ 1 crosses the aj-axis but once. 16. Find the roots between 1 and 2,-1 and — 2, of x^ -^ dx"^ - 2x - 6 = 0. 7. Find the root.between 2 and 3 of a!^ + dx^ - 80^ - 18 = 0. 8. Find the real roots of a^ -{- 2x^ — x - 1 = 0. 9. Find the real roots of «* + a;^ — a; — 2 = 0. 10. Find the real roots of x^ + 4x^ -j- Qx^ - Sx - i = 0, 11. If the terms of an equation are all positive, it has no positive root. 12. If the terms of an equation, with all powers of x present, are alter- nately positive and negative, it has no negative root. 13. The equation x^ -f a^x -{- b'^ = has one negative root and two imagi- nary roots. 14. Ifn is even, a?^ -J- 1 = ^ has no real root ; if 7i is odd, — 1 is the only real root. 15. The equation a^ -\- 2x^ -\- dx — 1 = has at least six imaginary roots. 16i x^ -\~ 12ic^ -\- 6x — 9 = has a positive root, a negative root, and two imaginary roots. 17. For n even, ir^ — 1 = has only two real roots ; for n odd. only one. 18. aj^ — 7a;* — 3«2 + 7 = has at least two imaginary roots. CHAPTER XIX. THEORY OF EQUATIONS. 171. The solution of the general quadratic equation (see § 31) was known to the Arabians as early as the ninth century. The solution of the cubic equation x^ -\- qx -\- 7' = was discovered by Scipio Ferreo, who imparted it to his pupil Florido, in 1505. The latter proclaimed his knowledge of the solution upon learning that Tartaglia had solved the analogous equation x^ -f mx'^ -f ^ = 0, But Tartaglia doubted the truth of his statement and challenged him to a trial in the year 1535, having succeeded in rediscovering Ferreo's. solution. Then Cardan appeared and solicited from Tartaglia his rules for solving various cubic equations and finally succeeded in getting them after giving the most sacred promises of secrecy. But in 1545 Cardan broke his promises and published the rules in his Ars Magna. . Tartaglia had intended to publish his rules in his own work, but died in 1559 before the part on cubic equations was reached, so that no mention is made by him of his discoveries. In time the rules came to be regarded as the discovery of Cardan and to bear his name. 172. Consider the cubic equation in the most general form (G) x^ + ax^ + hx + c=.0. Setting a; = y — —, the equation becomes o (R) f+[^-YF + ['-T + ^) = ''' II i8o Sec. 172] COLLEGE ALGEBRA. l8l an equation lacking the square of the unknown quantity, and called the reduced form of equation (G). It suffices to solve the equation in y, since the values of x are then given by the relation x — y — a/3. Consider then the reduced cubic equation (1) ^ ^'+i^i/ + ^==0. Making the substitution (2) y = z- P 3«' the cubic (1) becomes ,8 P' + r = = 0. Multiplying by 2;^ the latter becomes (3) 2-6 ^ ,.j;3 _ 27 0. Considering z^ to be the unknown quantity, equation (3) is a quad- ratic equation, the solution of which gives z^ and then (4) z - y^- '- ± ^R, R = '^+^- Kemembering that a quantity q has three cube roots * (§8), l/'q,^ 00 p'q,^ od'^ l/q' (ct) = — | + | / - 3, cy^ — 1) it would seem that the six values of z, given by (4), would lead to six roots y, given by (2), of the cubic equation (1). But (-i+^s)(-;--*'^)=T-^=-4 Hence, if the notation of two of the cube roots (4) be definitely fixed in such a manner that * The vertical bar at the right of the radical sign is used to mark the choice of a particular root. I«2 THEORY OF EQUATIONS, [Cn. XIX then the remaining four cube roots (4) may be paired so that the product of the two in each pair is — i?/3, viz., V-i p + i/R . oo^!/-L-^R =_-|, 'lJ-\ + VR .-^- ^=-f But the vahie of y given by (2) remains unaltered if we replace z by^; mdeed. 3z 'd ' [ dz J 'dz' Hence the two values of z in any one of the three pairs give the same value to y, and the value of y equals the sum of the two^ values of z in that pair. Hence the three roots y are (5) 3 /_ r /— ' ^ I — r /- ' Y-^^^'-'-'v 2 /i^, f |/ii + CB* / -^ - ^E. 2/3- -A/ 3 Example. Solve the reduced cubic equation ^^ _ 15^ _ 126 = 0. Set V = 2 — (^^^] = 2 + -. Then z^ — 12Qz' + 135 = 0. \ Sz J 2 ... (23 _ 1)(23 - 125) = 0. • . 2 = 1 fij, oi)^, 5, 5(», or 5(»^ Then y = z -^ — equals respectively ' z 6 00 H = GO -r 5gi3^ g?^ + 5GJ, GD 5gi? -|- gi?^, 5gi3' -|- go, Sec. 173] COLLEGE ALGEBRA, 183 giving only three distinct roots y. Since ca^ -[- &? -f 1 = (§ 8), they are 6, — 4a3 — 5, 4o!9 — 1. Since go = — i + ii^ — 3, they become 6, _3_2|/-3, -3-|-2|/-3. The two imaginary roots are conjugate (§ 7). 173. Formulae (5) for the roots are known as the formulae of Cardan. The method used for obtaining them is essentially that given by Hudde in 1650. According to the value of R^ — + ~, we have, when p and r are real, for R posiiive, one real root and tivo conjugate imaginary roots ; for R equal zero, all roots real and two of them equal; for R negative^ all three roots real and distinct. In proof, suppose first that 7? > 0. Let the radicals be A and B so that the roots are A-\- B, ooA -\- od^B^ od^A-{- goB. Since '=-i+i4/-3, G92^ -i-i4/-3, the three roots are A + B and - i{A + B) ±i{A- B)\/ - 3. Since A and B are now real, one root is real and the others are con- jugate imaginaries. I li R=0, the roots are A -{- B,- ^{A -^ B), - \{A + B), and hence are all real. If i? < 0, the radicals in formula (5) are imaginary. But (see Appendix) there exists a cube root of — - + \/R of the form a + ^\/ — I, a and /5 being real, the other two cube roots being QD{a -\- 15\/ — 1) and Go^[a + /?|/ — 1). The conjugate imaginary — - — \/ R will then have as its cube roots a — (3 \/ — 1, , fd^[a — /Jj/— 1), GD{a — /3\/ — 1). Hence the roots (5) may be written (since the product of the two terms in any root is the real number — ^/3): 1 84 THEORY OF EQUATIONS, [Ch. XIX (a + /^|/^=3) +{a- Ps/~^\) = 2^, G?(ar + /?!/- 1) + G^(ar - p\/~^^\) = - a- j3\/3, These real roots are readily seen to be unequal. Thus when the roots of a cubic equation are all real and unequal, Cardan's formulae present their values in a form involving imaginaries. In eliminating these imaginaries, it was necessary to express the cube root of a complex number in the form of a,-{- ^ ^ — \, If we attempt to do this algebraically, we find that a, /? depend upon equations similar to the proposed equation. For example, a depends upon a cubic equation all of whose roots are real, so that its roots would depend upon expressions as complicated as a itself. Hence the case when all three roots are real is called the '* irreducible case," since Cardan's formulae then have no prac- tical value. But this case is precisely the one which can be readily solved by trigonometry. [See Appendix.] 174. Since gj^ -{- oo -\- 1 = 0, we find by formulae (5) that 2/i + !/2 + !/3 - (1 + ^ + ^%^ + £) = 0, y^y^Vz = {^ + ^)(^' + ^' + oj'^B + gdAB) = A' + B^=^r, 3 / _ r ,— ^ I — r — ' where we have set J[ = a / -— 1- y ^, ^ ~ \ / -^ \/R. Hence in a cubic equation lacking the term y^, the sum of the roots is zero, the sum of the products of the roots taken two at a time is the coefficient of y, and the product of all the roots is the negative of the constant term. I Sec. 175] COLLEGE ALGEBRA, 185 EXERCISES. Solve the cubic equations 1. a? - l%x + 35 = 0. 2. x^ - 15a; + 4 = 0. 3. 0!^ - 27aj + 54 = 0. 4. x'' -f 6:r« + 3a; + 18 = 0. 5. 28ar* + 9a;2 - 1 z= 0. 6. a;^ _|_ 4^2 _ ^^j + 6 = 0. 7. Since a;, = .Vi — a/3, x^ — y^ ^ a/3, aJs = ^3 — a/3 are the roots of the general equation (G) if y, , ^2 , ys are the roots of the reduced equation (R), show that aJi + a^a + a^s = — a, a^iaja + a^ia^a + X'^x^ = b, XiX^Xz = — c. 175. The solution of equations of the fourth degree was at- tempted by Cardan, but was first accomplished by his pupil Ferrari. It is sometimes ascribed to Bombelli, who published it in his algebra in 1579. Other solutions have been given by Descartes in 1637 and by Euler in 1770 and by many later writers. 176. FerrarVs solution of the quartic equation (6) a;^ + px^ -{- qx^ -{- rx-\- s = 0. The equation may be written {7? + |:rj = {^^ - q^x^ ^rx--s. In case the second member is not a perfect square, add to each member [7? + ^Ay + -f-- Then (7)(.^+f.+i;=(.+^-.>'+(f-r).+(i^-.). The second member will be a perfect square (§ 31) if ■ Hence y must be a root of the cubic equation (8) y^ — qy'^ + {'pr — 45)y + ^^^ — jf'S — r^ = 0, Let y^ be a root of (8) and set 2/j,+ ^ •— ^^ = 1". The right mem- iS6 THEORY OF EQUATIONS, [Ch. XIX ber of (7) may be written ^L'+Kf---)/^]* Hence (7) factors and gives (9) \ Solving these quadratics, we obtain the roots x^, x^; x^, x^ of (6) We observe (§ 32) that Hence ^1 + ^2+^3+^4 = -P^ ^,^2^,^, = -^ I ^i' - \^~ - rj ~ T^ =8, ^1^2 + ^1^8 + -^1^4 + ^2^8 + ^2^4 + ^3^4 ^^ ^1^2+ (^1 + ^2/ (^3 + ^4) +^3^4 ^^ S'' XJX20!yo'~\*^i^2l~\ 1 Z i~i~ 2 Z A 12x3 "I 4/ ~t~ ^^^a\^i ~\ ^2) ~~~ For a general quartic equation (4), the sum of the roots is — p^ the sum of the products of the roots two at a time is q^ the sum of the products three at a time is — r, the product of the roots is s. Example. Solve the quartic equation x^ + 2a;' — 12ir^ — 10.^ -f 3 = 0. The auxiliary cubic (8) is here y^ -f- 12^^ — 32y — 256 = 0, one of whose roots is — 4. Taking y^ = — 4, we get T = 9. Hence equations (9) are x^ -\- Ax - 1 - 0, x^ - 2x - d = 0, whose roots are — 2 ± |/5 and 3,-1. P 177. If we set rr = 2; — -^ in (6), we obtain for z a quartic equa- tion lacking the term z^. Consider then the reduced form (10) z^+Qz^+Ez + S=^0. Sec. 178] COLLEGE ALGEBRA. 187 Descartes' solution consists in factoring tlie left member of (10) into (z^ -\-lcz^ l){z^ -Jcz+ m) = z^+{l + m- k^)z^ + h(m - l)z + Im. In order that this expression shall be identical with the left member of (10), the corresponding coefficients must be equal: Z + m - F = Q, ]c{m -I) = E, Im = S. Substituting the values of I and in derived from the first two rela- tions into the third relation, we get {Q + Jc^-\-B/k){Q-\-]c'- R/1c)=^4.S, We have thus a cubic equation for P. When it is solved, we derive the corresponding values of I and m. Then the roots of (10) are given by the solution of the two quadratic equations z^+kz + l^O, z^-kz + m = 0. For example, to solve z* — 'dz^ + 6;^ ~ 2 — 0, we have the con- ditions / + m - F ^ - 3, k{?n - /) = 6, Im = - 2. . •. k^ - 6k* + 17^- -36 = 0. The latter is satisfied by k^ = 4. Taking k = 2^ we get m — 2, Z= -1. • ... z* ^^z^ + 6z-2 = {z^ + 2z- 1)(^2 _ 2^ + 2). The roots are therefore — 1 ± |/2 and 1 ± |/— 1. 178. Elder's solution of tlie quartic equation (10) consists in assuming as the general expression for a root Squaring and transposing the terms free of radicals, we get z^ — u —V — tv — 2(|/w 4/^ + V'^^ V'^ + V'^ V^^ )• 1 1 88 , THEORY OF EQUATIONS, [Ch. XIX Squaring the latter and reducing, we get z^ — 2{u-\-v-\- 7v)z^ — s{u]/v \/w + v\/u \/w + 2Vi/u i/v)-{-A = 0, where A = {u -{- v -\- wY — 4:{tiv + tiw + vw). ,\ z'^ — 2{u + V + iv)z^ — Sz \/u \/v \/tv + ^ = 0. Comparing the latter with equation (10), we get u+ V -{- w = — IQy \/u \/v \/w — — \R, A — S. Hence, by Exercise 7 of § 174 or by the theorem of § 182, w, v, w are the roots of the cubic equation t' + W + {M' -\s)i- i^i^ = 0. Since the expression for z involves three square roots, it might seem that the quartic equation would have eight roots. But the signs of the square roots must be chosen so that 4/?^ \/v \/w = — IE, so that only two signs are arbitrary, giving four roots. EXERCISES. Solve the quartic equations 1. 2* - 22^ _ 82 - 3 = 0. 2. x^ - 2x^ - W + 8aj + 12 = 0. 3. 2* - IO22 - 2O2 - 16 = 0. 4. a^ - Sx^ + 9x' -{- Sx - 10 = 0. 5. If a quartic has two equal roots, the auxiliary cubic has two equal roots, and inversely. 6. If a quartic has two distinct pairs of equal roots, then two of the roots of Euler's auxiliary cubic are zero. 7. The roots of the quartic (6) are l{- p ± |/Z ± i^B ± |/c), all the signs being -f , or two signs — and one sign -f- , if ^, i?, C are the roots of W^ + {Sq-dp^) W + {Zp'^-X^p'q + 16^2 _|_ jg^^ _ ^4^) ^_ (^3_4^^ _|_ g^^2 ^ q. 8. Solve the general cubic by multiplying in a new factor x~\, and using A to make the auxiliary cubic solvable by inspection. 179. During the eighteenth century many unsuccessful attempts were made to solve the general equations of the fifth and higher degrees. In 1770 Lagrange analyzed the methods of his predeces- sors and traced all their results to one principle and proved that the Sec. 179] COLLEGE ALGEBRA. 189 general qiiintic equation cannot be solved in this way, since the auxiliary equation is of the sixth degree and of essentially general character. It was then proved * by Abel, Wantzel, and Galois that it is impossible to solve by algebraic means the general equation of degree n for n > 4:. But the equation of the fifth degree may be solved in terms of elliptic functions f (Hermite). The roots of cer- tain equations of the fifth degree may be expressed by radicals. X Example 1. Find all the roots of ^ — 1 = 0. The four j-oots different from the root 1 satisfy ^""" \ = a;* + a^ 4- a;2 + a? + 1 = 0. X — 1 If we divide by 01^ and set a? -| = y* we find that ... a.'^_|-(-l ± |/5") + l = 0. Hence there are four imaginary fifth roots of unity : i{_l-p |/5 ± |/l0-|-2|/5 i/-l!, il-1- 1/5 ±|/lO - 2|/5 V -^\ If we denote one of them by e, the others are e^, e^, e*. Indeed, since e^ = 1, then {e'^f = 1, (e^)^ = 1, {e^f = 1. Also no two of them are equal; for ex- ample, e = e^ gives e — ± 1, e = e* gives e^ — e'^ = 1. We notice that 1 + e + 6-2 + e^ 4- e* = 0, l-e-e^'e'^'e* = e^o = 1. Example 2. Solve the equation ^ -]-pf + Wy + ^ == 0. Substituting (11) y=^'- S' * See Serret, Cours d'Algebre Superieure, t. 2, sec. Y, ch. II. f On this and allied subjects, see Jordan, Traite des substitutions et deS equations algebriques, Paris, 1870 (pp. 370-382). if Concerning solvable quintics see McClintock, American Jour, Math., Vol. 6, p. 301. 190 THEORY OF EQUATIONS. [Ch. XIX the equation becomes We therefore obtain an equation of the second degree for z*. Hence Proceeding as in § 173, we obtain the five roots Y - y + v« + e^-'\/- f - i^e a = 0, 1, 2, 3, 4), where e denotes an imaginary fifth root of unity (Example 1). The same method is to be followed in working Exercises 9-13 below, the substitution being y — z — ~ for an equation of degree n, as in (2) and (11). Example 3. The equation y^ -f py^ -f qif' -\- qy'^ -\- py ^ \ — may be solved by algebra. It has the root — 1. Dividing out the factor y -f 1> we obtain the quartic equation y'-\-{v- 1)^/' + (?-;? + i)y' +(^- i)y + 1-0. The latter may be solved by the earlier methods or more simply by setting y A — z, whence y'^ A z=z z'^ — 2, so that y y z^A^{p-\)z-^q-p-l = 0. Then y = \{z -\- \^z^ — 4) gives the four values of y. 180. A reciprocal equation is one which is not altered by re- placing the variable by its reciprocal. The equations solved in ex- amples 1 and 3 of § 179 are reciprocal equations. Just as these special equations were solved by means of equations of lower degree, so the solution of any reciprocal equation may be made to depend upon the solution of an equation of lower degree. If the given equation is (12) y^ ^p^yn-^J^p^ yn-^ + . . . + Vn-^V^ +Pn-l V +Jt?„ = 0, I Sec 180] COLLEGE ALGEBRA, 191 the equation obtained by writing — for y and multiplying by 3/" is If the two equations are to be the same, we must have PnVx^Vn-X.pnl\-Vn-'l, ' • • , PnPn - 2=" P^ ^ PnPn -I ^ P^ , Pn = ^^ Hence jt?„ = ± 1, so that there are the two following types: (I) Pn^+^, Pl=Pn-l P^'-^Pn-%y P^=Pn-Z, ' ' ', (II) J0„=-1, P,^-Pn-l, P^——Pn-%, Pz= - Pn-Zy y" P Suppose that (12) is a reciprocal equation of type (I). Then (13) (r+l)+^,(i/^-^+2/)+i^.(2/"-'+^')+it^3(!/"-*^+^')+ . . . =0. If n is odd, 7i = 2t -{- 1, the hnal term in (13) is Pt{y^'^^ -f- y^). The terms are all divisible hjy-\-l and, if the quotient be Q{y)y the equation Q{y) = 0, of even degree tj — 1, is a reciprocal equa- tion. In fact, the dividend (13) and the divisor y -\- 1 remain un- altered, aside from a factor ?/"* and y, respectively, wlien — is written for y, so that a similar result must hold true for the quotient Q{y). Moreover, Q{y) = is of type (I) since the con- stant term is + 1, being derived from the division of ?/" -f 1 {n odd) by ^ + 1 [see formula (3), Chapter III]. As an illustration see Example 3 above. Hence Q{y) = is of the character next treated. I If 71 is eveTi, n = 2t, the final term in (13) is p^ y*. Dividing the equation by y^, we get {y'+^-)+p{y'-'+^)+ply'-'+^^+- ■ ■+p.-^{y+l)+P.=o. 8ety + ^=z. Then ^^ + i = ^^ _ 2, yB_^l_^^s_ 3^, rj,^^ general term may be derived from the identity (14) r+y = n. For the case r = 7i-\-l, we retain the above relations with the exception of the last two. The one preceding the last is no longer true, the correct result being now = (— lY{a/3y . . . /uv- + a/3y . . . yuV -{- . . . -|- a^/3y . . . /uv). \ Upon adding the latter to the n preceding relations, we get In a similar manner, we may prove the formula {^^)^n + yn+Pi^n^m-l+P2^n^m-2+' • ' +Pn-l ^m + 1+Pn ^m = 0. But the last formula is derived more simply as follows. Multi- plying the given equation (18) by x^, we get ^^n4-m_j_^^^n + m-l_|_^^^n^m-2_|_^ _ ^ ^^ _^ ^m + 1 ^ ^^ ^m _ q^ Setting x=a, /3, y, , , , , /A^ V in turn and adding the 7i resulting relations, we evidently obtain formula (*22). 190. Relations (21), which include relations (9), enable us to express 2^, 2^, 2^, . . • , ^n^^ terms of the coefficients. We have only to take r = 1, 2, 3, . . . , ^^ in turn, the respective equations giving JS'j, ^25 ^s» . . . , ^n in turn. For example, ^2= -Pl^l- ^P2 = Pi ~ ^P2^ ^Z^ -Pi ^2+ Pi P2 - 3^3 = - Pl-\-^PlP2 - ^PV ^4 = - Vl ^3 -'P2^2- Pz ^l - ^Pi = p,* - 4p^% + 4^j p^ + 2p,^ - 4p^. Then by employing formula (22) for m = 1, 2, 3, ... in turn, we may express -S'^^j, 2n + 2 9 ^n^-zf > - - in terms of the coefficients. 204 THEORY OF EQUATIONS, [Ch. XIX EXERCISES 1. If a, fiy y are the roots of a^ -f j^^ x^ -\- Pi^ -^ P% — ^, find 1^3, ^,, 2,, 2,, {a'fi^ + aV^ + /jy^), (a + ft){a + ^)(y5 + ;.). 2. If «:, /5, ;^, d are the roots of x^ -{- Pi^^ -\- Pi ^^ + it's ^ + i?4 = 0, find ^3, ^„ :S5, :S6, {a'Py + a^yj^ + . . . + /i;^^^), (a^^ + ay^s _^ _ ). 3. Find the sum of the reciprocals of the roots of the equations in Ex- amples 1 and 2. Find the sum of the squares of their reciprocals. 4. Show that for the general equation (18) n :Si (« - fif + {cz - rf -{- [fi - rf -\- . , , = n:s, - 2,^ = (^ - m^ - rYifi -ry + ^ n -^1 ^2 ^1 ^2 ^3 ^2 -^3 •^4 6. Solving equations (19) and (21) by determinants, prove that ;?i 1 Fl - ^ c ^3 -- 2^2 = ^1 1 I ^iV ^Ps Pl 1 2P2 Pl 1 3^3 Pi Pl ^\ 1 ^1 ^1 2 ^'3 ^^2 ^1 ^. 24p^: 2p, p, 1 ^Ps Pi Pl 1 4i?4 ;>3 ;?2 ;?i ' :^i 1 ^1 2, 2 ^^3 ^2 ^1 2, 2, 2, 3 6. Form the equations whose roots are (1) the squares, (2) the cubes, of the roots of ar^ + 3«2 + aj — 7 = 0. 7. If a, ^, y are the roots of x^ -[- p^x^ -\- p^x -{- p^ — 0, form the equa- tions whose roots are (1) a:^ 2 2 2 y^; (2) a-2, ^-2, -^-2. (3) ^^^ ^^^ ^^ . r f^ ,r ay /3 8. Form the equation of lowest degree with rational coefficients, one of whose roots is (1) |/2 + ^5 ; (2)|/2 + ^3 ; (3) i^S- |/^; (4) |^^+ V^^- : Sec. 190] COLLEGE ALGEBRA. 205 MISCELLANEOUS EXERCISES. . . + 71* ^ Jq ^(^^ + l)(2/i + l)(3n2 + 37i - 1). . . -I- 7^5 = tV^'(^ 4- Ifi^n' + 2/1 - 1). 1 . , . . .. 1 1* + 2*-+ , l^+2^ + . 4. The arithmetical mean of two positive quantities is greater than their geometrical mean. 5. According as ^ = 5 or a ^ &, a^ + 5^ = 2ah or > 2ab. 6. If (tt + Z> + c/ = 3(^2 _|_ ^,2 _^ ^2)^ tiieji ^ ^ 5 ^ ^^ 7. If (^1-1-^2+. . .-\-a„fz= n{a^-{-a^-{- . . .+an'0, then a^ = a^=z , . . = «„. 8. According as .t; -f ^ + 2 > , = , or < 0, a^ + ^^ -}- 2;3 > , = , or < 3a?^2. 9. If X, y, z are real and not all equal, then mxyz < {x^ y -\- zf <^{a^ -i- y^ -\-^). 10. Prove by mathematical induction that a + (« + ^) + (« + 2cZ) + . 11. Show that the roots of a;^ 1.3568958 . . . 12. 13. 14. 15. 16. , ^ {a -{- n - \d) = 7aj -h 7 = are 1.69 . . . , - 3.048. - 1 = are 1.24698. <-+"-T^4 The roots of x'^ -\- x^ — 2^; - 1 = are 1.24698, — 1 80194, — .44504. The positive root of Ix^ + 20;^^ -\- ^x^ - 16aj — 8 = is 0.91336. • a ■ 1) ■ c a b d -d ' -e -f :{af- be-\- cd)\ A + ri A + r2 r ^- (X Yx + «^i a +/? ^i + /A ^2 + P'l - 2 17. Factor .t* - 17^^ ^ 20aj - 6, x^ + 8.c3 + ^Ix" + 26« + 14. 18. a?* + «"^ — 2^2 -|- 4ic - 24 =: has two real and two imaginary roots. 19. Factor {a + 1)^ ~ d^ - \. 20. Solve the reciprocal equation 27a^^il — xf — 4t{l — x -\- x'^f. 21. li a, Py y are the roots of ic^ + a!^ — 2a; — 1 = 0, form the equation a^ + r'> P' + r'- whose roots are a^ + ^^, 22 Sum to n terms ttt + tt 4- 3! 4! Sum to infinity — + -^ -f- 5! 3^ 33 • 1^ 4- "^ "67 + ' + « 2o6 THEORY OF EQUATIONS. [Ch. XIX „^ 2 , 12 , 28 , 50 , 78 , 112 , ^ , ^ , ^^•ri+2-I+3!+4!+5-l+^ + -- = ^^ + ^- 23 33 43 2*-i' + r! + 2!+31 + ---=-^^^- 26. Find the number of shot arranged in a triangular pyramid, there being 25 shot along each edge. 27. If the sides of a right-angled triangle are in A. P., they are proportional to 3, 4, 5. 28. At tennis A and B play together with the respective winning proba- bilities a and h. If the score is deuce, show that A's, probability of winning the game is a^ + %d'h + 4a*6^ + ^a^h' + . . . = , ^ ^ ^ . APPENDIX. Argand's Diagram. In 1806 Argand gave a method of repre- senting complex numbers by points in a plane. The complex number x -\- y \/ — 1 is represented by the point {x, y) whose coor- dinates referred to two fixed perpendicular lines Ox and Oy are the real numbers x and y. To each complex number x + iy there cor- ^ Y P {x-^iy) / y /e X \ Fig. 14. responds a single point P, and to every point of the plane there corresponds one and but one complex number. If the point P moves in the plane, z^x -\- iy changes continually in value and is called a complex variable. The length of the line OP, viz.^ 207 2o8 APPENDIX. \/x^ + y'^, is called the modulus of x + iy. The angle 6 = XOP is called the argument oi x -\- iy; it is to be measured counter-clock- wise from OX, By trigonometry, the sine of 6 is the ratio of y to r, the cosine of 6* is the ratio of x to r. These relations are written t/ X sin 6 = —, cos = —. whence sin^ 6 + cos'-^ 6 = 1. r r .'. a; + zy — r(cos 6 -{-.i sin 6) . For example, \/2 + i\/2 = 2(cos 45° + ism 45°), - 1 _^ 1. 4/-T==cosl20°H-i sin 120°, i+i |/3"* =: cos 60°+z sin 60°, If the points Pand A represent x + iy and a^ -f ia, respectively, it is seen that the point which represents their sum {x^a)-\-i(y-\-a) is given by completing the parallelogram two of whose sides are OP and OA. It follows that the modulus of the sum (the length of the ' diagonal) of two complex numbers is less than (in a special case, equal to) the sum of their moduli. The point represepting the product of two complex numbers may be constructed by applying the following theorem : T/ie modulus of tM product of two complex numbers is the prod- uct of their moduli, it^ argument is the suin of their arguments. Let x-\-iy = r(cos /^-j- ^ sin ^), x'-\- iy' = r'(eos 6' + isin 6'). Then {x + *'y)(^' ~^ W')~ rr' {cos 6 -\- i sin ^)(cos 6' -\- 1 sin 6') = rr' ; (cos B cos B' — sin 6 sin &) + i (sin d cos 6' -\- cos 6 sin 6') \ r=. r/-'jcos(/9 + &) + i sin {B -\- &)\, l-y the formulae proved in trigonometry. Demoivre's Theorem. If n he any rational number and 6 any angle, (cos 6 -\- i sin BY =^ cos nB -\- i sin nB. We have just shown that (cos B-{-i sin 6')(cos 6' + i sin 6') = cos {B + B') + * sin ((9 + B'}. COLLEGE ALGEBRA. 209 Taking first 6' — d, we have {cos 6 + i8m 8y=: CO&26 + ism20. Taking next 6' = 26/, and applying the last result, we get (cos -\- i sin Oy = cos 36 -\- i sin 30. Taking next 0' = 30, and applying the previous result, we get (cos 6 -^ i sin 6y = cos 4:0 + ^ sin 40, Similarly, if the theorem be true for 71 = r, a positive integer, it is true for n ■= r -\- l. Hence, by mathematical induction, it is true for every positive integral value of ^. n Eeplacing by -, we get / oy COS — |- I sin - = cos + { sin 0, \ n nl - . • . (cos 4- i sin Oy = cos — \- i sin -, ^ 71 n as one of the fith. roots of cos -\- i sin 0. It follows that the theorem is true for positive fractional values of /^. Next, it 71 = — m, 771 being a positive fraction, (cos -{- i sin OY = 7 ^ , . . — 77- ^ ' ^ (cos 4- I sm ^)"* 1 cos 7nO — i sin mO cos mO -{- i sin 7nO cos^ mO -|- sin''^ mO = cos 7n0 — i sin mO = cos 7iO -\- i sin tiO. Corollary. Any root of a complex number may be expressed as a complex number. Thus ^x -j- iy = \/r (cos -\- i sin^)n. Hence — \- i sin -1 is one of the 7ii\i roots of x -\- iy, Eeplacing 6/ by 6/ + 360°, 6> + 2 • 360°, , , , , ^ {71 - 1) 360°, we obtain the remaining w — 1 roots. But -\- 71 360° gives the same root as 0, e + {n -\- 1) 360° gives the same root as (9 + 360°, etc. 2IO APPENDIX. Example 1. Find the three cube roots of unity. 1 = cos + 1 sin -= cos 360° + i sin 360° = cos 720o + i sin 720^ Hence the cube roots of unity are cos f + / sin I = 1, cos 120'' + i sin 120° = - i -f- i \/^ = co, cos 240° + i sin 240° = - ^ - J \/'^ = gd\ Example 2. Find the five fifth roots of — 1. -1 = cosl80°+^ sinl80°= cos 540°+^sin540°=...=cos 1620°+ z sin 1620^ ,-. i/Zl = cos 36°+isin36°, or cos 108°+^sinl08°, ..., or cos 324° -|-^ sin 324°. Solution of the cubic y^ -\- py -\- r = in the irreduciUe case (see §173): (1) ^-T + ^<«- . Set i^ = — p^ sin^ 6, — - = p cos 6. .-. p'^ = p2 (cos^ e + sin'^ 6)=z^ -E= -£^, . • . p z= 4/- f/27, cos e = - --^i/- py27. Notice that p must be negative by (1), so that p is real; also that — — is in numerical value less than 4/— p^/27 by (1), so that tos is numerically less than unity. Hence 6 may be found by means of a trigonometric table of cosines. V — — 4" 4/^ = j/p cos 6 -f- ip sin 6^ 3 f ^ _j_ M.360" , . . ^ + 77/^360°) 4/p -^ cos + I sm — ■ — l y/-^-i/B = Vp\ 6+ m.360' . . 6' + m.3()0° cos z sm 3 r where m =: 0, 1, or 2 and Yp ^^ ^ ^^^^ quantity. Substituting in Cardan's formulae, we obtain the three roots y = 2|/p cos — ! — {m = 0, 1, and 2). COLLEGE ALGEBRA. 211 We may proceed independently of Cardan's formulae. By trigonometry, cos 3:^: = 4 cos^ x — ^ cos x. Set z = cos x. .' . z^ — \z — \ cos 3.T = 0. To reduce the given cubic to this form, set y = kz. p 3 / 1_ 4/p Determine k so that f^ = ——, viz., take k = \/ ^ ^ . The two A;^ 4 V 3 equations will then be identical if 1 o ^ — - cos 3a; = ^^ 4 KT ~8 "^ Y "27^ Hence cos 3^; = — — ^ 4/— i?V27, the value found above for cos 6 Finding 3:^; from its cosine by means of a table, a root z equals cos X. But cos ?>x — cos {^x + 360^) = cos {Zx + 720°). Hence the three roots z are cos X, COS (2; + 120°), cos (x + 240°). Multiplying these by ^ = 2 4/— /?/3, we get the three roots y, FUNDAMENTAL THEOREM. Every equation z^ + p^z''~'^ + . . . + jf?„ = has a root. If we set w = 2;** + j^ja;""""^ + A^*""^ + • • • + A > we are to prove that there exists at least one complex number z \fhich will make w — 0, so that the point (called the w-point) in Argand's diagram which represents w will be the origin 0. Unless the theorem is true, there is a ^(^j-point, say P^, corre^onding to some value z^ of z and distinct from 0, such that no other z^-point is nearer to than Pj , so that there is no value of z which will bring the corre^ sponding lu-^omi within the circle about of radius OPy If we prove that the last statement is false, the theorem will be proved. 212 APPENDIX, Let us change z from z^ to z^ + ^i by adding a small quantity Zy Then «^ will change from w^ to ]^^, where w={z,-^z,Y+p,{z,+zy-^+p,{z,+z,Y-^~^,,.-^p^_,{z,+z,)+^^ upon expanding the powers by the binomial theorem and arranging the terms according to powers of Z^ The coefficient of Z^ is seen * to be* A = nz;^-' + {n - l)p^z^--'^+ (n - 2)p,z,^-' + ... +Pn-i* Since w^^z^+ VK~^ + . . . + jt?^ , we get W=w^'\- AZ, -f- BZ^^ -^ , , , + Z,\ Set A = a (cos a + t sin a), Z^— r (cos ^ + * sin ^). As above, AZ^ — ar [cos (^ + ^) + ^ sin (^ + o')]. .-. W= iv^+ar [cos(^+«') + ^sin {e-{-a)]-^C,r^-\-C^r^+, . . + (7„r% the e^act form of C^ ~ B (cos 6^ + ^' sin 6y^, G^^ . . . , (?„ ^ being immaterial. By taking r sufficiently small, the value of C^r^ + . . . + ^vT^ can be made as small as we please, and there- fore W will differ as little as w^e please from w^ + «^ [cos {6 -\- a) -\- i sin {6 -f- a)']. But the point representing this complex quantity will describe a circle of radius ar about the point P^ when the angle 6 increases gradually from 0° to 360°. ^ut this circle intersects the circle about with radius OP^ Hence by assigning suitable values to r and 0, the w-point may be brought into coincidence with some PT-point lying inside the circle of radius OP^ * The later argument assumes that A 4^ 0. Since there are at most n — \ values of z^ which make ^ = (§ 77), such values may be avoided by taking some new point for Pj on the circle of radius OP^, INDEX. (The numbers refer to pages.) Absolute value, 1*14 Annuity, 74 Area of triangle, 156 Argand's Diagram, 207 Argument, 208 Arithmetical progression, 64, 115 Axes of coordinates; 153 Bas e of logarithms, 18, 20, 142 Bino.raial Theorem, 90, 101, 130 Chance, 94 Characteristic, 20 Commensui-able, 58 Common diti*erenc(i, 64 loga>ithms, 20 ratio, 1^7 * Combination, 85 Complex quantity, y5, 207 , Composition and divi^^ion, 61 Compound Interest, 73 Conjugate, 7 Constant, 76 Convergent, 113 Cubic equation, 180, 210 Cyclo-symmetry, 33 Demoivre's Theorem, 208 Descartes' Rule of Signs, 176 Determinant, second order, 36 third order, 38 fourth order, 50 Divergent, 114 Double root, 194 Duplicate ratio, 59 Ellipse, 166 Equating coefficients, 78, 126 Equation, see Cubic, Quartic, Fifth Exponential equation, 26 Exponential Theorem, 137 Extremes, 60 Factor Theorem, 27 Fermat's Theorem, 103 Fifth-degree equation, 189 finite, 106, 113 First law of indices, 10„ 13 Fractional root, 198 Function, 76 Fundamental Theorem, 193, 2ld Generating fraction, 148 relation, 145 Geometrical progression, 67, 114 Graph, 154, 1^3, 168 Harmonical progression, 71, 116 Horner's method, 169 Identity, 31 (Note), 78 (Note) Imaginary quantity, 5 ' root, 199 Incommensurable, 58, 144 Independent events, 96 Indeterminate form, 106 InalV'es, 10 Infinite, 105, 113 Interest, 73 Interpolation, 14}^ Irrational, 2, 3 Irreducible case, 184, 210 Limit, 17, 69, 104, 110 Location of roots, 178 Logarithm, 18 Mantissa, 20 Mathematical induction, 100 Means, 60, 64, 67, 71 213 INDEX, Method of differences, 148 Minors, 38, 50 Miscellaneous exercises, 205 Modulus, 142, 208 Multinomial Theorem, 92 Napierian, 142 Natural logarithm, 142 number, 1 Negative number, 2 Order of determinant, .^6, 38, of surd, 2 Origin of coordinates, 153 Parabola, 163, 168 '' Partial fraction, 80 Pascal's Triangle. 88 Permanence of form, 132 Permutation, 85 Perpetuity, 75 Plot, 153 Positive integer, 1 Power series, 125 Prime number, 101 Probability, 94 Proportion, 60 Quadratic equation, 29 surd, 2 Quartic equation, 185-188 Ratio, 56, 67, 158 Kational number, 3 j^ Sol X Sqi / $> ^ Rational integral function, 77 Rationalizing factor, 29 Real number, 4 Reciprocal equation, 190 Recurpng series, 14^ Reduced cubic, 181 Remainder Theorem, 27 Reversion of series^ 128 Root, 30, 31, 194 Second law of indices, 10, 14 * ■^ ries. 113 limultaneous equations, 35, 54, 164 inequalities, 166 Solution of numerical equations, 168 Square root, 6, 8 Sum to infinity, 69 Surd, 2, 3 Surd root, 198 Symmetrical functions of roots, 200 Symmetry, 33 Synthetic division, 169 Table of logarithms, 24 Third law of indices, 11, 14 Third proportional, 60 Triple root, 194 Triplicate ratio, 59 Undetermined coefficie nts, 78 Vandermonde's Thcjorem, 132 Variable, 76 Variation, 61 Var}^ inversely, 61 .^.^/■. ^f & '■^■^lik ^ RETURN CIRCULATION DEPARTMENT TOh^ 202 Main Library 642-3403 LOAN PERIOD 1 2 3 4 5 6 LIBRARY USE This book is due before closing time on the lost dote stamped below DUE AS STAMPED BELOW Am 11977. . FORM NO. DD 6A, 12 m, 6'76 UNIVERSITY OF CALIFORNIA, BERKELE BERKELEY, CA 94720 d LD 21A-50m-4,'60 (A9562sl0)476B General Library University of California Berkeley THE UNIVERSITY OF CALIFORNIA LIBRARY