IL «£ Pi iiii. 1 ! in ill I ;! I 1H! II ■ FMI It ; H'J 4' OS* -3' :\ REESE LIBRARY UNIVERSITY OF Received Accessions Np..jL*$~~fy/'S~ Shelf No. x-^-P<&J t88*jC TREATISE ON ALGEBRA, FOR THE USE OF SCHOOL^" AND^(?ODLEGES. BY S. CHASE, PROFESSOR OF MATHEMATICS IN DARTMOUTH COLLEGE. row NEW-YORK: D. APPLETON & CO., 200 BROADWAY. PHILADELPHIA : GEO. S. APPLETON, 164 CHESNUT-ST. MDCCCXHX, Entered according to Act of Congress, in the year 1849, by S. Chase, In the Clerk's office of the District Court of the District of New Hampshire. I tiff ST DARTMOUTH PRESS, Hanover, If. H. PREFACE The following treatise is intended to exliibit such a view ot the principles of Algebra, as shall best prepare the student for the further pursuit of mathematical studies. The principles presented I have endeavored to enunciate as clearly and briefly as possible, to demonstrate rigorously, and to illustrate by strictly pertinent examples. Part of the examples are of the most elementary form, part are purely numerical, and a large part of the rest are expres- sions employed in the reasonings and investigations of Trigo- nometry, Analytical Geometry, Mechanics and other branches of mathematical study. Thus, the application of the principle is exhibited, relieved of all extraneous difficulty, and connected with the familiar ideas of Arithmetic ; and, moreover, the forms and operations employed in demonstrating truths of Geometry, and of other related sciences, are rendered familiar, and made ready for use when they shall be needed. This last consideration is of great importance. Much of the difficulty which students find in later parts of the course results from want of familiarity with the algebraic expre'ssions employed, and with the elementary operations performed upon them. At the same time, such expressions and operations are frequently among the most convenient illustrations of algebraic principles. The discussion of the theory of exponents and powers (§§ 11-24) is, so far as I know, original. The use and interpre- IV PREFACE. tation of the fractional and negative exponents is exhibited as a necessary consequence of the definition. The demonstration of the Binomial Theorem for negative and fractional exponents (§§ 291-294), and the development of the fundamental logarithmic formula (§§ 320-323) are substantially those of Lagrange. The nature of the Modulus (§§327-332), and some of the properties of logarithmic differences (§§333-336) are discussed more fully than I have seen them in any elementary treatise. Familiarity -with these principles is of great advantage to the student, and their discussion is, by no means, difficult. A table of the principal formulae of the book is placed after the table of contents, for convenience of reference and review. It has also the advantage of generalizing, and bringing into one view, principles exhibited, -with more or less fulness, in different parts of the book. For the suggestion of this table, I am indebt- ed to Mr. Richards, the able Principal of Kimball Union Acad- emy. I am also very greatly indebted to my associates, Professors Crosby and Young, for valuable suggestions and criticisms. In correcting the proofs of the last half of the work, I have had the assistance of Mr. Edward Webster, a recent graduate of the Col- lege, whose tastes and attainments qualify him to do excellent service in the cause of science. S. C. Dartmouth College, May 1, 1849. CONTENTS. INTRODUCTION. Symbols of quantity, — Signs, 13-i< Positive and negative quantities, .... 17-21 Factors, — Coefficient, — Exponent, . . . 22-25 Exponent ; fractional, zero, negative, — Reciprocal, . 26-34 Power, — Root, — Function, 34.-4* Degree, — Homogeneous, — Like, ... 40,41 Monomials, — Binomials, — Polynomials, ... 42 Reduction of Polynomials, 43-45 Equations ; identical, absolute, conditional, . . . 45, 4G Degree, — Axiom, — Transposition, .... 47-50 Clearing of fractions, — Solution, — Problem, . . 51-58 CHAPTER I. ADDITION AND SUBTRACTION. i. Addition, ii. Subtraction, — Combination of signs, 59-62 62-67 CHAPTER II. MULTIPLICATION AND DIVISION. in. Multiplication, — Monomials, — Signs, — Degree, - 68-69 iv. Polynomials, — Detached coefficients, . . . 70-75 Division, — Monomials, — Signs, .... 75-77 Polynomials, — Detached coefficients, .... 77-83 Synthetic division, — Infinite series, .... 83-86 Theorems, (a ±b) 2 , (a + b) (a — b), . . 86-89 Divisibility, (o« — b n ) -± [a — b), &c, . . . 90-92 Greatest common divisor, 92-96 Least common multiple, — Problems, %, #, . 96-100 *1 VI CONTENTS. CHAPTER III. FRACTIONS- . . . 101 Reduction, — Addition and Subtraction, . . . 102-105 Multiplication and Division, .... 106-108 CHAPTER IV. EQUATIONS OF THE FIRST DEGREE, WITH TWO OR MORE UNKNOWN QUANTITIES, . . . 109 Two unknown quantities, . . . 109-115 Elimination, — By Addition and Subtraction, . . 110-112 By Comparison, — By Substitution, . - . 112,113 More than two unknown quantitieSi . . . 115-119 « » p, 120-121 CHAPTER V. INEQUALITIES. . . 122-124 CHAPTER VI. POWERS AND ROOTS. Monomials,— Powers,— Roots, .... 124-127 Radical quantities,— Imaginary quantities, . . 127-131 Polynomials, — Powers, •• 131-133 Square root of a polynomial, — Of a number, . . 133-139 Cube root of a polynomial, — Of a number, . . 139-142 »»** root of a polynomial, 142 Square root of a ± 5*, 143-147 Binomial surds rendered rational, .... 147-149. CHAPTER VII. EQUATIONS OF THE SECOND DEGREE. Definition,— Complete equation,— Incomplete, 149-151 Solution of incomplete equation, .... 151-15 j Solution of complete equation, . • 156-160 CONTENTS. Til General Discussion, — Factors, — Roots, — Signs, . 161-164 *«» -j- Par" -|- Q = 0,— Radicals, . . . 167_lt>9 Two unknown quantities, — Variables, — Curves, . 169-174 CHAPTER VIII. RATIO AND PROPORTION. . . 17c Mean proportional, — Equal Products, . . . 176-178 Inversion, — Alternation, — Composition, — Division, . 178-180 Equimultiples, — Sums, — Powers, — Products, . 180-182 I u verse Proportion, — Variation, .... 182-18C CHAPTER IX. EQUIDIFFERENT, EQUIMULTIPLE, AND HARMONIC SERIES. i. EquidifFerent series,-r-Difl'erence, ... 186 Last term, — Sum, — Mean, — Interpolation, . . IS 7-1 90 II. Equimultiple series, — Multiplier, ... 190 Last term, — Compound Interest, — Mean, — Sum, . 191-194 m < 1 and n = co, — Annuities, . . 195-199 Interpolation, 199 in. Harmonic Series, — Proportion, .... 200 CHAPTER X. PERMUTATIONS AND COMBINATIONS. 203-207 CHAPTER XI. UNDETERMINED COEFFICIENTS. . 208-212 Development of series, ..... 210 Decomposition effractions, 211 CHAPTER XII. BINOMIAL THEOREM. i. Positive integral exponent, .... 213-218 ii. Negative and fractional exponent, . . . 218-226 Derived fuuctlon, or derivative, . . . 220-223 Ylli CONTENTS. CHAPTER XIII. DIFFERENCES. Orders, — n' h term, — Interpolation, — Sum, . . 227-234 CHAPTER XIV. INFINITE SERIES. . . 235-24*' CHAPTER XV. LOGARITHMS. Exponents, — Characteristic, — Base, . . . 241-244 Development, — Formulas, 246-250 Modulus, — Naperian system, — Differences, . 250-257 Computation, • . 257 Exponential Theorem and Equation, . . . 259-264 CHAPTER XVI. THEORY OF EQUATIONS. Degree, integral, fractional, 265 Divisibility, — Roots, — Number of roots, . . 266-271 Coefficients, — Form of the roots, fractional, imaginary, 271-275 Signs of the roots, — Variations and permanences, . 276-280 Transformation, — Change of roots, — Removal of terms, 280-289 Limits of the roots, ...... 290-293 Limiting equation, — Equal roots, .... 293-296 Sturm's theorem, 296-306 Numerical equations, — Integral roots, . . . 306-309 Incommensurable roots, — Horner's method, . . 309-318 Recurring, or reciprocal equations, .... 318-322 Biroinial equations, 322-325 CHAPTER XVII. CONTINUED FRACTIONS. . . 32C Convergents, — Error, — Lowest terms, . . 327-331 Reducing to a continued fraction, . . . 332-335 I TT I T Y FORMULA. § 6. a.) -}- a > ; — a < 0. That is, A positive quantity > ; a negative quantity < 0. § 7. a, b.) — (— a) =z -\- a. .: — [-— (— a)] =r — a ; — (—(—(— «))) = + «; &c. §§63; 68. a, — b) — b. §§89,90. (a±b)* = a 2 ±2ab-{- 6 2 . §91. (a+.Z>) 2 + (« — £) 2 = 2(a 2 + & 2 ). (a + 5) 2 — (a — 6)2 — 4a5. . §92. (a + b)(a — b) = a 2 — 6 2 . §96. a.) = a »-i-j_ a »-9J . . . -f-a&*- 2 + b"~ l . . , a n — a" „ , a — a §97. - "" , f"" = a 2 "- 1 — a 2 "- s 6+ . . 4- a 6 2 "- 2 — & 2 "-i. a-}-6 a 2»+l I £2«+l §98. 4-= = a 2 "— a 9t-lJ-l_. . _ a J2«-l I JSn, X FORMUL-ffi. §§ 109, 139, 140.) pr = co. — = 0. -, indeterminate. §151, c.) (a n ) m = a mn . (±a) 2 " = -f-(a 2 »). (±a) 2 "+i = ±(«2»+i). i 1 i § 152. (+ a)* n =± (a 2 ",). (— «) 2 ", imaginary. i i (±«) 2 ; i +i = ± (« 2 h-i). 1158. (-o 2 f = a(-l)^ffly-l. §162. (y-l) 2 =-l; (y-l)3=_y_i ( = »(n — 1) .... (n— p + l) = [«, w— i» + l]- $ 275. No. of combinations of w things, taken ^? and jp = ?i(?i— 1) . . (w— p+l) _ [w, n—j? + l] 1 . 2 . 3 . . . p |>] § 280. If M + JVSc + Pa; 2 + &c. = for all values of x, then if = ; ST= ; &c/ § 294. (x +y) n - x\ + je-^ + ^^-**- 2 3/ 2 + &c. $295. i.) (^±^=^(l±f|+^f^^± & c-) , „„~ ^ . n(n — 1) §300. D n :=±a l ^:na 2 ±-^- — — ^a 3 q:&c; taking the upper signs, if n is even ; and the lower, if it is odd. X" FORMCLjE. §301. „. = a , + (n - 1 )J>, + ( " ~ *> [ ( * ~ 2) J 8 + Ac. §304. j = ,.,+sfe^p, + M <"7; 1 ><7 2 > J,+to ^307. ? = I(± * ). §323. !^g T =Jf[y_l_|( 3 ,_l)2_j_ i ( y _i)3_ &c j. §§ 327-8. J/= —J- — U, a — 1— i(«— l) 2 +&c. Xa §§329,330. J^= .434 294 481. e = 2.718 281. §340. . = l + Za.* + I^: + (Mi|L +& c. §351. a; n 4-^4 1 x n ~ 1 . . -|-^ n = = (a; — aj . . (a; — a,,;. §355. A x = — (a 1 +a 3 +a„); i 2 =a 1 a 2 + a 1 fl 3 -l-&c. -4„=r±(a 1 a 2 . . a„). t§ 365-7. X—sf-^A^- 1 . .+^„:=0, and# = z — a/. T-y^^-B^-^ .... + J B„_ 1 y + .£„-0: or r=y:»+ f^*+f%py +/(*) = 0. The parenthesis with the sign of equality, it will be ob- served, is sometimes used as an explanatory expression. Thus (§ 18), " 10-i(=^)" is used for " 10" 1 (i. e. T V-" ALGEBRA. INTRODUCTION. § 1. Algebra" is that branch of the science of number, which employs general sy7nboh h of quantity*. a.) Arithmetic*, in its largest sense, includes the whole science of number ; but, in its popular use, the term is lim- ited to that branch of the science, which employs symbols of known and particular numbers only; as 2, 3, 10, 12. b.) Algebra, on the other hand, employs general symbols (for the most part, Italic letters of the alphabet), any one of which may represent any number whatever. Thus a rep- resents, not some particular number, but simply a number. Note. Such symbols are termed algebraic or literal", in distinc- tion from those of common Arithmetic, which are termed numerical'. A quantity expressed algebraically is often called an algebraic quan- tity or expression. c.) For convenience and perspicuity, certain classes of letters are usually appropriated to distinct uses. Thus, the first letters of the alphabet, as a, b, c, usually stand for known or given quantities, and the last, as x, y, z, for un- (a) A word derived from the Arabic ; the Arabs having been among the earliest cultivators of this science. (6) From the Greek av/ij3o?,nr, token, sign, (c) From the Latin quantus, how much, (d) Greek, upt&noQ, member, (e) Latin, littera or litera, a letter. (/) Latin, numerus, number. AI,G. 2 14 INTRODUCTION. [§ 2. known ox required quantities; while for exponents (§ 16), the letters near the middle of the alphabet, as m, n, p, are oftener used. Note. A quantity is regarded as known, when it may be assum- ed at pleasure; as unknown, when it cannot be assumed, but must be found from its relation to the known quantities. d.) A quantity is sometimes represented by the first let- ter, or by several letters of its name : thus interest is repre- sented by i ; sum, by S; difference, by D ; time, by t ; veloc- ity, by v ; radius, by r or R ; sine, by sin ; cosine, by cos } tangent, by tan 9 ; &c. e.) Different quantities of the same kind, or standing in the same circumstances, are sometimes represented by the same letter accented. Thus similar known quantities may be represented by a, a' (read a prime), a" {a second), a'" [a third), &c. ; similar unknown quantities by x, x J , x", &c. So, if the radius of one circle is represented by R, the radius of another may be represented by R', &c. A distinction is sometimes made, by using different forms of the same let- ter ; as x, X; u, U ; r, R. SIGNS. § 2. In addition to the symbols of quantity above mentioned, Algebra, in common with other branches of mathematics, employs certain symbols of opera- tions and relations, called signs*. Thus, the sign of a.) Equality, =, equal to ; as 1 foot = 12 inches ; a = b. b.) Inequality, 1. Superiority, >, greater than ; as 10>7. 2. Inferiority, <, less than; as 7<10; 5«<6a. Note. The opening of the sign of inequality is always towards the greater quantity. (g) Radius, sine, cosine, and tangent are the names of certain lines ;lrawn in or about a circle, and express quantities of great import- ance, and of continual use in the higher applications of Algebra (h) Latin, signum, mark, sign. § 2, 3.] signs. lo •.) Addition, -\-,plus\ or together zvith ; as 6+4= 10 ; x-\-a. d.) Subtraction, — , minus', or less ; as 7 — 3 = 4 ; la — 3a. Note. The quantities, which are connected by the signs + and — , are called terms*. e.) Multiplication, X, or . , into, or multiplied by; as 4X5 or 4.5 = 20; 4aXob = 20a,b = 20ab. Note. Between numbers and letters, and between letters them- selves, the sign of multiplication is commonly omitted. Thus 3abc is the same as 3XaX6Xc. Between numbers, on account of the local value of figures, the sign can never be omitted. Thus 35 i< not the same as 3X5. /".) Division, ■—, divided by ; as 8-^-2 = 4 ; 6a-|-2 = da. Note. Division is more frequently denoted by writing the divi- dend above, and the divisor below a fractional line. Thus a divided by b is written-; 8-^-2 — - = 4. g.) Inference, .*. , therefore, as a = 5, .\ 4a = 20. A.) Union. The parenthesis, (), or vinculum 1 , either hori- zontal, , or vertical, | , is used to show that several quantities, connected by the signs -4- or — , are to be tak- en together, or subjected to the same operation. Thus (3+4)X2, or (3-f-4).2, or 3+4.2, or 3 2, shows that 3 and 4 are to be added together, and their sum multiplied by 2. So (a+b) (a—b) ; 6— (4— 2) = 6—2 = 4. With- out the parenthesis, the last expression would be 6 — 4 — 2 = 0. Other symbols will be introduced and explained, as they are needed. § 3. It should be remembered that these signs are abbre- viations for words ; that they are, in fact, words and phrases of the algebraic language. (i) Lat. plus, more, (j) Lat. minus, less, (k) Gr.repfia, bound, limit; Lat. terminus, Fr. terme. (Z) Lat., tie, bond. 16 INTRODUCTION. [ § 3. a.) Translate the following expressions into common language. a\-b . a — b L — + — =«• Ans. The quantity obtained by adding b to a and divid- ing the sum by 2, together with the quantity obtained by subtracting b from a and dividing the difference by 2, is equal to a. Or, The half of a plus b, plus the half of a minus b, is equal to a. 3. («+&) (c-\-x) = ac-\-bc-\-ax-\-bx. 4. 7?Xsin(a+£)=sin a cos 6+ cos a sin b. See J. 4, below. 5. aa+aV+a'V+a"'^'^ (a+a'y-fa'") 1 - G. (100+40+4)12 = 144. 10+2 = 1728, <200X 10. 5.) "Write, in algebraic language, the following sentences. 1. 10 added to 4, and the sum diminished by 8, is equal to 3 times 4 divided by 2. Ans. 10+4—8 = 3x4-^-2. 2. a multiplied by b, and the product divided by e, is equal to x multiplied by a, and the product diminished by b. 3. The diiference between a multiplied by x, and h mul- tiplied by y, is equal to m multiplied by e. 4. Radius into the sine of the sum of a and b is equal to the sine of a into the cosine of b, together with the product of the cosine of a into the sine of b. See a. 4, above. 5. The sum of a and b is greater than c, and c is greater than the difference of a and b. The greater brevity and clearness of the algebraic lan- guage cannot fail to be observed. *4. ] POSITIVE AND NEGATIVE QUANTITIES. 17 POSITIVE AND NEGATIVE QUANTITIES. § 4. In finding the aggregate of any number of quantities, or terms (§2. d. N.), those, which tend to in- crease the amount, are called positive" 1 , and, as they must be added, are preceded by the sign + ; those, which tend to diminish the result, are called nega- tive", and are preceded by the sign — , to show that they must be subtracted. 1. A has Bank Stock, to the amount of $2000, Real Es- tate, S5000, other property, $1000 ; he owes to B $500, and to C 8300. What is the net amount of his property ? Here the items of property tend to increase the amount, and are, therefore, positive ; the debts diminish the amount, and are, therefore, negative. The former must, consequent- ly, be preceded, or affected by, the sign +, and the latter, by the sign — . Hence, we shall have, for the true ex- pression of the net value of the estate, +2000+5000+1000— 500— 300= + S7200. a.) The character of every term as positive or negath< . must, of course, be indicated in the expression. Quantities, however, are regarded as positive, unless the contrary is shown ; hence, if no sign is prefixed to a term, the sign + is always understood. Hence, when a positive term stands alone or at the beginning of a series of terms, its sign is usually omitted. Thus 5 is the same as +5 ; so 4 — 3 = +4 — 3 ; a = +a ; a-\-b = +«+&. 2. Let the items of property amount to $10,000, the debts, to $9000. "What is the aggregate, or the net estate ? 3. What is the aggregate, if the property be represented by a, and the debts by b? (m) Lat. positivus, from pono, to place, as placing or giving value, (n) Lat. negativus, from nego, to deny, as denying value. 2* 18 INTRODUCTION. [ §4. 4. Again, suppose a surveyor runs on one side of his field 20 rods east, and, on another, 15 rods west. What is his distance east of his starting point, i. e. his departure, as surveyors call it? Am. 20 — 15 = 5 rods, or E. 20 rods, W. 15 rods=E. 5 rods. The distance run east is positive, because it increases the distance east of the starting point; and the distance run west is negative, be- cause it diminishes that distance. b.) As each sign indicates simply the character of the term before which it stands, the order of the terms is obvi- ously immaterial, provided each retains the proper sign be- fore it. Thus 4 — 3 is the same as — 3+4. So, 10—8+6 = 10+0—8 = 6+10—8 = —8+6+10. 5. How far will a surveyor be east of his starting point, if he runs 10 rods west, and 50 rods east ? Am. —10+50 = 50—10 = 40 rods. G. A owes $5000, and holds property to the amount of $20,000. What is his estate ? 7. What, if he owes a dollars, and holds property to the amount of b dollars ? 8. What, if he owes $5000, and holds $5000 worth or property ? 9. What is his estate, if his property amounts to $5000, and his debts, to $6000 ? Am. 5000— 6000 = —$1000. or, property $5000, debt $6000 = debt $1000. In this instance, $5000 of the debt can be paid, and there will re- main $1000 to be paid afterwards, i. e. to be subtracted from any property, which may be afterwards acquired. 10. A surveyor runs 20 rods east, and 30 rods west. What is his distance east of his starting point ? Am. — 10 rods, or, E. 20 rods, W. 30 rods = W. 10 rods. 20 of the 30 rods run west can be subtracted from the 20 run east, nnd 10 remain to be subtracted. Thus, if he should afterwards run 15 rods east, his distance east of his first starting point would be —10 +15 = 5 rods. §4-6.] POSITIVE AND NEGATIVE QUANTITIES. 19 c.) If it bud been proposed to find his westerly distance from the lirst point, the easterly distances would have been negative, and the westerly, positive. In like manner, if we had proposed, in the examples above, to find the net indebtedness, we must have made debts positive, and property negative. § 5. Thus the contrary signs + and — show that the quantities, before which they are placed, are in precisely opposite circumstances; that is, that they produce opposite effects in respect to the aggregate result ; — that, as in the case of the distance east and west, they are reckoned in opposite directions. In other words, the sign — is the algebraic expression for contrariwise, or, in reference to distances, back- wards. Thus, if distance north be positive, distance south is neg- ative ; if, for instance, north latitude have the sign*-)-, south latitude must have the sign — . If distance upward be positive, distance downward is negative ; if future time be positive, past time is negative ; if velocity in one direction be positive, velocity in the opposite direction is negative ; &c. § 6. A negative quantity is frequently said to be less than zero. This expression is most conveniently illustrated by examples 8 and 9, above. In example 8, the net estate is ; in example 9, it is — $1000. But a man, whose prop- erty is as represented in example 9, is obviously poorer than he would be, if, as in example 8, he were worth sim- ply nothing. He is worth less than nothing. It is not meant, that the thousand dollars to be subtracted, is less than zero ; but, that it has less tendency to increase his estate, than zero would have ; that is, it has a tendency actually to diminish his estate. a. In like manner, if he had owed $2000, he would have been worth less than he is now, when he owes only $1000. 20 INTRODUCTION. [§G, 7. Hence, we say, that —2000 <— 1000. That is, the sub- traction of 2000 leaves a smaller remainder than the sub- traction of 1000. In other words, — 2000 tends to increase the debt more, that is to increase the property less, than — 1000, and is therefore said to be itself less. So, in example 5, — 10 gives a greater distance west and therefore a less distance east, than — 5 could have giv- en ; and either of them, a less distance east than 0. Hence, 0>— 1;— 2>— 3; — 5<— 4; +a > ; — a<0. b. Again, if we begin with 3 and subtract 1, we diminish the amount ; and we continue to diminish it, as long as we continue to subtract 1. Thus, 3—1 = 2; 2—1 = 1; 1—1 = 0; 0— 1 =— 1 ; — 1— 1 =— 2. Or, if, from the same quantity, we subtract continually greater and greater quantities, we shall obtain less and less remainders. Thus, 3—2 = 1; 3—3 = 0; 3— 4 = — 1; 3— 5 = —2; that is, the greater the quantity to be subtracted 1 the less the remainder. § 7. As a positive and negative quantity are reck- oned in opposite directions, the difference between l hem is greater than either, and is equal to the sum of the units in both. Or, as a negative quantity is less than zero, the difference between a positive and a negative quantity is greater than the difference between the positive quantity and zero ; and greater by just so much as the negative quantity is less than zero ; that is, by the number of units in the negative quantity. 1. A has $5000, and B owes $5000. What is the dif- ference of their estates ? i. e. by how much is A richer than B ? Ans. 5000+5000 = $10,000. «.) If they should combine their estates, the aggregate value would be 0. The difference between them is clearly § 7, 8.] POSITIVE AND NEGATIVE QUANTITIES. 21 $10,000, the sum which B must obtain, in order to be as rich as A. This difference is expressed thus, 5000 — (—5000). Hence, 5000— (—5000) = 5000+5000 ; or —(—5000) =+5000. So — ( — a) = +a. Hence, b.) The subtraction of a negative quantity has the same effect as the addition of an equal positive quantity. 2. The latitude of New Orleans is 30° North; that of Buenos Ayres is 34° South. How many degrees is the one place North of the other ? That is, what is the differ- ence of their latitudes ? 3. X has a dollars, and Y owes b dollars. "What is the difference between their estates ? Ans. a — ( — b) ~a-{-b, as in example 1. 4. At sunrise on the 20th of February, the thermome- ter stood at 30° below zero; at sunrise on .the 20th of March, it stood at 30° above zero. What is the difference in the temperatures ? 5. The reading of the thermometer on one day is — 10° (10° below 0) ; on another day, it is — 20°. Which indi- cates the greater heat ? How much? §G. a and b. § 8. The process of finding the aggregate of several quan- tities, regard being had to their character as positive or negative, is algebraic addition ; the process of finding the difference between quantities so considered is algebraic sub- traction. Arithmetical addition and subtraction, on the other hand, relate to numbers regarded simply as such, without distinguishing them as positive and negative. (a) The algebraic sum may be less than the algebraic difference (§7. a) ; and (b) the algebraic sum may be equal to the arithmetical difference (§4) ; or (c) the algebra!' 'Ference, to the arithmetical sum. 22 INTRODUCTION. FACTORS AND POTTERS. § 9. Quantities multiplied together are called, a? in Arithmetic, factors" in respect to the product. and are also called coefficients' 2 in respect to each other. Thus, in the expressions 3a, 2a, ba, bca, and ha, 3, 2, b, be and I are coefficients of a. In 3xy, 3 is the coefficient of xy ; ox. of y ; and 3y, of x. a.) The coefficient shows, how many times the quantity multiplied is taken as a term (i2. d. N). If the coefficient is positive, it shows how many times the quantity is added : if negative, how many times it is subtracted (§4). Thu-. 3a =■ a-\-a-\-a : 2x =. x-\~ . — 3X+a — — a — a — a = 3X — a = — 3a. So —aX-\-b = aX—b = —ab. —2X—a = —(—a)—(—o ) = a+a (§ 7. a, b) = 2a. Note. In the last example, — a is to be subtracted twice; and subtracting — a twice has the same effect as adding +a twice (§7 b). Hence, if two factors multiplied together are both posi- tive or both negative, the product is positive : if one is posi- tive and the other negative, the product is negative. Or. more briefly. Like signs give -{-, unlike, — . 1 . What is the product of 2a and — b ? of — 2ab and — c ? aX— xy = what? —3aX—.ry? —3aX—xy? —2 X— 3 ? h.) A letter, or combination of letters, used as a coeffi- cient, is called a literal coefficient ; a number, so employed, is called a numerical coefficient. Coefficients are also dis- tinguished as integral or fractional, &c. (o) hut., maker, producer, (p) L. productus, produced, i e. by the multiplication. (& ; ~ = -4. 1. _9«Z,_^_2a;=what? — 2ab-+2a? 2ax^-—a? 2. _iOx-| 10 = what? 60-; 10? — GO-; 10? § 11. When a factor occurs more than once in a product, it is usually ivritten but once, and the num- ber of times it is employed, is denoted by a number or letter placed over it at the right, called an expo- nent", or index™. Thus, instead of aa, aaa, bbbbbb, we write a 2 , a 3 , b G ; instead of 2.2, 2.2.2.2, 3.3.3.3.3.3, we write 2 2 , 2 4 , 3 e , the exponent, in every case, showing how many times the quantity over which it is placed 'is taken as a factor; in other words, how many equal factors the product contains. Thus, in the expression, (a~\-b) 3 , the exponent 3 shows that a-\-b is taken three times as a factor, or that the pro- duct consists of three factors each equal to a~\-b. So, the product a' 2 b 3 x 6 contains two factors equal to a, three equal to b, and five equal to x. 1. Write 2.2.3.2.3.2 with exponents. Am. 2 4 .3 2 . 2. Write aabcabac with exponents. 3. Write 2 3 .\0 3 .o±.b* without exponents. 4. Write a i b 3 c 2 x 5 y 6 without exponents. Note 1. These expressions may be read thus; a 2 , a taken twice as a factor; b 3 , b taken three times as a factor; &c. Also, a 1 , (§11. a), ao (§13), a taken once, a taken no times as a factor; a 2 " (§ 12) a taken half a time as a factor; a~ 2 (§ 14), a taken minus twice as a factor; &c. Or, if the teacher prefer, the student may examine § 22 and a under it, and use the expressions given there. Note 2. A negative quantity may obviously occur more than once as a factor; as (— a)(—a) = (— a) 2 ; (— b)(— 6)(— b) =(— b) 3 . In such cases, if the number of factors be even, the product will be positive; for, if they be combined two and two, the product of each (t>) Lat. exponens, setting forth, showing, (w) Lat. indicator' mark. ALG. 26 INTRODUCTION. [§12. pair will be positive (§ 9. a) ; and the product of these positive pro- ducts will, of course, be positive. If the number of factors be odd, the greatest even number will give a positive product, and this, mul- tiplied by the remaining negative factor, will give a negative pro- duct (§9. a). Hence, If the number of negative factors be even, the product will be pos- itive; if odd, negative. Thus, (— a) a =+a* ; (— a) 3 — —a 3 ; (— a;)* = -f-a?*; (— x)* — —x 5 . a.) When a quantity is taken as a factor only once, the fact may be shown by the exponent l ; but in this case, the exponent is usually not written ; and whenever no exponent is written, 1 is always implied. Thus a is the same as o 1 j ax=a 1 x 1 ; ax 2 =za 1 x 2 . § 12. b.) The fraction J shows that the unit is separated into two equal parts, and that only one of them is taken. i i So the exponent 2 , in the expression a 2 , shows that a is separated into two equal factors, and that only one of them is employed ; in other words, that a is introduced as a fac- tor, half a time. If this half-factor were introduced two, 2. a ± three, or four times, we should have a 2 , a 2 , a'-. Thus, £ i. x Li i 4 a 2 = a 2 .a 2 .a~.a 2 = (a 2 ) . If a were separated into three, four, or n equal factors, 11-. and one only employed, we should write a 3 , a*, «" ; if tioo were employed, a 5 , a 4 , a n ; &c. Hence, The denominator of a fractional exponent shows, into how many equal factors the quantity under the exponent is sepa- rated ; and the numerator shows, how many of these fac- tors are employed. Thus, 9^ = 3 ; 9^ = 9^9^=3.3 = 9; 9 * = 3.8.3 = 27. 8^ = 2; 8^ = 2.2 = 4; 8^ = 2.2.2.2 = 16. 1 . What is the meaning of (aa) 2 ? of i? 2 ? of a$ ? 2. (# 2 )* = what? 16*! 27 * ? 25 ^ ? 36 * ? 49 ^ ? § 13.] FACTORS AND POWERS. 27 e.) Otherwise, as f = i of 4, a 2 indicates, that one half of four factors each equal to a are introduced ; or that a had been introduced four times as a factor, and the pro- duct, so formed, had been afterwards separated into two equal factors, of which only one was actually employed. Thus, a $ = a* X 2 = (a 4 ) *= (aaaa) ?=zaa=a z . Hence, again, The numerator of a fractional exponent shows, how ma- ny times the quantity under the exponent has been em- ployed as a factor ; and the denominator shows, into how many equal factors the product so formed has been separated. 1 £ 3 ™ Thus a 3 , a 3 , a 4 , a n indicate, that a, a 2 , a 3 , a m have been separated, the first two into 3, the third into 4, and the fourth into n equal factors, of which only one is employed. Or that a is employed as a factor £, ~, f , " of a time. Thus, 9^ = (9 2 ) 2 " = (9.9)' = 9; 8^= (8 2 )^= (8.8)^ = 64*= (4.4.4)*= 4. (i? 2 )5— (R2.R2.Esy=(B.B.B.R.R.li.)i=(RG)?= 1. What is the meaning of (aa)*? of 2 2 ? of 3-? of3 2 ? ofxh 2. (£2) 2 — w hat? 16*? 27*? 5*? 2 2 ? (x 3 )*? § 13. d.) Any quantity, which is not found as a factor in a product, may be introduced with zero for an exponent. For this exponent will show, that the quantity, though writ- ten, still is not employed, or is employed no times, as a fac- tor ; and, of course, the value of the expression is the same as if the quantity were not written. Thus a°bx 2 is the same as bz 2 ; ax° = a; but oXl=a; that is, aX#° = aXl; .-. x° = l. Hence, 28 INTRODUCTION. [§ 14. Corollary I. Any quantity with zero for its exponent is equal to unity. Note. A corollary 1 is an inference from a preceding prin- ciple. §14. e.) "When a factor is introduced less than no times (§ 6.), i. e. -when instead of being introduced, it is taken out, the fact will be properly indicated by a negative exponent (§§4,5). But a factor is taken out by division (§10). Consequently, a negative exponent shows, that the quantity under it is to be employed as a divisor, as many times (§ 11), or parts of a time (§ 12. b, c), as there are units or parts of a unit in the exponent. Thus, in the expression a~ 1 x, a, instead of being multiplied into, is to be divided x out of x, and the expression is therefore equivalent to -. a Also, in the expression a 2 x, the negative factional ex- ponent 2 indicates, that a is separated into two equal fac- tors, and that one of these half-factors (§ 12. b) is taken out five times by division ; i. e. that the whole factor a is taken out five halves of a time. This is evidently the same thing as saying, that it is introduced minus five halves of a time. In other word3 a 2 indicates, that a product, containing a five times as a factor, is separated into two equal factors, and that one of these two factors is to be taken out by di- vision. The expression is, therefore equivalent to — . a" So, *-»*-*. 10-112-^- 2- 5'-ii-^- 9-*.6 = -^ = | = 2. See§§17, 19. ^ 3 1. 2~ 1 .3 = what? 3-V2? 10~ 2 .30? I5-1.80? (x) Lat. corollarium, something given over and above, from co- rolla, a wreath, a common present or mark of honer. § 15, 16.] FACTORS AND POWERS. 29 2. «- 3 5 = what? a°b~ l x? b~*x? aV^ar 1 ? a~ m x m ? § 15. /.) If a quantity be found any number of times in n multiplier and multiplicand, it will be found in the pro- duct as many times as in both the factors. For a 2 b 3 Xa i b=zaabbbXaaaab = aaaaaabbbb (§2. e. N) = a G b*. Hence, Cor. II. The exponent of any quantity in a product wil be equal to the sum of its exponents in the factors. 1. a 2 b 2 Xab = what? ax 2 Xa 2 x? a°bc*Xa*bc* ? a 3 x°X« 3 y? 2. 2 3 .3 3 X2.3 2 =what? Ans. 2*.3 5 = 16X243 = 3888. 3. 2 2 .34x2°.3 = what? 5 3 .2x5°2? 10 2 X10 3 ? 4. 100*X 100^ = what? Ans. 10.10 3 = 10* = 10,000. 5. 100*X100 = what? 25*25*? 27*. 27 1 ? 16*.16*? 6. a*Xa 2 = what? cfix<$? 16 2 .16 2 ? 10*. 10$?. § 16. g.) It is also evident, that the exponent of a quan- tity in one of the factors must be equal to the exponent of that quantity in the product, minus its exponent in the oth- er factor. Hence, Cor. III. The exponent of any quantity in a quotient is equal to the exponent of that quantity in the dividend, mi- nus its exponent in the divisor. 5. m a 5 aaaaa n 10 3 „.„ a 2 4 Thus— = — aa — a 2 ; -— =10 2 ; — = a* = a 3 aaa 10 3 a {aaaaf = a 2 , x 3 , _ a 7 b 3 r « 2 J 2 a 3 b 2 . ax 5 . 1. — - = Whftt? rr—i — ? 5 ? ? x 2 a*b ab ab 2 a°x x 3 2. — = what ? ^4w5. a; 3 - 3 =:x° = 1 (Cor. I). *3 30 INTRODUCTION. [§ 17, 18 3. cfi-r<$ — what? 10 2 -M0*? a+c$? a^~a 2 ? 4.—- = what? Ans. x 3 ~ * = x~ 1 , or — =z—. Hence x* x± x § 17. It.) We have ar-i =-. See § 14 x -/ , «? '/*'> IT 10 Y* In like manner, — , — , — , -— , give x~ 2 , x~ 3 , x~*, 1111 « — * = — , — , — -, — -, respectively. Hence, x 2 ' x 3 ' x* x 11 ' l J ' Cor. IV. A quantity xoith a negative exponent is equal to unity divided hy the same quantity with an equal positive exponent. § 18. ?'.) The quotient obtained by dividing unity by any quantity is calied the reciprocal 1 ' of that quantity. Thus 1 1 1 1 -o- -, — , 1-f-a, — -, -7-T, 1-i-a 2 , or the equivalent expressions x x 2 10 10" _i x— l , x—-, a -1 , 10 — x , 10 — 2 , a 2 ,are the reciprocals of x, x 2 , a, 10, 10 2 and a- respectively. Also the reciprocal of 10~ 1 (= T V) is jrpr = 1 -T- tV = 10 5* tlie reciprocal of a- 2 (=\) is -i^l-r- 4 = « 2 -* Hence, • \ a 2 / a — a- To express the reciprocal of any quantity, we have only to change the sign of its exponent. Write the answers to the following questions both by means of exponents with their signs changed, and under the fractional form. What is the reciprocal of 2 ? of 3 ? of 10 ? of J ? of \ ? of T V(=10-i)? ofi? of.01? ofx 2 ? ofar-3? f 0- 3 ? of9 2 ? of 8^ ? of 25"^ ? (y) Lat. reciprocus, returning upon itself, mutual. *Note. This is evidently true; for, if a unit be divided into 10 equal parts, one of them will be contained in any quantity 10 times as often as the whole unit is contained in the same quantity; and, if thi unit be divided into a 2 equal parts, one of these parts will be contained a 2 times as often as the whole umt. §10.] FACTORS AND POWERS. 3i L) Positive and negative exponents have the Bame rela- tion of contrariety or oppositeness as other positive and negative quantities (§5). Thus, an exponent shows, how- many times or parts of a time a quantity is introduced as a factor. The opposite to introducing a factor is taking it out. When therefore a quantity is said to be introduced minus three, or minus n times, as a factor, it is the same thing as saying that it must be taken out three, or n times (§14). Thus, in example fourth (§ 16), x can be taken out three times, and the fact, that it is to be taken out once more, is indicated by the negative exponent — 1 (§ 4. c). It is to be so taken out, whenever, in subsequent multipli- catiun, x shall be introduced. If the operation be represented and performed in the x 3 1 fractional form, we have — = — ; that is, three of the four x+ x factors of the divisor are cancelled out of the dividend, and one remains to be taken out, whenever x shall be introduced into the dividend. § 19. I.) As the factors under the negative exponent di- minish the whole number of factors in the product, there- fore, (1.) In combining the exponents of a letter in the factors, to find its exponent in the product, the negative exponents must be treated precisely as negative terms in making ud an aggregate (§4). Thus a 3 X« —1 =a 2 ; x 5 Xxr~ 4, = x y - —X. 1. x 8 X^- 6 =what? x 13 Xar 4 ? a 2 b 2 Xa~ 1 b? 2. x 2 Xx~~ 5 = what? Ans. x 2 ~ 5 = x~ 3 . x 2 But — — x 2 ~ 5 —x~ 3 . x b . . i*/ /\ \Aj — — *Aj ^ *A* . (2.) In like manner, if negative exponents be found in a divisor or dividend, they must be treated like negative terms in finding a difference (§7). Thus, 3S INTRODUCTION. [§20,21. a*-^a- 3 = a 2 - (- 3 )= a 5 . See § 7. a, h. But a 2 Xa 3 =fl 5 . a 2 -ra- 3 = a 2 Xa 3 . Hence (1, 2), Cor. IV. To multiply or divide by a quantity with a neg- ative exponent, is the same as to divide or multiply by the quantity with an equal positive exponent. Or, more generally, To multiply or divide by any quantity is the same at to divide or multiply by its reciprocal. Thus ab 2 cx 2 y-^-abc = ab 2 cx 2 y X a~ 1 b- 1 c~ i = bx*y. 2 2 .3 -± 2.3 = 2 2 .3 X 2-i.3-i = 2. 1. abc -f- a be z= what ? abcXa— 1 b~ 1 ~c 1 ? o 2 xX« -1 x? a-x~ax~ l ? 2. 2 2 .3 2 .4 2 -^-2.3.4rr: what? 2 2 .3 2 .4 2 -^2- 1 .3~i.4- 1 ? 2.3.10-^2.3 ? 2.3.10-^2- 1.8-1 ? 3. cAcf* = what? « _2 \a~ 2 ? 10 -2 \10~* ? x.x~h 4. a 2 -^-a~ 2 " = what ? z-^""^ ? cT^+cT^? 10^-10"*? § 20. m.) When a quantity is taken a9 a factor any num- ber of times, and the product so formed is again taken as a factor any number of times, the first quantity will evident- ly be employed a number of times equal to the product of the exponents. (See § 12. Examples.) Thus, (a 3 ) 3 =a 3 .a 3 .a 3 =a<>; (a^ = flW = o^; 1. («-4) 3 =what? (2 3 ) 2 ? (10 2 )*? ( x *y? (a s )~h (a m ) n ? (2 s ) 3 ? ?i.) Thus we see that the exponent may be either inte- gral or fractional, positive or negative, and it may be either known or unknoion. §21. 0.) The analogy, as well as the difference, between the coefficient (§ 9. a) and exponent, is very obvious. Both relate to the introduction of equal quantities ; the coeffi- § 21.] FACTORS AND POWEKS. 33 dent, of equal terms (§ 2. d. K) ; the exponent, of equal /ac- tors (§ 9). If positive, they ajw-m the introduction of the quantities; the coefficiently addition (§ 9. a); the exponent, by multiplication (§ 11). If negative, they aVfty the intro- duction, i. e. they affirm the removal or taking out of the quantities ; the coefficient, by subtraction (§ 9. «) ; the ex- ponent by division (§ 14). If fractional, they show the in- troduction or removal, by addition or subtraction, or by multiplication or division, as the case may be ; the coeffi- cient of equal fractional parts (§ 9. J) ; the exponent, of «0M«£ components" (§ 12) of the quantity. That is, they show, how many times or parts of a time, a quantity is introduced or taken out ; the coefficient, as a term ; the exponent, as a. factor. In other words, they show the introduction, positively or negatively (§ 4), of a term or factor, so many times as there are units in the coefficient or exponent. Thus +2X4 = 4+4 = +8; 4+ 2 = 4x4=16. _ 2X 4 = -4-4 = -8; 4-« = ^=Q=^ +£.16 = i(S+8) = 8 ; 16 + * = (4.4)* = 4. _i.l6-i(_8-8)=-8; 16"* = -^- = —^ = \ 16* (4X4) 2 * So, x-{-0Xa=.x J r = x; xXa°—xXl—^- 1. Write abbbccxxxx with exponents. ^4«s. a 1 & 3 c 2 x 4 . 2. Write in like manner, aayy, fefcc, (a-\-b)(a-\-b). 3. Write a-b 3 x 1 y° without exponents. Ans. aabbbx. 4. Write in like manner, a 4 j; 5 , (a+6) 2 , (a — 6) 3 . 5. Write with exponents 2X2x3x2x3x4. Ans. 2 3 .3 2 .4 1 . 6. Write with exponents 2X2X2X3X3X2X3X4X4. 7. 4 3 — what? Am. 4X4x4=64. (z) Lat. coinpono, to compose ; factors, which, multiplied together, produce a quantity, are called its components. INTRODUCTION. [§ 22. 8. 4* = what? 53? 7 2 ? 10*? 10 5 ? 21 2 ? 9. What is the difference between 10x4 and 10 4 ? 10. Show the difference between 3a and a 3 ? Arts. 3a — a-{-a-\-a ; a 3 =zaXaXa. 11. What is the difference between a° and aXO? be- •ween 10° and 10X0? between 1° and 1X0? between ■» u , 1° and 10° ? between 1°, l 1 and l 2 ? i 12. Show the difference between \a and a 2 . Ans. \a-\-la = a, crXa = a. i 13. What is the difference between l00XH and 1002 ? ween i of 9 and 9- ? between £.27 and 27 3 ? between §.16 and 16^? 14. What is the difference between — 3 2 , 3~ 2 , 3—2 and 3(-2)? ^s.-3 2 =-9; 3~ 2 =p = i; 3-2 = 1 ; 3(-2) = -6. 15. Write in like manner 6, 8, 10 and 15, and interpret the exnressions. 1G. What is the difference between 9" and 9 2 ? 8 3 and 8~ f ? 17. What is the reciprocal of 10 ? of 10 2 ? of 100? of— 10? ofl? of a? of-^r? of-? of a- 1 ? ofa ? ? 10 a '«-"? of 100*? of 27"*? of 8^? 18. a 2 -^a=what? a*+a°? a*+a 2 ? a 2 -~a 3 ? 19. « 3 -^-« = what? o 3 H-a°? a 3 -^" 1 ? a 3 -H*- 2 ? 20. Substitute 10 for a in the last two examples. 22. If a is employed m times, and b, n times, what is the expression for their product ? § 22. Any quantity ivith an exponent, is called a i'ower of the quantity under the exponent. § 22.] FACTORS AND POWERS. 35 Note. The quantity under the exponent is called the base,, of the power. a.) A power is designated by its exponent. Thus, x~ 2 is read x minus second power; x~ 1 , x minus first power. '" " a: zero power; x 1 , x first power. x 2 " x second power or square* ; re 3 , x third power or cube . 1 2. X s " x one half power ; x 3 , x two thirds power. _i x 2 " x minus one half power, &c. b,) It will be observed, that the term power, as used here, has a wider signification than is attached to it in Arithmetic. In Arithmetic, the term is applied only to a product of equal factors. As here defined, it includes a single factor (§ 11. a), unity (equal to the zero power (§ 18) of a factor), and all products and quotients formed by mul- tiplying and dividing (§ 14, 17) unity, any number of times, by the factor, or by any of its equal components (§ 12, 14). c.) We have therefore several classes of powers, distin- guished by the characters of their exponents. Thus, there are (1.) Powers with positive integral exponents (§ 11), the same as ordinary arithmetical powers ; (2.) Powers with positive fractional exponents (§12), consisting of equal components and their combinations ; (3.) Powers with negative integral exponents (§ 14), the reciprocals of the first class ; and (4.) Powers with negative fractional exponents (§14), the reciprocals of the second class. d.) Powers of these several classes are sometimes called positive, negative, &c, powers ; meaning, not that they are positive or negative, integral or fractional quantities, but (a) Gr. /3aaif, foundation, (b) Lat. quadra, Fr. quarre'; be- cause the second power of a factor represents the surface of a square, whese side is represented by the factor (Geom. §§124, 171, 177). (c) Gr. Kii/3oc; because the third power of a factor represents the solid content of a cube, whose edge is represented by the factor. INTRODUCTION. [§ 23; that they have such exponents. So a power, whose expo- nent is an even number, is frequently called an even pow- er ; one whose exponent is an odd number, an odd power. § 23. One of the equal factors (§12) of a quantity is called its root. a.) A root is called the second or square, the third or cube, the fourth, the nth, according as it is one of two, three, four, or n equal factors, which produce the given quantity ; i. e. into which the given quantity is separated. Thus 2 is the third or cube root of 8, because it is one of three equal factors, which produce 8. So a is the third root of a 3 , the fourth root of a 4 ; a 2 is the second or square root of a 3 , because it is one of the two equal factors, into which a 3 may be separated (§ 12. c). b.) A root of any quantity is properly expressed by writ- ing the quantity under a fractional exponent, whose numer- ator is unity, and whose denominator is equal to the num- ber of the root (§ 12. c). For this denotes, that the quan- tity under the fractional exponent is separated into so many equal factors, as there are units in the denominator, ami that only one of them is taken. Thus, x i The second or square root of a is cr ; that of a 2 is (« 2 ) — a - (§ 20) — a ; the third or cube root of a is a 6 ; that of a 2 is 2 )*=cA c.) The principle of § 12. b, c may, therefore, be express- ed as follows : A fractional exponent shows, either that the root of the base, denoted by the denominator, is raised to the power de- noted by the numerator (§ 12. b); or, that the base being raised to the power denoted by the numerator, the root de- 5 noted by the denominator is taken (§ 12. c). Thus, a 2 ex- presses the fifth power of the second or square root of a ; or the square root of the fifth power of a ; so 8^ is equal § 23.] FACTORS AND POWERS. 37 to the square of the cube root of 8 ; or to the cube root of the square of 8. d.) A root is also frequently indicated by the radical* sign, */ c , placed before the quantity, with a number over the sign, to show the number of the root. In expressing the second or square root, however, the number is more frequently omitted ; and, accordingly, wherever the sign stands without a number over it, it must always be under- stood to denote the square root. Thus, x .y 4 = 4 2 = the second or square root of 4. V8 = 8* = " third or cube « « g. «V« ==■«* = " fifth " " a. Note. Either of these forms of expressing the root, may be used at pleasure, and both should be made familiar. The fractional ex- ponent is, however, generally, more convenient than the radical sign; and is, besides, to be preferred because it exhibits roots as a class of poivers, and enables us to refer the operations upon roots to tho gen- eral principles, which govern the operations upon powers. Quanti- ties written under a radical sign are frequently called radical quan- tities. e.) As the product of an odd number of positive factors is positive, and of negative factors, negative (§11. Note 2) ; hence, an odd root (i. e. a root denoted by an odd number) of any quantity must have the same sign as the quantity it- self. Thus, (+a) 3 — +« 3 > and ( — a) 3 = — a 3 . (+a 3 ) * or V+a 3 = +a ; JL and (— a 3 ) 3 , or 3 y— a 3 = — a. /.) Again, since the product of an even number either of positive or of negative factors is always positive (§11. Note 2) ; therefore, (1.) Every even root (i. e. every root denoted by an (d) Lat. radix, root, (e) A modified form of the letter r, the initial of radix. ALG. 4 38 INTRODUCTION. [§ 24. even number) of a positive quantity may be either positive or negative. This character of the root is denoted by the double sign ± (read plus or minus). Thus, (+a)(+a)=:-(-a 2 , and (—«)(—«) = -(-a 2 . (a 2 ) 2 or ya 2 == ±a. (2.) An evera root of a negative quantity, can be neither positive nor negative, and therefore does not really exist, and is said to be imaginary. For neither (-\-a)(-\-a),nor ( — a)( — a) can produce — a 2 . § 24. It is evident, from the definition of a power, that whatever has been demonstrated of quantities with expo- nents is true of powers. Hence we have the following rules. RULE I. a.) To multiply powers of the same quantity to- gether. Add their exponents. § 15. Cor. II. a 4 .a 3 = what? a~\a G ? Ac 2 ? aAaf*? 3*.3~ 3 ? RULE II. b.) To divide a power of a quantity, by any pow- er of the same quantity. Subtract the exponent of the divisor from that of the dividend. § 16. Cor. III. _=what? — ? ^? gj? -t? ~? RULE III. c.) To find the reciprocal of a power. Change the sign of the exponent. § 18. § 25, 26.] FACTORS AND POWERS. 39 What is the reciprocal of a ? of a 4 ? of 10 ? of 10 2 ? of lO-i? ofi^? ofa 2 ar 2 ? ofx^? ofa^cc -5 ? RULE IV. d.) To find any power of a power. Multiply the exponent of the given power by that of the required power. § 20. 3 •- 1. What is the second power of a 2 ? of a 1 ?' of 16 2 ? of 16"^? of— a? 2. (a 2 )- 2 — what? («*) e ? (— 10) 3 ? (or*)*? 3. (a*)*=what? (10g) 2 ? (R 2 )~h (xrf? (lO^h § 25. e.) The last rule obviously applies equally to the finding of a root ; i. e. a power, whose exponent is unity di- vided by the number of the root (§ 23. b). But to multiply by such a fraction is the same as to divide by its denomina- tor. Hence we have the common rule for finding a root of a power : Divide the exponent of the power by the number of the root. What is the third root of a 3 ? of a 2 ? of a ? of 10 6 ? What is the second root of 10 4 ? of x 3 ? of x 6 ? of 2 ? What is the third root of 10 2 ? ofa^? ofa^? of af" 3 ? Note. It should be borne in mind, that the word power is used, !n all these cases, in the ividest sense; and that the rules are equally applicable to all the classes of powers specified in § 22. § 26. A quantity, whose value is determined by the value assigned to another quantity, is said to be a function 7 of that other quantity. Thus, a 2 , a 3 , a*, are functions of a, because their value depends upon, and is determined by, the value assigned to (/) Lat. functio, from fungor, to perform, as depending on the performance of certain operations upon another quantity. 40 INTRODUCTION. [§ 27, 28. a. Thus, let a = 1, then a 2 = l; if a = 2, then a- = 4; if a = 10, then a 2 = 100. So, if 2« = a; 2 , or u — 2x, or u — 3x, then m is a function of a; ; or, as it is usually expressed, u = F(x), or u =zf(x) ; where F and f are not factors, but mere abbreviations for the words function of A power is a function of a quantity, expressed by an exponent written over the quantity ; i. e. an expo- nential function of the quantity. § 27. A power is said to be of such a degree^ as is indicated by the exponent. Thus, a 3 is of the third degree; « 2 , of the second; a of the JL first; «~ 4 , of the minus fourth; and a 2 , of the one-half de- gree. § 28. The degree of a term is equal to the sum of the exponents of its literal factors. Thus, a, x, 2x, 3a 2 :*:- 1 , a 3 b°x~ 2 are of the first degree. L I 3 _X. 2. 1 So a-x-, a-.x -, a' s x' s are of the first degree. \ 3 2ax, 2px, y 2 , a*b~ 2 , p 2 x 2 are of the second degree. 3a-x is of the third, and Aa 3 x, of the fourth degree. i a a 2 is of the one half, and a 3 , of the two thirds degree. a~ 2 x~ 2 , and a 3 x~ 7 are of the minus fourth degree. 1. 9c< 5 6 4 c- 3 is of what degree? 15x 2 y 2 ? 5a 3 &X&cy? 5 3 2 7 5 _2 1 _2_ «-&%? 3a%3? a 3 x 3 ? a 3 « 3 ? or 3 * ? Note. A term ia also sometimes said to have as many dimen- sions 71 as there are units in its degree. (g) Fr. degre', from Lat. gradus, step, (h) Lat. dimensio, from dimetior, io measure. The use of this word resulted from taking a factor to represent a line, and, consequently, a product of two fac- tors to represent a surfaco, and one of three factors, to represent a solid. The factors were therefore regarded as the dimensions, or measures of the magnitudes. See Geom. §§3, 170,177. The word is, of course, not strictly applicable to any term of a degree higher than the third (Geom. § 2. a), or lower than the first. § 29, 30.] FACTORS AND POWERS. 41 a,) In estimating the degree of a fractional term, the exponents of the letters in the denominator must, of course, be regarded as negative (§ 14, 16), and subtracted from the sum of the exponents of the letters in the numerator. Thus, 3 n - h * ft I) y — is of the first degree ; — -r- and , are of the second. a 2 a c b.) A term is said to be of the first, second, third or nth degree with respect to a particular letter or letters, when it contains the first, second, third, ^or nth. degree of the letter or letters. Thus, 3a 2 x, and a~ 1 x are of the first degree with respect to x. b 2 x 2 and ax 2 are of the second " " x. x abx 2 and +/x are of the one half " " x. a 2 x°, and abc are of the zero " " x. axy is of the first degree with respect to either a, x or y ; and of the second degree with respect to x and y, or any two of the letters ; while it is of the third degree with respect to all the letters. § 29. Terms of the same degree are said to be ho- mogeneous'. Thus,^, 2x, and \a 2 x~ x are homogeneous. So a 3 , Sax-. . ... ab B 2 x" xyz ; m like manner, y, — , and . 2 f / x, 1. Are A 2 y 2 and B 2 x 2 homogeneous? x 3 , 2y 2 and x'r JR 2 and sin a sin b? § 30. Terms, which consist of the same literal fac- tors, with the same exponents (i. e. each letter being of the same degree in the several terms), are called similar or like terms. Thus, 2xy, 8xy, and 3yx are similar terms ; so 3x 2 y, and \x 2 y. But 3x 2 y and 3xy 2 are not similar, because, though the letters are the same, they have different exponents in the two terms. Are 3x 2 y and 3xy 2 homogeneous ? (i) Gr. buoyevris, compounded o( 6/jor, like, and yevoc, kind, *1 J 42 INTRODUCTION. [§ 31-33. Are a 2 b 2 and x 2 y 2 similar? Are they homogeneous? a.) Thus terms may be homogeneous without being sim- lar, but they cannot be similar without being homogeneous. b.) Terms, in which the same letter, with the same expo- nent, enters, are sometimes said to be similar with respect to that letter. Thus the terms ax, obex and c*x are similar with respect to x. MONOMIALS AND POLYNOMIALS. § 31. A quantity consisting of one term, is called a monomial*; of more than one, a polynomial'. A polynomial of two terms is called a binomial'"; one of three terms, a trinomial". Thus, 2ax, a, a 2 b 2 , abc are monomials ; so aby.xy-^z ; a-\-b, a — b, x- — y 2 are binomials; a-\-b-\-c, a 2 ±2ax-\-x 2 are trinomials. § 32. A polynomial is said to be homogeneous. when all its terms are homogeneous (§ 29.). Thus, a 3 ±3a 2 b-\-3ab 2 ±b 3 , A 2 y 2 -{-B 2 x 2 —A 2 B 2 are ho- mogeneous polynomials. 1. Is x 2 -\-y 2 — R 2 homogeneous? x 5 ±ox 4 y-\-lQx :i y- ±\0x 2 yZ-\-oxy±±y 5 ? b 3 b 3 2. Is a 2 A-b 3 homogeneous? a 2 1 ? a 3 -\ ? c ' a a §33. When the several terms of a polynomial contain different powers of any letter or letters, it is generally con- venient to arrange the terms according to the powers of some one letter ; that is, to write the term containing either the highest, or the lowest power of the letter first, and the other terms successively, according to the order of their ex- (fc) monome, from Gr. fiovoc, alone, and bvojia, Lat. nomen, name; as being expressed by a single name or term. (I) poly- nome from Gr. -oA<'< , many, and oroiia, name, (m) Lat. bis, twice, and nomen, name, (n) Lat. tres, three, and nomen, name. § 34.1 REDUCTION OF POLYNOMIALS. 43 ponents ; from highest to lowest, or from lowest to highest. If the highest exponent is placed first, the terms are said to lie arranged in a descending series, or according to the des- cending powers of the letter ; if the lowest is placed first. the arrangement is said to be in an ascending series, or ac- cording to the ascending powers of the letter. Thus, a 2 -\-2ab-\-b~ is arranged according to the ascend- ing powers of b, and according to the descending powers of a. 1. Arrange oa Q b-\-3ab 2 -\-b 3 -\-a 3 according to the des- cending powers of a ; of b. 2. Arrange r qx n ~ 2J r x n -\-px"- 1 -\-rx n - 3 according to the descending powers of a?. Note. The letter, according to whose powers the terms of a po- lynomial are arranged, is frequently called the letter of arrange- ment. When there is no special reason for a different order, it is generally convenient to write the letters of each term in the order of the alphabet; and also to take the first of those letters, as the letter of arrangement. REDUCTION OF POLYNOMIALS. § 34. A polynomial, which contains similar terms. can be reduced to a simpler form. This is done according to the principles of § 4. Thus, in the polynomial 4a — 6a-f-9a — 3a, 4a and 9a are to be added, and 6a and 3a are to be subtracted. It is usually- most convenient to brino; together the terms which are to be added, and also the terms which are to be subtracted, and then take the less from the greater. If the quantity to be added is greater than that to be subtracted, the result is to be added ; i. e. is positive. If the quantity to be sub- tracted is greater than that to be added, the result is to be subtracted ; i. e. is negative (§ 4. a, b). Hence, for reduc- ing or simplifying a polynomial containing similar terms, we have the following 44 INTRODUCTION- [§ 3-i. RULE. Add together the coefficients of such similar terms as have the sign +; and then the coefficients of such as have the sign — ; take the less of these sums from the greater ; and prefix the remainder, with the sign of the greater, to the common letter or letters. Thus, 4a — 6a-\-9a — 3a — 4a-f-9a — 6a — oa = 13a — 9a = 4a. a.) Terms of a polynomial, which are not similar, will, of course, remain as they were ; each being preceded by its own sign. Reduce the following polynomials to their simplest form. 1. a 2 —ab—ab-\-b 2 . Ans. aj—2ab+b 2 . 2. a 2 -\-ab—ab—b 2 . b.) There may be several sets of similar terms in the same polynomial. In that ease, the above method must, of course, be applied to each set separately. Reduce, 1. 5 a _|_G6— 7x— 8J-f3a— 4a+2ic+9a— 3x. 2. «*_ 3 a 3 x -{-3a 2 x 2 — ax 3 — a 3 x-\-3a 2 x 2 — 3ax 3 -{-x*. 3. l-\-x—l-\-x. 4. 1-4-x+l— x. 5 . y 2 -\-x 2 —px-\-\p 2 — x - —px — \p 2 . 6. 2bx-\-2x 2 — b 2 — 2bx— x 2 . 7. a 3 +a 2 b+ab 2 —a 2 b—ab 2 —!> 3 . c.) If a polynomial contains several terms similar in res- pecl to a certain letter (§ 30. b), the same principle will ob- viously apply. Thus, the terms ax-\-bx — 2cx, are similar in respect to x. Now, a times x, 2)lus b times x, minus 2c times x is evidently the same as x taken a-\-b — 2c times, which (§ 2. h) is expressed (a-\-b—2c)x. Hence, we may write the coefficients, whether numerical or literal (§ 9. b), of the common letter or letters in the several terms, in or- der, with the signs of the terms ; enclose the whole expres- sion, so formed, in a parenthesis, or put it under a vinculum ; } 35-37.] EQUATIONS. 45 and write the common letter or letters, without the paren- thesis or vinculum, as a separate factor. Eeduce, 1. A 2 x 2 — c 2 x 2 . Ans. (A 2 — c 2 )x 2 . 2. 2px"-\-px — px". 3. j^ y z+B 2 x 2 —A 2 B 2 —A V 2 — B-x" 2 -\-A 2 B 2 . § 3-5. The numerical value of an algebraic expres- sion is the result obtained by assigning particular values to the letters, and performing the operations indicated by the symbols. Thus, Let a = 10, and b = 5, then a-f-5 = 15; (a-\-b) 2 =z 15 2 = 225 ; a 2 -\-2ab+b 2 = 10 2 -f 2.10.5+5 2 — 225. 1. Let« = 10 and 5 = 4, and find the value of a 5 -\- 3a 2 b+3ab?-{-b 3 ; (a—b) 2 ; a 2 —b 2 ; (a+b)(a—b). 2. Find the value of the same expressions, when a = 8, and b = 3; when a = 20, and 5 = 5; when a = 10, and 5=10; when a = 10, and 5 = 9 • when a = 1, and 5 = 1. 3. Find the value of y^-2x— 4, when ?/ = 10, and # = 3 ; when y = 8, and :c = 2 ; when # = 4, and x = 0. EQUATIONS. §3(5. The expression of equality between two quantities constitutes an equation" ; as, 5+4 = 10—1 ; d m Xa" = a m + n ; 3x = 15 ; ax = b. a.) The two quantities themselves are called the mem- bers 1 ' or sides of the equation. The member on the left of the sign is styled the Jirst, and that on the right, the sec- ond member. b.) Most of the investigations and reasonings of Algebra are carried on by means of equations. § 37. c.) The simplest form of equation is that, in which (o) Lat. sequatio, from icquo, to make equal, (p) L. membruin. limb. '16 INTRODUCTION. [§ 38, 39. the two sides are precisely alike ; as 10 = 10 ; x-\-2 ±= x~\-2. These are called identical' 1 equations. d.) Another class of equations we have already employ- ed, in expressing the results of operations, or the truth con- tained in such results. Thus, a 3 X« 4 = « ; ; a m a n = a'"+" ; l-7-x=. x~ l (§17). These may be called absolute equa- tions ; inasmuch as their truth has no dependence upon the value assigned to a, x, m or n. The second member necessarily results from the operation indicated in the first. § 38. e.) In another class of equations, there is no abso- lute and essential equality between the members ; but they are equal only on the condition, that some particular value or values be given to one or more of the quantities involv- ed. Equations of this kind may be called conditional equa- tions. Thus, 2x =10 is a conditional equation, in which the equality of the members depends on the condition, that x shall be equal to 5. If 4 were taken as the value of x the two members would not be equal ; we should have 8 on one side, and 10 on the other. But taking x=^5, then 2X5 = 10, or 10 = 10. f.) A conditional equation, moreover, itself furnishes the means of investigating and ascertaining the value which must be given to x, in order that the members may be equal ; that the equation may become absolute or identical. For lx is obviously half as much a3 2x ; if then, we have 2x=zl0, we shall have a: = ^ of 10 = 5, the necessary value of x, as above. Conditional equations may therefore be called equations of investigation. § 39. Any quantity, to which a particular value must be given, in order to render the members equal, is called an unknown quantity (§ 1. c, N). That value of an unknown quantity, which renders the members equal, is called a root (q) Fr. identique, from Lat. idem, the same. §40,41.] EQUATIONS. 47 of the equation. "When this value is substituted for the un- known quantity, it is said to satisfy or verify the equation. The process of finding the root of an equation is called solv- ing the equation. Note. When equations, without any specification, are spoken of, or when the subject of equations is spoken of, as a branch of alge- braic science, the expression must, in general, be understood to im- ply conditional equations. § 40. Conditional equations are distinguished into orders, according to their degree. a.) The degree of an equation depends on the degrees of its terms with respect to the unknown quantity or quanti- ties (§ 28. b) ; and is determined by the range of those de- grees from lotvest to highest. b.) The full consideration of this subject would involve the consideration of equations containing negative and frac- tional powers of the unknown quantities. c.) For the present, however, it is sufficient to consider those equal ions only, in which the exponents of the un- known quantity or quantities are all integral, and in which the least of those exponents is zero. d.) In this, case, the degree of the equation is ihe same as the highest degree of its unknown quantity or quantities. Thus, ax ~b, 2x— 10, and x-\-y — 10 are of the first degree. ax 2 =b, x' 2 -\-3x = 10, and xy = 20 are of the second degree. § 41. We shall, at present, confine ourselves to the con- sideration of equations containing but one unknown quanti- ty ; subject also to the limitation mentioned above (§ 40. c). These equations are said to be of the same degree as the highest power of the unknown quantity which they contain. Thus, 3x= 18 ; ax — b are equations of the first degree. I INTRODUCTION. [§ 42, 43. x- — 9 ; ax 2 -\-bx = c are equations of the second degree. ax*-\rbx*-j-cxz=hi x 3 =S « third " a^+6x 3 =rc; x 4 = 16 « fourth « x n +^"- 1 +&c. = 7« is " nth « Note. Equations of the_/irs£ degree are sometimes called simple equations ; those of the second degree, quadratic*; those of the third, cubic ; and those of the fourth, biquadratic*. \ 42. All reasoning by means of equations pro- ceeds upon a single axiom", or self-evident truth ; viz. Equal quantities, equally affected, remain equal. Geom. 20. The meaning of this axiom, which, though not always expressed in words, is assumed in all mathematical opera- tion, may be illustrated by a few familiar examples. Thus, 3x5 = 15 is an equation. Adding 2 to both sides, we have 3X5+2 = lo-f-2. Subtracting 4 from both sides of the first equation, we have 3x5 — 4=15 — 4. In like manner, we might multiply or divide both sides by the same quantity, and obtain equal products or quotients. Hence, if both members of an equation be a. increased by the addition of, h. diminished by the subtraction of, \ equal quantities, the re- c. multiplied by, [ suits will be equal. d. divided by, J § 43. I.) 1. Given x— 3 = 7 ; to find the value of x. Add 3 to each side ; then a;— 3+3 = 7+3. § 42. a. or x = 10, the value required. 2. Given x — 5 = 4, to find the value of x. Ans. x = 9. 3. Given x— 1G = 20, to find the value of x. (s) Lat. quadra, square, (t) Lat. bis, tivice,and quadra, square. (») Gr. utjiu/ia, from dftou, to deem worthy, suppose, take for grant- ed. 1 44.] EQUATIONS. 49 II.) 4. Given x-{-3 = 7, to find x. Subtract 3 from each side ; then x+3— 3 = 7—3. § 42. b. or x = 4, the value required. 5. Given x-|-10 = 15, to find a\ ^4ns. x= 5. 6. Given 2a; = 10-f-#, to find x. Subtract x from each side ; then 2x— x = 10-\-x— x. § 42. b. or a? = 10, the value required. 7. Given 3x — 10 = 10-(-2x, to find the value of x. Note. To verify or prove these results, we have only to intro- duce, into the given equation, the value found for the unknown quan- tity in place of the unknown quantity itself. Thus, in example 1 above, substituting for x its value found, we have 10 — 3 = 7, an absolute equation. See §39. Verify the otber equations in like manner. § 44. Thus we see that the application of § 42. a and b causes any term, which stands on one side of an equation, preceded by the sign either of addition or subtraction, to disappear from that side, and to reappear on the other side with the opposite sign. Thus, in § 43. 1, by adding 3 to both sides, and reducing, —3 is canceled in the first member, and -f-3 appears in the second ; so, in § 43. 4, -f-3 is canceled in the first member, and —3 appears in the second. This is called transposition 1 '. For the same effect would obviously have been produced, if we had simply removed the term from the one side, and written it with the oppo- site sign upon the other. In fact, removing — 3 (i. e. ceas- ing to subtract 3) from the first member (§ 43. 1) increases that member by 3 ; 3 must, therefore, be added to the sec- ond member, to preserve the equality. So (§ 43. 4), re- moving -f-3 (i. e. ceasing to add 3) diminishes the first (t>) Lat. transpositio, from transpono, to place beyond, carry ovi r, ALG. 5 i ) INTRODUCTION. [§ 45. member by 3 ; 3 must, therefore, be subtracted from the second member. Hence, Any quantity may be transposed from one side of an equation to the other, if, at the same time, we change sign. If we transpose all the terms of an equation, the signs will all be changed, and the members will still be equal. Hence, Corollary. The signs of all the terms of an equation may be changed at pleasure, without affecting the equality of the members. It is also evident from § 42. a, b, that the same quan- tify, with the same sign, occurring on both sides of an equa- •. may be suppressed. c.) The object of transposition is, in general, to bring all the terms containing the unknown quantity to stand on one ^ide of the equation ; and all the known terms, upon the i ither. The polynomials so formed should, of course, be reduced to their simplest form (§ 34). 1. Given 8a+4 = 72+12, to find the value of x. 2. Given loy — 3 = 12+5?/ — 3+9?/, to L find the. value of y. Ans. y= 12. 3. Given 2x-{-a-\-b = ox-\-2a — 2x, to find x. Ans. x = a — b. 4. Given 4a+-3a+25 = 4a+3a+S, to find x. 5. Given 2>x— 10 = 5+2x— 15, to find x. Ans. x = 0. 6. Given 2x — 10 = x— 15, to find x. Ans. x = — 5. x ^ 45. 1. Given -+3 = 8, to find x. 4 Transpose 3 ; then i — 5 - § 44> 4 Multipl}' by 4 ; then x = 20. §42. c. § 46.] EQUATIONS. 51 To verify this equation, substitute 20 for x, and we hav< 20 _-J-3 = 8; or 5+3 = 8, an absolute equation. 4 2. Given ^—5 = 3, to find x. Ans. x = 24. o X 3. Given - — ^x— 2 = 0, to find x. Multiply by 2 ; then x— %x— 4 = 0. -'. c. Multiply by 3 ; then 3a;— 2x— 12 = 0. Reducing, a;— 12 = 0. § 34. x = 12. H4- 4. Given - — - = 5, to find x. 6 4 § 46. Thus, if a quantity in an equation be divided by any number, the application of § 42. c enables us to free it from its divisor, i. e. to clear the equation of fractions. The terms of an equation may, therefore, be freed from divisors, or, in other words, an equation may be cleared of fractions, by multiplying- all the terms of the equation by the denominators of the fractional terms. Note. The equation is to be multiplied first by one of the denom- inators, and then the resulting equation by another, and so on, till all the terms containing the unknown quantity become whole numbers In this process improper fractions may always be reduced to whole numbers, whenever it can be done; and no more multiplication.- should be performed, than are necessary to clear the equation of fractions. a.) The same effect would obviously be produced by mul- tiplying all the terms of the equation, by any common mul tiple of the denominators ; i. e. by any number which th< denominators will all divide without a remainder. For ii ^nominator will divide the multiplier, it will necessarih 52 INTRODUCTION. [§ 47. divide the product of its own numerator into that multipli- er. Thus, Let - + -4-^ — £^_i_i Multiply by 30 ; en 1 ox+Gx+ox = 25a;+30, §42. e X=r30. §§ 44, 34 Given --j-- + -__-__f-i, to find x. Note. A common multiple may easily be found by trial. Thus, in the above equation, try 8, the largest denominator, and see if the other denominators will divide it without a remainder. We find, that 3 and 6 will not so divide it. Then multiply it by 2; still we do not obtain a multiple of those denominators. Multiply 8 by 3, and all the denominators will divide the product; 24, therefore, is a multi- plier, which will clear the equation of fractions. It is important to employ the smallest multiplier, which will accomplish the object. b.) By clearing of fractions, the coefficients of the un- known quantity all become integral ; and the polynomial, formed by collecting all the terms containing the' unknown quantity into one member, is the more easily reduced to a simpler form (§34). Note. Whether transposition or clearing of fractions be first per- formed, is indifferent. Any course may be taken in this respect which is found convenient. See §45. 1, 3. The whole process of clearing effractions, transposition, and reducing the polynomial mem- bers to their simplest form, is sometimes called the reduction of tin equation. 1. Given — -f- — = — -)-5, to find x. Ans.x — ^- 3 4 3 2. Given - + - = — +4, to find x. Zoo 3. Given x-f-10 — — + ^ +50, to find x- § 47. 1. Given 2x— 7 = 9— Gar, to find x. Transpose ; then 8.*= 16. §44. § 48.] EQUATIONS. 53 Divide the terms by 8 ; then x = 2. § 42. d. 2. Given 3x-\-5 = x-\-20, to find x. Ans. x — l\. 3. Given 4x — 8 — 40 — 2x, to find x. Thus it is obvious, that, when, by any means, a single term containing the unknown quantity is made to constitute one member of an equation, while the other member consists wholly of known quanti- ties, the root of the equation will be found by divid- ing both members by Vie coefficient of the unknown quantity. Note. If the coefficient is unity, there will, of course, be no need of dividing. § 48. Bringing together the principles above explained (§§ 43-47), we have, for solving equations of the first de- gree, containing but one unknown quantity, the following RULE. Clear the equation of fractions, and bring alt the. terms containing' the unknown quantity upon one side, and all the known terms upon the other. Reduce tht two members to their simplest form, and divide them both by the coefficient of the unknown quantity. 1. Given - +2 = - -|- - -|-3, to find x. Ans. x = 20, 2. Given 6|+^-3 = ^+^+2H, to find *. 6£ and 2\^ are obviously the same as 6-J-£ and 2-f-}£. Either form may be used. In this instance, the latter form will be found more convenient for reduction. 3. Given x—%xz=33—3x, to find x. *5 54 INTRODUCTION. [§49-51. § 49. Many equations, which are not of the jirst degree. can be so easily reduced to that form, that they may, pro- perly enough, be briefly considered in this place. I. An equation may contain higher or lower powers of the unknown quantity, which may be canceled by transpo- sition, so as to leave no power higher than the first, or low- er than the zero power. Thus, Let *"+? + -£ +3 = x+af. Canceling a", we have — -| — = — [-3 = x, an equation of the first degree, o o Equations of this form, or which, on reduction, take this form, need no farther remark. § 50. II. An equation may contain only the zero and mi- nus first powers of the unknown quantity. This may pro- perly be called an equation of the minus first degree. But, if we multiply by the unknown quantity, we shall evident- ly reduce the equation to the common form of the first de- gree. Thus, Let x- l +2x~ 1 +3x-i = 2 : or 6a;- 1 — 2. Multiply by x ; then 6x° — 2x; or &=2x. < 12. - x=-2>. Otherwise, --j [-- = 2. '< xx x Clearing of fractions, 1+2+3 = 2x ; or 6 = 2x. §46. x — o. Hence, Equations of the minus first degree can be reduced to the first degree by multiplying by the unknown quantity. §51. III. Any equation containing only tico powers of the unknown quantity, provided their exponents differ by 52.] EQUATIONS. 00 unity, may evidently be reduced to the common form of the first degree, by dividing by the lowest poicer of the unknown quantity. Thus, Let x 2 — lOx = ; orx 2 = 10./ . Divide by x ; then a;— 10 = 0; or x =10. § 42. c£ So also if .T n =5a: n -i, then, dividing by a.-" -1 , x — 5. 1. Given 3x 5 -f-2a; 4 — x 3 = lx 5 -{-llx' 1 , to find x. 3 1 3 X 2. Given x?-\-2x'- = ±x 2 -\-5x 2 , to find a:. a.) The principle of § 51 obviously includes that of § 50. inasmuch as dividing by x~ 1 is the same as multiplying by h.) The whole class of equations included under § 51, are actually of the first degree, according to the more general definition of the degree of an equation. For, the range of the degrees of the terms with respect to the unknown quan- tity, from loicest to highest, is expressed by unity (§ 40. a) ; as is found by subtracting the lowest from the highest. § 52. IV. An equation may contain, besides the zero power of the unknown quantity, only a simple root of the unknown quantity ; i. e. it may contain only the zero and the one half, one third, or ith powers of the unknown quantity. The equation, in this case, is of the one half, one third, or Ith degree. Thus, \ Let x = 5 ; or^Ac — 5. Squaring both members (§ 42. c), we have x = 25. So, if we had x n — a, or n */x =z a, we should find x = a". Hence, An equation of the 1th degree can be reduced to tbe first degree, by raising both members to the nth power. 56 INTRODUCTION. [§ 53-55. Note. This operation evidently comes under §42. c, for the members being equal, multiplying them by themselves is multiply- ing them by equal quantities. So, if they be separated into the same number of equal factors, one factor on one side will be equal to one on the other; i. e. any fractional power or root of one side is equal to the same power or root of the other. For this is formed by divid- ing all the factors but one out of each member (§ 42. d). Hence, If both members of an equation be raised to the same power, whether integral or fractional (§22), the results will be equal. i i 1. Given Jar 3 +2 = ^-\-o, to find x. 2. Given y 4 +2 z= -^- -f-8, to find y. o § 53. We have classed equations with reference to their unknown quantities. They are also sometimes distinguish- ed, with reference to the form in which their hioion quan- tities are expressed, as numerical or literal. A numerical equation is one, in which the known quanti- ties are all expressed by numbers ; as x- ■=. 10x-(-24. A literal equation is one, in which a part or all of the known quantities are expressed by letters ; as ax 2 -{-'2bx=. c. § 54. A conditional equation is the algebraic expression of a problem™ ; i. e. something proposed to be performed or discovered. Thus, the equation x — 3 = 7 (§ 43. 1), proposes this prob- lem ; viz. To find a number such that if it be diminished by 3, the remainder shall be 7. So the equation Ja:-|-3 = 8 (§45. 1), proposes this prob- lem; viz. To find a number, whose fourth part, increased by 3, is equal to 8. State, in like manner, the problems involved in each of the equations of §§ 43-52. Compare § 3. a. §55. As we bave seen, an equation is the algebraic ex- pression of a problem ; and the solution of the equation gives the solution of the problem. Hence to solve a proL- (w) Gr. npoffiriua, from Trpoj3u?J,8 qcO "• " ^-T+T-T + T-T+T-T + IT' ^» 2 ^y»3 T*^ 9" ^ / } r> ® ^/*^ -1*8 CP^ and -^-T-T""T _ T— 6— y— s~ T 12. Add a 2 -j-2a5-fJ 2 , and a 2 — 2ab+b 2 . 13. " sin a cos £>-|-cos a sin b, to sin a cos b — cos a sin b. 14. " 2a + a 2 x- 1 , 8a*~x ""*, 6aar°, lOa 5 ^, — 15a°a:, — 12a~-x^, 9a- 1 ;*; 2 , lOa 2 ^- 1 , lla^aP*, 8«ar° ? 1 3 — 5^/a^/x-|-5a a:-f-2a 2 x 5 , and 18a x. 15. " ay-\-bx, and a'y — J'.t. +v Ans. (a-\-a')y-\-(b — V)x, or a +a>' 1 G. " ay — bx-\-cz, a'y-\-Vx — c'z, and — a"y-\-b"x — c"z. 17. " y 3 -\-ay 2 -\-aby ; by 2 -\-acy, and cy 2 -\-bcy-\-abc ; and arrange the result according to the descending powers ofy (§§33, 34. c). § 58.] PROBLEMS. 61 18. Add, member by member (Geom. § 22), the equa- tions —7x-\-5y = 19, and 10a;— 5y = — 10. Arts. 3#— 9. .-. x = what? PROBLEMS. § 58. 1. The sum of 2x— 10 and 4x— 20 is equal to 3a;. What is the value of x ? 2x— 10+4»— 20 = 3x. Qx— 30 = 3x. .-. 6x— 3a; = 30 ; or Sx = 30. .-. x = 10. 2. The sum of 5a; — 8, 2x — 20, and x — 10 is equal to 10 — 4a;. What is the value of x ? 3. The sum of %x — 1, 2 — | x, l-\-x — §x, and x — 2 is equal to ar-f-5. What is the value of x ? 4. The sum of 2x, 7x, fa:, and — 6 is — 23. What is the value of x ? 5. The sum of 13| — \x and — 2a?4-8§ is nothing. What is the value of a;? 13|— %x— 2x-f8| = 0. .-. 22£ z= 2\x. .-. x= 9. 6. A's property is 3a dollars, and his debts 2a ; B's property is 5a, and his debts 3a ; if they make common stock of their property, what is their net capital, x ? Let a = 100, 500, 1000, 10,000, and find the value of x in each case. 7. An estate was divided among three sons. The eld- est received $4000 less than one half; the second received one third ; and the youngest received S2000 more than one quarter of the whole. What was the estate, and what did each receive ? Let the estate be represented by x. Then we shall have the share of the first —^ — 4000 ; x " second = -, and o ALG. 6 62 SUBTRACTION. [§ 59. share of the third = - -j-2000. The sum of the shares 's, of course, equal to x. 8. Let the first receive a less than half; the second, one third ; and the youngest, one half of a more than one quar- ter of the estate. What was the estate, and what did each receive ? 3C 3C CC Ct Here the shares are - — a, -, and j + -^ 2 . # 2 x 3 ic* a? 2 e 3 13. From x — -4— p subtract — .r — — a: 4 T 14. From 1 Ox— ly — 30, subtract 8a:— ly = 20. 15. From^V-he^ 2 ^:^ 2 ^ 2 , su b tract ^V 2 +^ 2 x" 2 =A 2 B 2 . Rem. A 2 {y 2 —y" 2 )+B 2 {x-—x" 2 ) = 0. §61. It is sometimes important to indicate the subtrac- tion of a polynomial, without actually performing it. Thus, a-\-x — (b-\-c — d) which, when performed, : a-\-x — b — c-\-d. As, in performing a subtraction which has been ind ed, we change all the signs of the quantity within the pa- renthesis ; so we may return from a performed, to an indi- cated subtraction, by re-changing all the signs of the quan- tity whose subtraction is to be indicated, and enclosing the terms hi a parenthesis, with the sign — before it. We may, therefore, put a polynomial under different forms, without affecting its value. Thus, J2_2J c _j_ c 2 — J2_(2J C — c 2 ) = —(—b 2 -\-2bc—C' ) . cf—A 2 -\-B 2 = cy"—{A 2 —B 2 ) . § 62, 63.] COMBINATION OF SIGNS. 65 1. _ (a: 2 — _4 2 ) = what? R 2 — (cos «coa b— sin a sini)? 2. Indicate, in every way possible, without changing the order of the terms, the subtraction of r—s-\-t—u from a. § 62. We have already found, in several instances, two signs combined before a single term (§§7. a, b, 60. b). There is nothing to hinder any number of signs from being thus combined. It is proper therefore to consider the ef- fect of such a combination. a.) In the first place, as addition is simply the bringing of quantities together in their proper character (§ 54), the sign -\- can never change the previously existing sign of a term. "Whether employed once or oftener, it simply leaves the sign of the term as it was before the sign -f- was pre- fixed. Thus, a-\-( — b) [i. e. a together with — 5] = a — b. Hence, in estimating the effect of any number of signs, the positive signs may be disregarded ; the sign of the term Upends upon the negative signs. b.) As subtraction, on the other hand, always changes the sign of a term, the sign — always reverses the charac- ter of the term to which it is prefixed. Thus, -\-a = a (§4. a); .-. — (+«) = — «• Again — ( — a) = -\-a ; § 7. a, b. —(—(—«) = — (+a) ——a. Hence, § 63. If the number of negative signs before a term be evexN, the resulting sign is +; if odd, — . Com- pare §11. Note 2. Note. This includes the case, in which the signs are all positive. For then the number of negative signs is represented by 0, an even number, being less by unity than 1, which is an odd number. 1. What is the value of — (4-'— 26c+c 2 ) ? Ans. — (b*— 2bc+c*) =— (&»)— (— 2bc)— (+c 3 ) ——{,< -\-2bc— c 2 . *6 66 SUBTRACTION. [§ 64, 65. 2. What is the proper sign of — ( — « 2 )? of — ( — ( — a 3 )? - which is 10. x— (60— x) = 10. 2. The sum of two numbers is 100, and their difference is 20. What are the numbers ? 3. The sum of two numbers is S, and their difference i* D. What are the numbers ? ci_ n Ans. The greater is — - — ,or hS-\~hD, and the less, -^-, or }&-*!>. Note. The 1st, 2d, and 3d examples propose the same question under different forma. But, in the 3d, the quantities employed re- main in the result (§ 55. 3. N.), and show how they are employed to obtain that result. Thus S denotes the sum of any two numbeis. and X), their difference ; and we find the greater by adding the differ- ence to the sum, and dividing by two; and the less, by subtracting the difference from the sum, and dividing by two. (Compare § 57. 2, 3, §60. 3, 4, and Geom. §22.) Thus, Let the sum of two numbers be 50, and their difference, 6 ; and find the numbers. Here #=50, and D = G : S+D 56 no 3 S-£> 44 an ^ = T = 28,and— = T = 22. And we find 28+22 =50 = S, the sum ; and 28—22= 6 = D, the difference. Let the sum be 75, and difference 25 ; 12, " 2; " 12, « 3; " " 19 « 7 • « « 75°27', " 13°15' what are ► the num- bers? CHAPTER II. MULTIPLICATION AND DIVISION. I. MULTIPLICATION. § 66. Multiplication is the process of combining factors into a product (see § 10) ; in other words, it is the process of taking as a term, one quantity called the multiplicand 10 , as many times or parts of a time, as there are units or parts of a unit, in another quan- tity called the multiplier. Thus, if 6 dollars be taken as a term 3 times, the result is 6x3 = 6— f— 6— f— 6 =■ 18 ; if 6 dollars be taken as a term § of a time, the result is 6X| = (2+2+2)§ = 2+2 = 4. Note. It is obvious, that, in numbers, either factor may be made the multiplicand, and the other, the multiplier, without affecting the result. See Geom. § 172. MULTIPLICATION OF MONOMIALS. § 67. All multiplication resolves itself, as we shall see, into the multiplication of monomials. We shall, therefore, consider that case first. Numerical coefficients are, of course, subject to the prin- ciples of Arithmetic, and must be multiplied accordingly. Letters, we hare seen, are multiplied by writing them to- gether (§ 2. e. N.) ; and powers, by adding their exponents (w) Lat. multiplicandus, to be multiplied, from multiplico, com- pounded of multus, many, and plico, to fold ; as if the quantity were folded on, or added to, itself. § 68.] MULTIPLICATION OF MONOMIALS. 69 i 24. a). Hence, we have, for the multiplication of mono- mials, the following RULE. § 68. Multiply the numerical coefficients as in Arith- metic; and annex the letters of the factors, giving- to each an exponent equal to the sum of its exponents in the factors. a.) We have shown (§ 9. a), that the product of two fa< - tors of like signs is positive, and of unlike signs, negative ; and (§11. N. 2), that the product of any even number of negative factors is positive, and of any odd number, negative. We have also shown (§ 62. a), that positive signs have no effect to change the sign of a term ; but that the sign de- pends upon the negative signs. Hence, whatever be th< number of factors, If the number of negative factors be even, the product is positive ; if odd, negative. b.) The sign of each factor obviously produces its effect upon the whole product (§ 9. a). Hence, we may write the signs of all the factors before the product, and determine the resulting sign by § 63. c.) When one only of the factors has a double sign (± or :f ), the sign of the product will, of course, be double ; and will be either the same as that of the factor, or inverted, according as an even or odd number of the remaining fac- tors may be negative. Thus, ±aXb=±ab; ±aX — b=^:ab; ^:abX — c — ±abc. (±o) ( — b) ( — c) — ±abc ; ±a. — b. — c. — x = ^abcx. if two factors have each a double sign, and if it be un- derstood, that the upper signs must be taken together, and the lower signs together, the sign of the product will, obvi- ously, be single; and, if the signs of the factors be alike, the product will be positive ; if unlike, negative. Thus, ±'>X±b = -\-b; ±aXTb = —'-d. [±a)[±b){^:c)— qpaSe. 7(1 MULTIPLICATION. [§ 69, 7<». d.) The degree of the product of several monomial fac- tors is, evidently, equal to the sum of the degrees of those factors (28). 1. Multiply together 2a-b, —3ab' 2 , Aa~ l b~ 2 , and —\b. Product 12« -'/;•-'. 2. SaX—bX—cX—2hy = what ? 3 . a'" X a-" X b" X« Jr 1 = what ? Jns. a"-"+ * 6" - J . 4. Multiply together |, — J, i? -3 , — a; 2 , and — x 2 . 5. (±a#X±#) — what? ax 2 (±x)? ( — «x 3 )(±.r) r MULTIPLICATION OF POLYNOMIALS. ■ 69. First, let one factor only be a polynomial. Thus, Multiply together b-\-y and a. (b-\-y) times a is the same as a times (b-\-y) [see § 66, X.] ; i. e. a times the sum of b and y ; which is, obviously, the same as the sum of a times b, and a times y. (b-\-y)a, or a (&-[-#) = «&-}-«#• Hence, the product of a polynomial into a monomial con- sists of the aggregate of the products of the monomial into the several terms of the polynomial. See Geom. § 178. 1. 1. Multiply a 2 — 2ab+b* by a. Prod a 3 —2a*b-\-ab*. 2. (a 2 ± 2«/;+6 2 ) X ± b = what ? .4ms. ±aVj-\-2ab 2 ±l 3. (j2_L^2_ a 2) X _J22 — w hat? 4. Multiply l+Ja- 2 u; 2 — |o-***+A«- 8a:e by a. § 70. Again let there be two polynomial factors. Thus. Multiply a-\-b by c-\-y. (c-j-y) times a-\-b is evidently the same as c times cir\-b, added to y times a-\-b ; i. e. [a-\-b)[c-\-y) = («-f-%-R«+% = ac-j-Je-f-ay+Jy. S. i §67. §71.] OF POLYNOMIALS. i Hence, we have, for the multiplication of polynomial- the following RULE. § 71. Multiply each term of the multiplicand by each term of the multiplier, and add the products. See Geom. § 178. Cor. III. a.) This is precisely the method employed in Arithme- tic. Thus, to multiply 84 by 25, we have 34 = 30+ 4 25= 20+ 5 170= 150+2o 68 = 600+80 850= 600+230+20 1. (a 2 +2b)(a~ 2 — b 2 ) = what ? (a+b){a+b) = (a+b) 2 ? a 2 +2b a+b a- 2 —b* a+b a"+2a~ 2 b a 2 +ab —an 2 — -2b 3 ab+b 2 l+2a~ 2 b—a 2 b 2 —2b 3 a 2 +2ab+b 2 2 . ^+ay+y*)(a2-ay+y*) = what? Ans. a*+a 2 y 2 +y±. (a*—b 2 ) (a—b) = what ? yircs. a 3 — a 2 b— ab 2 +b 3 . {a+b)(a—b)(a—b) = what ? («+£>)(«— 6)(a+6) ? {a 3 +3a 2 b+Sab^+b 3 )[a 2 +2ab+b 2 ) — w h a t ? (2a 4 —' 3« 2 Z> 2 +4&4)(x— ^/)2x = what ? {a+b—c)(a—b+c) = what ? (a-+^)(a;*— y*) ? o. 4. 5. 6. 8. (x — a)(x-\-b) = what ? -4ras. x 2 — a 9 . (x+a)(x+b)[x — c)(x—e) = what ? 10. (a2±6z 2 +cz 3 )(l±H-z 2 ±z 3 ) = what? cc — ab. Ans. az±a z 2 +a z*+b ±b +i ±b +c +c ±c z 5 ±cz r -. § 72-74.] MULTIPLICATION. 72 § 72. If two polynomials are each homogeneous, their product will be homogeneous also. For the degree of any term in the product is equal to the sum of the degrees of a term in each factor (§ 68. d) ; and those degrees being the same throughout, their sum must be always the same, and therefore all the terms of the product will be of the same degree. Hence, if, in multiplying homogeneous polynomials to- gether, we observe that the degree of one term is greater or less than the degree of the other terms, we may know that some mistake has been made. This remark is the more important, because so many of the investigations of Algebra, especially those relating to Geometry, give rise to homogeneous expressions. § 73. If the product of polynomials contains similar terms, it may, of course, be simplified by § 34. But it is apparent that, if the factors themselves were reduced to their simplest form, there will always be some terms of the product unlike all the others, and, therefore, incapable of any reduction except the partial reduction explained in § 34. c. These are, 1. The product of the terms containing the highest powers of any letter, in each of the factors ; and 2. The product of the terms containing the lowest pow- ers of any letter. For these two terms must contain that letter, the one with a greater, and the other with a less exponent, than any of the other terms or partial products ; and, consequently, can- not be similar to any of them. Hence, no product, involv- ing a polynomial factor, can consist of less than two terms. § 74. If there are no similar terms in the product of two polynomials, the whole number of terms in the product will be equal to the product of the number of terms in the mul- tiplicand by the number of terms in the multiplier. For, if there be four terms in the multiplicand, and one § 75, 76.] DETACHED COEFFICIENTS. 73 in the multiplier, there will be four terms in the product ; another term in the multiplier will give another four terms in the product, and so on. Also, if we introduce another factor, the same reasoning will apply to the product of this factor into the former pro- duct. Hence, in general, if there is no reduction, the num- ber of terms in any product is equal to the continued pro- duct of all the numbers of the terms in the several factors. § 75. The multiplication of polynomials is frequently in- dicated, without being performed. Thus, a(y+A) 2 = a(y*-\-2yh+h*) = ay 2 +2ayh-\-ah*. (P — ^) X (p — c) ; a-\-b — c . a-\-c — b ; is(f s — a). When a multiplication, so indicated, is performed, the expression is sometimes said to be developed. MULTIPLICATION BY DETACHED COEFFICIENTS. § 76. In multiplying polynomials arranged according to the powers of any common letter or letters, that letter or those letters may be omitted in the operation, and the pow- ers supplied in the result ; the product of the highest or lowest powers being placed in the first term, and the pow- ers then regularly descending or ascending through all the terms. This is called multiplication by detached coeffi- cients ; and will be best explained by a few examples. Thus, To multiply x 2 +2.z+l by x 2 — 2x+l, we write the co- efficients, and multiply, as follows : 1+2+1 1—2+1 1+2+1 —2—4—2 1+2+1 1+0 — 2+0+1. Supplying the powers of x, we have a^+Oz 3 — 2z 2 +0a+l =x 4 — 2x 2 +l. ALG. 7 74 MULTIPLICATION. [§ 77. Multiply « 2 +2a&+6 2 by a-\-b. Here the polynomial factors being arranged with respect to both the letters, both may be omitted, and afterwards supplied, one with descending, the other with ascending powers. Thus, 1+1 1+2+1 1+2+1 1+3+3+1. Supplying the letters, we have a 3 +3a 2 &+8«& 2 +& 3 . a.) In adding the coefficients of the partial products in the first example, we obtain zero in the second and fourth places. The cypher must be written, to occupy the place of the term, and show what powers of the letters fall out. In like manner, if any power of a letter, between the high- est and lowest in any factor, be wanting, zero should be re- garded as its coefficient, and written in its place. This will fill out the series, aud will, obviously, cause the coefficients of similar terms to stand under one another. Thus, 3. Multiply a 2 +2a?/+# 2 by a 2 — y 9 . 1+2+1 1+0—1 1+2+1 0+0+0 —1—2—1 1+2+0—2—1. The product is a 4 +2a 3 #— *2ay 3 — y*. 4. Multiply z 3 — 3z 2 #+3z?/ 2 — y* by z 2 — 2z#+# 2 . 5. ( a +&) 3 = what? (a+5) 4 ? (a+6)*? C. 3 +z VH# 2 +3/ 3 ) 0— y) = what ? PROBLEMS. §77. 1. Given *— i(2x+l) = £(a+8) to find x. Ans. x =13. 8.] division. 75 2. Given ^i- +2x = ^=^ +16, to find x. 5 o an- i« . « O+14)(36x+10) 3. Given 16x + 5 — Q \ Q1 » to fi nc * x - JX — f—oi. Ans. x = 5. 4. Given («c+fo-) 2 +5 2 x:= 2«6er-f-(a 2 +£ 2 )c 2 , to find -4ws. x = e 2 — r 2 . 6. Given cc 2 -f-x~ 2 = (x— x" 1 ) 2 -fx, to find x (§ 49). Ans. x = 2. 7. A's age is to B's as 2 to 3 ; and if they live 15 years, A's age will be | of B's. What are their ages ? Let x = B's age ; then %x = A's age. Moreover x-f-15, and §x-}-15 will be their ages after 15 vears. ••• fx+l5 = |(x+15). Ans. A's age, 30 ; B's, 45. 8. A's age is \ of B's ; and 18 years ago, A's age was B's. What are their ages ? H. DIVISION. § 78. Division is the process, by which, having a product and one of its factors, ive find the other fac- tor (see § 10) ; in other words, it is the process of finding hoiv many times, or parts of a time, one quan- tity is contained in another. Thus, if 12 be a product, and 3 be one of its factors, the other factor is 4; or 3 is contained in 12, 4 times; if 12 be a product, and 24 be one of the factors, the other factor • or 24 is contained in 12, \ a time. ' 76 division. [§79,80. DIVISION OF MONOMIALS. § 79. As in multiplication, so in division, whatever be the quantities involved, the operation is actually performed upon monomials only. "We shall, therefore, consider first the division of monomials. Numerical coefficients are, of course, subject to the prin- ciples of Arithmetic, and must be divided accordingly. Let- ters, we have seen, are divided by suppressing in the divi- dend the letters of the divisor (§ 10. b) ; i. e. by subtracting the exponents of the letters in the divisor from the expon- ents of the same letters in the dividend (§§16, 24. b). See also § 13. Hence, we have, for the division of mono-* mials, the following RULE. § 80. Divide numerical coefficients as in Arithme- tic ; and annex all the literal factors, which remain after suppressing in the dividend those of the divisor, a.) If the exponent of any letter be greater in the divi- dend, than in the divisor, its exponent in the quotient will be positive ; If equal, it will be zero ; i. e. the letter will disappear ; and If less, it will be negative. Or, in the last case, the division may be expressed, as we have seen, by placing the letters, with positive expon- ents, as the denominator of a fraction, of which the remain- ing factors of the dividend constitute the numerator (§ 10. c). b.) In case of a single division, we have shown, that, as in multiplication, like signs give -\-, unlike, — . In case of successive division by several divisors, the same rule, of course, applies to each operation. Or, bringing the signs together, as in subtraction (§ 63) and multiplication (§ 68. a) we may regard only the negative signs. If the number ot these be even, the quotient is positive ; if odd, negative. $81.] MONOMIALS AND POLYNOMIALS. 77 c.) The law of the signs may be otherwise demonstrated, as follows. To divide by any quantity is the same as to multiply by its reciprocal (§ 19. Cor. IV.) ; and the recip- rocal of a quantity evidently has the same sign as the quan- tity itself (§ 18). Therefore, to divide by any number of divisors is the same as to multiply by the same number of multipliers having each the same sign. Hence, the law of the signs is the same in division as in multiplication. d.) One quantity is commonly said to be divisive by another, when the division does not give rise to fractional coefficients, or to negative exponents. Note. Any quantity may be said to be divisible by any other. For, whatever be the dividend and given factor, another factor can always be found, which will produce the dividend. It is, however, convenient, in many cases, to distinguish as perfect or exact, the di- visibility above mentioned which does not give rise to fractional ex- preisions. 1. 20a 6 b 3 c —- 4a 3 5 3 c 3 = what ? Ans. 5a 2 c~ 2 . 2. a 2 &~ 2 -i-a- 1 5 = what? a^x^-^-c^x-^y ? — a ^~-a~^? 3. a m b- m -~a"b- n = what ? (ar\-x) %-±(a+x)~% ? TO DIVIDE A POLYNOMIAL BY A MONOMIAL. § 81. In multiplying a polynomial by a monomial, we multiply each term of the polynomial by the monomial, and add the products (§ 69). Therefore, reversing the process, we have, for dividing a polynomial by a monomial, the fol- lowing RULE. Divide each term of the dividend by the divisor, and add the quotients. Thus, (ab ± ay) ~ a = b ± y. 1. Divide r?y-\-xy 2 by xy. Quotient, x-\-y. #7 78 division. [§82. 2. {ax±x 2 )- : rx = what? {rs—s)~s? 3. (2rx—x 2 )~x=\rhat? {A 2 B 2 —B 2 x 2 )-±B 2 J 4. (— R cos b cos c -\- R sin b sin c) -J i? = what ? 5. (a— ;c) -^ a z= what ? Ans. 1 — a -1 ^ or 1- 6. Divide Rz—x* by R*. 7. « R — \R-^x 2 — \R-*x* by R. 8. " a _t — §a~^c a + f cT^ar* by x a a 3 . TO DIVIDE ONE POLYNOMIAL BY ANOTHER. § 82. Divide 3a& 2 -f3« 2 H-a 3 -H 3 by a+5. a.) This dividend beiDg regarded as the product of the divisor aud quotient (§ 10), the terms containing the highest and the lowest powers of a and b must consist of the unre- duced products of the highest and of the lowest powers of those letters in the two factors (§ 73. 1, 2). b.) If, therefore, we divide the term of the- dividend which contains the highest power of a, by the term of the divisor which contains the highest power of the same let- ter, we must obtain the corresponding term of the quotient. c.) If, now, we multiply the divisor by the term of the quotient, which we have found, we shall have one of the partial products whose sum is the dividend. d.) If, then, we subtract this partial product, there will remain the sum of the other partial products, viz. of the di- visor into the other terms of the quotient. e.) There will, of course, be a highest power of a in this new or remaining dividend, which term divided by the term containing the highest power of a in the divisor, as before, will give a term containing the highest power of a in the remaining terms of the quotient ; and so on. /. And, as the sum of the products of all the terms of the divisor by each term of the quotient must make up the § 82. J POLYNOMIALS. dividend, if we subtract those partial products, one after 1 another from the dividend, they must exhaust it ; and the remainder, after the last subtraction, will be zero. g.) If we obtain a remainder equal to zero by simply di- viding the first term of each remainder by the first term of the divisor, the division is said to be exact, and the dividend is said to be divisible by the divisor (§ 80. d, N.). h.) If, however, after exhausting the given terms of the dividend, wo still have a remainder, the division may be immediately completed by writing the whole remainder over the whole divisor, for the last term of the quotient ; or the division may be still farther continued (§ 87) accord- ing to the rule, and terminated, whenever we please, by a fractional term, as above indicated. ■i.) These operations will be more conveniently 'perform- ed, if the dividend and divisor be first arranged with res- pect to the powers of some one letter (§ 33. a). This arrangement may be according to either the ascend- ing or the descending powers of the letter. The descend- ing order, however, is most commonly employed. k.) The polynomials above being arranged with refer- ence to a, and placed in order for division, will stand thus : the divisor being placed at the right of the dividend, and the quotient under the divisor. a3-\-3a 2 b-\-3ab 2 -\-b* a+b a 3 -\- a-b a 2 -\-2ab+b* 2a 2 H-3a& 2 +£3 2a 2 5+2«5 2 ab 2 +b s ab 2 +bs From the reasoning above, we deduce the following gen^ eral DIVISION [8 88. RULE. § 83. 1. Arrange both dividend and divisor accord- ing to the powers of some common letter, cither as- cending, or descending in both. 2. Divide the first term of the dividend by the first term of the divisor (§ 80), and set the result, with its proper sign, as a term of the quotient. 3. Multiply the divisor by this first term of the quo- tient, and subtract the product from the dividend. 4. Divide the first term of the remainder by the first term of the divisor, set the result in the quotient with its proper sign, multiply, and subtract as before, and continue the process as long as the case may require. 1. Divide a*-\-3a 2 x+x 3 +3ax 2 by a+x. 2. Divide x^Qy 2 x^ix^i-4:Xy s -\-t/t by x *-\-2xy-\~y n -. a.) It is not necessary to write all the remaining terms of the dividend, after each subtraction. Indeed, none need be written, except those which change their form by sub- traction and reduction. It is convenient, however, to bring down one additional term of the dividend, at each subtrac- tion. This is the method commonly practised. 1. Divide a 6 — Sa^x-{-16a^x-— 20a 3 x 3 -\-15a*x*— Sax 5 -\-x z by a — x. 2. Divide a±-\-a*zz+ z* by a*-\-az-\-z*. 3. Divide x 5 -\-Sx*— 10x 3 — 112oc 2 — 207x— 110 by ar 2 +7x4-10. Quot. x 3 — x 2 — 133—11. 4. Divide a G — 3a 4 x z -\-3a 2 x' i — x* by a 2 — x~. 5. Divide 1— a~n by a?— J-. Quot. a"^-f a~i$. 6. Divide ^a 3 — \a*b— ^ab 2 -{-^b 3 by \a-\b. Quot. la'—lb" 1 . 7. Divide x 3 -\-ax 2 — bx*-\-cz 2 — abx-\-acx— box— abc by x --\-ax — bx — ah. 84.] Is* Rem. POL ^NOMIALS. x 3 -\-a x 2 — ah x — ahc x 2 -\-a x — ah —h -\-ac —he —h +c x-\-c ( Quotient x 3 -\-a x 2 — ahx —h cx 2 -\-ac x — ahc —he cx 2 -\-ac x — ahc —he 81 b. The operation may be still further shortened. Ar- range, divide and multiply, as directed ; but, instead of writing the product under the dividend, subtract each term mentally, as it is formed, and write the reduced remainder (§ 83. a). Thus, a 4 — 4« 3 x+6a 2 x 3 — 4:ax 3 -\-x* 1st Rem. 2d Rem. -2a 3 x-\-5a 2 x 2 — iax 3 a 2 —2ax-\-x 2 a 2 — 2ax-\-x' a 2 x 2 — 2ax 3 -\-x 4: 1. Divide x 2 — 7x-\- 12 by x — 3. 2. Divide 2a 2 '"-\-2a'"h p — 4a'V— 3a m b— 3h p +*-\-§hc n by 2a m —3b. Quot. a m -\-h i> —2c n . § 84. c.) We need only the first term of each remainder (§ 83. 4). The other terms are simply reserved till we sub- tract from them the terms of the next product, and so on. Instead, therefore, of performing these successive subtrac- tions, we may write the similar terms of the several pro- ducts under one another, and subtract the aggregate of each set, when the corresponding first term of a remainder is re- quired for division. Or we may change the signs of the several terms of the products as we write them, and add each column as we come to it. If we adopt this course, we shall be less liable to mistake, if we change the signs of the divisor, all except the first, which should remain unchanged, to prevent mis- takes in the signs of the quotient ; and which can occasion 82 division. [§ 85. no mistake in subtracting, as its product always cancels the term above it, and need not be written. Divide a*— 4a 3 a:+6a 2 a: 2 — 4aa; 3 -jr-a; 4 by a 2 — 2ax-\-x 2 . a 4 — ±a 3 x+Qa 2 x 2 — 4ax 3 -f-.T 4 -\- 2g3x— a 2 x 2 — 2a 3 x —±a 2 x 2 +2ax s a 2 -\-2ax — cc 2 a 2 —2ax-\-x 2 -\-a 2 x 7 -\-2ax 3 — x* d.) The last method is conveniently written as follows. Write the terms of the divisor under one another, on the left of the dividend, changing the signs of all but the first. Write the terms of the partial products, except the first of each, diagonally under the corresponding terms of the divi- dend. Below, in a horizontal line, write the first terms of the remainders as they are formed, each under the column from which it is produced. Write the quotient also in a horizontal line below the last, each term under the term of the dividend, from which it was formed. Thus, a 4 — ia 3 x-\-6a 2 x 2 — iax 3 -\-x i \-2ax -\-2a 3 x—Aa 2 x 2 -\-2ax 3 —a 2 x 2 +2ax 3 —x* — x 2 — 2a 3 x-\-a u x' Quotient, a~ — 2 ax -j-x 2 . Notes. (1.) If any term in the series of powers be wanting, its place should be filled with a cypher (§76. a); or the given terms should be placed at such distances from each other, that like terms of the partial products may stand under them. (2.) Each term of the partial products will stand against that term of the divisor from which it is formed. 1. Divide a 6 -\-2a 3 z 3 -\-z 6 by a 2 — az-f-z 2 . 2. Divide a 3 -\-a 2 b — ah 2 — b 3 by a — b. DIVISION BY DETACHED COEFFICIENTS. § 85. Division, as well as multiplication, may be perform- ed by DETACHED COEFFICIENTS. Thus, § 86.] SYNTHETIC DIVISION. 83 1. Divide a 3 — 3a 2 £+3a& 2 +&3 by a—b. 1—3+3 -1 1—1 1—1 1—2+1 —2 —2+2 1 1—1 Supplying the letters, by dividing the first term of the dividend by the first term of the divisor, we have a 3 — 2c +&*. 2. Divide a 4 — 6* by a 2 — 6 s . 1+0+0+0—1 1+0—1 0+1 1+0—1 1+0—1 1+0+1. .-. Quot.=za*+b* SYNTHETIC DIVISION. 86. Synthetic 2 division is division with detached co- efficients, performed by the method of § 84. cL With de- tached coefficients, however, the method admits of simplifi- cation, when the first coefficient of the divisor is 1. For, in this case, the coefficient of each term in the quotient will be the same as the corresponding coefficient of the first term of the dividend or remainder ; and may, therefore be found by simply adding the coefficients above it. Thus, to divide a*— 4a 3 a+6« 2 ^ 2 — 4aa: 3 +a: 4 by a 2 — 2ax+« 2 . 1 +2 — 1 1—4+6—4+1 +2—4+2 —1+2—1 1—2+1+0+0 .-. Quot. = a B — 2ax-\-x-. Moreover, if the fir3t coefficient of the divisor be not 1, it can evidently be made so, by dividing both divisor and dividend by the given first coefficient. (x) Gr. cvv&ecic , composition, putting together; each term of the quotient being formed by adding the like terms of the dividend and of the pan 'acts with their signs changed. 84 division. [§ 87. 1. Divide x 3 — 3x 2 +3;r— 1 by x 2 — 2a+l. 2. Divide 4a 4 — 9« 2 5 2 +6a6 3 — 5* by 2« 2 — 3a&+5 2 . Solve the examples of §§ 83, 84 by this method, observ- ing, when the series of powers is not complete, to fill the place of the missing terms with cyphers (76. a). INFINITE SERIES. § 87. When an exact division is impossible, the opera- tion may still be carried to any extent, forming what is called an infinite v series. The process is similar to the pro- cess of approximation in the division of decimals in arith- metic. Thus, to divide a by a-\-x. a a-\-x a-\-x 1 — ar l x-\-ar 2 x 2 — a~ 3 x 3 -\-a~ 4: x 4: — &c. — x — x — a -1 a? 2 a l x- a~ 1 x 2 -\-a~ 2 x 3 -a~ 2 x 3 -a~ 2 x 3 — a~ 3 x- a~ 3 x* Or (§ 86), thus, 1 — 1 1 -1+1— 1+1— &c. 1—1+1— 1+1— &c. .-. Quot. = l— a- 1 x-^-a~ 2 x 2 — &c "We have, therefore, a = 1— a-*x-\-a- 2 x 2 — ar 3 x 3 -\-a-±x*— a" 6 a: B +&c.(l.) a-j-x or, in another form (§ 14), ff / y T 1 -* >y»3 t»4 «, they will diverge. The series of c. 4; will converge, when ^( .-. Adding and subtracting the equations (a-f-Z>) 2 +(«— *>y = 2a 2 +25 2 = 2(a 2 +5 2 ). Geom. § 199. (a-f-J) ^— (a—b) 2 = Aab. Geom. § 184. Hence, Cor. (1.) The square of the sum, plus the square of the difference, of two numbers, is equal to twice the sum of their squares. (2.) The square of the sum, minus the square of the difference, of tivo numbers, is equal to four times their product. § 92. Multiply a~\-b by a—b. We have, («+&) («— &) = a 2 — b 2 . Hence, Theor. III. The product of the sum and difference of two numbers is equal to the difference of their squares. See Geom. § 185. Cor. ix. 1- (H-f)(*-f) = what? (A+x)(A-x)? (y-h/O (2/-y) ? ((i+x) h +(i-x)*)«i+xy-(i-x)h 2. (R+x) (R—x) = what ? (AB+BC) (AB—BG) ? (* 2 -h/ 2 )(* 2 -2/ 2 ) ? (a ! 8_|^8).( a .a_ Sf a) ? 3. (sin a cos &-(-sin 5 cos a) (sin a cos b — sin b cos a) — what? Ans. sin 2 «cos 2 £> — sin 2 £cos 2 a. Note . Sin 2 a denotes the square of the sine of a. This notation is more precise than sin a%, which might mean the sine of the square of a; and is less cumbrous than (sin a)-. The same remark applies to cos'a, tan 2 a, &c 4 93, 94] THEOREMS. — SUM AND DIFFERENCE. 89 4. (a+J+c) (a-\-b—c) [i. e/the sum of a-\-b and c, into the difference of a-\-b and c] = -what ? ^ws. (a+b) 2 — c 2 = a 2 +2a5+5 2 — c 2 . 5. (a+b—c) (a—b+c) ( = (a+5— c)(a— J— c)) = what ? So in Arithmetic ; 12X8 = (10+2) (10-2) = 10 2 — 2 2 = 100—4 = 96. 19X21( = (20— 1) (20+1)) = what? 103x97? 51X49? 101X99? 1004X996? 1000£X999£? § 93. The same formulas, read with the second member first, give the converse" of the above theorems ; and enable us to resolve several classes of polynomials into their fac- tors (§ 75). Thus, (I.) The sum of the squares, plus twice the product, of two numbers, is equal to the square of their sum. 1. aj 2 +2a;?/+3r = what? Ans. {x-\-y) 2 . 2. 2/ 2 +2yy / +y /2 = what? l+2?i+?i 2 ? 9a 4 +24a 2 6 2 +166 4 ? 169 (= 100+2.10.3+9 = 10 2 +2.10.3+3 2 ) ? (II.) The sum of the squares, minus twice the product, of two numbers, is equal to the square of their difference. 1. x 2 — pa+£p 2 = what? Ans. {x— ^p) 2 . 2. b-— 2Z>c+c 2 = what ? 1— 4ra+4n 2 ? a 2 — 12ab +366 2 ? 81( = 100— 2.10.1+1 =10 2 — 2.10.1+1 2 )? (III.) The difference of two squares is equal to the pro- duct of the sum and difference of their roots (§ 23). 1. i? 2 — z 2 = what? Ans. (H-\-x)(E—x). 2. sin 2 a— sin 2 6 = what? x*—y*? {AB) 2 —[BC) 2 ? „G_&6 ? c 2_£2_|_2fo— c 2( — a 2_(J_ C )2 [§§ 63. 1, 90.]) ? £2_|_26c+c 2 — « 2 ? l_cos 2 i< = i 2 — cos 2 *) ? x 2 — x" 2 ? § 94. 1. Divide a 2 — b 2 by a—b. Quot. a-\-b. (a) Lat. conversus, turned about. Of two propositions or sen- tences, each is said to be the converse of the other, when the condi- tion of the first is the conclusion of the second, and the conclusion of the first is the condition of the second; or when, in like manner, subject and predicate change places. See Geom. §32. Note n. *8 90 division. [§ 95. 2. Divide a 3 — b 3 by a— b. Quot. a*-\-ab-\-b 2 . 3. « a 4 — 5 4 by a— i. " a 3 +a 2 6+a& 2 +6 3 . 4. " a n — b n by a— 5. Employ tbe method of § 84. d; thus 6 , aa n —b' c . +& Guo*. = a^ 1 +a ,, - 3 H-« ,, ~ 3i2 + -\-ab n - 2 +b n -K Now, in the successive terms of the remainders, the ex- ponents of a diminish, and those of b increase by unity. The terms of the quotient, of course, follow the same law ; and the sum of the exponents in each term of the quotient is n — 1. Hence we shall find a term containing a and J* -1 . This term, multiplied by b, will give b n , which ad- ded, will cancel — b n in the dividend, and leave a remain- der equal to zero, showing a perfect division (§ 82. g). § 95. a.) Otherwise, a n — b n a — b a—ib — b n =(a n - 1 — b"~ l )b. Now, if a—b will divide (§ 82. g) this remainder, it will, obviously, divide the whole dividend. But it will, evident- ly, divide this remainder, if it will divide one of its factors, a »-i_ fr'-i. Hence, if a n ~ l — b"~ l is divisible by a — b, a a _ ^ ig ^ divisible hy a — b. That is, if the difference of like powers is divisible by the difference of their roots, the difference of the powers of the next higher degree is divisible in like manner. But we have found a 4 — & 4 divisible by a—b; .: a^—b 5 is divisible by a—b; so a e —b e , a"—b 7 ; and so on, with- out limit. Or, a 1 — b l is divisible by a—b; .: a 2 —b 2 is di- visible by a—b; and so on. Hence (§§ 94, 95), (b) Here, the sign of b being changed, the second term of each product is, without any reduction, the first term of the corresponding remainder, and, of course, need be written but once. We cannot write all the terms of the quotient, unless we assign a particular val- ue to n. The whole number of terms is, obviously, equal to ?». §96-97.] THEOREMS. — DIVISIBILITY. 91 § 96. Theor. IV. The difference of any two posi- tive integral powers of the same degree is divisible by the difference of their roots. Notes. (I.) This method of proof is of great utility in Algebra, and should be perfectly understood. It consists in showing, that, in how many soever instances a principle has been found true, it will be true in one instance more. If it be true in n — 1 cases, it will be in n; if in n, then in n+l ; if it be true in one instance, it will be true in the second; if in the second, then in the third, and so on. (2.) The limitation to "positive integral powers" is necessary; for the principle has been applied to such powers only. And, if n — 1 is a positive integer, n, obviously, cannot be either negative or frac- tional. a^-Jf _ an _ i ^ a>i _ 2b + _j_ a5 »-2 _}_ J«-i. a — b a » b n b.) If a = b, we have = m" _1 ; a result which a — a will be considered hereafter. c .) a _n_J-n_( a _i)»_(J-l)». §2±.d. a~ n — b~ n is divisible by ar~ 1 — [b— 1 . § 96. Or, which is the same thing, -— — 7- is divisible by -7. a n b" a b d.) ci^m — b^m ( = (o^)"- (fi^m)") is, evidently, di- ±- z±- visible by a m — 6 »<• §97. e.) a2"— i 2 "( = (o 9 ) n — (5 2 )") is divisible by a 2 — b- (§ 96), i. e. by {a~\-b)(a— b). a- n — b 2n is divisible by a + b. Now 2m, being divisible by 2, is an even number. Hence, Cor. 1. The difference of any two even positive integral powers is divisible by the sum of their roots. To divide a- n —b- n by «+&, employ the method of § 86. -1+1-1+ -1+1-1+1 1 IK — 1 1_1_L.1_1+ _l_|_l_l_|-0 ( a i"—b 2n )-±-{a+b) = a 2 "- 1 — a 2 "- 2 & + a 2 *- 3 Z> 2 — . . . + «3&2»-4 _ a 2Z,2"-3 _L. ab~ n ~~ — b 2n ~ l . "2 division. [§ 98-100- Thus, if 2n = 4, (a* — b±) + {a-{-b) = a3—a 2 b-\-a?>S — b3, * 98. /.) Divide a 2 "+i + & 2 «+i bya + b, a 2«+l _|_J2«+1 a-f - ^ a 2 "— &c. But & 2n — a 3 " is divisible by a + 5 (§ 97). a s«+i _j_ J2H-1 i s divisible by a + b. Moreover 2w-f-l is an odd number, being greater by uni- ty than 2n, an even number. Hence, Cor. ii. The sum of any two odd positive integral pow- ers of the same degree is divisible by the sum of their roots. a 2>4-l_l_52>4-l i- = « 2n — a 2n ~ib + a 2n ~*b 2 — a 2 "- 3 £ 3 4- . . H-G 4 6 2H -* — a 3 5 2 "-3-f «233»-2_ a J2 ; »-l_|_J2r. > Thus, if 2n+l = 5, § 99. The principles of § 94-98 obviously enable us to resolve another large class (§ 93) of polynomials into their factors. Thus, a 3 +b3 = (a-\-b)(a*+ab+b*). What are the factors of a 4 — 6 4 ? of a 5 — b r >? of a 5 +3 5 ? of* 3 — 27(=x 3 — 3 3 )? of a G —x*? ofa: 3 +64? THE GREATEST COMMON DIVISOR. $100. A factor common to two or more quanti- ties is called a common divisor ; and the greatest common factor, i. e. the product of all the common factors, is called the greatest common divisor, or mea- sure. Thus, of the quantities 18abx and 20a^by, 2, a and b are common divisors ; and the greatest common divisor is evi- dently 2ab, the product of all the common factors. § 101, 102.] GREATEST COMMON DIVISOR. 93 Notes. (I.) The term divisor is used here with reference to perfect divisibility (§ 82. g). (2.) A single factor, as a, which has no integral divisor but itself and unity, is called, as in Arithmetic, a prime factor. (3.) Quantities which have no common divisor but unity are called incommensurable, or prime to each other. a.) One of two or more quantities may be either multi- plied or divided by any factor not found in all the other quantities, without affecting the greatest common divisor. For the factor so introduced or taken out, not being com- mon to all the quantities, can form no part of their greatest common divisor. Thus, the greatest common divisor of \'2ax and 20ay is the same as that of I2ax and 20ayX5b or 20ay- : r oy. § 101. The greatest common divisor of several monomi- als must evidently consist of all the common literal factors, multiplied by the greatest common divisor of the numerical coefficients. What is the greatest common divisor of 12a 3 a: 4 and 2«2 X 5? of A-x"y" and B 2 x"y"? of ax and a'x? of A -cy" and c *xf'y" ? of nx n ~ 1 0+1)" and nx n (x+l)"- 1 . § 102. The process of finding the greatest common divi- sor of two polynomials is substantially the same as that employed in Arithmetic, and depends on the following principle ; viz. The greatest common divisor of two quantities is the same as the greatest common divisor of either of them, and of the remainder obtained by dividing one by the other. To prove this, let the two quantities be A and B, and di- vide A by B. Let the integral quotient resulting from this division be Q, and the remainder R. Then A — BQ — R, ox A — BQ+R. Now every divisor of Bis, of course, a divisor of BQ. Therefore every common divisor of A and B is a common divisor of A and BQ. Also, every'such common divisor is and i?, and, of course, ^^+i? or A. That is, every com- mon divisor of A and B is a common divisor of B and i?. Hence, the greatest common divisor of B and R is the greatest common divisor of A and B. § 103. By the same reasoning, if we proceed to divide B by R, and obtain a remainder R', the Greatest common di- visor of R and i2' is the greatest common divisor of B and R, and, therefore, of A and i?. Thus, the greatest common divisor of any of these divi- sors and its remainder is the greatest common divisor of all the preceding remainders, and also of the original quanti- ties. If then we find a remainder, which divides the pre- ceding remainder, it is the greatest common divisor re- quired. a.) If the first term of any dividend be not divisible by the first term of the corresponding divisor, we must (1.) suppress any factor of the divisor, not found in the divi- dend ; and (2.) we may, if necessary, multiply the dividend by any factor not found in the divisor (§ 100. a). Note. If we suppress in the divisor a factor found also in the dividend, that factor, originally common, will cease to be so, and the common divisor will be less than it ought to be. If, on the other hand, we introduce into the dividend a factor already found in the divisor, that factor, not originally common, will become so, and the common divisor will be greater than it ought to be (§ 100). Hence, to find the greatest common divisor of two quan- tities, we have the folio wing RULE. §104. Divide one quantity by the other; then di- vide the first divisor by the first remainder^ the second (c) If, for instance A-—D and BQ-hD are both whole numbers, their difference or their sum must be a whole number. § 104] GREATEST COMMON DIVISOR. 95 divisor by the second remainder, and so on ; alwetys rendering the first term of the dividend divisible by the first term of the divisor (§ 103. a). The divisor which gives no remainder, is the greatest common divisor required. a.) If the first remainder which divides the preceding remainder be unity, the quantities are said to have no com- mon divisor, but to be incommensurable, or prime to each other. b.) If the greatest common divisor of more than two quantities be required, we must first find that of two of them, and then of that divisor and a third, and so on. 1. Find the greatest common divisor of 98 and 112. 112 98 98 ~T 7 .'.14 is the greatest common 98 98 divisor required. Note. In Arithmetic, it is, of course, proper to divide the great- er number by the less. In Algebra, the quantity containing the high- est power of the letter of arrangement will be the first dividend. If the highest power is the same in both, either may be made the divi- dend. 2. Find the greatest common divisor of cc 2 -f-5rc-{-6 and x*+2x— 3. x*+ 5x-f-6 x 2 +2x— 3 ar 2 +2x— 3 i 3x+9 = 3(aH-3). Reject 3 (103. a). x 2 -{-2x—3 x*-\-3x x+3 x— 1 -x— 6 -x—3 x-{-3 is the greatest common divisor. 3. Find the greatest common divisor of a 2 -{-2ax-\-x- and a 3 — ax 2 . Ans. a-\-x. 96 division. [§ 105, 106. 4. Find the greatest common divisor of 9x 3 -\-53x 2 — 9x —18 and x 2 -\-l ls+SO. Ans. x+6. 5. Find the greatest common divisor of 2x 3 -\-x 2 — 8rtr-}-5 and 7a; 2 — \2x-\-5. Ans. x — 1. 6. Find the greatest common divisor of a 3 x-\-2a-x- -}-2ax 3 -}-^ 4 and 6a 5 -\-\0a*x-\-oa 3 x' 2 . § 105. c.) The application of the above rule (§ 104) to polynomials is simplified in various ways. Thus, before applying the rule, 1. Any factor obviously common, may be taken out, and reserved, as a factor of the common divisor required (§ 100). 2. Any factor, found in a part only of the polynomials, may be rejected (§ 100. a). 3. If one of two polynomials contain a letter not found in the other, the common divisor, obviously, cannot contain that letter, i. e. must be independent of it, and must there- fore be the common divisor of the coefficients of the seve- ral powers of that letter. In this case it is often best to arrange the polynomials with reference to that letter, and to find the greatest common divisor of its coefficients. Note. 1 and 2, above, are most easily applied to monomial factors of the polynomials; for such factors can always be found by inspec- tion (§§ 69, 81). But they are equally applicable to polynomial fac- tors, when we can discover them (§§ 93, 99). 1. Find the greatest common divisor of a 3 -\-2a*x-{-ax- and 5ab 3 — oabx-. Ans. a(a-\-x). 2. Find the greatest common divisor of x 3 -\-ax^-\-bx- — 2a-x-\-abx — 2« 2 5 and x' 2 -\-2ax — bsc — 2ab. COMMON MULTIPLE. §106. A common multiple (§46. a) of two or more quantities is a quantity which each of them will divide (§ 80. d). The least common multiple is the least quantity which they will divide. § 107-109.] COMMON MULTIPLE. 97 § 107. A quantity is evidently a multiple of any other quantity of which it contains all the factors ; if it contain the factors of each of several quantities, it is their common multiple ; if it contain each factor no often er, than some one of the quantities, it is the least common multiple. That is, The least common multiple of several quantities consists of all their factors, each tvith the highest exponent which it has in any of the quantities. Thus, the least common multiple of Qab-( = 2.3ab 2 ) and 9a 2 c(=:3 2 a 2 c) is 2.3 2 a 2 6 2 c = 18a 2 6 2 c. § 108. The least common multiple of two quantities con- sists of all their prime factors, each with its greatest expon- ent (§ 107) ; and the greatest common divisor consists of the common prime factors, each]with its least exponent (§ 100). Therefore, The least common multiple of two quantities is equal to their product divided by their greatest common divisor. Thus, the greatest common divisor of x-y and xy 2 is xy ; their product is x 3 y 3 ; and their least common multiple is x 2 y 2 =zx 3 y z -—xy. Note. Every algebraic factor of the first degree, whether mono- ;r,ial or polynomial, is a prime factor. PROBLEMS. X 2 § 109. 1. Given 4a4-x = - — — , to find x. 4a-[-x x x 2. Given — — -j = 2, to find x. a-\-b a — b ax , a : to a — x a-\-x a 2 +a 3 — 1 3. Given — —a = 1 — , to find x. a 2 — x* a — x a-\-x Ans. x = a — a' J 4. A sum of money, x, is divided among several per- sons, so that A receiving $1000 lesg than half, and B, alg. 9 98 MULTIPLICATION AND DIVISION. [§ 109. $1000 move than one third, of the -whole, find their por- tions equal. What is the value of x ? 5. Let A receive a less than half, and B, a more than one third, and let their portions be equal. What is the value of x ? G. Two couriers are traveling on the same route, and in the same direction. A is 100 miles in advance of B, and travels 10 miles an hour, while B follows at the rate of 1 2 miles an hour. In how many hours will they be together ? Let x = the number of hour.:. Then 10cc =z the distance A will travel , and 12x = the distance B will travel, before they come together. Now, if B overtake A, he must travel as many miles as A, and the distance between them, 100 miles, in addition. 12.r=10x+100; or 12x— 10z=:100. x ■=. 50 hours. 7. In what time will they be together, if A goes 10 miles an hour, and B 1 1 ? 8. In what time, if A goes 10 miles an hour, and B 10^ ? A 10, and B 10^? A 10, and B 10^? A 10, and B 10 r io? A10,andB10 TIJ W? A 10, and B lO^Voo ? 9. In what time, if A goes 10 miles an hour, and B 10 ? In the last case, we have lQx — 10x=z (10 — 10)x= 100. _ 100 _100 x ~ io=io -"o~" a.) How shall this result be interpreted ? If we divide 100 by .01, .001, .0001, .00001, &c, the quotient obviously increases as the divisor diminishes, and in the same pro- portion. Consequently, if the divisor becomes numerically less than any quantity whatever, or 0, the quotient must become greater than any quantity whatever, i. e. infinite. For no number can be assigned or conceived, so great as, when multiplied by 0, to produce 100. Hence i#£, or, in general, ~- (a being any quantity whatever, numerically § 109."| PROBLEMS. ' greater than 0), is infinite, i. e. greater than any assignable quantity; and is expressed by the symbol oo. Now, as the difference of the rates, in the preceding problems, be- came less (i. e. as B gained less in an hour), the number of hours required for him to overtake A became greater. When the difference of tho rates is nothing, the time will be infinite (i. e. B will never overtake A). In other words, if B gains nothing in one hour, no number of hours can enable him to gain 100 miles. 10. Again, suppose that A travels 10, and B 8 miles an hour, when will they be together ? Here we have 8x— 10* = — 2x— 100. .-. x~— 50 (§ 5). That is, A and B xvere together 50 hours ago. Note. Had it been proposed to find when they had been togeth er, the answer would have been positive (§4. c). 11. Let A be a miles in advance of B ; and let A trav- el n, and B m miles an hour. When will they be togeth- er ? Ans. In hours. m — n b.) The last problem is the generalization of the preced- ing (6-10). We shall evidently have, if «>0 (§ 6. a), when ?»>», m — n positive, and, of course, the result positive ; when m = n, m — n — 0, and the result, infinite ; and when }«<«, m — n, negative, and the result negative (§ 10. d). If a ■=. 0, and m>, or <«, the result is (i. e. they are togeth- er now) ; and if m = n, the result is % (i. e. they are togeth- er now, and must always remain together [§ 109. c]). 12. Let them travel towards each other, A, n, and B, m miles an hour. When will they meet? Ans. In — ; — hours. m-yn Let a — 100, m =■ 12, and n = 8 ; &c. Note. The formula of 12, above, includes this case also. For the rate or velocity of one, being positive, and represented by m, that of the other must be negative (§5), and may be denoted by — n; and the difference of the rates will be properly expressed by m — ( — n) = m + n. Hence, we have [12], a a ~ rn — ( — n) m-\-ri 100 MULTIPLICATION AND DIVISION. [§ 10^. 14. The age of a father is 36 years ; that of his son is 12. In how many years will the age of the father be just double that of the son. Ans. 12. 15. In how many years will it be triple? Ans. (i. e. it is triple now). 1 6. In how years will it be quadruple ? Ans. — 4. That is, it tvas quadruple 4 years ago. If we had inquired, hovr long since it had been quadruple, the result would have been posi- tive. 17. In how many years will the ages be equal ? Ans. co (i. e. they will never be equal [§ 109. «]). 18. Let A's age be a, and B's, b years ; in how many d — ?2 J years will A's age be n times B's ? Ans. . J n—\ Here, if «>1, the result will be positive (i. e. the event will be future), when a^>nb; negative (i. e. the event will be past), when a^nb; and zero (i. e. the event will be present), when a = ?i6. If n = 1, the result will be ± oo, when a> or <^6 ; and the result will be Sl, when a^=.b. If 7i» «»££■» <»»» 104 FRACTIONS. [§117. , „ , A*B°—B«~cx" . . 4. Reduce -rs — r, Q ,. „ to its lowest terms. A 2 cy' — c 2 x"y" Ans. — :.. „ n , nx*-U14-x)"—nx n (l-\-x) n - 1 5. Reduce ji—. — r-=- to its lowest terms, and simplest form. .4ws. #3 I 6. Reduce — ~ — — to its lowest terms. X" — 1 (l+x)*+* a~c 7. Given (a-f-a:) (5-f-a;) — a(b-\-c) =— ^--J-a: 2 , to find x. ^dns. a: = — . b § 117. To reduce fractions to a common denominator. The value of the fraction must remain unchanged. Con- sequently, in effecting this reduction, we must either multi- ply the terms by a common multiplier, or divide them by a common divisor (§ 118. 3). If then the given fractions be already in their lowest terms, the common denominator must be a multiple of each of the given denominators. Hence, the following RULE. Multiply all the denominators' together for a new denominator, and each numerator by all the denomi- nators except its own, for a new numerator. a.) Otherwise, Multiply both terms of each fraction by the denominators of all the other fractions. 2 1 5 1. Reduce ^, - and - to a common denominator. o 4 7 2X4X7_56 1X3X7_21 > 5X3X4_60 3X4X7 - 84 ; 4X3X7~"84 ; 7X3X4~~ 84* a x oc 2. Reduce T , - and T to a common denominator. be b • 118.] ADDITION AND SUBTRACTION. 105 aXcXb __ahc xXbXb __b 2 x xXbXc __ hex bXcXb~~b*c ' 'cXbXb~~~b^c' !>XbXc~~ b 2 c ' u 1 3. Reduce to a common denominator — and -. v J v x x 4. Reduce — r-r and r to a common denominator. a-\-o a — b XXX X 5. Reduce -, -, -, and -= to a common denominator. 2 o 4 b b.) It is evident, that the results obtained by the rule may often be reduced to lower terms, and still have a com- mon denominator. This will be obviated by taking, for a common denominator, the least common multiple of the de- nominators (§ 106). In that case, we must divide the least common multiple by each of the given denominators, and multiply the corresponding numerator by the quotient. XX X 0. Reduce r , — - — ^r- and — r-p to fractions with the a — o a 2 — o 2 a-\-b least common denominator. ADDITION AND SUBTRACTION. RULE. § 118. Reduce to a common denominator; and then add, or subtract the numerators. Note. The resulting fractious in the following examples should be reduced to their lowest terms. a c a ± c b ± b~~b~' I . Add -, T , - and -. a o c r a a+b a+b . a 3 -{-bx 2 a 3 -\-bx 2 Am. — cn—T — "V <. • x{a — x){a-f-x) a J x — x 3 = what ? Ans. 2(z— 4) 2(x— 2) a; 3 — Cx+8" 106 FRACTIONS. [§119,120. a 3 1 2 (x—a) - ~ 4 (x—a) n " 4(a;-fa) B 2 x" —y" A 2 y"~c"—x" ■=■ what ? 6. — — . , ■ = what r MULTIPLICATION AND DIVISION. § 119. To MULTIPLY A FRACTION; RULE. Multiply the numerator, or divide the denominator, by the multiplier. § 113. 1, 2. _, ax c ax ax Ihus, — Xc = — - = — ; or, :=axc~^ i c=zaxc~ 1 =. — c 2 c i c c I 14). x /ax\ aX- ( — ) ox c \ c J ax , , , ax So T X- = — r~ = — r~ =7— 5 or, =«6- 1 a;e- 1 =:-r— . b c b b be oc a.) The application of the rule to the last example gives the common rule for multiplying fractions together ; Multiply the numerators together for the numerator of the product, and the denominators for its denominator. Note. To multiply by — is to multiply by "z and divide by c. But the fraction is multiplied by multiplying the numerator, and divided by dividing the denominator. , -«r i • i aArx , a — x 1. Multiply — — by — . JL *C n a+x x , , B*x" —y" 9 2. -Z-x— j— =what? — -r^T/X^-7/ ? a 2 -4-aW-fl 9 a— b . . 3. t_ L -X— r-r = what? : 120. To DIVIDE A FRACTION ; RULE. Divide the numerator or multiply the denominator, by the divisor. § 113. 1, 2. § 120.] division. 107 a v c/ a ab— l a .- " br T ~e = =—(§14) b ' b be' b • c be Also ^-c=7^^^ (§113 - 3); or ' &X- ^* = aJ-i jL.a-c-1 — ab-^x~^c=:~ (§ 19. Cor. IV.). Z» ' c ox a.) Hence, the common rule for dividing by a fraction ; Invert the divisor, and multiply. Note. To divide any quantity by x divided by c, is the same as to divide c times that quantity by x. b.) The last rule is otherwise demonstrated thus ; a x a t\ \ _ fa m 1\ b + ~c~b~ \c XX ) - \b~cJ ■x. -™ a . -, a a . 1 ac rc .- T _ d» 6 c b *~ ac bx a . x fa . 1\ ac . Apply the same reasoning to | ■— f- c.) Dividing either term of a fraction has the same effect as multiplying the other term. Hence, to divide one frac- tion by another, we may divide the terms of the dividend by the corresponding terms of the divisor (i. e. numerator by numerator, and denominator by denominator). Note. This course is convenient, when the divisions can be ex actly performed (§§ 80. d, 82. g) ; and it amounts simply to invert- ing the divisor and canceling equal factors. 1. Divide -z r- by . Quot. —?-. : — „. a 3 — x 3 a — x a--\-o.x-\-x- 2 u a*-p>±x* h «-* QuQfl a 2 — x- a-\-x 9 3 , , 10 2, 1.2.3.4.5x5 1.3.5x 2 , a : — whit 3 : ? - — ? 12 • 4~ IS • o" 10.9.8a" ' 9 then a < b, ab-\-ak < ab-\-bk, and y < ~!~ . o 6 6-j-a; If t->1, then a > 5, a5-j" a ^' > «H~^"> an< l y > 7T~7~' If -=• = 1, then a — b, ab -\-ak = ab -}- &£, and 7 — ,7, . £* o b-\-k Hence, k being any positive quantity, a ^ «+& ,. « ^ , r<» >> or = , according as T <, >, or = 1 ; i. e. as o o-f-/c o a <, >, or — b. That is, If any positive quantity be added to both the terms of a fraction, the primitive fraction will be less, great- er than, or equal to the new fraction, according as it is less, greater than, or equal to unity. Add 100 to the terms of the fractions f, §, and f. CHAPTER IV. EQUATIONS OF THE FIRST DEGREE, CONTAINING TWO OR MORE UNKNOWN QUANTITIES. TWO UNKNOWN QUANTITIES. § 122. I.) Let x-\-y — 10, x and y being both unknown. Here the only condition (§ 38) is, that the sum of the unknown quantities shall be 10. Hence we may have ;e = 0, y=10; x=zl, y=9; x =. 7, y =z 3, &c. ; orrcrr — 1, y=ll; rc:= — 2, y=:12, &C ; or y=z — 1, 0, &C., re = 11, 10, &c. ; or x — ^, y — 9%; x = — |, y = 10f,&c. II. . Again, let x — y = 4. Here the only condition is, that the difference of the numbers shall be 4. Hence we may have x = 4, y = ; re = 5, y = 1 ; re = 7, y = 3, &c. ; or a? = 0, # = — 4, &c. ; or x — 20, y = 1 6, &c. a.) Either of these equations is, by itself, obviously in- determinate (§ 109. c); and may be satisfied (§39) by any one of an infinite number of values of x, with correspond- ing values of y. b.) But the conditions may be united. That is, it may be required, (1.) that the sum of two numbers shall be 10, and (2.) that their difference shall be 4. We shall then have, at the same time, rc-f-^ = 10, and x — y = 4. And the same values of x and y must satisfy both equations. c.) Now, if x — 9, we have, by the first condition, y = 1 ; and, by the second condition, y = o. Thus, the same val- ue of re satisfies both conditions, but the values of y are dif- ferent. Again, if y =. 6, we have, by the first condition, x = 4 ; and, by the second, re — 10. Here, the same value ALG. 10 110 EQUATIONS. [§123,124. of y satisfies both conditions, but the values of x are diffe- rent. The values in both these cases are said to be incom- patible. d.) But the same values of both x and y must satisfy- both conditions (b). And, in fact, among the values found above (L, II.) there is one set common to the two equations, viz. x = 7, and y = 3 ; thus 7+3 = 10, and 7—3 = 4. e.) The solution of the problem consists in finding these common, or compatible values of x and y. § 123. The union (§ 122. b) of the two conditions is alge- braically expressed by the combination of the equations, treating x and y as symbols of the same quantities in each. Note. It is obvious, that, if x represents the same quantity in the two equations, the sura of x in the first and x in the second, will be 2x, and their product, a: 2. But if x in the second equation de- noted a different quantity from x in the first, it might be distinguished ■ ', and the sum of the two quantities would be x+x', and the" product, xx 1 . The same remark, clearly, applies to y. ELIMINATION. BY ADDITION AND SUBTRACTION. § 124. Combining (§ 123) the equations x+y= 10 x — y — 4 adding them, member by member (Geom. § 22), we have 2a; = 14; .-. x = 7. Substituting, in the first equation (x+y — 10), for x its value, we have 7+y = 10; .-.y=3. These values of x and y introduced into the second equa- tion, x— y = 4, satisfy it; thus 7—3 = 4, an absolute equation (§ 37. d). ,tes. (1.) The value of y might with equal propriety have m obtained by the substitution of x in the second equation. Thns § 125, 126.] TWO UNKNOWN QUANTITIES. 1 1 I 7_y = 4. ... y = 3. (2.) If we had subtracted the second equa- tion from the first, we should have found the value of y; and then, by introducing it in either of the equations, we should find x. § 125. The solution of the problem in § 124, it will be observed, is effected by removing one of the unknown quan- tities, till the value of the other has been found. This is called elimination'' ; and the method employed above is called elimination by addition and subtraction. 1. Given x-{-y=lo, (1) 3x+4?/=54 (2) .Multiplying (1) by 3, Zx-\-5y = 45. Subtracting the la3t from (2), y — 9. Then from ( 1 ), x+9 — 15 ; .-. x = 6. 2. Given Sx— %y = — 27, . . (1) 4z-Hy = 24. . . Multiplying (1) by 2, and (2) by 3, we have 6as— # = — 54, . . (3) 12x-hy=x72. . . I Then, by adding (3) and (4), 18x=18. .\*=l,y=60. Eliminate x, by dividing (1) by 3 and (2) by 4, and subtracting iho first quotient from the second. § 126. (1.) When one of the unknown quantities has the same coefficient in both equations, it can be eliminated, if the signs of the equal coefficients are alike, by subtraction ; and if unlike, by addition. See §§ 57. 18 ; 60. 14. (2.) We may cause one of the unknown quantities to have the same coefficient in both equations, by suitably multiplying or dividing one or both of the equations. 1. Given 5x-\-Gy = 40, Zx-\-2y = 20, to find x and y. Ans. x = 5, y = 2 \. (d) Lat. elimino, to turn out of doors. 112 EQUATIONS. [§ 127, 128. 2. Given 7aH-10y = 72, 9x+ 3y =± 63, to find x and y. 3. Given 2x— dy = 7, x-\-2y = 14, to find x and y. BY COMPARISON. § 127. 1. Resuming the equations, x-\-y = 10, . . . (I) *— y = 4» ... (2 we have, from (1), x= 10— y, and from (2), x = 4rj-y. Equating 6 these values of x, we have 10— y = 4-{-y. .: y = 3, and x == 7. 2. Given 2# — 3y = 7, and a;-J-2y = 14. From the first, %y = 2x—l ; .-. y ■=. fa;— $ . From the second, 2^=14 — x; .-. y=z7 — hx. %x — £ = 7 — \x. .: x = 8, y = 3. In this method, we find from each equation the value of one of the unknown quantities, in terms of the other un- known, and of the known quantities. We equate these tws> values, and from this new equation find the value of the oth- er unknown quantity ; and substitute as before. Note. This is called elimination by comparison, because we compare the two values of the unknown quantity. BY SUBSTITUTION. § 128. 1. Taking again the equations x+y=lO, ... (1) x—y = 4, ... (2) we have, from (1), y = 10 — x. (e) Lat. tequo, to make equal. Quantities are said to be equated, or made equal, when they are made to constitute the members of an equation. * 129.] TWO UNKNOWN QUANTITIES. 113 Substituting in (2), x— (10— x) = 4. .-. x = 7, y = 3. 2, Given 2z— 3y = 7, . . (1) x+2y=U. . . (2) From (2), we have y = 7—lx. Substituting in (1 ), 2x— 3 (7— \ x) = 7; | or 2x— 21+|a; = 7. x = 8, y = o, as above (§127. 2). "We here ./mrt*, as m the last method, the value of one of the unknown quantities from one equation, and substitute it in the other equation. Notes. (1.) This is called elimination by substitution. (2.) Ei- ther of the above methods may be employed at pleasure. Sometimes one will be found most convenient, and sometimes another. Practice will enable one to fix upon the best method in each case. It will be useful for the learner, at first, to solve each example by all the meth- ods. $129. 1. Given %x J r\y = 11, \x-\-\y = 8, to find x and y. 2. Given 3z— 4y = — 13, 7x — by = 0, to find x and y. Ans. x=.b, y—1. 3 . Given — Sx-{-iy — 8, Sx-\-5y — 3^, to find x and y. Ans. x — — £, y — \. 4. Given y = 2x — 4, y = — Bx-{-8, to find x and y. Ans.x=2%,y — ? f . 5. Given y = ax-\-b, y = a'x-\-U, to find x and y. b—V a'0—aV Ans.x=— , y — — . a! — a a! — a With what values of a, a', b and V will x and y, in this result, become zero ? negative? infinite (§ 109. a) ? inde- terminate (§ 109. c) ? *10 114 EQUATIONS. [§ 129. 6. Given ax-\-by = c, a'x-\-Vy = d, to find x and y. _ Vc — bd _ad — a'c Ans - x -^ZdV y-av^Tb- 7. Given x-\-y = S, and x—y = D,to find x and y. Am. x = !(£+■#)> y = K^— D )- See § 65. 3. 8. A horse and saddle are worth $175 ; the horse is worth six times as much as the saddle. "What is the value of each ? Solve the problem by means of one, and of two unknown quanti- ties. 9. Let the horse and saddle be worth a dollars ; and let the horse be worth m times as much as the saddle. a „ , , ,. iua . nil Ans, value of the saddle ; — ; — , that oi the horse. 1+m l+»» 10. A bill of SI 65 was paid in dollars and eagles, the whole number of pieces being 70. How many were there of each ? Let x = the number of dollars, y = " eagles. Then x-\-y = 70, and x+10y = 1 65. Or, if x = the number of dollars, then 70 — x = the num- ber of eagles ; &c Or, let x = the number of eagles ; &c. 11. a coins of one kind make a dollar, and b of another kind. How many of each kind must be taken, in order that c pieces may make a dollar ? Let x = the number of the first. Then either y or c — x will be the number of the second. ™ x , V -• x , c — X 1 Then - + | = l;or-H — 7— = 1. a b a b a(e — b) _ , „ , . - b(a — c) „ , Am. — — t*- of the first kind ; -s — —- 01 the second. a — b a — Let a = 20, 5 = 10; and c = 12, 13, 20, 10, 21, 9. Leta=10, 5=6; andc = 8, 9, 10, 6,5, 11. §130.] TWO OR MORE UNKNOWN QUANTITIES. 115 a.) The nature of the question requires whole number for the answers. Such values, therefore, should be assign- ed to a, b, and c, that the numerical values of the above re- sults may be integral. Which of the values above comply with this condition ? b.) With what values of a, b and c, will the above results, or either of them, be positive ? negative ? zero ? infinite ? indeterminate ? How shall these several results be inter- preted ? 12. Find a fraction, such that if 1 be added to its nume- rator, the value will be |- ; and if 1 be added to its denom- inator, the valiu will be £. Let x be the numerator, and y the denominator. 13. A certain number is expressed by two digits whose sum is 9 ; and if it be increased by five-thirds of itself, tli order of the digits will be inverted. What is the number ? Let re = the left hand digit, and y — the right hand digit. Then 10x-\-y = the number, &c. 14. A places a sum of money at interest; B invests 1000 more than A, at 1 per cent higher interest, and finds his income $80 more than A's. C invests $1500 more than A, at 2 per cent higher interest, and receives an in- come greater than A's by $150. What are the three sums invested, and at what rates ? Let x =■ A's sum, and y =■ his rate of interest per cent. xy Then Too" *" s mcome ? *» c - MORE THAN TWO UNKNOWN QUANTITIES. §130. I. Let x+y+z = 10 .... (1), x, y and z being all unknown. Here we may assign any value we please to any one of the unknown quantities, and still have an infinite number of values for the other two ; or we may assign any values 116 EQUATIONS. [§18L whatever to two of them, and find a corresponding value for ihe third. Thus, the problem is doubly indeterminate. II. Again, let 2x— y-\-3z = 7 ... (2) This equation is equally indeterminate as the first. And if we unite the two conditions, W6 may still assign any val- ue we please to one of the unknown quantities, and deduce corresponding common values for the other two. Or, elim- inating one of the unknown quantities, we shall have a sin- gle equation with two unknown quantities. Thus, adding (1) and (2), 8x+±z = 17 . . . (a) Hence, the problem is still indeterminate (§ 122. a). III. But again, let 3x+2y+4z=z27 . . . (3) Multiplying (1) by 2, 2x+2y+2z = 20 . . .(b) Subtracting (b) from (3), x+2z = 7 . . .(c) Combining (a) and (c), [§ 123], we find x=z3, y = 5, and, then substituting in (1), (2) or (3), z = 2. Note. We might, obviously, have employed either of the other methods of elimination (§ 124-128). § 131. Hence, to find the common values of three un- known quantities, from three equations, we eliminate one of the unknown quantities from all the equations, thus forming two equations with two unknown quantities. We then solve these equations by § 124-128. 1. Given &H4rK* = 12, *— y+ z = 12 > 2x-\-3y — 4s = 12, to find x, y and z. 2. Given x-\- ly+^s =27 x-\-\y-\-\z=z 16, to find x, y, and z. Ans. x = 1, y = 12, z = 60. § 182-134.] SEVERAL UNKNOWN QUANTITIES. 117 § 132. "We have found one equation with tivo unknown quantities, and one or two equations with three unknown quantities to be indeterminate. In like manner, if we had three equations with four unknown quantities, by eliminat- ing one of the unknown quantities, we should have two equations containing three unknown quantities, and, of course, indeterminate. By like reasoning, we shall find, that any number whatever of equations must be indetermin- ate, if the number of unknown quantities is greater than the number of independent equations. Notes. (1). Independent equations are those, of which no one is implied by the rest. Thus x+y=B, and 3x+3y = 9 are not independent, because one is a necessary consequence of the other. (2.) When a number of equations containing several unknown quantities are spoken of, they must be understood to be independent equations, unless the contrary is stated, or clearly implied by the connection. § 133. If we have four equations involving four un- known quantities, the elimination of one of the unknown quantities will result in three equations containing three un- known quantities, which may be solved by § 131. The same reasoning will obviously extend to any number of equations containing an equal number of unknown quanti- ties. § 134. Hence, to find the value of any number ol unknown quantities from an equal number of equa- tions, we eliminate one of the unknown quantities from all the equations, thus diminishing by one, at the same time, the number of equations and of unknown quan- tities contained in them. We then eliminate, from the new equations, another unknown quantity, and so on, till we arrive at a sin- gle equation containing one unknown quantity. 1. Given 7x—2z+3u = 17, 4&—2z+t = ll, 5y—3x—2u=:8, •118 EQUATIONS. [§ 134. Ay— 3u+2t=0, 3z-\-8u — 33, to find x, y, z, u and t. Ans. x = 2,y = 4:,z = 3,u=z3,t=zl. 2. Given 2x—3y+2z = 13, 2u — x=. 15, 2y+z = 7, 5y-\-3u — 32, to find x, y,"u and z. Ans. x = 3, y=l, m = 9, z = 5. 3. The sum of four numbers is 25. Half of the first number is equal to twice the second, and to three times the third; and the fourth is four times the third. What are the numbers ? Let u, x, y and z represent the numbers. Also let x represent one of the numbers, and solve the problem with one unknown quantity. 4. Find three numbers, such, that the sum of the first and second shall be 15 ; the sum of the first and third, 16 : and the sum of the second and third, 17. Solve the above problem by one, by two, and by three unknown •quantities. 5. A, B and C form a partnership. A contributes a cer- tain sum ; B contributes a times, and C, b times as much as A ; and the whole stock is c. How much did each con- tribute? See §55. 4. c etc ■Ans. — — —, A's part; , , , , , B's part, &c. I-\-a-f-b l-\-a-j-o 5. A and B can perform a piece of work in 8 days ; A and C, in 9 days ; and B and C, in 10 days ; in how many lays could each person, alone, perform the same work ? Let x, be the number of days required by A ; y, by B ; and z, by C. Then, in 1 day, A will perform - of the work ; B, - ; * y md C. -. But A and B together perform, in 1 day, - of rhe^work; &c §135, 136.] SEVERAL UNKNOWN QUANTITIES. 119 1,1 1 1,1 1 , 1,,1_1 x y 8 $x z 9 y z 10 Note. Instead of clearing of fractions, regard — , — and — as the x y z unknown quantities; and from tbeir values, when found, find the values of x, y and z (§ 50). Ans. A in 14§§ days; B, in 17|| ; and C, in 133 7 T . 6. Let A and B perform the work in a days ; A, and C, in b days ; and B and C, in c days ; and find the general expression for the time in which each person, alone, would perform the work. . 2abc . , . 2abc _, Ans. -= — ; 7, A s time ; -= — ; — 5 , B s ; and oc-\-ac — ao oc-\-ao — etc 2abc „. ~rr a - ' Cs - ab-\-ac — be § 135. We have seen, that, when the number of unknown quantities is greater than the number of independent equa- tions, the problem is indeterminate. "When, on the other hand, the number of unknown quantities is less than the number of independent equations, the equations are incon- sistent in their conditions, and cannot all be satisfied by the same values of the unknown quantities. For, if the values found from two equations containing two unknown quanti- ties would satisfy a third, this would be implied by the rest, and, of course, would not be independent of them (§ 132. N. 1). E. g. the equations x-\-y=. 10, x — y — 4, and 2x-{-y = 40, are obviously inconsistent. § 136. When a single equation containing more than one unknown quantity is considered by itself, the unknown quantities are frequently called variables ; and one of them is said to be a function (§ 26) of the rest. a.) Thus, in the equations 2y-\-3x = 10, y = ax-\-b, y is a function of 'x, and x of y ; or, as it is usually expressed, y — F(x), and x = F(y). For, if we give any value what- ever to one of these quantities, we can deduce a correspond- ing value for the other ; and, if we vary the value of the first, the value of the second undergoes a corresponding change. 120 EQUATIONS. [§ 137. h.) If an equation contain more than two unknown quan- tities, each of them is a function of all the rest. Thus, in the equation 2x-\-3y-\-z = 75, we have x == F(y, z) [i. e. x a function of y and z], y — F(x y z), and z = F(x, y). Notes. (1.) Of two quantities, that of which the other is said to be a function, is called the independent variable. (2.) Either of the unknown quantities may, obviously, be made the independent variable, and the other will be the function. (3.) If there are more than two variables, one may be regarded as a function of all the rest, they being all independent; or one may be a function of the second, the second, of the third, and so on, the last only being independent. § 137. Arithmetic, in its ordinary applications, furnishes only positive and definite solutions. It is, therefore, some- times said, that negative, infinite and indeterminate results do not furnish a proper answer to a question. The answers which they furnish would indeed not be intelligible to one unacquainted with the algebraic language. But to one fa- miliar with that language a negative result answers a ques- tion as directly and intelligibly a3 a positive ; an infinite, as a finite. See §§ 4. 9, 10 ; 109. 10, 16. Thus, when we inquire, how long it will be before a cer- tain event will take place, we equally answer the question by saying that it will take place in 12 years (§ 109. 14), or in no years (i. e. now, § 109. 15), or that it took place 4 years ago (§ 109. 16), or that it will never take place (§ 109. 9, 17), or that it is taking place all the time (§ 109. c). In like manner, if we inquire how far east a certain point lies, we equally answer the question by saying an infinite distance, a finite distance (as 10 miles), no distance, a dis- tance west, or that such a point exists every where in a line running east and west. Arithmetic does not ordinarily take cognizance of infin- ite or indeterminate results ; and, regarding numbers sim- ply as such, without respect to their character as positive or negative (§ 8), its questions must be proposed in such a manner, that an answer may be expressed by a number simply. § 138-140.] SEVERAL UNKNOWN QUANTITIES. 121 § 188. It will be observed, that between the positive and negative values, we always have a value equal to zero or to infinity, i e. equal to or %. See §§ 4. 6-10 ; 109. 8-10, b, 14-16. That is, between the positive and negative re- sults, there is one, either equal to 0, or whose denominator has become 0. § 139. We have % — cc (§ 109. a). That is, (1.) A finite quantity divided by zero is equal to infinity. Also (§ 42. c), a = OX co. That is, (2.) Zero multiplied- by infinity is equal to a finite quan- tity. Again (§ 42. d), — = 0. That is, CO (8.) A finite, divided by an infinite quantity is equal to zero. Note. We arrive at the idea of infinity by continually diminish- ing a divisor, and thus finding a greater and greater quotient (§ 109. a). Hence is sometimes said to denote an infinitely small quan- tity, or an infinitesimal (i. e. a quantity less than any assignable quantity [§ 109. a]). § 140. The expression $ is not always indeterminate. For, instead of the whole numerator and denominator, a common factor may have been reduced to 0. If then this common factor be removed (§ 113. 3), the expression will no longer be indeterminate. Thus, when b = a, a 2 — b* _ «2_ S 3 (a+J)(a— J) r=7v But -— v ^ JK — J -=a-\-b — 2a. a — b a — b a—b ' when b=.a. oji jf o So, if bz=z a, r—7;- But, performing the division. a — b ft « jf and then making b=i a, we have -z^na"- 1 (§ 96. b). a — b x s i !• - — — = what, when x = 1 ? Ans. 3. x— 1 2. -z z- = what, if x = y ? Ans. —. — = 0. x~—y 2 v x +y ALG. 11 122 INEQUALITIES. [§141-144 o. -. ^ = what, if x = y ? ^tns. — ^L = o>. CHAPTER V. INEQUALITIES. § 141. Two quantities, connected by the sign < or > (§ 2. b), constitute an inequality. An inequality may be cal- 1 ed increasing, or decreasing, according as the second mem- ber is greater or less than the first. When two inequalities both increase, or both decrease, they may be said to have the ne tendency, or to subsist in the same sense or direction ; otherwise, they are of contrary tendency. § 142. Operations upon inequalities are similar to those upon equations, and depend chiefly upon an analogous ax- iom (§ 42) ; viz. Unequal quantities, equally affected, remain unequal. Hence, if equal quantities be (1.) added to, or (2.) sub- tracted from, both sides of an inequality, or if both sides be (3.) multiplied, or (4.) divided by equal quantities, the results vdll be unequal. § 143. In transforming an inequality, however, we must not only preserve the inequality, but we must, at every step, determine which way it tends (i. e. which member is the greater). Hence, the necessity of observing the following obvious principles. § 144. a.) If equal quantities be added to, or sub- tracted from, both members of an inequality, the tenden- cy of the inequality will always remain unchanged. Thus, 10>6; 10±8>6±8 145-147.] INEQUALITIES. 123 — 10<— 6; — 10±12<— 6±12. See § 6. a. Note. Hence, transposition applies to inequalities, in like man- ner ae to equations. Thus, 10— 5>12— 8. .-. 10>17— 8. So, ify 3 +* 2 — E 2 >0, theny 2 -fa: 2 >i2 2 , andy 3 >i2 2 — x*. § 145. b.) "With still greater reason, If two inequalities, having the same tendency, be added, nember by member, there xvill result an inequality of th same tendency. Thus, 9>7 and — 1>— 3. .-. 8>4. So if a~>b, and m>n, then a-\-m>bA^n. Note. If one inequality be subtracted, member by member, fr«fn ■nother of the same tendency, the result will not always be an ine- quality ; nor, if it be, will it necessarily have the same tendency. § 146. c.) If the members of an inequality be subtracted from the same member, the tendency of the inequality will be "hanged. Thus, 8>6, and 10— 8<10— 6. In like manner, — 8<0 — 6 (i. e. — 8< — 6). Hence, d.) If the signs of both members be changed, the tendency will be changed. Note. This results directly from the principle, that, of negative quantities, that which is numerically the greatest is absolutely the least (i. e. leaves the least remainder). See § 6. a. § 147. e.) If both members of an inequality be multi- plied or divided by the same positive number, the re- sulting inequality will have the same tendency ; if by the same negative number, the tendency xoill be changed. Thus, 6>— 8 ; and 6X3>— 8x3, or 18>— 24. But 6X— 3<— 8X— 3, or — 18<-f 24. So 6-^2>— 8-r-2, or 3>— 4. But 6^ 2<— 8-i 2, or — 3<+4. Also, if a > b, ah > bk (§ 121), but — ah <— bt !24 POWERS AND ROOTS. [§148-151- § 148. /.) Hence, an inequality may always be cleared of fractions. For, if we multiply by a positive denomina- tor, the tendency remains the same ; if by a negative, it is changed. Or, if the denominator is negative, we may place its sign before the fraction, and then multiply by the posi- tive denominator (§§ 68. b, 80. b). § 149. g.) If the members of an inequality be positive, and be both raised to the same positive integral power of any degree whatever, the tendency of the inequality tvill remain unchanged. Thus, 7>3; 7 2 >3 2 ; Note. This holds equally of fractional powers or roots (§23. />), so long as we confine ourselves to their positive values (23./. 1). If we regard the negative values of an even root, the tendency is, of course changed. § 150. h.) Whatever be their signs, if the members of an inequality be both raised to the same odd positive potver. the tendency will remain unchanged. Thus, — 3<2; (— 3) 3 <23« CHAPTER VI. POWERS AND ROOTS. MONOMIALS. § 151. To raise a monomial to any power; Multiply the exponent of each factor by the expo- nent of the required power. (§24. d). a.) This rule depeuds on the obvious principle, that a § 151.] MONOMIALS. 125 power of a product is equal to the product of the same pow- ers of the several factors. Thus, {abc) n —a n b n c n \ [pbc)* = a 2 b 2 c 2 ; («J)^a¥. b.) This rule applies equally to numerical and literal factors ; and, so far as Algebra is concerned, it is sufficient. It is proper, however, to perform upon the numerical coef- ficient the arithmetical operations indicated by its expon- ent. Thus, if its exponent be positive and integral, raise the coefficient to the arithmetical power denoted by the ex- ponent ; if the exponent be positive and fractional, raise the coefficient to the power denoted by the numerator and ex- tract the root denoted by the denominator ; if the exponent be negative, perform the same operations as if it were pos- itive, and place the result in the denominator of a fraction, of which the other factors of the monomial constitute the numerator. c.) The sign of an even integral (§ 22. d) power is posi- tive ; the sign of an odd integral power is the same as thai of its base (§ 22. N.). See § 11. N. 2. 1. What is the fourth power of 2ab^x^y~^? Ans. 2*a*b*x%~%= 16a*h*x*y~$. 2. {—Za-b- 1)3 — what? (— aWcx) 2 ? (aV 3 ) 2 ? 3. (na- 2 x~?)^ = whsLt? (a~*x~%)~*? (10&) 6 ? (a n x r )t? (^) s -i? - - - i r 4. (a n x ,J ) s = Yfhsit? (3a- 2 b*x2y~?) 5 ? (fi 2 x- 2 )~% ? d.) In determining the sign of a fractional (§ 22. d) pow- er, its exponent should be reduced to its lowest terms. Then, if the numerator of the exponent is an even number, the power is positive; if the denominator is even, the pow- er of a positive quantity is ambiguous (i. e. ±), and of a negative quantity, imaginary (§ 23./) ; and if both numera- tor and denominator are odd numbers, the power has the same sign as the quantity itself. *ll 126 POWERS AND ROOTS. [§ 152. e.) A power of & fraction is found by raising both nume- rator and denominator to the required power (§§ 119. a, 120. c). Or, all the factors of the denominator may be carried into the numerator, if we change at the same time the signs of their exponents (§§14, 17) ; and then the quantity may be treated like any other monomial. G)' : ROOTS. § 152. From the preceding rule (§ 151), we deduce the following specific rule, in which the term root is used in the same 6ense as in Arithmetic. To extract any root of a monomial ; Extract the root of the numerical coefficient as in Arith- metic ; and divide the exponent of each literal factor by the number of the root. a.) This rule is obviously included in the preceding (§ 25). But for convenience, and on account of the rery frequent necessity of extracting the square and cube roots, it is given here in a distinct form. b.) An odd root of a positive quantity is positive; of a negative quantity, negative (§ 23. e). c.) An even root of a 2 } ositive quantity is either positive or negative (§ 23./. 1). d.) An even root of a negative quantity is imaginary (22./. 2). 3 1. "What is the square root of 2oaHc- y x z '? Ans. J{25a*bc-ix3)= (25a 2 bc~^x^ — 5a5-c~M. 2. y(49a 3 J 2 a:-3)=what? (100«- 4 5 m a; 2M )- ? (x^? 3. V-^ = what ? (*£)*? fe£L)*P 646 4 y 12 \2(b z zJ \**/c*/r/ [§ 153-150. KAD1CALS. 127 § 153. Any root of any monomial can be algebraically expressed, but it is not always possible to perform exactly the arithmetical operations upon the coefficient. Thus the jl 1 square root of 2ab- is 2' 2 a 2 b; but the exact arithmetical computation of 2 2 cannot be attained. Such a root is cal- led incommensurable^ irrational or surd h . A numerical quantity whose root can be exactly found is called a •per- fect power. Note. It will be shown hereafter, that, if a root of a whole num- ber is not a whole number, it cannot be expressed at all except by approximation. § 154. The use of the term perfect power, as applied to algebraic monomials, is sometimes restricted to the cases in which the numerical coefficient is a perfect power, and each exponent is divisible (<§ 80. d) by the number of the root. The roots of all quantities which are not perfect powers are called irrational, radical (§ 23. d. N.) or surd quantities. § 155. A radical quantity can frequently be reduced to a s impler form. Thus, (192a 3 fr 3 c)S=(64a 3 5 2 X3ac)^ = 8a5(3ac)^. (I08a^x)^= (27a 3 £ 6 X4aVp = 3a5 2 (4a 2 x)*. We here separate the root into two factors, one of which is rational (i. e. expressed by integral exponents), while the other is radical (i. e. expressed by fractional exponents). This can, obviously, be done, whenever, after the extrac- tion of the root, any of the exponents are improper frac- tions; or, when, before the extraction, any of them are greater than the number of the root, and not exact multi- ples of it. § 156. We shall, evidently, effect this simplification, it, (/) Lat. in, not, con, together and mensura, measure; having 520 common measure (§100) with unity, (g) Lat. in, not and ra- tio, relation, ratio ; whose ratio to unity cannot be exactly expressed, (/i) Lat. surdus, that is not heard; because it cannot be expressed. 128 P0WER9 AND ROOTS. [§ 157, 158. in extracting the root, we divide the exponent of each let- ter by the number of the root, and set the integral part of the quotient as an exponent of the letter in one factor, and the fractional part as an exponent of the same letter in another factor. If the root has been extracted, we have only to reduce all the improper fractions among the expo- nents to mixed numbers, and set each letter under its integral exponent in one factor, and under its fractional exponent in another. 1. Eeduce y60a 3 6 4 a; to its simplest form. Ans. (60aH*x)^=: (4.l5)%aah*x^=2ab*(ttaxft= 2ab" +/loax. 2. Reduce (7oa 2 b 5 x 7 ) 2 to its simplest form. 3. Reduce also 3 ^/5 4a 8 x 3 ; ^/32x 2 y 5 ; *Ja s bpc~ 2 x; J(2p)x 2 ? 4. Separate a 2 b 3 c 4 x » into rational and radical factors. CT fin _^n 3r> JL2>'» Ans. aWcxXahh*x n =aWcxXa^ n b 12 V- n x i '- n = a 2 b 2 cx l a V« 6 "& 4 "<^ 12 "'— a*b*cz{a an b**c* n xi ■»)A» See § 1G0. § 157. a.) In simplifying an irrational fraction, it is gen- erally best to multiply both numerator and denominator by a multiplier which will make the denominator rational. /3\- Thus, we may simplify the fraction f-J 2 , as follows : /8\i_3*_**7*_(8J7)*_l, sl a W -^-^|~~7~~7 ( ^ * Note. If the sum of the exponents of each letter in two mono- mials be an integer, the product will, of course, be rational. § 158. b.) Every negative quantity can, obviously, be re- §159-161.] RADICALS. 129 garded as containing the factor, — 1, together with a posi- tive factor. Thus— a — a{— 1) ; — a 2 = a 2 (— 1) ; — 25 = 25(— 1). Hence, (— a) 2 — a 2 (— 1)^ ; (— a 2 ) 2 = (a 2 ) 2 (— 1)^ = a*/ — 1. 1. (— ^ 2 ) 2 "=what? Am. Bj— 1. 2. (— 2o« 2 ^) 2 " = what? Ans.halPx^J—\. Hence, every even root of a negative quantity consists of a real quantity multiplied by */ — 1. Note. Such expressions as the above must not be regarded as having any actual value whatever. One factor is real, but the other is imaginary; and the product is, of course, imaginary. § 159. Addition, subtraction, multiplication and division are, of course, performed upon irrational quantities accord- ing to the general rules. In addition and subtraction, it is frequently more convenient to separate the quantities into their rational and radical factors, and reduce the resulting polynomials by § 33. c. § 1 60. After the separation of the rational and radical factors of a monomial, it is frequently convenient to reduce all the fractional exponents to a common denominator, and, writing only the numerator of each exponent over its letter, enclose the whole in a parenthesis under the reciprocal of the denominator as an exponent; or, if preferred, place the whole under a radical sign with the common denominator over it. See § 156. 4. Notes. (1.) Radicals which have the same quantities, both nu- merical and literal, under the same fractional exponent or radical sign, are called similar radicals. (2.) The rational factor, multi- plied by a radical, is, of course, properly called the coefficient of the radical. § 161. The rational factors may be placed under the rad- ical exponent or sign, if their exponents be reduced to frac- tions having the common denominator. This is commonly 1#0 POWERS AND ROOTS. [§ 162. called carrying the coefficient of the radical under the sign. Thus, 2 1 i xja=z&«p = (ax-y, or J (ax-). 1. x(2JRxy = what? Ans. (2Rx*y, or J (2Rx*). 2. x(2ifo)^:=:what? Ans. (2Rx*)$, or V(2ifa 4 ). This transformation is particularly useful in finding an approximate root of a number. Thus, 7y5 = 7X2 (the nearest unit) = 14. But 7^5 = 7*5*= (7*.5)*= (49.5)*= (245)^ = 16 (the nearest unit). Note. In extracting the root of 5 and multiplying by 7, we mul- tiply the error in the root by 7. In the other process, we avoid this source of inaccuracy. Remark. In some, especially of the earlier treatises, the radical sign is used almost to the exclusion of fractional exponents. The exponent, however, is much more convenient, and many of the diffi- culties connected with the calculus of radicals, as it is called, dis- appear, when the exponent takes the place of the sign. Hence, if it is intended to u«e the radical sign in expressing the result, it is still generally best to employ the exponent in the operations by which the result is obtained. IMAGINARY QUANTITIES. § 162. The expression */ — 1 may be taken as the repre- sentative of all imaginary quantities. The treatment of imaginary quantities will be best illustrated by considering soine of the powers of J — 1. Thus, (y-i) 2 = (-i)^.(-i)^=-i. (y-l)3 = (-l)^-_l)* = ^l(-l)* = -^-l. k/-l)*=(-l)*=(-l)» = l. U-l)o-^_l. (y_l)G-_i ; (y_l)7__ v/ _ 1; (^-1)8 = 1; &c. Hence, (,/— a 2 ) 2 = (aj—l)° = a 8 X— 1 = —a 2 . y—a° y— 5 3 = ay— lx V— 1 = abX— 1 = — ab. §163, 164.] POLYNOMIALS. BINOMIALS. 131 Notes. (1.) Caro must be taken not to confound imaginary with irrational expressions. A numerical surd, as^/2, cannot be exactly expressed in units or parts of a unit, but we may approximate as near as wo please to its true value. An imaginary expression, on the other hand, as ^/ — 1, has no actual value, and we can, of course, make no approach to its value; nor can one quantity be said to come any nearer to its true value than another. Thus, no quan- tity can be conceived, which, multiplied into itself, will produce — 1; and the expression / — 1 is merely a symbol of an impossible opera- tion; a symbol, to which there exists no corresponding quantity. (2.) It may be thought, that such symbols, not representing quanti- ty, can be of no utility, and should have no place in investigation? relating to quantity. But some of the most remarkable and useful results of algebraic reasoning depend upon the presence of imaginary symbols. (3.) An imaginary result generally indicates, that we have, in some way, introduced inconsistent conditions into our inves- tigation; and demonstrates the impossibility of finding, under the cir- cumstances, such a result as we, at first, proposed to find. POLYNOMIALS. v 163. "We shall consider here only the positive integral powers, and simple roots of polynomials. It is evident, moreover, that if we can find such powers and roots of a polynomial, we can find all powers. For the formation of the power denoted by the numerator, and the extraction of the root denoted by the denominator will give any positive fractional power; and the proper combination of those processes with division will give all negative powers. § 164. The most obvious method of finding a positive in- tegral power of any quantity is by continued multiplication of the quantity by itself; taking it as a factor as many times as there are units in the exponent of the power. Thus we have already found (§ 89) (a+x) 2 = (a+x)(a-\-x) = a 2 +2ax-f-x 2 . So (a-f-x) 3 = {a-\-x){a-\-x){a-\-x) — a 3 +3a 2 x+ 3ax*+x*. (a+x) * = (a+ar) 3 (a+x) == a*+4a 3 a:-f-6a 2 a: 2 +4a2; 3 -|-a; 4 . (a+x) 5 = a5+5a*x+10a 3 2; 2 -f 10a 2 z 3 -f 5ax*+ x°. 132 POWERS AND ROOTS. |J 165-168. § 165. We find that, in these instances, (1.) the first term of each power of the binomial, a-\-x, is that power of the first term of the binomial; (2.) that the exponents of the first or leading quantity, a, diminish, and those of x increase by unity in the successive terms ; (3.) that the exponent of a in the last term is zero, and that of x is the exponent of the recpiired power ; (4.) that the numerical coefficient of the second term is the same as the exponent of the recpiired power ; and (5.) that the numerical coefficients at equal dis- tances from the two extremities of the series are equal. Note. It will be shown hereafter, that these principles apply to all positive integral powers of a binomial, and that all but the third and fifth apply to every power of a binomial, whether the exponent be positive or negative, integral or fractional. §166. "We have enunciated these principles as proved only so far as we have found them true by actual multipli- cation. Let us suppose, that we have found the law of the first and second terms, given above (§ 165. 1, 4), to be true to the nth. power. See § 95. N. 1. Then we have (a-\-x) n = a n -\-7ia n ~ l x-\-&c. Multiplying by a-\-x, («4-;r) n +i — a n +i-\-(n+l)a n x-\-&c. If then the principles 1, and 4 of § 165 are true for the nth power, they are true for the n-\-l power, and so on, without limit, n being any positive integer whatever. § 167. If we substitute, in the above expressions (§ 164), — x for -\-x, we shall, evidently, obtain the powers of a — x. This substitution will, obviously, cause all the terms con- taining the odd powers of x to become negative, and will occasion no other change. Thus, (a—x) - — a 2 —2ax+x*. See § 90. (a— x y = a 3 — 3a 2 a:-f-3aa; 2 — z 3 . (a—x)* = a 4 — 4a 3 a:-{-6a 5 a: 2 — 4aa; 3 -fx 4 . $ 1 68. (a-f-z) 2 = a 2 -j-2rtcc-Hc 9 . Substitute b-\-y for x $ 169, 170.] squart: root of a polynomial. 133 Then («+H-3/) 2 = « 2 + 2 «(H^)+(Hiy) 2 . . (1) or (a+H-y) 2 = («+£) 2 +2(«+%-h$/ 2 • • • (2) Developing, (a+b+y) 2 = « 2 +2a5+& 2 +2ay+2fy+2/ 2 . That is, The square of the sum of three members is equal to the sum of their squares, plus twice the sum of their pro- ducts, taken two and two. Note. By increasing the number of terms, we might find similar expressions for the square of any polynomial. Thus, (a+b+c+z) 2 = (a+b+c)2+2(a+b+c)z+z*. Hence, The square of any polynomial is equal to the sum of the squares of the terms, plus twice the sum of their products, taken two and two. §169. (a+x) 3 = a 3 +3a 2 a;+3ax2+a: 3 . Substitute b-\-y for x. Then («+Hi/) 3 = « 3 +3« 2 (H^)+3«(H^) 2 +(H^) 3 • W or(a+6-f-y)3 = (a+5)3-L.3(a+5)2 y+ 3(«+% 2 4 1/ 3 . (2 ) .-. (a-L.J-fy)3 =a 3 -{-3a 2 b-{-3ab 2 -\-b 3 +3a 2 y-\-Qaby+3bSy +3ay 2 +3fy 2 +#3 = a s+b*+yS+Za* (b+y)+3b* (a+y) +3y*(a+b)+6aby. That is, The cube of a trinomial is equal to the sum of the cubes of the terms, plus three times the square of each term into the sum of the other two, plus six times the product of the three terms. Notes. (1.) We might find, in like manner, expressions for the higher powers of a trinomial. (2.) If one of the terms of the tri- nomial becomes zero, the formulae of §§ 168, 169 give the square and •ube of a binomial. SQUARE ROOT OF A POLYNOMIAL. § 170. Find the square root of a 2 +2ab-\-b 2 . a.) The polynomial being arranged according to the des- cending powers of a, we know, that a 2 must be the square ©f one term of the root (§§ 73. 1 ; 82. a, b). b.) We know, moreover, that the polynomial containa, ALG. 12 134 POWERS AND ROOTS, {^§ 171. besides the square of the first term, twice the product of the first term by the second (§ 168), and so on. If, therefore, we divide the next term of the arranged polynomial by 2a, we shall find another term of the root. c.) If now we subtract from the given polynomial the square of the terms of the root already found, the remain- der, if there be one, will contain the terms which resulted from the multiplication of the remaining terms of the root by each other, and by the terms already found (§ 168. 2, N.). d.) We may, therefore, find another term of the root, by dividing the first term of the arranged remainder by twice the first term of the root ; and so on (§ 82. c). Thus, a 2 +2a5+& 3 a* 2ab+b°- 2ab+b 2 a+b 2a+b Notes. (1.) It will be seen, that we have subtracted the square of the two terms of the root found (§ 170. c). For., (a+b)^ = a^+2ab+b- =za.2+(2a+b)b. Now we subtracted a 2 at first, and afterwards subtracted (2a+b)b. (2.) Also, after each subtraction, we shall have subtracted the square of the whole root then found (§171. Ex. 1, a). (3.) As there is no remainder, there can be no other terms in the root. And whenever we find a remainder equal to zero, the work it completed (§82. g), and the given polynomial may be said to be a perfect power. (4.) If, however, after exhausting the given terms of the polyno- mial, we still have a remainder, the root cannot be exactly found by this process. (5.) We may, however, continue the process, and develop the root in an infinite series, as in division (§87). From the reasoning above, we deduce the following RULE. § 171. 1. Arrange the polynomial according to the powers of some letter. 2. Extract the root of the first term for the first § 172.] SQUARE ROOT OV A POLYNOMIAL. loO term of the required root; and subtract its square from the given polynomial. 3. Double the part of the root already found, for a partial divisor ; and divide the first term of the re- mainder by the first term of the doubled root; setting the quotient, with its proper sign, as a term both of the root and of the divisor. 4. Multiply the divisor thus completed by the new term of the root, and subtract the product. Continue the process as long as the case may require. 1. (Ox*— 12x; 3 +16a:2— 8x-f-4)^ = what? 9z 4 — 12z 3 -j-l&c 2 — 8a>f4 | 3;r 2 — 2a:+ 2 9a:* — 12a: 3 j 6a: 2 — 2 x — 12or 3 4-4a; g 12a; 2 j 6z 3 — 4x-f-2 12a; 2 — 8*4-4 a.) We must be careful, at each step, to double the whole of the root already found, for a divisor. For (a+i+c) 2 =z (a+b)*-\-2{a+b)c+c 2 . § 168. 2. Also, (a-f-S+c-fa-) 2 = {a+b+ c) 2 +2{a+b+c)x+x 2 ; and eo on. § 168. N. 2. What is the square root of a*-f-4a 3 J-f-6a 2 5 2 +4ai' v +b* ? Am. a*-\-2ab+b*. 3. (16x*+24a; 3 4-89a; 2 +60a;+100) * = what ? 4. (a± 2a?$+b) * = what ? -4»s. a 2 ±5 2 , or c 2 ±2c5 2 -|-5=ry. X y — c 2 b b 2 = ±£ — (§ 42. o, c?) ; that is, an irrational, equal to a rational quantity, which is absurd ($ 175). See Geom. § 23. Hence, Two binomials, consisting each of a rational and of an irrational term, cannot be equal, unless the rational terms are equal to each other, and also the irrational. § 184. Let (a-\-b*y=.x *-\-y , one or both of the quan- x i tities x' J and y' 2 being irrational, and x and y monomial. Then, squaring, a+5* = x+2x 2 y 2 -\-y ; or a+^b = x +2^(xy)+y. a = x+y, and b 2 = 2x^y 2 (§ 183). Hence, subtracting, a— b 2 = x —2x 2 y^-\-y = (x^—y^) 2 . (a— b 2 )^ = x^—y 2 (§ 52. N.). That is, If +/{a-\-jV) zrz^x-^y/y, then +/(a — +/b) = +/x — +/y. Thus, (3+5 2 ) 2 = 9+6x5^+5 = 14+6X5 2"; and (3— 5 2 f = 9— Gx5*+5 = 14—6x5^. .-. y (14+6^5) = 3+^5, and y(14— 6^/5) = 3—^/5. (2^±3 2 ) 2 = 2±2x2i3*+3 = 5*2x2^.3^. ^/(5±2y (2X3)) = y2±y3. Note, x and y being monomials, the squares of Jx and^/y must be rational, and will, of course, combine by addition, into a tingle rational term a; while their double product, being equal to 5 ^-J SQUARE ROOT OF a±b*. /b (§ 181), is irrational, and will be positive or negative according as Jx and Jy have the same or different signs. § 185. Now assume (a-\-$)% = x^+y^ (1) ; then (<*-&*)*= x^-y^ (2). § 184. Squaring (1) and (2), a-\-b^ — x-\-2x^y 2 -\-y, and a— $=x— 2x^-\-y. Adding, and dividing by 2, we have a = x+y{3). Again, multiplying together (1) and (2), we have (a 2 — b)^ = x—y{4).* §92. Hence, from (3) and (4), ,. ti= ( fi±±V)\ and ,i = (fc^zS*)* Hence, substituting in (1) and (2), ( ^i =( f±rf ) i +( f=(^ ) 4 Or, putting (a 2 — b) 2 = c, we have (a+J l)l=(2f)i + (2=?)* or y (o+y*) = y-f- +^^- : (a+62)i( a -6l)l = [( a +&2-)( a -&2)]£ [§ 151. a ] -6)2 [§92]. ALG. IS 146 POWERS AND ROOTS. [§ I («- h -)--\rY) -vt; ' a-\-c a — c or */(«— x/o) = y— ^~2~ ' Note. These expressions will, evidentty, not reduce to a con- venient form, unless (a 2 — b)'$ is rational, i. e. unless a 2 — b is a perfect square. The above results may be verified by squaring. Thus, J f - C ^ = «±(« 2 — c 2 ) 2 " = «±(a 3 — (a 2 — J))^ = «±^. 1. (3-f-2.y2)* = what? Here a = 3, and &* = 2(2)*= (2 2 .2)* = 8* (§ 161). c = (a 2 — ft)* = (9— 8)* = 1* = 1. ••■ C4 c ) i +(?) i =(^ i ) i +( 2 i i ) i = 2 -+ 1 - Wc may verify this result by squaring 2 2 -j-l. Thus (2M-1) 2 = 2+2(2)^4-1 = 3+2^2. (9±4X5*)* = what ? Ans. 2±oK 3. (7±2XlO-)-=what? Ans. 5-±2 J . (£±(f)"*)* = what? Ans.^±ne of the terms of a fraction, or one of the members of an equation, of irrational expressions. Thus, let it be required 2 73 to reduce j— j- — - to a fraction having a rational denomina- tor. We have (W3) X ^~^ - 191-194.] EQUATIONS OF THE SECOND DEGREE. § 191. /.) Again (§ 98), a+b is divisible by « 2 ' i+1 -[-//" Thus, 1 I 2n 2»— 1 ] (a+o)-T-(a +6 )=a — a o 4~ . . . 1 o n — 1 2«+l 7 2-1+1 , 7 12" 2 " (« -j-6 ) (a — -\-b ) — a-\-b. Thus, {J+$)(a?— a^4-#) = «4-o- So, (7*4-4*) (7^— 7*4*4-4?) = 7-}-4 = 11. CHAPTER VII. EQUATIONS OF THE SECOND DEGREE. S 192. We shall, at present, consider only equatio which the exponents of the unknown quantities art - gral. With this limitation, an equation is of the se< degree, when the difference between the highest a the lowest degrees of its terms with respect to \ known quantity or quantities (§28. b) is two (§40. § 193. An equation containing but one unknoivn quantity is, therefore, of the second degree, when difference between the greatest and least exponent the unknown quantity is two. §194. Notes. (1.) We shall, at present, confine our atl to equations containing but one unknown quantity; and shall sup c •18 150 EQUATIONS OF THE SECOND DEGREE. [§ 195-197, them to be arranged according to its descending powers (§ 33), and to be reduced to the simplest form in respect to each of those pow- ers (§34. c). (2.) Then, each power of the unknown quantity, to- gether with its coefficient (whether monomial or polynomial), will constitute a term of the equation. Thus, Let x 2 +2ax+6 2 — mx 2 — 4x = 7ix+r — q — 3x 2 . Then (1+3— m)x 2 +(2a— 4— n)x+bz+q—r = 0. §§ 33, 34. c, 44. Or, making .# = 1+3— m, B = 2a— 4— n, and C = b 2 +q— r, Ax^+Bx+C~=0. § 195. An equation of the second degree, containing but one unknown quantity, its powers being all integral, may contain any three consecutive powers, and no more. For, if there were more than three consecutive powers, or if there were three powers not consecutive, the differ- ence between the greatest and least exponent must be mor than two. Thus, Ax3-{-Bx*-\-Cx = 0, Ax n --{-Bx+C=0, Ax+B+ Gc- 1 ( = Ax+Bx°+Cxr *) = 0, and Ax~ r -\-Bx---\- Gx~ 3 — are all of the second degree. § 196. Hence an equation of the second degree, when reduced as above (§ 194), can consist of only three terms (§ 194. 2) ; anil therefore, an equation of the second degree. consisting of three terms, is called a complete equation. §197. Let Ax 3 -\-Bx*+Cx = 0. Dividing by x, Ax*-\-Be+ G— 0. Again let Ax+B+ Cher » = 0. Dividing by x~ y , or multiplying by x, Ax*+Bx-\-C=0. Or, again, let Aar+Ba*- 1 -f Car- 2 = 0. Dividing by a;"- 2 , Ax s -\-Bx-\-C= 0. Hence, Every complete equation of the second degree, containing only one unknown quantity, can be reduced to the form Ax°-+Bx-\-C=0, in which the coefficients, A, B and G, may be either post- $198,199.] INCOMPLETE EQUATIONS. 151 five or negative, integral or fractional, numerical or alge- braical, monomial or polynomial. Reduce the following equations to the above form. 1. ax 2 -{- bx-\-c— (mx 2 -\-nx— p) = 5x 2 — 8#-|-7. 2. a 2 -f-2ar cos i>-fr 2 cos 2 r+& 2 +25r sin v+r 3 sin- u = i2' 3 ; r being the unknown quantity. Sx— 3 __ 9 .3x— 6 a? — o / § 198. As the coefficients may have any value whatever, they may be equal to zero. But if the coefficient of a term becomes zero, the term itself becomes zero, and disappears from the equation. The equation is then sometimes called incomplete. Notes. (1.) If all the coefficients become zero at once, the equa- tion will, of course, disappear. Also, if A and B become zero, we shall have C=0, and the equation will be annihilated. But, if A and C become zero, we shall have Bx = 0, and x = 0. Again, if B and C become zero, we shall have Ax% r=: 0, and x = ±0. (2. ) Again, let A = 0. Then we shall have Bx+C = 0. Now this is no longer of the second degree. It is of the first degree, and must be treated accordingly (§4S). Neither of the above supposi- tions needs any further consideration. §199. Now let B — Q. Then the equation becomes Ax 2 +C=0. C C\ x 2 — — j-=q 2 (putting q 2 = — — J. x=(—^)*=.(g')*=±q. See §52. N. In this case, we find the values of the unknown quantity by reducing the equation to the form x 2 = q~, and extract- ing the square root of both sides. Thus, let x 2 — 49. Then x = ^49 = ±7. Note. The term, incomplete equations of the second degree, i* sometimes applied exclusively to equations of this form. They are also sometimes styled pure equations of the second degree, or pure quadratics (§41. N.). 152 KQUATIONS OF THE SECOND DEGREE. [§ 200. a.) This form of equation will, evidently have two roo (' 39) numerically the same, hut with opposite signs (> 23. /• 1). b.) Also, if x n - = q-, then x-—q* = 0. (x+q) (x—q) = 0. § 93. 111. Now it is evident, that a product will become zero, only when one of its factors is zero. The last equation, there- fore, will be true, when either of its factors is equal to zero 1 in no other case. Hence we may have, either oc-\-q = 0, or x — q = ; and, in either case, we shall have the product (x+q)(x— q) — x 2 —q* = 0. But, if x ~\-q — 0, then x = — q, I if x — q = 0, then x = -\-q. So a; 2 — 49 — gives (x-\-7)(x— 7) = 0. Whence, #-J-7 = 0, and a? = — 7 ; or a; — 7 = 0, and a; = +7. Either of these values of a; will satisfy the equation, and is consequently a root of the equation (§ 39). c.) If the equation, x- r= q 2 , or x- — q- = 0, be pu< un- der the complete form, thus, x 2 +0a:— «7 2 = 0, we shall have -\-q — q = 0, the coefficient of x ' ; and (~H?)( — ?) — — l"i tue coefficient of So, in the equation, a? 2 -|-0a; — 49 — 0, Ave have +7-7 = 0; (+7)(-7) = -49. . 200. tZ.) We find here certain results, which will here- ter be shown to hold of all equations of the second de- ■ee, when placed under the form, x 2 ±2px±q 2 = 0, viz. 1. The equation can be resolved into two binomial s; of which the first term of each is the unknown quan- tity, and the second term, with its sign changed, is a root the equation. § 201, 202.J INCOMPLETE EQUATIONS. — PROBLEMS. l5u 2. The equation has two roots. 3. The algebraic sum of the roots, with their sign? changed, is equal to the coefficient of x 1 . 4. The product of the roots is equal to the coefficient of x°. Note. The student should illustrate and test these principles by applying them to the roots of every equation which he solves. § 201. e.) If the equation be of the form x 2 -{-q 2 = 0, we shall have x 2 = — q 2 , and, consequently, x =■ ±^/ — q 2 = ±q+/ — 1, an imaginary result (§§ 23./. 2, 158). Thus, let z2_|_49 — q. Then x 2 = —49 ; .-. x = y— 49 = ±7^—1. Notes. (1). These expressions do not indeed represent any ac- tual value, but they are called roots of the equation, because, when substituted for x, they satisfy the equation (§39). (2.) This imag- inary result indicates an absurdity in the conditions of the problem. It is here proposed to find a number, whose square added to another square shall be equal to zero. That is, the sum of two positive (§11. N. 2) quantities is required to be zero, which is, evidently, impossi- ble. See § 162. N. 3. /.) The results, xz=z-\-q^/ — 1, and x — —qj — 1 give x— qj— 1 = 0, and x+qj—l = 0; § 199. b. and .-. (x— qj— 1) (x-\-qJ— 1) = x 2 -\-q 2 = 0. § 200. So (a:— 7 y— 1 ) (x+7y— 1 ) = x 2 +49 = 0. §202. 1. Given 5(x 2 — 12) = (a: 2 +4), to find x. Ans. x=-±A, . _,. a; 2 — 50 , x 2 — 25 . „ , 2. Given — \-x = \-x, to find x. z o Ans. x=. ±10. 3. In a right angled triangle, the square of the hypot- enuse, or side opposite the right angle, is equal to the sum of the squares of the other two sides (Geom. § 188). If then the base is 4 feet, and the perpendicular 3 feet, what is the hypotenuse ? Let x = the hypotenuse. Then x 2 = 3 2 +4 2 , &c. 154 EQUATIONS OP THE SECOND DEGREE. [§ 202. 4. A rope 50 feet long is extended from the top of a flag staff 40 feet high, in a straight line to the ground on the east of the flag staff, and on a level with its foot. How far from the foot of the staff will it strike the ground ? Ans. ±30 feet (§5). .">. How far, if the rope be 45 feet long? Ans. ±20.615 &c. feet. 6. How far, if the rope be 40 feet long? Ans. ±0 (i. e. it will strike the ground a1 the foot of the staff). 7. How far, if the rope be 32 feet long? Ans. ±y— 576 = ±24y— 1. In this case, the rope, evidently, will not reach the ground; so thai there is manifest absurdity in inquiring how far from the foot of the «taft" it will strike the ground. This absurdity is indicated by the im- aginary result (§201. N. 2). 8. Let the perpendicular drawn from any point of the circumference of a circle to the horizontal diameter be represented by y ; and let the distance from the foot of the perpendicular to tbe centre, measured on the horizontal diameter, be denoted by x ; and the radius of the circle, by B. Then we shall have, for every point of the circumfer- ence, x 2 -\-y 2 = R 2 ; ovy 2 =R 2 —x 2 . §202.3. Or, if the radius be 10 feet, we shall have R- = 100, and y* = 100— a; 2 . What now is the length of y, when x = ? Ans. y=z-\- 10, or —10 (§ 5). 9. "What is the length of y, when x = ±l, 2, 3, 4, 5, 6. 7, 8, 9, 10 ? 10. What is the length of y, when x = ±11 ? . his. y = */ — 21. In this case, the distance measured on the horizontal diameter from the centre, being greater than the radius, extends beyond the circumference; and, of course, no perpendicular to that line at it» extremity can meet the circumference. Hence the imaginary result, indicating an absurdity (§201. N. 2). 203.] INCOMPLETE EQUATIONS. L55 9. It is required to lay out 10 acres of land in a square. What must be the length of one side ? 10. The product of two numbers is P, and the quotient of the greater by the less is Q. What are the numbers ? P Let x zr the greater ; then — = the less ; &c. ° x 03. Again, resuming the complete equation, Ax 2 -{-Bx+0=Q, if we suppose C=0, we shall have Ax^-\-Bx = 0. Dividing by x (§ 51), we have an equation of the first de- gree (§51. b), 7) Ax-\-B =. ; and .\ x — — . a.) If, however, we divide by A, we shall have 7? B \ x 2 -\- —x = 0, or x 2 -{-2px =■ (putting 2p z=z — J . Separating the last expression into factors, we have x(x-\-2p)'==:Q ; an equation, which will be satisfied, either when x=z0, or when x-\-2p = 0; i. e. when x=-0, or when x=z — 2p. The roots, therefore, of this equation regarded as of the second degree, are 0, and — 2p (§ 200. 2). b.) In this case also, the sum of the roots with their signs changed is equal to the coefficient of x 1 , and their product, to the coefficient of a; (§ 200. 3, 4). The two bi- nomial factors (§ 200. 1) are x — and x-\-2p. Note. This form of equa/ion is frequently classed and solved a* a complete equation of the second degree (§ 206). 1. Given 2Rx — x 2 = 0, to find x. .Ins. xz=0, or 2R. 2. Given r- — 2B cos v r = 0, to find r. Ans. r = 0, or 2R cos '.'. 3. Given x- — lOx =. 0, to find x. 156 EQUATIONS OF THE SECOND DEGREE. [§ 204, 205. § 204. Returning now to the complete equation, JB C and dividing by A, we have x 2 -\~ —x-\- — — ; B C or, putting 2p = — , and q 2 = — , x*-\-2px-\-q 2 = 0. a.) This is a complete equation of the second degree (§196); and it is perfectly general, since every complete equation can be reduced to this form by dividing by the coefficient of a: 2 , and substituting convenient symbols for the coefficients of a; 1 , and x°. b.) This is also the form, to which the principles of § 200 apply, and will, therefore, be commonly employed in our future discussion of the subject. § 205. In solving the complete equation, x 2 -\-2px-\-q 2 = 0, we may happen to have q =p. In this case, the equation becomes x 2 -\-2px-]-p 2 = 0, or (§ 93. L), (x+P) (x+p) = 0. We have here the equation resolved into two binomial factors (§ 200. 1), either of which may be equal to zero. But in this case, the factors are equal ; and, consequently, the values of x, found from them, will be equal. The equation is said, in this case, to have equal roots, viz. —p and —p. Thus, let » 2 +20ar+100 = 0. Then (x-f-10)(ar+10) = 0, and x — —10, or —10. If we had x 2 — 20x-4-100 = 0, we should have x = +10, or -[-10. a.) The sum of the roots, with their signs changed is still equal to the coefficient of x 1 , and their product, to the co- efficient of x°. § 206, 207.] COMPLETE EQUATIONS. 157 § 206. But suppose that q is not equal to p ; i. e. that q 2 , the coefficient of a; , is not equal to p 2 , the square of half the coefficient of a: 1 . Then the equation, x 2 -\-2px-\-q 2 = 0, gives x ~-\-2px = — q 2 . If now we add p 2 to hoth sides of the equation, the first member will, evidently, become a trinomial perfect square (§§ 89, 172), and we shall have x 2 -{-2px-\-p 2 =p 2 — q 2 . x-\-p = J(p 2 —q 2 ); §52. N. and x = —p-\-*/(2* 2 — 9 2 )' or x = ^p — */{p 2 — ? 2 )- Thus, let x 2 +8:c+15 = 0. Then cc 2 +8;c = — 15. Adding 4 2 (=j» 2 ), x *-+8x+l6 = — 15+16 = 1, Extracting the root, a+4 = ±1. x = — 4±1 = — 3, or — 5. x+3 = 0, or x-\-o = (§ 199. b) ; and (a+3)0+5) = a: 2 +8.z+15 = (§§ 200, 208. b). Also (—3) 2 +8(— 3)+15 = 9—24+15 = ; and (— 5) 2 +8(—5)+l 5 = 25—40+15 = 0. The process of rendering the first member a perfect square, is commonly called completing the square. Hence we have, for solving a complete equation of the second degree, containing but one unknown quantity, the following RULE. § 207. 1. Reduce the equal ion to the form x 2 ±2px ±q- — 0. Transpose the coefficient of x° to the sec- ond member, and add the square of half the coefficient ofx 1 to both sides. 2. Extract the square root of both members, ana solve the equation of the first degree thus obtained. ALG. 14 158 EQUATIONS OF THE SECOND DEGREE. [§ 20B'.- 1. Given x-+ix— 60 = 0, to find x. 2. Given x-— Oa+lO = 65, to find x. 3. Given 3a: 2 — 3a:+9 = 8£, to find x. _4hs. a; = §, or J* 4. ^2_ia;+30^ = 52i, to find a;. Ans. x = 7, or — 6 -*-. § 208. a.) The same effect, obviously, will be produced, if, without transposing the coefficient of x°, or the absolute term as it is sometimes called, we add to both sides a quan- tity, tvhich together with that coefficient shall be equal to the square of half the coefficient ofx 1 , (i. e. p 2 — q 2 ). Thus, x 2+2px+q 2 +(p 2 -q 2 ) = p 2 -q* ; or x 2 -\-2px-\-p 2 =p 2 — q 2 . x+p = ±^/(p 2 —q 2 ). §52. N. x = — p++/(2^— ? 2 )> or x = — p— «/(p 2 — q*). b.) These values (§§ 206, 208) give the equations x+p—(p 2 —q 2 ) 2 =-Q, and x+2>Mp-— ? 2 ) 2 =- 0. [x+p-ljjS-q^lx+ji+iP*-? 2 )^ = (a+p) 2 — [{p-— ? 2 ) 2 ] 2 (§ 92) = xJ+2px+p 2 —p 2 +q 2 = x 2 -\-2px+q 2 = 0. § 200. 1 . Also p-(p 2 -q"f+p+lP°—V 2 f=2p-> §200.3. and [- p—{p n — <7 2 ) 2 ][— p+(j* 2 — 7 2 )"] — H) = a; 2 +6aH-8 = 0. §200.1. Also 2+4 = 6 = 2;?, and 2X4 = 8 = q*. §200.3,4. 2. Given x 2 — <6x— 40 = 0, to find a\ Ans. x = 10, or — 4. Here ^2— $ 2 = 9_(_40) =49. § 209.] COMPLETE EQUATIONS. 159 3. Given x- — 16a;-f-63 = 0, to find x. 4. Given ar 2 +16x+63 = 0, to find x. § 209. c.) Resume the equation Ax 2 -\-Bx-\-C=0 ; or Ax 2 -\-Bx = — C. -Multiplying by A, A 2 x 2 -\-ABx = —A C. Adding (%B) 2 , A 2 x 2 +ABx+\B 2 = \B 2 — A C. Extracting the root, Ax+\B = (IB 2 — A 0)K Hence, to complete the square, Reduce the equation to the form Ax 2 -\-Bx-{-C= ; trans- pose the coefficient of x° ; multiply by the coefficient of x 2 ; ■and then add to both sides the square of half the primitive coefficient ofx 1 . 1. Given 5x 2 +4x— 204 = 0, to find x. 5x 2 -\-lx=2Qi. 25x 2 +20o:4-4=:1024. ox+2 = ±32 ; and .-. 5x =± 30, or —34. x = 6, or — 6f . 2. Given 2x 2 -\-8x— 90 = 0, to find x. Ans. x = o, or — 9. d.) Or, Ax 2 +Bx+C=z0. Multiplying by A, A 2 x 2 +ABx+A C— 0. Adding ±B 2 —AC, A 2 x 2 +ABx+\B 2 — \B 2 —AC. Hence, to complete the square, Multiply the equation, Ax 2 -\-Bx-\- O=0, by A, and add ■to both sides \B 2 —AQ. Given 3a: 2 -f-2x— 85 = 0, to find x. 9x 2 +6z— 255 = 0. Here \B 2 —A C= 1— (-255) = 256. 9x 2 +6x+l = 256. .-. 3as-|-l = ±16; and.-. 3x = — 1±16 = 15, or— 17. 2 x ■=. 5, or — 5^ Note. When ^ = l, this solution (§209. c, d) is, evidently, the same as that of §§ 207, 208. 160 EQUATIONS OP THE SECOND DEGREE. [§ 210. §210. e.) Again, Ax*+Bx-\-C=0; or Ax 2 -\-Bx = — 01 Multiplying by A A, 4A*x 2 +4ABx = —4A C. Adding B*, 4A 2 x 2 -\-4ABx-\-B 2 = B^—iAC. 2Ax-{-B — (B »—4A C ) *. _ — B±^(B 2 —4AC) .*. x — — — . 2A Hence, to complete the square, Reduce the equation to the form Ax 2 -\-Bx-\-Cz=i ; trans- pose the coefficient of x° ; multiply by four times the coeffic- ient of x 2 ; and add to both sides the square of the primitive coefficient of x 1 . 1. Given 3a;2— 3a+§ = 0, to find x. 36a; 2 — 36a; = — 8. 36*2— 36a+9 = —8+9 == 1. 6a;— 3 = ±1 ; .-. Gx = 3±1 = 4, or 2. x = | , or £. 2. Given \x 2 — : \x— 22£ == 0, to find x. Ans. x = 7, or — 6J» /.) Or, multiply the equation, Ax 2 -\-Bx-\- (7=0, by 4A Then 4A 2 x 2 +4ABx-{-4AC=Q. Adding B 2 —4AG, 4A 2 x 2 -\-4ABx-{-B 2 = B*—4A C ; as in e above. Hence, to complete the square, Multiply the equation, Ax 2 +#£+(7=0, by 4A ; and add to both sides B' 2 —4AC. Given x 2 — 5x — 24 = 0, to find x. 4x 2 — 20a;— 96 = 0. Here B 2 — 4A C = 25— (—96) = 25+96 = 121, 4a; 2 — 20a+25 = 121. 2a:— 5 = ±11; .-.a; = 8, or — 3. Note. When A — 1, this solution (§210. e,f) is, obviously, the same as that of §§ 207, 208. §211-213.] GENERAL DISCUSSION. 161 § 211. Let x 2 -\-2px-{-q 2 = be any equation whatever of the second degree, containing but one unknown quanti- ty ; let also a be one of its roots (i. e. such a quantity as, being substituted for x in the given equation, will make the members equal ; or, in other words, will reduce the first member to zero). See § 39. 1. Since a is a root of the equation, we have x = a, and x — a — 0. Divide the given equation by x — a. Thus x 2 -\-2px-\-q< x* 2 — ax x — a x-\-{a-\-2p) (a-\-2j))x (a-{-2p)x — a 2 — 2pa a 2 -\-2pa-\-q 2 = 0, because the remainder is simply the first member of the given equa- tion with a substituted for x ; which, by hypothesis, redu- ces it to zero. The division is therefore perfect (§ 82. g). Hence (x—a) (x+a+2p) = x 2 -\-2px-\- q 2 = 0. §200.1. 2. And the equation will be satisfied, if we take x — a = 0, or x-\-2p-\-a =: (i. e. if x — a, or x = — 2p — a = b (by substitution). § 200. 2. 3. We have also — a-\-(2p-\-a) =z 2p. § 200. 3. 4. Moreover, since a 2 -\-2pa-\-q 2 = 0, (see 1, above), a(— a— 2p)[ = —a 2 —2pa'] = q 2 . § 200. -J. § 212. 5. It is also evident from § 211. 1, that, if a is a root of the equation x 2 -\-2px-\-q 2 =. 0, this equation is di- visible by x — a, and will give a quotient of the form x — l>, of which the second term is the other root with its sign changed. § 213. Hence, universally (§ 200), 1. Every equation of the second degree, of the form x 2 ±2])x±q 2 = 0, containing but one unknown quantity, can be resolved into two binomial factors, of the first degK e *14 162 EQUATIONS OF THE SECOND DEGREE. [§ 214-216. in respect to x (§ 28. b) ; either of which, being put equal to zero, gives a root of the equation. 2. Every such equation has, of course, two roots. 3. The algebraic sum of the roots, with their signs changed, is always equal to the coefficient of a; 1 . 4. The product of the roots is always equal to the coef- ficient ofx°. 5. Every such equation, of which a \s a root, is divisible by x — a. § 214. a.) Hence (§ 213. 3, 4), Cor. I. (1.). If the coefficient of x 1 be equal to zero, the roots must be numerically the same, but with opposite signs (§ 199. a). (2.) If the coefficient of x° be equal to zero, one of the roots must be zero (§ 203. a). §215. b.) Also (§§213. 4; 9. a; 213.3), Cor. ii. (1.) If the coefficient of x° be positive, the roots must have like signs ; (2.) if negative, unlike. (3.) If the two roots have the same sign, it will be unlike the sign of the coefficient of x 1 . (4)7/' they have different signs, the sign of the root which is numerically the greater will be unlike that of the coefficient of x 1 . c.) It is obvious, that, if the roots have like signs, the co- efficient of x 1 will be numerically equal to their arithmeti- cal sum ; and, if they have unlike signs, to their arithmeti- cal difference. § 21 6. d.) If q- be positive and greater than p 2 , the pro- duct of two numbers is required to be greater than the square of half their sum. This will be shown to be impos- sible ({ 220. b), and as no real numbers can satisfy thia condition, the roots will be imaginary (§§ 201 ; 217. I.). Hence, Cor. in. If the coefficient of x° be positive and greater than the square of half the coefficient of x 1 , the roots must be imaginary. § 217, 218.] SIGNS OP THE ROOTS. 163 § 217. e.) The above principles may be otherwise de- monstrated ; thus, I. Let q 2 he, positive. Then x 2 ±2px-{-q 2 = ; and x — ^p±^{p 2 ^q 2 ). Now, evidently, *J(p 2 — ? 2 )p 2 , «/(p 2 — q 2 is imagina- ry(§§28./.2; 216.). It is also manifest, that, if one of the roots is imaginary, both must be. II. Again, let q 2 be negative Then x 2 ±2px — q 2 = ; and x = ^p±«/(p 2 -\-q 2 ). Here, obviously, .y(.P 2 4~!7 2 )>/> 5 and, therefore, one root must be of the same sign as p ; and the other, different (§215. 2). Also, the root which is of the same sign as p (i. e. of a sign different from 2/> on the other side}, will, of course, be numerically the greater (§ 215. 4). § 218. f.) Determine whether the signs of the roots in the following equations are like or unlike ; if like, whether positive or negative ; and, if unlike, which is numerically the greater. Also determine whether any of these equa- tions have imaginary roots. 1. x 2 +21x+110 = 0; x 2 — 20+75 = 0. 2. x 2 — 23x4-130 = ; x 2 4-23x-fl30 = 0. 3. x 2 ±60x-|-1000 = 0; x 2 ±60x— 1000 = 0. 4. x 2 ±G0x— 11200 = 0; x 2 ±10x = 200. g.) 1. Write the equation, of which 3 and 4 are the roots. Ans. (x— 3) 0—4) = x 2 — 7x4-12 = 0. 2. Write the equation, whose roots are — 3 and — 4; —11 and 4-20; -4-11 and— 20; —10 and -4-10 ; —10 and — 10; 10 4V— 5 and 10— y— 5; — 64~5y— 1 and —6— 5y— 1. 1 C4 EQUATIONS OF THE SECOND DEGREE. [§ 219, 220. h.) In the last example, we have (§§ 92, 162) ( T+6 _5y_l)(.r+6+5y-l) = (r+6) 2 -(5y-l) 9 = ^._[_6)2_j_52 = 0; which is, evidently, impossible (§201. N. 2). § 219. ».) Again (§ 210), we shall have the value, e = — ^Li — -, real, when B- — 4 J. C is positive ; 2A and imaginary, when i? 2 — 4J. C is negative. That is, the roots will be real and unequal, when B- — AA C>0 ; real and equal, " B°— \A C—0; imaginary, " -B 2 — 4-4 (7<0. PROBLEMS. § 220. 1. Given a; 2 — 2x— 24 = 0, to find the values of x# Ans. x = -4-6, and — 4. 2. Given a: 2 -f-l 2x4-35 = 0, to find x. Ans. x = — 5, Or — 7. 3. Given 3x 2 +2x— 10=75, to find a-. _4ws. jc = 5, or — of. 4. Given x°— x— 210 = 0, to find a. _4rcs. a; = 15, or — 14. 5. Given £x 2 — £x+6f = 7, to find a:. _4ws. a; = 1-i-, or — {?. 6. Find two numbers whose sum is 100, and whose pro- duct is 2100. Let x = one of the numbers. Then 100 — x = the other ; and x(100 — x) =± 2100, by the second condition. a; 2 — 100x = — 2100. We might have formed this equation immediately by considering, that the sum of the required numbers taken with a contrary sign must be equal to the coefficient of x ' ; and their product, to the coefficient of a: . Thus x*— 100x+2100 = 0. § 220.] PROBLEMS. 165 x 2 — 100;r-{-2500=:400. §208* x = 70, or 30. Otherwise, let x = the excess of the greater number above 50 (i. e. half the sum of the numbers) ; then 50-}-x = the greater, and 50 — x = the less. Hence [(50-f-.r)(50— x) — ]2500— x 2 = 2100. x 2 = 400; and a: = ±20. 50-f-a: = 70, or 30 ; and 50— x = 30, or 70. 7. Find two numbers, whose sum is 100, and whose product is 24Q0. 8. Find two numbers, whose sum is 100, and whose product is 2500 (§ 205). 9. Find two numbers, whose sum is 100, and whose product is 2600 (§216). Am. 50+10y— 1, and 50—10^—1. 10. Find two numbers, whose sum is S, and product P. Am. fK?~P)* and 4- (f-p)* a.) In what case will these values be imaginary ? Am. When P > ^T = (-^) '1. See 9, above. Hence, The product of two numbers can never be greater than the square of half their sum. b.) This principle can be proved otherwise ; thus, Let S be the sum of two numbers, and D, their differ- ence. Then l2 Q. Whence, The square of the sum of two numbers cannot be greater than twice the sum of their squares. Note. As either of the numbers may be negative, this applies equally to the square of the difference. 16. The sum of two numbers is 25, and the sum of their cubes is 8125. "What are the numbers? Ans. 20 and 5. 17. A rectangular field contains 20 acres, and one side is 40 rods longer than the other. "What are the dimensions of the field ? Ans. 80 rods long, and 40 wide. 18. A rectangular park, 60 rods long and 40 wide, is surrounded by a street of uniform width, containing 1344 square rods. How wide is the street? Ans. 6 rods, or — 56 rods. d.) The second value, — 56, is clearly not a proper solu- tion to the problem ; but it is a root of the equation, and. §221.] EQUATIONS OF THE 2)l th DEGREE. 1G7 in some sense, satisfies the conditions of the problem. For we find the area of the street by multiplying its width by each of the sides of the park, and adding to the sum of these product 5 the squares formed at the four corners. Thus, 6X60 ;xG0+Gx40-}-6x4(4-4xC 2 = 1344. So 2(- +- ? (—56x40)+4(— 56)2 = 1344. It freqi lappens, as we have already seen (§137 . that the a>_ expression of a problem is more general, and admits of more solutions, than the problem itself as expressed in ordinary language. x* n +Px n -\-Q=0. § 221. The preceding methods apply not only to equa- tions of the second degree, but to all equations of the form x*-"+Px n -\-Q = 0, in which the unknown quantity appears in only two term? : and its exponent in one of the terms is double that in the other. This equation may be put under the form, (x n ) 2 +/V-f Q = 0. Completing the square (§ 207), O") 2 -f P^'-fiP 2 = i?2_ Q. x n = —^P±(\P^—Q)^. x = t—hP± (IP-- Q) 2 ]". § 52. N. 1. Given x*— 52x 2 +576 = 0, to find x. Jlns. x= ±6, or ±4. 2. Given Xx — \*/x =-\\, to find x. Ans. */x = o, or — H; .-. x = 9, or 2|. In verifying these results, Jx must be positive for the first valua, and negative, for the second. A similar remark applies to the fol- lowing example. 3. Given (ic-f-12)*+(x-f-12)^= 6, to find x. (*+12)- 2 -+(*-hl2)H-i - 1+6 = ~. il 168 EQUATIONS OF THE SECOND DEGREE. [§ 222. 1 OH-12) 4 =— \±%—% or — 3. x+12 = lG, or 81. §52. x = 4, or 69. 4. Given x^+xS = 756, to find x. Ans. x — 243, or (—28)*. 3 5. Given x 3 — x 2 = 56, to find x. ^4ws. x = 4, or ( — 7)^. Note. We have seen (§ 213. 2) that every equation of the sec- ond degree has two roots. It will be proved hereafter, that every equation has as many roots as there are units in its degree. Sec 1, above. The above process, however, does not always exhibit all the roots. § 222. If an equation contain radicals which cannot be treated by the method of § 221, it may frequently be re- duced by properly arranging the radical terms containing the unknown quantity, and raising both members to the requisite power. There is frequently great advantage al- so in rendering a binomial surd rational (§§ 186, 187). The radicals, which most frequently occur, are radicals of the second degree (i. e. expressions of the square root of quantities). 1. Given x-\-J (2ax-\-x 2 ) = a, to find x. We have */(2ax-\-x-) = a — x. Then squaring 2ax-\-x 2 = a 2 — 2ax-\-x 2 . Aax z=z a 2 ; and x =l \a. Clearing of fractions, x-\-a-\-2 ls /(ax) z= b 2 x. Extracting the square root, •v/r-j-^/a — ± b^/ x ; or (lzfb)^/xz= — „/«. (lj:b) 2 x = a. a a -(i^by-'^iy' ' §223.] RADICALS— TWO UNKNOWN QUANTITIES. 169 3. Given 2x+2y(a*+x*) = _-JL_, to find x. Ans. x — \a. _. Jx-\-J(x— a) n-a 4. Given — — ] -^-j {■ = , to find x, Jx — +/{x — a) x — a If we render the denominator of the first member ration- al (§ 187), multiply by a, and extract the square root, we shall have ±na Clearing of fractions and transposing, */(x 2 — ax) = a±na — x = (l±n)a — x.. Squaring, x*— ax=z (l±n) 2 a*—2(l±n)ax+x*. { l±ny-a l±2n ' k ^. +/ (a-\-x)-\-+/ (a—x) , 5. Given ,; ( ,; (=J, to find *. V («+#) — y (a— a:) 2a5 6. Given ^+il±^M = 9, to find,. 7 . Glven _______ i __ = __, t ofind the value of a:. Ans. x — ±\. § 223. Every complete equation of the second degree, containing two unknown quantities, and having only posi- tive integral powers (§ 22. c, d), is, obviously, of the form (§197) A^+Bxp-\-Cx 3 +J)i/+JEx-{-F= 0. That is, it contains terms of the zero, the first, and the second degree with respect to both and each of the un- known quantities. a.) A single equation of this kind is, of course, indeter- minate (§ 122. a) ; and will give, for any value whatever of alg. 15 170 EQUATIONS OF THE SECOND DEGREE. [§ 224. itlier of the unknown quantities, two values of tlie other 13. 2). § 22 4. h.) Such equations are of continual use in tha ;er applications of Algebra, in expressing the relation rreen two variable (§ 136) quantities which are so con- nected, that a change in the value of one, in general, in- volves a change in the value of the other; i. e. between two variables, which are functions, one of the other (§§ 2G. 136. a). c.) Thus, let x denote the distance from any point in the circumference of a circle to a given straight line, and y the distance from the same point to another line perpendicu- lar to the first. Then the relation between these distances will be such, that, if one of them be given, the other will be determined ; and if another point be taken at a different distance from the first line, it will also, in general, be at a different distance from the second. That is, a particular value of x requires a corresponding value of y ; and a change in the value of x involves, in general, a correspond- ing change in the value of y. d.) An equation, expressing some known relation be- tween these distances, is called an equation of the curve. By means of such an equation, the properties of the curve are easily and rapidly deduced. e.) The equations of the circle, ellipse, parabola and hy- perbola are of the second degree, and contain two variables. Thus, y 2 -\-x 2 — R- = is the equation of the circumfer- ence of a circle, when the distances x and y are measured from two diameters at right angles to each other. For in that case these distances for any point of the curve, togeth- er with the radius drawn to that point, form a right angled triangle, of which the radius is the hypotenuse. Whence x*-+y 2 = B* (Geom. §188). Note. A straight line is represented by an equation of the frst degree, between two variables. Tliua, y = ax+b; a and b being either positive or negative. j 2*25, 226.] two unknown quantities. 171 f.) The employment of equations of this kind for the d co very of geometrical truth helongs to Analytical Geome- try and to the Differential and Integral Calculus. 1 this, at the same time, furnishes one of the most important applications of the principles, already demonstrated, of equa- tions of the second degree. § 225. The ordinary algebraic treatment of equations of the second degree, containing two unknown quantities, sup- poses two equations (§ 122. d, e) ; and deduces values the unknown quantities, which will satisfy hoth equatio Let there be given the two equations, Ay*-\-Byx+Cx*+Dy-\-Ex+F= 0, and A T jr»+Bjtx-t- C'x 2 -\-D'y-\-F'x-\-F' = 0. If now one of the unknown quantities, as y, be found in terms of x and known quantities, and this value be sub tuted in the other equation, there will, of course, result an equation containing but one unknown quantity. If this equation be solved, and the values found for x be substitu- ted in one of the primitive equations, corresponding values of y may be found. But it is sufficiently evident, that the equation so obtain- ed by the elimination of one of the unknown quantities will be of the fourth degree, which, in its general form, we not yet prepared to solve. § 226. Though we are not prepared for a general sol tion of two equations of the second degree containing two unknown quantities, yet certain classes of such equation.-!, can be solved by applying the principles already demon- strated. This is true of all those equations, in which the elimina- tion of one of the unknown quantities results in an equa- tion either of the second degree, or of the form, x in ±Px ±0 = (§221). 1. Given x 2 +r = 100, x°~—7ji = 28, to find x and y. 172 EQUATIONS OF THE SECOND DEGREE. [§ 226. Adding, subtracting, and dividing by 2, we bare • a: 2 = 64, .\*=±8;. and ^2 = 36, .-. y=±&. 2. Given z 2 -fy 2 = 100, xy = 48, to find x and y. From tbe second equation, /48\ s Substituting in tbe first, ••-©' ^+^=10* a:*— 100a: 2 := — 2304. §221. a 2 = 64, or 36 ; and x = ±8, or ±6. y = ±6, or ±8. a.) Tbe last example may be more conveniently solved witbout elimination. Tbus, adding and subtracting twice tbe seeond equation to and from tbe first, we bave attd *»-2ay+y 9 = 4J x-\-y = ± 1 4 ; and ic— y = ± 2. a: = ±8, or ±6 ; and y = ±6, or ±8 ; as before. 3. Given x 2 -\-xy-\-y 2 = 112, ar 2_ a: y_|_y2 — 43^ to find x and y. Ans. x = ±8, y = ±4. 4. Given a: 2 +a# = 180, xy-\-y 2 = 45, to find ar and y. ^4ns. a: = ±12, y=:±3. 5. Given 4ry = 96— x 2 y 2 , x-\-y — 6, to find x and y. Ans. ar = 4, or 2, or 3±y21 y=2, or 4, or 3:fV2L 6. Given ar 2 +a;4-y = 18 —y 2 ' a:y = 6, to find a? and $, Ans. x = 3, or 2, or — 3± ; # = 5, or 4. 3. Given x-\-y = 10, a; 2_j_^2 = 50 5 to find a; and y. $ 228. c.) It is sometimes convenient to employ auxilia- ry unknown quantities, such as the sum and difference, or the sum or difference and product or quotient. Note. If one or both of the equations be of a higher degree, the problem can frequently be solved by an equation of the second de- gree. 1. Given x-\-y = a, x 3 -\-y 3 = b, to find x and y. Let x = s-{-t, y = s— t, and .*. (§ 57. 3) s = ^(x-\-y) = \a. Then x 3 +y 3 = (s+t) 3 -f(s— t) 3 = 2s 3 +0si 2 = b. „ b—2s 3 , , /&— 2s 3 \i Hence £ 2 = — — , and t = ± ( ) . 6s ' V lis -/ ... * = S ±(_-) , and^ = Sq :(— -) . .•., introducing the value of s, , _ «. /^-^« 3 \i _ «. /46-« 3 x i ■ C_ 2 ± v _ 3^ ) -^vi^r) ' _ « /Ab — a 3 \l and ^2ni^-J- Let a = 10,and 6 = 370; a = 12,and 6=1008; a = 7 } and b = 217. *15 174 EQUATIONS OP THE SECOND DEGREE. [§228. 2. Given x-\-y = 8, x*-\-y* = 70 6, to find x and y. Let x = s-^-t, y = s — t, and .*. s = l(x-\-y) = 4. Then z*+y* = ( 5 +*)4-f( fi _*)4 — 2s*-f-12s 2 i! 2 +2^ = 706. Or, as s = 4, 512+192* 2 +2^ = 706. * 4 +96* 2 r=97. < 2 = 1, or — 97 ; and t = ±l, or±y— 97. x =z 5, and y = 3; or a? = 3, y = 5 ; or xz= 4±y— 97, and y = 4qV— 97. 3. Given 4x*—2xy = 12, 2y 2 -{-3xy = 8, to find a; and y. Assume a; = zy, i. e. substitute zy for x. Ans. x = ±2,ot ±fv/7. y=±l, = sin a-\-e'm b : sin(a-\-b). 2. What proportion from the equation sin b sin G = sin c sin B? a.) Also, if x 2 = a 5, then a : a; = a: : b. Hence, evidently, Cor. If the 'product of two numbers be equal to the square of a third, this last is a mean proportional behceen the other two. Thus, if 12 2 = 2x72, then 2: 12 = 12 : 72. So, if y 2 = R 2 — x-, then R-\-x : y = y : R — x. Transform the following equations into proportions. 1. y 2 = 2Rx — x 2 . Ans. x : y — y '• 2R — x. 2. y 2 = 2px. Ans. x : y = y : 2p. 3. R 2 = tan a cot a ; A 2 =x"x. § 234. Let a:b — 7c: I, or y = t- b I I. Multiplying by b, and dividing by Jc, a b 7 7/ - = - ; or a : « = b : I. Mis C Or, multiplying by /, and dividing by a, - =- ; or I : b-= k : a. Hence, b a The means or the extremes of a proportion may exchange places. Thus, if 2 : 3 = 8: 12, then 2 : 8 = 3 : 12. Note. The interchange of the means is called alternation'; (I) Lat. alterno, to interchange ; hence alternando, by inter- changing. §235, 236.] inversion. — composition. — division. 179 and the quantities are said to be in proportion alternately, or alier- nando. a Jc §235. II. Again 1-~ = 1-^- -. — — — ; or b : a = l : Jc. Hence, a k The terms of each ratio of a proportion may exchange places ; i. e. the antecedent may be made consequent, and the consequent, antecedent. Thus, if 2 : 3 = 8 : 12, then 3 : 2 = 12 : 8. Note. This is called inversions ; and the quantities are said te be in proportion by inversion, or inveriendo. §236. III. Adding ±1 to each side, .-. (§114. a) C ~- =-T~> or a±b : b = Jc±l: I. (1) b I b I Again 8 235) -=-. .M± : =l±j. a±b Jc±l , 7 ,,77 = — — ; or a±o : a = k±l : Jc. (2 a k Hence, The sum or difference of the first and second is to either the first or second, as the sum or difference of the tJd-d an ! fourth is to the third or fourth. Thus, if 7:5 = 14: 10, then 7±5 : 7 = 14±10 : 14. Note. In this case the quantities are said to be in proportion by composition," or componendo, when the sum is taken; and by divis- ion or dividendo , when the difference is taken. a.) Also a-\-b : k-\-l— a : Jc; § 234. and a — b : k — I = a : h. a+b : k-\-l = a—b : k—l. or (§ 234) a-\-b : a—b = fc+Z : k—l. (?ft) Lat. inverto, to invert; hence invertendo, by inverting, (n) Lat. compono, to compound , hence componendo, by compounding. (o) Lat., from divido, io separate ; by separating. 180 RATIO AND PROPORTION. [§ 237, 233. Hence, Cor. The sum of the first and second is to their differ- ence, as the sum of the third and fourth is to their difference. Thus 3:2 = 6:4; .-. 3+2 : 3—2 = 6+4 : 6—4. Hence (f§ 234-236), § 237. If four quantities he in proportion, they will be in proportion by alternation, by inversion, by composition, or by division. § 238. Let a : b = k : I, or % = T . b I Adding ±n (§ 42. a), a , k , a±nb k±nl a±nb : b = k±nl : I. (1) Again (§ 235), - = t ', and - ±m = -±m; Ct /C (X fc b±ma l±mk or = —j—. a k b±ma : a = l±mk : k. (2) We have also (§ 234) a:k = b:l; and from (1), a±?ib : k±nl= b : l=za : k; and from (2), b±ma : l±mk =z a : k = b : I. .'. (§ 231) a±nb : k±nl=b±ma : l±mk. (3) Now (§ 230. c) ma and wft have the same ratio as a and k ; also w6 and nl, the same as b and /. Hence, Jf either both antecedents or both consequents be increased or diminished by quantities having the same ratio as either consequents or antecedents, the results will be in proportion with either the antecedents or consequents, or with each other. Thus, if 2 : 4 = 6 : 12 ; then 2±3 : 4 = 6±9 : 12 ; and 2 : 4±1 = 6 : 12±3 ; 2±3 : 4±1 = 6±9 : 12±3. Notes. (1.) ma and mk are called equimultiples** (i. e. products by a common multiplier) of a and k. (2.) If m and n be (p) Lat. a;quus, equal, and multiplico, to multiply (§66. Note § 239-241.] EQUIMULTIPLES. — SUMS. — POWERS. 181 each equal to unity, the formula (1) and (2) of this section become identical with (1) and (2) of §236. § 239. Let a : b = k : I. ma mk r a k , ma mk The " T=T ; ni, = M-^nb = nV H2 ' '' * ma : b = mk :l; a : nb — k : nl; and wzrc : nb = m& : n?. Hence, Equimultiples of the antecedents and of the consequents of a proportion will be in proportion, either with the origi- nal antecedents, or consequents, or with each other. Thus, if 2:4 = 6:12; then 2Xo : 4X7 = 6X0 : 12X7. Or, 2X5:6X5 = 4X7:12X7 = 4:12 = 2:6. Note. We may, obviously, multiply both terms of a ratio (§230. c) or both the antecedents, or consequents (§42. c, d) of Ta propor- tion, by a common multiplier, without destroying the proportionality. §240. Let a : b = e :f=g : h = k :l. Then ab = ab; and (§ 232) af= be; ah = bg ; al = bk. a(b+f+h+l) = b(a+e+g+k). .-. (§ 233) a-\-e-\-g+k : b-\-f-\-h-\-l z=a:b = e:f,&c. Hence, In any number of equal ratios, the stem of all the antece- dents is to the sum of all the consequents as any one of the antecedents is to its consequent. Thus, if 1:2 = 3:6 = 4:8 = 5:10, then 1+3+4-1-5 : 2+0+8+10 = 1:2. § 241. Let a : b = k : I. Then (§ 52. N.) ~ = ^- ; or a\ : b n — k\ : l\ Hence, Like poxoers of proportional quantities are proportional. Thus, if 1:4 = 64:256, then I 3 : 4 3 = 64 3 : 256 3 ; and 71:74 = ^4: ^256. Note. The ratio of the squares of two quantities was formerly called the duplicate ; that of the cubes, the triplicate ; of the square ALG. 16 182 RATIO AND PROPORTION. [§ 242-244. a:id cube roots, the subduplicate and subtriplicatc, ratio of the quan- tities themselves. The ratio of the square roots of the cubes (i. e. of the three half powers) is sometimes called the sesquiplicate ratio of the quantities. § 242. Let a :b — k : I; e :f=g : h ; and r : s = x : y. ml a e r k q x aer kqx Then - . - . - = - . | . _ ; or — = -f- ; u f s Ihy bfs Ihy or aer : bfs = hgx : Ihy. The same will evidently hold of any number of propor- tions. Hence, The products of the corresponding terms of any number of proportions are proportional. Thus, if i : 3 = 6 : 18, and 10 : 6 = 15 : 9, then 1X10:3X6 = 6X15:18X9. Notes. (1.) When the terms of two ratios are thus multiplied together, the ratios are said to be compounded. (2.) If equal ra- tios are compounded, we obtain the ratio of the powers of the quan- tities (§241). § 243. The following exhibits, very briefly, most of the principles above demonstrated (§§ 232-242). If the truth of any of these expressions is not self-evident, write the ra- tios in the form of fractions. 1. ar:a = br: b ; or — = ^(§§113. 1; 114; 230. a). a b 2. abr — abr. §232. 3. ar:br = a:b. §234. See 113. 3. 4. a : arz=b : br. § 235. 5. ar±a : a = br±b : b ; or a (V±l) : a = b (r±l) : b. §236. 6. a (r+1) : a (r— 1) = b (r+1) : b (r— 1). § 236. Cor. Note. Other principles may be exhibited in like manner. § 244. When the first of four quantities is to the second a3 the fourth is to the third (i. e. as the reciprocal (§ 18) of the third is to the reciprocal of the fourth), they are said to be inversely (§ 235) or reciprocally proportional. § 245, 246.] variation. 183 Thus, if, on a railroad, a freight train runs 15, and a pas- senger train 30 miles an hour, their times of passing over equal distances on the road will be inversely or reciprocal- ly proportional to their velocities. That is, Time of 1st : Time of 2d = Vel. of 2d : Vel. of 1st = ovT: T'= V : V— ■* Vel. of 1st, ' Vel. of 2d. ' ut " ' ^ ~ • — y • y" If, however, they run equal times, as 3 hours, then the distances will be directly proportional to their relocitiee. VARIATION. § 245. These relations are sometimes concisely express- ed by saying, that one class of quantities, or, still more concisely, that one quantity varies directly or inversely as another. This form of expression is denoted by this symbol go, or ==, placed between the quantities. Thug, x&y, or x = y, (read x varies as y). Thus, in the examples of the last section, the time is said to vary (or to be) inversely or reciprocally, and the distance directly, as the velocity. Or, I 7 go — ; D go V. So, the number of men required to accomplish a work in a given time varies directly as the amount of work ; if the amount of work be given, the number of men varies in- versely as the time allowed. § 246. If x==y, then we shall have, obviously, x : x 1 = y : y> • ov x :y = x' :y'. x x' - — -, = m, a constant numbei*. (1 ) Also, x =z my ; and y = — x. (2) m Hence, When one quantity varies directly as another, (1.) the ratio of the numbers by which they are expressed is con- stant; and (2.) each is equal to the other multiplied by some constant number. 184 EATIO AND PROPORTION. [§ 247. §247. Let a: go -. Then x : xfz=- : - = */: y. y y y' y u xy =. x'y' =: m, a constant number. (1) • , otto Also, 3? = — ;andy = ^-. (2) y a x v Hence, If one quantity varies reciprocally as another, (1.) the product of the numbers by which they are expressed is constant ; and (2.) each is equal to a constant quantity di- vided by the other. a.) The converse of the principles in this and the last section is evidently true. Hence, (3.) any equation, containing variable quantities, way be written as an expression of variation ; and may be simplified by dropping any constant factor on either side. Also, (4) if all the factors on one side be constant the other side is constant (§§ 246. 1 ; 247. 1). Thus, if we have the area of a circle —tiR' 2 , n being constant 9 , then the area varies as the square of the radius ; or area = 2i 2 . So, S representing the space fallen through by a falling body, and T, the time of its descent, i£ S ~ mT 2 , m being constant, then the space varies as the square of the time ; or S = T 2 . Again, if the area (A 2 ) of a rectangle — its base (x) X its altitude (y) ; i. e. if A 2 = xy, then .. , the area . A 2 1 1 the base = -; — — — ; or x = — = A 2 - ; and x w - ; or the altitude * y y y the base varies inversely as the altitude. b.) In the last example, the area varies as the product of the base and altitude. So the solidity of a parallelopipe- don varies a3 the product of its length, breadth and thick- ness. (q) it, Greek letter pi, Eng. p ; the initial (§ 1. d) of Trepujtipeta, periphery, circumference. In common use, 7r = 3.14159 &c the circumference of the circle whose diameter is unity. § 248.] VARIATION. — PROBLEMS. 185 c.) I£xzc-, then x varies directly as y, and inversely as z. Thus, the weight ( W) of a body above the surface of the earth varies directly as its mass {M), and inversely as the square of its distance [D) from the centre of the earth. rrr M That is, W co -r-^. JJ~ § 248. 1. If, above the surface of the earth, the weight of a given body (i. e. its gravitation towards the" earth) va- ries inversely as the square of its distance from the centre of the earth, how high must the body be raised, that its weight may be only half what it was at the surface ? Let x r= the height above the surface of the earth ; r = the radius of the earth ; and w ■==. the weight of the body at the surface. 1 1 Then w : %w — — : -—. — — = (r-\-z) 2 : r 2 ; r- (r-f-x)- or 1 : l=(r-\-x)* : r 2 . i(r-\-x) 2 = r 2 ; or x n --\-2rx = r 2 . x = — r±r^/2. Note. Taking the upper sign, and finding ^y2 approximately, we have x — JtJULr. The lower sign gives the distance, measured downward (§5) through the centre. 2. How far must the body be removed from the surface, that its weight may be w' ? Here we have w : w' '= (f-\-x) 2 : r- ; or «/w : */io' = r-\-x : r w x — — r±r J y— r . w' 3. How much weight will the body lose, if it be remov- ed a given distance (D) from the surface ? and what will be its weight there ? Here . w : w> = (r-f-Z>) 2 : r 2 w : w-w' — (r+D) 2 : (r+D) 2 -r 2 . § 236. *16 186 EQUIDIFFERENT SBRIES. [§ 249. , (2rZ>4-D 2 )w , , „ . , w-w> = ^ 2+ 2rjD+ j a . tlie loss of weight ; r* (r+X»)2 If D is very small compared with r, D^ may be neglected, and we shall have 2D w — w' = — — — -w. r-\-2D Let D = l, 2, 5, 10, 100, 1000 miles, w=l pound, and r — 4000 miles ; and find the values of w' and w — iv'. CHAPTER IX. EQUIDIFFERENT, EQUIMULTIPLE AND HARMONIC SERIES. I. EQUIDIFFERENT SERIES. § 249. A series of quantities such that each differs from the preceding by a constant quantity, is called an equidifferent series ; and sometimes an arith- metical series or progression. a.) Such a series can, of course, be continued to any ex- tent ; and its character is determined, if we know any one of its terms and their common difference. Thus, if 7 be one of the terms, and 3 the common differ- ence, we shall have the series, .... —5, —2, 1, 4, 7, 10, 13, 16, ... . Or, if 8 be one of the terms, and — 2 the difference, we shall have .... 12, 10, 8, G, 4, 2, 0, —2, — 4, . . . . §250,251.] LAST TERM. — SUM. 187 b.) If the common difference be positive, the series is called increasing ; if negative, decreasing. The first of the series in a above, is an increasing series ; the second, a de- creasing series. c.) Though every series may be continued without limit, we ordinal ily have occasion to consider only some definite number of terms, of which the two extremes are called the first and last terms. § 250. If a be the first, and I the last of n terms of an equidifterent series, and D their common difference, we shall have 1st, 2d, 3d, (n-l)th, nth, a, a-\-D, a-\-2D, . . a-\-(n— 2)D, a-\-(n—\)D or /; whence, obviously, l=z a-\-(n — 1)D. (1) That is, The last term is equal to the first term, plus the product of the common difference by the number of terms less one. Note. Of course, the common difference must be taken positive or negative, according as the series is increasing or decreasing. 1. What is the 7th term of the series 1, 3, 5, &c. ? Here a = 1, D — 2, and n — 7. l = a+(n—l)D=l-\-Gx2 = 13 2. Given a = 25, D = —2, and n = 14 ; to find I. Ans, — 1. 3. Given a ■==. 0, D — 1, and n = 100 ; to find I. Ans. 99. § 251. If s represent the sum of n terms of a series, we shall have s = a J r(a+ B)+(a+2I)) . . +{ [ a -f-0-l)Z>]( = Z) y 5 and, writing the terms in the reverse order, obviously s = l+(l— D)+(l— 2D) . +{ [?— (n— l)Z>)]( = a) }. /. Adding the equations, 2s=(a+l)+(a+l)-\-(a+l) . . +(a+l) = n («+?). 188 EQUIDIFFERENT SERIES. [§ 252. s= 9 ^pl. (2) That is, The sum of any number of terms of an equidifferent se- ries is equal to the number of terms into half the sum of the extremes. 1. What is the sum of 20 terms of the series 1, 3, 5, 7, &c? Here a = l, D=z2, and n = 20. l=a-\-(n—l)Z> = 1+19X2 = 39. s — bi(a+l) = \ . 20(1+39) = 400. 2. Given a ■= 1, D= 1, and n =. 10 ; to find £ and s. Arts. 1= 10, s == 55. 3. Given a = 20, Z> = — 2, and « = 21 ; to find ? and s. Ans. I = —20, s = 6. 4. Let a = 20, D = — 2, and w = 11 ; and find I and 5. § 252. a.) It is obvious from the addition of the two series above (§ 251), that the sum of any two terms equidis- tant from the extremes is equal to the sum of the ext) ernes. Or, beginning with a, the mth term = «+(?» — \)D; and, beginning with I, the mtla. term = / — (m — 1)D. Now the sum of these two terms, equidistant from the extremes is a-\-l. b.) Hence, if the number of terms be odd, the middle term is half the sum of the extremes. c.) Such a term is called an equidifferent mean, and sometimes an arithmetical mean. d.) The equidifferent mean between two quantities is found by taking half their sum. Thus, the equidifferent mean between 1 and 2 is 11-, or 1.5 ; between 1 and 1.5, 1.25; between 5 and 15, 10. e.) The middle term is also equal to the sum of all the terms divided by their number. For 5= \{a-\-l)n ; .'. -= \{a-\-l) = the middle term. Tit § 253-255.] FORMULAE. — INTERPOLATION. 189 Note. A mean of several quantities, whether they be equidiffe- rent or not, is found by dividing the sum of the quantities by their number. The mean or average temperature for a week or month is found in this way from the several daily temperatures observed dur- ing the given period. § 253. /.) If (§§ 250, 251) we substitute in (2) the value of / in (l),'we shall have s in terms of a, D and n. Thus, s = nh(a+l) — na-\-hi(n—l)D = w[a-{-£(w— 1)2>]. § 254. The formula?, I ==■ a-\-(ii — 1)Z>, and s = fyi(a-\-l), should be carefully remembered. They contain, it will be observed, five quantities. If any three of these be given, we shall have two equations containing two unknown quan- tities which may therefore be determined (§§ 124-128). a.) In fact, from the first, a = I — (n — 1)D ; (3) n — 1 (4) a »=£+i. (5) b.) In like manner, from the second, 2s a-\-l (6) 2s a = 1 ; n (7) n (8) § 255. c.) From formula (4) we can interpolate* any number of equidifferent means between two given extremes. For let it be required to interpolate m intermediate terms between a and b. We shall have the whole number of terms, n, equal to m-\-2. Hence, n — 1 = m-j-1, and _ /__ I — a \ b — a \n~—\)~m+l\ Hence we have the series," (r) Lat. iaterpolo, Fr. interpoler, to insert. 190 EQUIMULTIPLE SERIES. [§256,257. a, a-\ — , a-\-2 — — , . . a4-(m4-l) — — (=b). m-\-Y ' m-f-1 ' v ' J m-\-\ K ' 1. Interpolate 8 equidifferent means between 1 and 10. 2. Find 6 equidifferent means between 1 and 15. §256. 1. Given a = 1, D = l, and » = 100; to find 7 and s. Ans. 1=100, s = 5050. 2. What is the rath term of the series of example 1 (i. e. the rath term of the natural series 1, 2, 3, 4, &c.) ? Ans. n. 3. What is the sum of n terms of the series 1, 2, 3, &c. ? Ans. ^±L } . 4. What is the nth. term of the series 1, 3, 5, 7, &c. ? .Jras. 2rc — 1. Substitute for n, 1, 2, 3, 4, 5, &c. 5. What is the sum of n terms of the above series of odd numbers, 1, 3, 5, &c. ? Ans. n 2 . Substitute for n as above. 6. Suppose a body, falling freely to the earth, descends m feet the first second, dm the second'second, bin the third, &c. Now if its fall occupy T seconds, how far will it fall in the last second? Ans.. (2T—V)m. 7. How far will it fall in the whole T seconds ? i. e. what is the sum of the series, m, om, 5m, &c, to T terms ? Ans. m T 2 . Substitute, in these two examples, for T, 5, 6, 7, 8, 10, &c. Al- so find the value of the expressions thus obtained, on the supposition tbat m — 16^. II. EQUIMULTIPLE SERIES. § 2-57. A series, such that each term is formed by multiplying the term immediately preceding by a con- stant multiplier, is called an equimultiple series ; sometimes also a geometrical series or progression, or a progression by quotient. §258.] LAST TERM. 191 Note. The constant multiplier has been sometimes called the ratio. For convenience and distinctness, however, we shall call it the common multiplier, or simply the multiplier. a.) Such a series can, of course, be continued to any ex- tent ; and its character is determined, if we know any one of its terms and the common multiplier. Thus, if 7 be one of the terms, and 3 the common multi- plier, we shall have the series, TJ 7 , y, 6h, i, -1, OO, .... So, if 8 be one of the terms, and | the multiplier, we shall have ... 32, 1G, 8, 4, 2, 1, i, \, . . . b.) If the common multiplier be greater than unity, we shall have an increasing series ; if less, a decreasing series. The first of the two series in a, above, is an increasing, the second a decreasing series. c.) Though every series may be continued without limit, we ordinarily have occasion to consider only some definite number of terms, of which the two extremes are called flhe first and last terms. § 258. If a be the first, and I the last of n terms of an equimultiple series, and m the common multiplier, we shall have 1st, 2d, 3d, 4th, 5th, (?i— l)th, Tith, a, am, am 2 , am z , am*, . . . am n ~ 2 , am"' 1 or/. Whence, obviously, / — am n_1 . (1) That is, to find the nth term of an equimultiple series, Multiply the first term by the (n — \)th power of the com- mon multiplier. 1. What is the 6th term of the series 1, 2, 4, &c ? Here a =z 1, m = 2, and n z= 6. l(= am"- 1 ) = 1x2 5 = 32. 2. Given a = 3, m = 2, and n = 10 ; to find /. Ans. 1536. 192 EQUIMULTIPLE SERIES. [§258. 8. Given a = 64, m = h, and n = 8 ; to find I. 4. Given a = $100, m = 1.06, and n = 10; to find/; i. e. to what will $100 amount in 10 years at 6 per cent, compound interest? Ans. 1=- $179.09. 5. "What is the amount (A) of p dollars, at compound interest for t years, at the rate r ? Here we have 1-J-r = the amount of one dollar for one year. p(l-\-r) = " p dollars " p(l-\-r)(l+r) = " ^(1+r) " " &c. Or, p = the amount at the beginning of the jirst year ; p(l-\-r)= « " wconrf « ■ • • • • jp(l-f-r) re - 1 = " « rath " p (l-j-ry= « « (<+l)th " i. e. at the ewe? of t years. The successive amounts constitute, obviously, an equi- multiple series ; in which we have given a=p, m=- 1-J-r, and n = t-\-\ ; to find 1= A. Ans. A =zp(l-\-r) ( . G. What is the amount of $50 at 6 per cent, compound interest for 12 years? Ans. $100.61. 7. What sum, at the rate r, will amount to A dollars in t years ? . A _ . , J Ans. p = ■ , ,, . See 4, above. (i-K) 8. What principal at 6 per cent will amount to $1000, in 10 years? Ans. $558.37. 9. At what rate of compound interest will p dollars amount to A dollars in t years ? . /A\L J A?is.r=( — U— 1. Let p = 100, A = 150, and t — 8 ; &c. a.) If we have m < 1, and n — &, then putting m =— y /IS § 259.] LAST TERM. — INFINITE SERIES. 193 («t' being of course > 1), we shall find I ( = am"- 1 ) = /l v 00 -! a am' am! rt# . ,« o\« >tm ^ • «[— l = = = — =0 (§ 138. 3Y. That is, The last term of a decreasing infinite equimultiple se- ries is zero. Notes. (1.) Of an infinite series, there can be no last term. And, on the other hand, in forming the terms by multiplication, we can never arrive at zero; though we must evidently approximate to it. Hence, the inconsistency in speaking of the last term of an in- finite aeries is compensated by placing it beyond any finite number of terms; i. e. at an infinite distance. (2.) In the two series, 1> b h h &c - 5 and 2» 1> b h & c-> any term whatever of the first is half the corresponding term of the second. Hence, the last terms are said to be in the same ratio. Now this comparison can, obviously, be made only between terms at some definite distance from the beginning. That distance, however, can be taken as great as we please; and the terms, consequently, can be brought as near zero (and, therefore, as near equality) as we please, while the ratio remains constant. Thus infinitesimals, though regarded as equal to zero, may, like finite quantities, have any definite ratio to each other. § 259. b.) It is evident, from the formation of the sev- eral terms of an equimultiple series, that the product of any two terms equidistant from the extremes must be equal to the product of the extremes. In fact, if a be the first of n terms, the term which has p term3 before it will be am? ; the one which has p terms after it, being the (»—/>) th, will be equal to am n ~ p ~' t . Hence their product am p Xam n -p-i — a 2 m n-i _ flXflm n-i — a i (§ 258). c.) Or again, if a be the first term, and m the multipli- er, we shall have the (jo-j-l)th term = amP. (s) It is evident that, since m'>l, a finite number of factors, each equal to m',may be taken sufficient to produce any finite num- ber whatever. Hence, if we combine an infinite number of these factors, the result will be infinite. ALG. 17 !94 EQUIMULTIPLE SERIES. [§260,261. But, if we begin with I, the multiplier is, obviously, — ; m f 1 \ p I and the (»4-l)th term = l( — ] = — . v/ J \m ) m" am p xl — = al. m p d.) Hence, if the number of terms be odd, the product of the extremes tvill be equal to the square of the middle term (it being equally distant from the two extremes). e.) Such a term may be called an equimultiple mean. It is sometimes called a geometrical mean, and is simply a mean proportional between the extremes (§§ 231. b ; 232. b). § 260. If s represent the sum of n terms of an equimul- tiple series, we shall have s =. a -\- am -j- am - — am 3 -\- . . -f- am n ~ l . Multiplying by m, ms — am -\- am - -\- am 3 -f- am 4 -f- . -f- am n . Subtracting the first of these equations from the second, ms — s z=z anf — a ; or (in — l)s = a(m" — 1). ^«K-=1). (2) ' That is, m — 1 To find the sum of n terms of an equimultiple series, Raise the midtiplier to the nth power, and subtraot 1 / multiply the remainder by the first term, and divide the pro duct by the midtiplier diminished by unity. r am n — a § 2GI. a.) We have s = — , § 260. ' m — 1 and l = am n ~K §258. s _lrnr-a That is, m — 1 To find the sum of n terms of an equimultiple series, Multiply the last term by the multiplier, subtract the first term, and divide by the midtiplier less one. § 261.] SUM. — INFINITE SERIES. 19"' k) If m < 1, both m n —\ and m—1 will be negative. In that case it is convenient to change the signs and the order of the terms, thus ; «(1— ra n ) a—lm s = —^ , or s = - . 1— m 1 — m 1. Find the sum of 20 terms of the series 1, 2, 4, 8, &c. Here a=.l, m — 2, and n = 20 ; 8= -(^-i) = i(^°-i) =1>048)WS , m — 1 2 — 1 2. Given a = 243, m = | 3 and n = 7 ; to find Z and s. Ans. ?= i, s — 3641. 3. Given a = 1, m = 4, and n = 5 ; to find ? and s. Ans. l=25Q; s = 341. • 4. Given arrl,m= J, and ?z = 6, to find Z and s. M.11S. I ^^ J4TT 5 ^ ~~ 23T3' c.) If m < 1, and n = oo, we should have, reasoning as in § 258. a, am" = am™ = 0. s = " (4) 1 — m That is, The sum of a decreasing infinite equimultiple series is equal to the first term divided by the difference between uni- ty, and the common multiplier. We might obtain the same result by substituting the value of I (§ 258. a) in formula (3) of § 261. Notes. (1.) If n is infinite, n — 1 is infinite also. For, if n — 1 were finite, n being greater by unity than a finite number, must be finite also. In like manner, if any finite quantity whatever be sub- tracted from infinity, the remainder is still infinite. (2.) Hence, we have co=^r oo±a. That is, an infinite quantity is not affected by the addition or subtraction of a finite quantity. 1. Given a = 1, m = ^, and n = oo ; to find the sum of the series. Ans. s = 2. 2. What is the sum of the infinite series, whose first term is 1, and multiplier ^ ? Ans. 1£. 196 EQUIMULTIPLE SERIES. [§262. 3. Given a and s in a decreasing infinite series, to find m - . s — a „ , a Ans. m = , i. e. 1 . s s 4. Given a = 1, 5 = 3, and n = oo, to find m. Ans. m = §. 5. Given m and s, when n = oo, to find a. _<4ns. a = (1 — /n)s, 6. Given m = -t, s — 10, and « = oo, to find a. Ans. a = 8. § 262. (£.) Suppose that at the end of one year from the present time, and also at the end of each succeeding year, a man invests a dollars at r per cent, compound interest. What will be the whole amount of his investment and in- terest at the end of t years ? "We shall have the amount of the Jirst investment for t — 1 years = ^(l-j-?-)'" 1 „ " second " t—2 " = a(l+r)'- 2 ; • • • • • ■ last but one " 1 year = a(l-f-?") ; last " " =a. Hence, if A 1 = the ivhole amount, we shall have ^ = a[(l+ry-i+(l-h-)'- 2 . . +(HV> 2 +(l+'-)+l)]; or (§260) A r =a>-^-± . (1) Note. This is the amount of an annuity* of a dollars, which has been forborne (i. e. left unpaid) t years. 1. Given a = $100, r = .06, and t = 10 years ; to find A'. Ans. .4' = $1318.08. 2. Given a = $200, r = .05, and£ = 8 years, to find A 1 . Ans. .4' = $1909.82. e.) The present xoorth of an annuity for any number of years is, evidently, the same as the present worth of the amount of the annuity (§ 262. d); i. e. it is such a sum, as, (t) Fr. annuite', yearly payment ; from Lat. annus, a year. §262.] INTEREST. — ANNUITIES. 197 put at interest now, will produce that amount in the given time (§ 258. 7). If p> — the present worth, we shall have (§ 258. 7 ; 262. d] ,_ A' _a (l+r)«-l _q/ lx g . ^-( 1+r )« — t r (1+r)' VV 1+r)'/ 1. Given a = $100, r=.06, and f = 10 years; to find p>. Ans.p' = $7S§.Q\. 2. Given a = $500, r = .06, and £ = 12 years ; to find .4' and/. -4ns. ^£' = $8434.97 ; ^' = $4191.92. /.) If the annuity be a perpetuity 1 ' (i. e. if it last forever), we have t = cc, and (§ 258. N. s) t — 0. .«, j>/ = -, the sum, evidently, whose annual interest is a. y.) These formulae, as well as those relating to com- pound interest (§ 258. 5-9), will be the same, whether the interest and annuity be payable at the end of each year, or of each half year, quarter, month, day, hour, or other peri- od ; r denoting the interest of SI for the given period, and /, the number of the periods. h.) Or, if r= the interest of SI for a year, t = the number of years, and n == the number of periods in a year, we shall have r - = the interest of SI for the given period ;* and n nt = the number of periods. .-.(§258.5) A=p(l+?) nt . (3) Also (§ 262. d) ] ^ = v[( 1+ D" Ll ] ; < 4 > and (§262..) ^=~\}~ i}+$T]' <«) Given p — $100, r = .06, n = 4, and t = 3 years ; to find A. Ans. A = $119.52. (u) Lat. perpetuitas, that which lasts forever. *17 198 EQUIMULTIPLE SERIES. [§ 263, 264. ?'.) The interest may be conceived to be payable at each moment as the use of the money is enjoyed. In that case, n becomes infinite, and formula (1) reduces to a peculiar form, which will be considered hereafter. j.) If, while the interest is payable annually, the com- pound interest for a part of a year be required, the value of t in the formula of § 258. 5 becomes fractional. Thus, the compound annual interest for half a year is i I ^(1-fr)- ; for one third of a year, p(\-\-r) 3 ; &c. So, for two and a half years, we have A=zp(l-\-r)" 2 . § 263. k.) This last result corresponds to the case in which n becomes fractional in the formula, Z=am n-1 of \ 258. Nothing prevents our assigning a fractional value to n either in the equidifferent or equimultiple series. Thus, in the series 1, 3, 5, 7, &c, if n =z 3-|-, we have /(— a +(n—l)D= 1+2^X2 = 6. So, in the series 1, 2, 4, 8, &c, if n = 3^, we have l( — am"- 1 ) = 1X2 2 *= 2^= 32*= 5.65685&C § 258. Note. This, it will be observed, is equivalent to interpolating a single mean, equidifferent or equimultiple, as the case may be, be- tween the third and fourth terms of the series (§§255; 265). /.) Again, n may, obviously, become zero, or negative. Thus (§ 250), let a = 1, D — 2, and n = 0. Then 7[ = a+(n— 1)2)] = l-f-(0— 1)2 =— 1. If n = —3, then I = l+(— 3— 1)2 = —7. Also (§ 258) let a = 1, m — 2, and n = 0. Then l( = am n ~ 1 ) = 1X2 " 1 = 2" 1 = | (§ 17). Ifw = — 3, then 1= 1X2- 3 - 1 = 2~ 4 = T \. ci(m n 1^ §264. The formula?, l=am n - 1 (1), s = -± - J (2) tit J. and s = (3) should be carefully remembered. m — 1 They contain, it will be observed, five quantities, from any § 265, 206.] INTERPOLATION. 109 three of which the other two may, obviously, be found (§ 254). a.) Thus, from the first, a = ■■ (4) m m =(=)*■ v b.) To find n, we have m n ~ x = -. That is, n — 1 is the exponent of the power to which m must be raised, to pro- duce -. An equation, in which the unknown quantity is Ctl an exponent, is called an exponential equation ; and i^ solved by a peculiar process, which we are not yet pre- pared to investigate. § 2G5. c.) From formula (5) we can interpolate (§ 255. N. r) any number of equimultiple means between two giv- en extremes. For, if it be required to intei'polate p terms between a and b, we shall have the whole number of term?, n, equal to p-\-2. Hence, n— 1 =zp-\-l, l=b, and m = (-J »-i = \-) p+1 • Hence we have the series, 1. Interpolate 2 equimultiple means between 3 and 81. Here ^ = 2,^+1 = 3, a = 3, and J = 81. m = (- V+ 1 = 27 J = 3. Hence the series is 3, 9, 27, 81. §266. 1. Given a = ^ wz = 2, and w = 10, to find / and s. Ans. l=z 32, s = 63if. 2. Given a = l, m = l, and rc = 100; to find / and s. Jns. 1=1, s = 100. Note. Here we have _ a(m n —l) ^ 1(1—1) _ / _ a\nr—L)\ __ i^i— J A m — 1 / 1—1 0' 200 HARMONIC SERIES. [§ 267. apparently indeterminate (§ 109. c). But when m = l, we have (§140) " m » — I - — - = nm n - I = 100x1" = 100. m — 1 3. Given a = 1, m = — x (where x < 1), and n = ce, to find s ; or, in other words, to find the sum of the decreas- ing infinite series, 1 — x-\-x 2 — x 3 -\-&c. Ans. s =■ — — • l-\-x If we had x > 1, we should find the same result, but by a different process. 4. Of four terms of an equimultiple series, the product of the two least is 8, and of the two greatest 128. What are the numbers? Ans. 2, 4, 8, 16. 5. "What is the vulgar fraction equivalent to the repeat- ing decimal .121212 &c. ? Ans. -fa. Notes. (1.) This is the same thing as finding the sum of the se- ries .12+.0012+&C. to infinity; where a = .12, 7/i = .01, and n = co. (2.) In the same way, the value of any repeating deci- mal may be found. Thus, we have .1111 &c. = .1+.01+ 001+&C. = .l-j-(l— -1) = .H-.9 = I 6. Find the vulgar fraction equal to .IOIO&c. ; 222&C ; .456456&C : 74357435&C. HARMONIC SERIES. i 267. 1. Three numbers are said to be in harmonic al v proportion, when the first is to the third, as the difference of the first and second is to the difference of the second and third. Thus, if a : c = a — b : b — c, then a, b, and c are in harmonical proportion. So 2, 3 and 6 are in harmonical proportion, because 2:6 = 3 — 2 : 6 — 3. 2. Four numbers are said to be in harmonical propor- tion, when the first is to the fourth, as the difference of the (») Gr. dp/tovia, joining, harmony. §268-270.] RECIPROCALS, EQUD1FFERENT. 201 first and second is to the difference of the third and fourth. Thus, 6, 8, 12 and 18 are in harnionical proportion ; because 6 : 18 = 8-6: 18—12. £ 268. Let a, b, and c be in harmonical proportion. Then a : c = a — b : b — c. ab — ac = ac — be; (1) or (a-\-c)b = 2ac. b — — — ; (2) and c = -. (3) § 269. A harmonic series or progression is one in which any three consecutive terms are in harmonic proportion. Thus, 6, 3, 2, 1.5, 1.2, 1 form a harmonic series, as will be readily seen by forming proportions as in § 267. 1. §270. Let «, b, c,f, g, h, &c, be consecutive terms of a harmonic series. Then (§ 268. 2) a+P 6-h/ c+ff Dividing unity by both sides of each equation, l_a+o l_l+f l_c+ff b ~ 2ac 4 ' c~ 2bf f~2cg ' C * o ;i=I(I + Iy ---(-+-V i-ifi+Iv&c b 2\c^J' c~2\f^b)' f~2\g^ c) ' 1 1 1 1 1 „ •'• -, r, -, -;, -, &c. are terms of an equidiflerent se- tt o c j g x ries (§ 252. d). That is, The reciprocals of the terms of a harmonic series consti- tute an equidifferent series. a.) This principle may be shown otherwise. Thus, ab — ac = ac — be. §268. 1. 1111 Dividing by abc, = T . § 249. c b b a b.) Conversely, it can be readily shown, that the recip- rocals of the terms of an equidifferent series constitute a harmonical series. 202 HARMONIC SERIES. [§271. c) In order, therefore, to interpolate any number of har- monic means between two quantities, or to continue a har- monic series, of which two terms are given, we have only to interpolate a like number of equidifferent means between the reciprocals of the given terms ; or to extend the equi- different series, of which those reciprocals form a part ; and take the reciprocals of the terms so found. Thus, to insert two harmonical means between 60 and'15> Ave must insert two equidifferent means between ^ and ,' .,. This will give the equidifferent series ^ ^, - b %, £$. I fence, the harmonic series is GO, 30, 20, 1-5. The succeeding terms of the equidifferent serieswill be n fi_ _7 . 7 > +l], by a notation analogous to that of § 273. N. a.) We have, obviously, That is, the number of arrangements of n individuals ta- ken p at a time is equal to the whole number of permuta- tions of n individuals, divided by the number of permuta- tions of n — p individuals ; (i. e. by the number of permuta- tions which can be made with the individuals left out of each arrangement). 1. How many arrangements can be made with the 10 Arabic numerals, taken 2 at a time ? Ans. 90. 2. How many, if they be taken 3 at a time ? Ans. 720. 3. How many arrangements can be made from the 72 numbers of a lottery, taking 3 numbers upon each ticket ? \ Ans. 357840. b.) I£p — n, we shall have simply the permutations of n things = n(n—l) .... 2 . 1, as in § 273. § 275. Combinations are the groups, that can be formed of individuals without reference to the order of arrangement ; in other words, groups such, that no two of them shall be composed of the same individ- uals. Thus ab and ba form two permutations or arrangements, and but one combination. And, in general, whatever be the number of things, on- ly one combination can be formed by taking them all at alg. 18 206 PERMUTATIONS AND COMBINATIONS. [§ 275. once. For two combinations are not different, unless they differ in, at least, one of the individuals contained in them. Hence, each combination of n things may be subjected to 1 . 2 . 3 . . n permutations, without affecting the combi- nation. So, if we combine n things,^? at a time, each com- bination admits of 1.2.3 p permutations or ar- rangements. Hence we shall have only one combination for every 1.2.3 p arrangements. If then we divide the number of arrangements by the number of permutations in each arrangement, we shall have the number of combina- tions. That is, if we combine n things, p at a time, we shall have _ T ... No. arrangements of n things taken » &» INo. combinations — — . No. permutations of p things. ." • the number of arrangements of n things taken p at a time is (§274) n(n — 1) (ii—p-\-\) ; or [n, n— £>+l] ; and the number of permutations of p things is 1.2.3 . . . . p ; or \_p~\. Therefore the number of combinations of n things taken p at a time is n(n — 1) . . . (n—p-\-l) _ [n, n—p-\-\~\ 1.2.3 7~. \ .~~ p " [p] a.) If the letters denote algebraic quantities, the number of combinations of n letters taken p at a time is, evidently, the same as the number of distinct products of the quanti- ties taken p at a time. b.) If n things be taken p at a time, then (§$ 274. a ; 275) n(n— 1 ) . . (n— H-1 ) (n—p) ..3.2.1 No. of arrangements = No. of combinations = 1.2.3 . . {n—p) »(n — 1) . . (n— p4-l)(n— p) . .3.2.1 1.2.3...^. 1.2.3... {n—p) ' § 275.] COMBINATIONS. — PRODUCTS. 207 Again, if n things be taken n—p at a time, we have 72(71—1) . . (Jl— p-\-l) («'— p) . . 3 . 2 . 1 No. of arrangements =: No. of combinations 1.2.3. . . . p n(»— 1 ) . . {n—p-\-l ) (n—p) ..3.2.1 1.2.3 . . . p .1 .2.3 . . . (n — p) Hence, the number of combinations of n individuals, is the same whether they be taken p, or n—p, at a time. Thus, the number of combinations of 10 letters will be the same, whether they be taken 3 and 3, or 7 and 7. c.) The last principle is evident also from the fact, that, for each combination of p things taken, a combination of n—p things must be left. 1. How many products (§ 275. a) can be formed of the 6 quantities, a 15 a 2 , a 3 , « 4 , a 5 , a 6 , x by taking them 1 by 1, 2 by 2, 3 by 3, 4 by 4, and 5 by 5 ? Ans. Q, 15, 20, 15, and 6. 2. How many products of 4 quantities taken 1, 2, 3, and 4, respectively, at a time? Ans. 4, 6, 4, and 1. d.) The number of combinations must, of course, be a whole number. Therefore ' •% is a whole number. (x) These numbers, 1} 2 , 3 , &c, are used as accents (§ 1. e). CHAPTER XL UNDETEKMINED COEFFICIENTS. § 276. Let the equation Aa?-\-BaP+ <7af-f-&c. = A>xi»-\-B>x*'+ CV+&C. (1) be true for all values of x (i. e. whatever value may be as- signed to x) ; A, B, G, &c., and A 1 , B', C, &c, being any quantities whatever not equal to zero or infinity, and each member of the equation being arranged according to the ascending powers of x. It is required to determine the relation existing between the exponents, and also between the coefficients of x in the corresponding terms, on the two sides of the equation. Dividing both members of the equation by x p , we have A+Bx*-*-t- Cx r -P+&c. = A'x p '- p -\-Bx v - p + G'x r '-*'+&c. Now, as this equation is true for all values of x, it i3 true when x = 0. But if x = 0, the first member reduces to A (the exponents of x in all the terms, but the first being > 0, i. e. positive) ; and the second member evidently, becomes zero, if jt/> p ; infinite, it' p'p, we shall have A — 0, which is contrary to the hypothesis : and, if p')x s -$-&c. = 0. (4) Making x = 0, A— A' = ; and .-. A — A'. Then (B—B')x+(C—C r )x*+&c. = Dividing by x, B—B>-\-{ C— C')x+&c. = 0. Making x = 0, B—B' = ; and B = B f ; and so on. Represent A— A' by 31; B—B' by If; C—C by P; &c Then if we have, for all values of x, M-\-Nx+Px-+&g. = 0, we shall have also M=Q; y=0; P=0; &c. Hence, § 280. If any polynomial of the form, M+Nx+Px 2 -\-SfC, be equal to zero for all values ofx, each of the coefficients of the several powers of x, must be sepa- rately equal to zero. *18 210 UNDETERMINED COEFFICIENTS. [§281. c.) An equation, which is true for all values of a varia- ble, is said to be true independently of the variable. Such an equation is an absolute equation (§ 37. d). Thus, the equation, (a-\-z) 2 = a 2 +2ax-{-x 2 , is true independently of x. On the other hand, the equa- tion, \-\-x- = 2x—x 3 , may be true ; but its truth depends on the value given to x (§38). § 281. The above principle (§§ 276-280) is the founda- tion of the method of undetermined coefficients ; a method of very great utility in the development of func- tions and the investigation of principles. 1. Develop — - — into a series. l-\-x Assume ■——=Ax~ 1 -}-Bx +Cx+Dx 2 +&c, Then, if x — 0, we have 1 = oo ; which is absurd. Again, assume = Ax-{-Bx 2 -\-Cx 3 -\-&c. Then, if x = 0, we have 1 = 0; which is absurd. Assume then = A-\-Bx-\- Cx 2 -\-Dx 3 -\-&c. Clearing of fractions and transposing. o> = A +A x+B x 2 +C a;3+&c. —1 +5 +<7 +Z> .-. .4—1=0; A+B=0; B+C=0; (7+2)= 0; &c. A = l; B——A — —\; C=—B=l; Z> = — C=— 1; &c. Introducing the values of A, B, C, &c, we have — 1— x+x 2 — a; 3 +x 4 — z 5 +&c. 1+a: ~ n Note. The results of the several suppositions, in this instance, indicate the method of ascertaining the form of the series to be as- sumed. We may generally determine this, before writing the seiies, §281.] SKR1ES. PARTIAL FRACTIONS. 211 by making x=:0 in the function to be developed. The series must be taken, so as to become finite, infinite or zero for x- = 0, according as the function becomes finite, infinite or zero for the same value of x. 2. Develop (a—x)- 1 . See §87. c. 2. Assume -±- = A-{-Bx+Cx 2 -\-Dx 3 +&c. a — x Then(§4G) l=Aa—A -\-aB x—B -\-aC x n —C -\-aD X 3 — &c. Aa — 1; aB—A — 0; a C—B = Q; aD—G=0; &c. A = 1 -, B = ± = L., C= B - = K; B: a a a- a (I'- ll -; «fec. ... _J_=i + la+4« s +&c. = -(l+- + ^-h&c.) a—x a a- a 3 a\ a a- / Or (a— x)- 1 =«- 1 +a- 2 «+a- 3 x 2 +«- i x' 3 +&c. = a-i(l+a-ia:+a- 2 a; 2 +&c.) i 3. Develop (a — x) 2 . Assume (a—x) * = A-{-Bx-\- Cx 2 -\-Dx 3 -{-&c. Squaring, a—x = A 2 +2ABx+B 2 x 2 +2BC ■\-2AG +2AD A 2 — a; 2AB = — 1; B 2 +2AC=0; 2BC J r 2AD = 0; &c. a; 3 +&c i 11 A = a 2 ; B = --= -: Z 2a- (a — xy 2 — a" — 1 x lx 2 C= ±a^ 2.4a 2 ; &c. 1 x- 2 a 2 2.4a 2 2.2. ! 1 x* &c. 1 X 2~a~Y74~a 2 2.2 3a;— 5 ^-&c.Y a* y 4. Decompose into fractions, whose sum is x 2 —Gx+8 the given fraction, and whose denominators are the factors of the given denominator. a;2_ 6a; _|_ 8 — (x—4)(x— 2). § 213. 1. 212 UNDETERMINED COEFFICIENTS. [$ 281. Therefore assume 3x— 5 A . B x 2 — 6x+8 x— 4 ^ a;— 2 ' 3x-5 _ ^(,;-2)+i?( a :-4 ) .-. 3a:— 5 — A{x— 2)-\-B(x— 4) = (J+-B)*— (2^4+45). .-. ^+5 = 3, 2A+iB = 5. .-. ^=|, B- — \. 3a:— 5 7 1 X 2_q x+s ~2(x^ 2^-2)' ^ee§118. 3. Otherwise ; as the equation, 3a:— 5 = A[x— 2)-\-B(x— 4), is true for a# values of a:, it is true, when x = 2 (i. e. when a:— 2 = 0). Introducing'this value of x, we have 6—5 = 5(2—4) ; and .-. B = — £. Again, if a: = 4 (or a: — 4 = 0), we have 12— 5 = J(4— 2) ; and .-. A = l,aa before. j. _ a 3 4-bx 2 / a 3 -\-bx 2 \ o. Decompose —^ -( =—. - w , ). a 2 x — x 3 v x{a — x)(ci-\-x/ . a , a4-b a-\-b Am --* + Z^)-2^)- See §118. 2. 6.^yelop(l^)-(=^ ) -= iT A_) in an infinite series. Ans. 1— 2a:+3x 2 — 4a: 3 -f&c. Compare § 87. c. 5. 3x 2 — 1 7. Decompose a:(a;+l)(x— 1) ' Ana. x a:-|-l a: — 1 l—x 3. Develop — — in an infinite series. 1-j-x Ans. 1— 2a;+2a; 2 — 2a;3+&c. CHAPTER XII. BINOMIAL THEOREM. § 282. Any positive integral power of x-\-a can be found by multiplying x-\-a into itself the requisite number of times (§ 164). The proper combination of this process with division and with the extraction of roots will give negative and fractional powers (§ 163). But this process, when applied even to positive integral powers, beyond a few of the lowest, becomes tedious ; its application to negative and fractional powers would be ex- tremely inconvenient. The Binomial Theorem enables us to find im- mediately any power of a binomial, whether the ex- ponent be positive or negative, integral or fractional. § 283. Let it be required to find the nth. power of x-\-a. I. Let re be a positive integer. Then the nth. power of x-\-a is the product of n factors each equal to x-\-a ; i. e. (x-\-a) n = (x-\-a)(x-\-a)(x-\-a) . . . to n factors. To find how the terms of these factors are combined in the terms of the product, multiply together n unequal fac- tors, x-\-a x , x-\-a 2 , x-f-«3j #+«»• Then (x-\-a 1 )(x-\-a 2 ) = x- J r a 1 x-\-a x a. 2 . -\-a 2 {x-\-a l ){x-\-a 2 ){x-\-a^)^=x^-\-a l x 2 -\-a l a 2 x-\-a^a 2 a y -\-a 2 -\-a 1 a 3 -\-a 3 -j-« 2 «3 214 BINOMIAL THEOREM. K 28^ ( x +^i)(^+a 2 )(x+a :i )(x-\-aJ = + « 4 + «1«3«4 -}-« 2 a 3 +a:) 3 = what? (a:-(-a) 4 ? (1+ar)*? § 287. d.) If, in the general formula (§ 285. c), we put — x for -\-x, we have («— x) n = a n — na n ~\x-\- \.~ * a*~*x\— &c. ; the terms containing the oc?c? powers of aj being negative, 1. (a— a;) 5 := what? (a— x) 2 ? (a— x) 3 ? 2. (a 2 — x 2 ) 5 = what? Jws. (a 2 )s— 5(a 2 )*x 2 +10(a 2 ) 3 (x2) 2 — 10(a 2 ) 2 (a: 2 )3+ 5(a 2 )(x 2 ) 4 — (a; 2 ) 5 ; or, reduced, a i °— 5« 8 a; 2 H-10a G a; 4 — 10a 4 a; 6 -l-5a 2 a; 8 — a; 1 ». 3. (a; 2 ±2oa?) 3 =what? Jns. a; e ±6aa; 5 -j-12a2a;*±8a 3 a: 3 . Make x 2 = 6, and 2ax = c; develop (6+c)3, and substitute the values of 6 and c. e.) The formula? (§ 285. c ; 287. rf) may be put under another form. For a±x=za(l±-Y .-. (a±x) n =a n (l±~y, ALG. 19 218 EIX03IIAL THEOREM. [§288,289 , j_ \n nft^l x i n ( n — 1) x2 . n(n—l)(?i—2) X~' .'. (a±x) n = a"[ l±n-4- — — — y — ± -^ y -i L — ; V aT 1.2 a 2 1.2.3 a- : w(??— l)(n— 2) . . (»— 2jp-f l) « 2 ^ n H 1.2.3 . .:(2p~l)2p ^ / Note. The above demonstration proceeds upon the supposition that n is a positive integer ; and is, of course, applicable to that case only. § 288. II. Whatever be the value of n, whether integral or fractional, positive, or negative, let it be assumed, that (x+y) n = A+By+ Cy?+Dy*+JEy*+&c. ; (1 ) .4, B, G, &c, being functions (§ 26) of aj and w, and entire- ly independent of;?/. Note. There can be no negative powers of y, because (x+y) n is not necessarily infinite when y = 0. There must, moreover, be a term containing y®, because (x+y) n is not necessarily zero, when y = (§281. N.). § 289. 1. Let n be a positive fraction, - (p and q being both integers). Welrave (x-\-y)* = w '(l+ -V '. § 287. e. Assume (l-f-^V = l+P'-+&c, for all values of -. \ x/ a; a; Then (l+^) P =(l+p|+&c.)*. § 52. N. .-. (§ 285) l+;>^ +&c. = 1+?P- +&c. ; the remaining terms on both sides containing only higher c y powers or -. JO .-.(§277) p — qP', or P= ? - = n. (x+y)« = xi (1+*- - +&c.) = xi +t-xz y+&c v *' q x q Hence, in the /rsi ftw terms, the same law prevails with § 290, 291.] NEGATIVE EXPONENT. — INCREMENT. 219 the positive fractional, as with the positive integral expo- nent {§ 285. 1, 3, 4). 2. Let n be negative. Then (^i-y) — ( x _|^)n — x n -{-nx"- 1 y-\-&c.' (x-\-y)- n = x~ n — ?ix~"- 1 y-\-&c, by division ; tJie remaining terms, evidently, containing successively low- er powers of a; and higher powers of y. Hence, again, the first two terms follow the same law with the negative, as with the positive exponent. Hence, universally, whatever be the value of n (i. e. whether it be positive or negative, integral or fractional), ; 0. The first two terms of (x-\-y) n are x n -\-nx n ~ 1 y. We have, therefore, in the series (1) of § 288, A =. x n , and B = ?ix n ~ 1 ; and the series may be written (x-{-y) n = a;"-f?2x"-iy+ CyS-\-Dy*-\-Ey*-\-&c. § 291. Let now each of the quantities x and y be suc- cessively increased by any quantity whatever k. The function (x-\-y) n will, obviously, undergo an equal change in each case ; i. e. Note. The quantity, h, by which x and y are increased, is cal- led an increment^ of a; and y. 1. Adding h to y, series (1) of § 288 becomes [»+(*+*)]" = A+B(y+h)+ C(y+h) *-\-D(y+h)3-h& C . ; or ix+(y+h)Y = A+By+Cy*±By*+By±+& C . (2) -\-Bh-\-2 Cfyk+8Ih/ 2 h-l-4By 3 h-\-&c. ; &c. &c. &c. writing only the terms containing A and h 1 . 2. The substitution of x+h for x will, of course, produce no change iny, or in the manner in which it enters into the ly) Lat. inerementum, an increase. 220 BINOMIAL THEOREM. [§ 292. expression ; but it will produce a change in each of the co- efficients, A, B, O, &c. For, as these coefficients are func- tions of x, they will, in general, change their value when- ever the value of x is changed. Therefore, the powers of y remaining as they are, their coefficients will be what A, B, C, &c. become, when x is changed into x+h ; i. e. they will be the 3ame functions of x+h, as A, B, C, &c. are of x. Representing, then, by A x + h , B^, C^ h , &c, the values assumed by A, B, C, &c, when x becomes x-\-h, we shall have [(»+*)+*? = A^+B^jrt- C x ^?,"-+n x+ft y 3 +&c. (3) $ 292. Now we have already found A = x", and B =. nx n ~ 1 . A x+h = (x+hy ; and B x+h = n(x+h) n -K (4) But we do not know, what functions C^, JD x + h , &c, are of x-\~h, because we do not know what functions G, D f &c. are of x. In other words, we do not know what C, D, &c. will be, when x-\-h is substituted in them for x, because we do not know what they are now. Let it be assumed, then, that C x+h = C+ C'h+&c. ; D x+h = D+iyh+&c. j (5) and so on ; and assume, for symmetry A x+h [=z (*+*)•] = A+A'h+&c. ; and B x+h [= n (x-\-h) n ~ * ] = B+B 'h+&c. Notes. (1.) This supposition, evidently, involves no absurdity (§281. N); for, when A = 0, the expressions (5) severally reduce to C, D, &c, as they ought, being then simply functions of x as at first (§288). (2.) It will be observed, that A, B, C, &c, in these assumed values, are the primitive undetermined coefficients, functions of x (§288); and that A', B>, C, &c are the coefficients of h* in the several expressions, when x+h is substituted for x. (3.) If the variable of a function is increased, and the function developed, the coefficient of the first power of the increment is a quantity very much employed in analytical investigations; and is cal- led the first derived function, or the first derivative, or derivate, or ' 293.] DERIVED FUNCTIONS. differential coefficient, of the primitive function. Thus A 1 is the first derived function, or derivate of A; B', of B; &c. (4.) In like manner, if x+/i be substituted for x in A', B' , &c, the coefficient of At is the first derived function, or derivate of A',B', &c; and may be called the second derived function, or second de- rivate of A, B, C, &c; and may be represented by A", B", &c The same process deduces from the second derivate, a third, A'", B'", &c; from the third, a fourth, A"", B"", &c; and so on. § 293. Substituting for A^, B x + fl , O x+ft , &c., the val- ues (5) assumed above (§ 292), and writing as in (2) of § 291 only the terms containing h° and h 1 , we have t(x+h)+j/] H = A+A>h+&c.+(B-\- B'k+&c.)y-\-( C-\- C'h +&e.)y 2 +&c; or, arranging according to the powers of h, I(»+*)+yr=^+ i H-^»+^'+^*-f*c- (6) +(A'+B'y+CY+D'y 3 +&c-)h -f&c. Equating (§ 291) the second members of (2) and (6), A+By-\-Cy'*+&c. ) (A-\-By+Cy*+&c. (7) +(B+2Cy+&c.)h f = -j +(A'+B'y+C'y*+&c.)h -f&c. j <• +&c. As this equation is true for all values of h, the coeffic- ients of like powers of h are severally equal $ 277). The coefficients of h° are identically (§ 37. c) the same. Passing then to the coefficients of A 1 , B+2 Cy+ZDij 2 +&c. = ^'+5^f C'y 2 +Z>y -f&c. (8) Again, as this equation is true for all values of y, the coefficients of like powers of y are severally equal (§ 277). .-. B — A; 2C = B'; 3B= C> ; ABzzzJD'; &c. or B — A>; G—\B'; D = \C> ; E—\D; &c. (9) That is, a.) B is found by substituting x+h for x in A and tak- ing the coefficient of A 1 , viz. A'. *19 222 BINOMIAL THEOREM. [§ 294. Cis found by substituting x-\-h for x in B (i. e. in A 1 ), and taking half the coefficient of A 1 , viz. \B'(=.\A"). See § 292. N. 4. D is found by substituting x-\-h for x in C (z=z \B' = J^t"), and taking one third of the coefficient of h 1 , viz. / B" A'" \ %0'(=-— = 9— 5) ; an( l so on - Thus, substituting these values of B, C, &c. in (1) of § 288, A" A'" A 1V (x+y)* = A+A>y+ —y 2 +— y 5 + ^-^-f&c. (10) b.) Or, in otber words, B is the first derivate of A ; C is half the first derivate of B, i. e. half the second derivate of A (§ 292. N. 4) ; D is one third of the first derivate of G, i. e. one sixth of the second derivate of B, i. e. one sixth of the third derivate of A ; and so on. § 294. Now we have (§§ 290) 292) A = x n ; and A x + h = (x-\-h) n ; or A-\-A'h-\-&c. = x n +nx n - l h+&c, A'zrznx"-*. §277. B{=.A!) = nx n -^ i as we found it before (§ 290). In like manner, B x+h = n(x-\-h) n - 1 ; or B-\-B'h+&c. = nx n ~ x -\-n{n— l)x n - 2 h+&c. B' = n(n—l)x n - 2 . §277, C{ = \B< = \A') = 5&±LV*. _ n(n—l) t 1.2 So C x+h =-±^(x+h)»-* ; or n(n — 1) „ . n(n — l)(n — 2) „ ._ , , J . _ j .. X>(_}0 - 2<3 ^ -2.3" 1 ' 1.2.3 &c. § 295.] DERIVATIVE. — COEFFICIENTS. 223 a.) It will not be necessary actually to make the substi- tution of x-\-7i for x. For the coefficient of each power of y is of the form Mx n ; and the two first terms of M(x-\-K) n are Mx n -\-Mnx n ~ 1 h (§ 290). Hence, The derivative of each coefficient of y in the series is found by multiplying that coefficient by the exponent of x, and diminishing that exponent by unity. Therefore, To find the coefficient of any power of y, §295. b.) Multiply the coefficient of the preceding term by the exponent of x in that term, diminish the exponent by unity, and divide by the number of terms preceding the required term. Thus, the coefficient of y p (i. e. of the term which has p terms before it) will be — - — - ^ ' — - — £-- — J -x n - p . (12) 1.2.3 . . . . p c.) The term -i 7 ; ' 1 — LJLJ x »-r.yP 1.2.6 p J is called the general term of the series ; because if we make in it p = 1, 2, 3, &c, successively, we shall have the cor- responding terms of the series. d.) If n is a positive integer, the series tvill terminate, as we have seen (§ 285. a). But, if n is negative, the subtrac- tion of unity will numerically increase the exponent with- out limit ; and, if n is a positive fraction, the subtraction of whole units will first render the exponent negative, and then numerically increase it in like manner. Hence, if n be either negative, or fractional, the series will be infinite. e.) The sum of the exponents of x and y in each term is, evidently, constant, and always equal to n. f) If — y be substituted for +y,the terms containing the 224 BINOMIAL THEOREM. [§ 296. odd powers of y will, of course, be negative (§ 237). Thus, (x—y)l = a »—na?- 1 g+ n ^~^ x n -*y*-&c. (13) g.) Let se = l, and y = 1, in formula (11). Then , rcfn— 1) , w(re— l)(rc— 2) . . Hence, in an_y poiver whatever of the si«n of two quanti- ties, the sum of the coefficients is equal to that same power of 2- h.) Let x = l, and y = l, in formula (13). Then , n(n— 1) w(w— l)(w— 2j . . ( 1-1)" = = l-n+ A_ J j-^- +&c. Hence, in any power whatever of the difference of two quantities, the sum of the coefficients is equal to zero; i. e. the sum of the positive, is equal to the sum of the negative coefficients. i.) Let n ■=■-. Then = as±^~Vf^r^»~V± & c- (14) g 1 . z . q -u.i-py i pip— 9) Y~ ± p(p—q)(p— 2 4 , &c. we shall have D x — a 2 a i > D n --±a^na 2 ±-^-^a z ^. 1 £ 3 ' a 4 ±&c.;(2) D 2 = a 3 — 2a 2 -\-a 1 ; D 3 = a 4 — 3a 3 -j-3a 2 — a x ; Z> 4 = « 5 — 4a 4 -f-6a 3 — 4a 2 -\-a 1 ; and, obviously, «(w— 1) n(n— l)(n— 2) , A = «„+!—««»+ 1>2 «*-i 17273 — " a "- 2 + &c. ; (1) the coefficients of a n + v a n , Sic., being the coefficients of the wth power of a — x. Or, reversing the order of the terms, n(?i — 1) ^?i(n — l)(w — 2) I taking the upper signs throughout, when n is even ; and the lower signs, when n is odd. Hence the first terms of the several orders of differences may be found without finding the remaining terms. 1 . What is the first term of the third order of differences of the series, 1,3, 6, 10, 15? Here a 1 = l } a 2 •=. 3, a 3 z=.Q, a^=z 10, and n = 3. ... D 3 ( = a 4 — 3a 3 -f 3a 2 — x a) = 10— 3X G+3X3— 1 = 0. So we should have D 2 =: a x — 2a 2 -\-a 3 ==. 1 — 2x3-[-6 = 1. 2. Given the series 1, 8, 27, 64, 125, to find B^JD^ D ? and Z> 4 . Ans. D x — 7, D 2 = 12, D 3 = 6, and Z> 4 = 0. § 301. From the values of D lf D„, &c. in § 300 we have a 3 = « 1 +22) 1 +Z) 2 , a 4 = a 1 +32> 1 +8Z> 2 »+i> s . and, obviously, . / i^n i (w-l)( n-2) n a.= a l +(n— 1)D^ — D 3 + (tt-l)(tt- 2)'(n-3) n , (n -l)(n-2)(n-3) n . — ni — *»+ — r^3 — ^+ &c - j (2) §301.] DIFFERENCE SERIES.— ORDERS. 229 the coefficients of the terms being the coefficients of the (n — l)th power of a-\-x. 1. What is the fourth term of the series of squai'es, 1,4, 9, 16, &c.? Herea^l, D l= :3, D 2 =i2,D 5 — 0, and n — 4. ... a 4 (= fl 1 +3D 1 +3Z> 2 +D 3 ) = 1+3X 3+3x2+0 = 16. 2. What is the twentieth term of the same series ? Ans. 400. 3. What is. the nth term of the same series ? a n =a l +(n-l)D 1 + MK^ j^ == l+3(?*— l)+(w— 1) (n— 2). == 1+3 (»— !)+«(«— 1)— 2 (n— 1). = 1+m — l+?z(n — 1) =n 2 . 4. What is the nth term of the series, a, a+D, a+2Z>, «+3Z>, &c. ? ^4ns. «+(n — 1)Z>, Notes. (1.) The problem contained in the last example has been already considered (§250). In fact, the whole subject of equidiffe- rent series, there treated, is only a particular case of the more gene- ral subject of differences ; viz. the case, in which the first differences are constant (§ 249); and, of course, the second, and all higher dif- ferences are equal to zero. (2.) It is proper to remark here, that an equidifferent series, hav- ing its first differences constant, is called a difference series of the first order ; a series whose second differences are constant, is said to be of the second order; and so on. Thus, we have the series, 1, 2, 3, 4, 5, 6, of the first order. 1, 3, 6, 10, 15, 21, second." 1, 4, 10, 20, 35, 56, third " (3.) These, which are only particular examples of the various or- ders of difference series, have also this property; viz. the nth term of each series is equal to the (« — l)th term of the same series plus the nth term of the preceding series. And, consequently, the nth ALG. 20 - ■ N.-r : sr vcmfcas: teemse tbe ■■iirtu « saberical bodies, at fe» ic tr ««; tiara-. the naber of balfe ea eacb ale >«ns ejy,u«.d bj tke urn yn—TiL. torn of tbe aatnal series -?tbenaili rffe riesaracaMa^rosMfeJ "-- - i J - - — - ~ .: :^ '"- : : ..-.-"... :: \-'-a:r:z :ir r - : r -:: ; Wt ia tbe fewest l — ui.i boaf expressed by tbe tant-y dag Sena Wii: is li* fifteenth term of the ser Am *. Wbii is tbe *± term of the se: ?. €, 1» -- J f. — - " ITIju iszlxMik torn oe -: — . ; . — - - __;_ — Sc: = . - • • - • - - 7;; :=." ; " •■-.—:;::: : - : — '. r_r i-- .-h? ::" fn.:i r'-r : . \- . '-1 : — I.rad — 2ad - «■?*". -h : fraud to be eaek ercsl to 0. Ttos wiB be 9- - VH.r tbe saceesaea ef difieraKce :' as ef die I t -- 7 1 :.i - -- —I, •, I, «» £», 4 5,] «, l. i, l. 1,3 i» § 302.] INTERPOLATION. 23 1 In the corresponding series of the fourth order, 1, 5, 15, 35, 70, 126, we should find four terms equal to zero, and the terms, corres- ponding to the negative local- indices beyond, positive; and so on, t'ie number of terms each equal to zero being equal to the number of the order ; and the terms, corresponding to the negative indices beyond, being positive or negative according as the number of the order is even or odd. § 302. The formula (2) of the preceding section had pri- mary reference to those terms only whose place in the se- ries is expressed by whole numbers ; i. e. to those denoted by integral local indices. *\Yq have found, however, by taking ra = ^, f, &c. in the general solution of examples Cth and 7th, terms corresponding to those fractional local indices, and still conforming to the general law of tile- ries. "We shall find, in like manner, that the above formula applies in general to such intermediate terms correspond- ing to fractional local indices, equally as to terms whose local indices are integral ; only giving a suitable value to n (§ 263). Note. This is simply a more •general form of the problem of in- terpolation ; and applies to all series, whose differences of any or- der become either zero, or so small that they may be neglected. 1. Given 2 2 = 4, 3 2 = 9, and 4 2 = 16 ; to find (2£)*. Here a 1 =4:,D 1 ~ 5, D 2 — 2, D 3 = Oj and n — it a n = a^ = 4 + 1X0 — $X2 = 6£. 2. Given (2500)* = 50, (2501)* = 50.009,999,8, (2502)* =50.019,999,6; to find (2500.5)*. Here a 1 = 50, D 1 = .009,999,8, D a = ; and n == 1> . .-. a n = (2500.5)* = 50-H r x.009,999,8 = 50.004,999,9&c. 3. Given 64* = 8, 66*= 8.124,038, 68* = 8.246,211, and 70*= 8.3666; to find 65*. (z) Lat. locus, place. Local indices indicate the place of the term in the series. 232 DIFFERENCES. [§ 303. Here a t =8, D^ =.124,038, D 2 = — .001,865, & 3 = J(.000,076-i-.000,081) = .000,078 ; and n = l£. Ans. (65)^ = 8.062,257. 4. Interpolate 3 terms between the fourth and fifth terms of the series, 4, 8, 12, 16, 20. Here a x = 4, D x = 4, D 2 = ; and n = 4^, 4£, 4|, « n =4 + 3iX4 = 17; &c. Or « x = 16, D x = 4, D 2 = ; and » = 1J, If, If. « ?1 =16 + 1X4 = 17; &c. Ans. 17, 18 and 19. See § 263. 5. We find, in a table of natural sines, sin 30= = .5, sin 30= 10' = .502,517, sin 30= 20 7 = .505,030, sin 30° 30' = .507,538. What is the sine of 30° 1' ? of 30° 2' ? of 30°, 3' ? of 30' 4' ? Ans. sin 30= V = .500,252 ; sin 30= 2' = .500,504 ; sin 30= 3' = .500,75 6 ; sin 30= 4' = .501,007. § 303. a.) In finding a term of the series by § 301, n being a whole number, the formula (2) will always termi- nate, because the coefficient n(n — 1) (?i — ?i) = 0, But, in interpolation (§ 302), the formula will not terminate, unless we find an order of differences equal to zero. For n being fractional, none of the factors, n — 1, n — 2, &c, can become zero ; but they will become negative, and then in- crease numerically (§ 295. d). Iii this case, the required term can be found only by an infinite series. b.) It will have been observed, that Ave have found terms, whose places are expressed both by integral and by frac- tional local indices, without knowing the law of the series into which they are introduced ; knowing, in fact, nothing of the series but a few terms ; or even a single term with the successive differences. §304.] sum. i c.) Hence, obviously, the differences, together with u single term, determine the character of the series. Tht-v enable us to continue the series to any extent (§ 301), to supply intermediate terms (§ 302), and, as we shall sec (§ 304), to find the sum of any number of terms. §304. Let it be required to find the sum of n term- oi the series, flj, Ct 2 ) Cl s , ft 4 , 65 5 , .... Ct n . Assume a series, whose first differences shall be the terms of the given series. Thus, 0, ctj, a x -\-a 2 , a 1 -{-a 2 J j-a 3 , . . a x -\-a 2 -\-a z . . -f-c*,,. Now the (?j-(-l)th term of this last series is,, evidently . the sum of n terms of the given series; and the (n-\-l)th differences of the last series are the nth. differences of the given series. Hence, marking the terms and differences of the assume series with the accent ', we have, in formula (2) of § 301, a? 1 = 0, D\—a x , D'z = I) 1 ,&,c.; and, putting 7i-\~l in place of n, and denoting by S the re- quired sum of n terms of the given series (i. e. the (n-\~l)tL term, ci' n + 1 , of the assumed series), we find 5( = „<„ +I ) =na 1+ ^i> 1 + " ( "7 1 »7 2) O a + 1 . What is the sum of n terms of the series, 1, 2, 3, 4, 5, G, &c. ? Herea 1 = l, D l =zl, and D 2 =z0. ■ n(n—l) n(»+l) b = n-\ — ^ — — -=— —^ — -. See § 2o(j. 3. 1 1.2 2 2. "What is the sum of n terms of the series, a, a-\-D, a-\-2D, &c. (i. e. an equidifferent series)? Ans. na+\n(n—l)D. See §§ 253 ; 301. N. 1. *20 234 DIFFERENCES. [§ 304. 3. What is the sum of n terms of the series, 1, 3, G, 10, 15, 21, 28, &c. ? Here a x = 1,D 1 = 2, D 2 = 1, and D 3 = 0. a . t in ■ n(n-l)(n-2) _ n(n+l)(n+2) ... S =n+n(n-l)^ r0 — 1<2 3 . 4. What is the sum of « terms of the series, 1, 4, 10, 20, 35, &c. ? n(w+l)(n+2)(n+3 ) ^ nS * 1.2.3.4 ' 5. What is the sum of n terms of the series, 1, 3, 5, 7, 9, 11, 13, &c ? Ans. n 2 . See §256. 5. *?. What is the sum of n terms of the series, 1 2 , 2*, 3*, 42, 5 2 , &c? w(n +l)(2H-l ) ' 4nS - 17273 ' 7. What is the sum of n terms of the series, 1 3 , 23, 3 3 , 43, 5 3 , &c? j^«wv=(&*£y. Notb. From the result of examples 1st and 7th, we have 13 + 23+33 ..+713 = (1 + 2 + 3 ..+*)*. CHAPTER XIV. INFINITE SEEIES. § 305. An infinite series, we have seen, may arise from an imperfect division (§ 87. a) ; or from the extraction of a root of an imperfect power (§ 170. N. 5) ; or by the continuation of an equimultiple (§ 261. c) series to infinity* Infinite series of various forms are also developed by the method of undetermined coefficients (§ 281), and by the bi- nomial Theorem (§ 295. d) ; and by many other processes, which we are not yet prepared to investigate, and some of which are beyond the reach of elementary Algebra. § 306. As the processes of developing infinite series are so various, the methods of summing them are equally vari- ous. Even of those which are summed by the elementary processes of Algebra, we shall consider here only one or two of the simplest. a.) The method of summing a converging infinite equi- multiple series has already been investigated (§ 261. c). b.) The true sum of an infinite series resulting from di- vision, or from the development of a fraction by undeter- mined coefficients, is the fraction from whose development the series originated ; and this, whether the series be con- verging or diverging (§ 87. d.f). We may, moreover, approximate to the value of a con- verging series by the actual addition of a small number of 236 INFINITE SERIES. [§ 307. the terms (smaller or greatei', according to the greater or less rapidity of the convergence). But the doctrine of infinite series proposes to find con- venient expressions for the sum of any part, or the whole of a series, without the labor of adding the several terms. §307. We have ■£- 7- = -7 ~Vv § 118 - .-. (§42. d) _4— =!(£ £-). (1) Thatis, A fraction of the form — ; — ; — - is equal to - of the dif- m{m-\-p) p fcrence between the two fractions — and — ; — . Now, as J m m-j-p this is true of any fraction of this form, it is^true of each of the terms of a series composed of such fractions. Hence the sum of such a series will be equal to - of the difference between two series, one consisting of terms of the form — , fit 9 and the other, of the form m-\-p I. Let it be required to find the sum of the series, 1 1 h&c, to n terms, and also to infinity. 1.2~2.3 o.4^ Here we have m{m+p) =1.2, 2.3, &c. = l(l-j-l), 2(2+1), &c. .-. q=z 1, p=zl, and m = 1, 2. 3 . . n, successive ly. Represent also the sum of n terms by S n , and, by gy, the sum of an infinite number of terms by S^. Taen n 1 — 1 == . 1 if" «+l ~ "+ 1 s" _i * * ' ' " n n+1 J If n = 00, we have (§ 138) —r-r = ; and .-. Sa> = 1. § 307.] SUMMATION. Otherwise; when n = cc, we have (§ 261. 1 ■ = - = !; and .•.£„=! n-\-\ n 1. What i3 the sum of the series, - — - -\- — - -f- — -- -f- 1 . c> O . O Hit &c., to il terms, and also to infinity ? IIere o(=i(i^)' o(-3^T2))' &c ' are of t!l! form (2n-l>[(2n-l)- r -2]- We have, therefore ^ — 1, j*> := 2, and m r= 2>i — 1 . Hence, S n —-< 1+1+* • ••• + 3 ' 5 2n— 1 1 [ 3 5'"'' 2«— 1 2n+l J — ? * rt — 2 V 1 2n+l)~ 2n+l' Also, making ?j z= oo, we have n n n t r» = °> or orr-r = nr ; and .-. >S' O0 — - 2«-J-l 2w-hl 2?i ' CO — r> 8. Find the sum of n terms of the series, +J 1 , 1 , » + _+&c; 1.4 ' 2.5 ' 3.6 ' 4.7 and also of the whole series to infinity Here IoW 2(2^3)' &C - Sive q = 1, p — 3, and rani, 2, 3 n. „ lf+2 + 3+I- + » ^« — *i l i ° i 1 i > ! n n-\-\ n-\-2 n-f-3 J 238 INFINITE SERIES. [§ 307. S - Vi + i + i I I ±J\ "3\ r 2" r 3 w+1 n+2 n+3J' s( w-j-1 «+2 rc+3 , n n . n re+1 ' 2n+4 ' 3w+9. n , /i 1 w 3w+3 ' 6^+12 ' 9»+27 AIM :fl.S=i(l+H-*) = H- 4. Find the sum of the series, 1+H~^4-tV+ &c - 5 (the denominators being the terms of a differential series of the seeond order, viz. the triangular numbers ;). Dividing the series by 2, we have by example 1st, above. S - 2 o. "What is the sum of the series, 1 1.1 +&c. 3.8 ' 6.12 ' 9.16 Take out of each term the common factor ^, by divid- ing the second factors of the denominator by 4, and the first factors by 3. as * 6. Sum the series, A _ _ + _ _ _ +&c. Here p — 2,q — n-fl, and m — 2ra+l ; n being taken = 1, 2, 3, &c.j successively. \"2 3 4 5 w-fl 1 -iis - ^ - ^- " ,:f 2»+i •'• ^» — 2] _2 3 4 w w+1 (' I 5 + 7 O" 1 "' ' ' T 2n+1 ' 2?i+3j Note. If n is infinite, i-i+i-i+&c.=q^=*. § 308.] SUMMATION. 239 The sum of n terms of this series will be equal to 1, if n is finite and even ; to 0, if it is finite and odd. Hence we have S 0B =*[§-(i-i+i-i+&c.)] = Ki-i) = tV _1/2_1 1 n+1 x .1(1 /1_ n±l_\} n ~ 2V3 2 T 2 2w+3 / 2 1 G T V2 2re+3/ j ' _1/1 L-Wf-L * > ~2VG :F 2(27i+3)y' Vl2 :F 4(2rt+3)>/• 7. What is the sum of the series, Am. S„ = - i -^ x + ~); S„=j. ' 8. What is the sum of the series, 1 1.1 — — + ~ — &c. ? 1.3 2.4^3.5 Am. S^ = -. 9. What is the sum of the series, 1.5^5.9^9.13^13.17^ * Here q = 4, p = 4, and m = 4»+l; « being taken =0, 1, 2, 3, &c, successively. rp t § 308. Again, as we evidently have (§ 118) q 9 I p i m(m-\-p) .. [m-\-(r—l)p) (m+p) {m+2p) . . {m+rp) I , (2) m{m-\-p){m-\-2p) . . . (wi+rp)' a series of terms in the form of the second member of this equation is equal to — of the difference of two serie3 of terms in the form of those in the second factor of the first member. 240 INFINITE SERIES. [§ 308. A K C 1. Sum the series, —- ^ .+ ^-^. +r — g +&c. Here 1 1T3T5T7 "^ 3T57T79 n ~ 5T7T9T1 1 5 ' Given i-2^r4 + 2-^^ + 3T4^' t0 fmd S». , « - 89 ^In*. /Sao — gj> CHAPTER XV. LOGARITHMS. § 309. All finite, positive numbers may be regard- ed as powers of any finite, positive number except unity. Thus, if 10 be taken as the base (§ 22. N.), 1, 10, 100, 1000, &c, xVs xfoy> t^Vo' & c -> Wl ^ De expressed as integral powers of the base; those above 1, positive ; those below, negative. Moreover, it is obvious, that all numbers between the in- tegral powers can be expressed as fractional powers, either positive or negative. That is, the base can be separated into factors so small, that^ a certain number of them multi- plied together (§ 12), or divided out of unity (§ 14), shall produce, at least to any degree of approximation, any given number ($319). a.) It is evident that 1 cannot be taken as a base of such a system of powers, because every power of 1 is 1. b.) It is also evident, that, if a proper fraction, as y 1 ^, be taken for the base, fractions will be expressed as positive, and integers, as negative powers. c.) The base must be a positive number ; for if it were negative, only such positive numbers could be expressed as should coincide with its even powers ; and only such nega- tive numbers, as should coincide with its odd powers. alg. 21 242 LOGARITHMS. [§310-312. d.) Again, of a positive base no negative number can be a power, unless the denominator of its exponent be even,, and the numerator odd (§§ 11. N. 2 ; 23. e,f). Hence the limitation to positive numbers. § 310. If all numbers, with the limitations above explained, were thus expressed as powers of a sin- gle number, the labor of multiplication and division would obviously be reduced to the adding and sub- tracting of the exponents (§§ 15, 16). Thus, since 100 = 10 2 , and 1000 = 10 3 ; 100X 1000 = 10 2 X10 3 = 10 5 = 100 000. Also, iooo = io;>, T i ¥ = io- 2 . 10(KH- T io = 10 3 -M0- 2 — 105 — ioo 000. 8 01 30 JL9897 ^O 2- =z \Ql'0O0'S'U0 ft — i()T oo OVISTS^ •2x5 = 10- 3oloso Xl0- G * S07O = 10 l . ) 311. When numbers are thus expressed as pow- of another number, the exponents of those powers are called logarithms" of the numbers so expressed; and the number whose powers are thus employed, is ed the base (§22. N.) f and sometimes also the radix (§23. d), of the system. r . fence, for a given base, § 312. The logarithm of any number is the expon- ent of the power to which the base must be raised, to produce that number. Tims', 2 is the logarithm of 100 to the base 10 ; because ; the exponent of the power to which 10 must be raised to produce 100. So, because 2 = 10- 301030 , .301030 is the logarithm of > the base 10 ; for .301 030 is the exponent of the pow- er to which the base, 10, must be rais ed to produce 2. t) dr. ?oy»r, ratio, &pt&ftbg, number ; number of the ratio. §313,314.] CHARACTERISTIC. 243 or.) Cor. I. The logarithm of the base of the system is al- ways 1. § 11. a. b.) Cor. II. The logarithm of unity in every system is zero. § 13. §313. The base of the system of logarithms in common use is 10. "We have, therefore, log 10,000 = log 10 4 = 4, log 1000 = log 10^ — 3, log (100 = 10 2 )=: 2, log (10 = lO 1 )^, log(l=10°) = 0, log ( T V = 10-i) = _ 1} log (.01 = 10-2) = —2, log (.001 = 10-3) _ _ 3j & c. a.) Hence, obviously, the common logarithm of any number between 1 and 10 is a proper fraction ; that of any number between 10 and 100 is 1 -f- a fraction; be- tween 100 and 1000, it is 2 + a fraction ; &c. b.) Again, the common logarithm of any number between ■fo and 1, as .3454, is between —1 and 0, and therefore it is — 1 -\- a fraction ; of a number between .01 and .1, as .0205, the logarithm is —2 -j- a fraction ; of a number be- tween .001 and .01, the logarithm is —3 -f- a fraction. § 314. c.) The integral part of a common logarithm is called its characteristic ; because it characterizes the logarithm by showing, where in the series of the powers of 10 the number of which it is the logarithm falls. TIio characteristic of the logarithm of a number greater than ten is positive; of a number less than unity, negatire (§ 309). d.) Moreover (§313), the characteristic of the common logarithm of any number is always equal to tho exponent of the integral poioer of 10 next below that number; and hence, in the common system, (1.) If a number be greater than unity, the characteristic of its logarithm is one less than the number of its integral places; (2.) if less than unity, the negative eharacteristii 244 LOGARITHMS. [§315,316. is numerically one greater than the number of cyphers be- tween the decimal point and the first significant figure on the left, in the decimal expression of the fraction. e.) Otherwise; the characteristic of a logarithm of a number is equal to the number of places from the unit place to the highest significant figure, including the latter ; posi- tive, if that figure be on the left of the unit place ; negative, if on the right. § 315. /.) We have log 20 = log 10 + log 2 = 1 + log 2 : log200=log 100 + log2 = 2 + log 2; log 525 = log 10 + log 52.5 = 1 + log 52.5 ; = log 100 + log 5.25 = 2 + log 5.25. So log .525 = log 525 — log 1000 = — 3 + log 525. But adding whole units to a mixed number cannot afi'ec^ its fractional part. Hence, the decimal part of the com- mon logarithm corresponding to a number expressed by any figures whatever, is the same, whether those figures stand all on the right, or a part or all on the left of the dec- imal point. Thus, we have log 25 = 1.397 960 ; log 250 = 2.307 9G0 ; log 25 000 = 4.397 9G0 ; log .025 =—2.397.960. g.) The principles of §§ 313-315 result from the employ- ment of the base of our scale of notation as base of the sys- tem of logarithms. * On account of this peculiarity, the common, or Briggs's 1 logarithms are much more convenient than any other for numerical computations ; and are, there- lore, in universal use for that purpose. §310. The following principles, resulting from the na- ture of logarithms as exponents (§§ 309-312), are formally stated here for reference. (&) So called from Mr. Henry Biiggs, who first suggested to Lord Napier, the inventor of logarithms, the employment of 10 as a base; and who completed the computation of the first table of logarithms \vith that base. §317,319.] INTERPOLATION. 245 1. The sum of the logarithms of any number of fact is equal to the logarithm of their product {§ 15). 2. The logarithm of a dividend minus the logarithm of a divisor, is equal to the logarithm of their quotient (§ 16 . 3. The double of the logarithm of a number is equal to the logarithm of the square of that number ; the triple of ids logarithm, to the logarithm of its cube, &c. ; the half to the logarithm of its square root; one third, to the logarithm of its cube root, &c. ; and, in general, n times the logarithm • r •»*■ h l& le „ c) Also, since ly — M. Ly, M= ~ = — = — &c. ( 1 6) Ly La Le That is, Cor. in. The modulus of any system is equal to any log- arithm in that system divided by the Naperian logarithm of the same number. d.) Again, since a is the base, la=z 1. 31=1. „„ That is, Cor. iv. The modulus of any system is the reciprocal of th-e Naperian logarithm of the base of the system. Thus, the modulus of the common system is = — -. J Log 10 Hence (§ 328) e.) We have f(a) = Log a. (18) That is, The denominator of the second member of equation v 8) in § 323 is the Naperian logarithm of the base. Note. This is evident also from § 324. 3, and § 326. (14). /.) Again, if e = the Naperian base, we have Le — 1. M=le. (19) That is, Cor. v. The modulus of any system is equal to the log- arithm of the Naperian base, taken in that system. Thus, the modulus of the common system is the common logarithm of e. -'■">2 LOGARITHMS. [§329,330. §329. ^0 f(a) — La — nLa n . §316.3. Also [§326. (14)] Zr^=a"-l-K«"- 1 ) 2 +K« ;; -l) 3 -K«"-l) 4 +&c. (20) 11 1 ... f(a) [= La — nLa n ] z= n[a n — 1— \{a n —1) 2 +&c] ; =:7i(a"-l), (21) when n is very large. / _1_ _ J. _ L ^ — 1 M Xf{a) ~~ La~~ nL{ n +/a) ) ~n("\/a—l) ' =i.^-. (22) n — a n —\ And, for the common system, taking n = 2 60 , we have approximately (21) /(a) = Zl0 = ??Z10 ;r =2Go.(i02 6O _l). 1 _1_ _L % _1_ 1 £10~2« * -L- 2co- 2 co •JQO6 i */ IV 1 Now we have — = 0.000 000 000 000 000 000 867 361 737 988 &c., and 2 |; ° j 10 26O = 1.000 000 000 000 000 001 997 174 208 125 &c _ 867 361 737 988 &c. — 1 997 174 208 125 &c." M{— S- = h) — 0.434 294 481 903 251 &c. (2-3 § 330. k.) If we take /(a) = 1, then (§§ 326. b; 328./) a — e, and [§ 329. (21)], approximately, i 11 11 \ =in(e" — l); or- = e n — 1; orl-f- = c. 1\" (1+-). .-.(W4J § 331, 332.] NAPERIAN BASE AND MODULUS. 253 . = 1+B * + »fcL>(i) Vi!!=i}(^!)(I) 3 +& „, 1 n 1.2 \w/ 1 .2 .d w or, reducing, and neglecting all the terms, which have n, n-, &c. in the denominator, inasmuch as (n being = 2 60 ) they cannot affect the first sixteen places of decimals, e = 1 +i+iVi^s+nnrT+ &.■]. p,7) I i i log y»=M[y»—l—l s (y»—iyj t & Q .] ; ( 28) 1 j_ .i logy» = M(y n —l), (29) when n is very large. Also (§327) logy = MIuogy. (30) - i § 333. Let a n l = y n , n being a very large positive num- ber, as 2 eo , and y being a small number greater than uni- ty ; so that y n may differ but very little from tm%, and -, n from aero. Then [§§ 312 ; 332. (30), (29)] x I 11 - = \og y» = M. lag y* = M(yn— I). fit x' 1 I re" 1 i So - = ly'» = M{y<» — 1) ; —z= ly"n — M(y">j:—l) ; &c. n n 1111 I 1 - ly n :ly' n : ly" n — y n —l : y' n —\ : y" n —l. l l »r log y n =y n — 1. §245. That is, approximately, 1. The logarithms of numbers very wear ttm'Jy are to :jach other as the differences of the numbers from unity. Thus, log 1.000 001 = .000 000 434 294 ; log 1.000 002 = .000 000 868 588 ; and .000 000 434 294 : .000 000 868 588 =.000 001 : .000 002. Also (§238) ii. 1 JL 1 1 1 1 lyin—ly* : ly" n —ly n =y ,n —y n : yH«—y\ That is, approximately, 2. The differences of the logarithms of numbers very near unity are to each other as the differences of the numbers. See example under 1, above; and under § 335. b, c. § 334. a.) Let y be not <2, and w = 2 G0 , so that y"may § 335.] DIFFERENCES OF LOGARITHMS. 253 be a very large number ; also let D = a small number, as 1, 2, &c. Then (§§ 31G. 2 ; 115 ; 333. 1) ii' y n is constant (5 247. 3). That is, approximately, If large numbers differ by quantities very small in com- parison with themselves, the differences of their logarithms will be as the differences of the members. Thus, in the common system, if the logarithms are car- ried to only seven places of decimals, we have n0000 = 4; 1 10 001 = 4.000 043 4; HO 002 = 4.000 080 8 &c. ; where, for equal increments of the number, we have equal increments of the logarithm. Note. We must not extend the series far, because the difference? of the numbers would cease to be very small compared with the num- bers themselves. t § 335. b.) We have, by (26) and (29), 11 11 Ly n = y n —1, and Ly ln = yi"—\. 1 III Ly' n —Ly n = y" l —y>\ That is, approximately, The difference of the Naperian logarithms of two num- bers very near unity is equal to the difference of the num- bers. Thus, Log 1 =0; Log 1.000 001 = 0.000 000 999 999, or 0.000 001 ; Log 1.000 002 = 0.000 001 999 998, or 0.000 002. That is, the numbers differing by .000 001, their Nape- s-ian logarithms differ, within an extremely small fraction. by the same quantity. c.) In any system whatever, we have I I i i ' . log y» = M(y»—1 ) ; log y ln — M{xj n —\ ). 256 LOGARITHMS. [§336, I 1 1 i log y> "-log y« = M(y>»-f). That is, in any system, approximately, The difference of the logarithms of two numbers very near unity is equal to the difference of the numbers multi- plied by the modulus of the system. Thus, in the common system, log 1 =0; log 1.000 001 = 0.000 000 434 29 ; log 1.000 002 = 0.000 000 868 58. That is, having cliff, numbers = .000 001, we have diff. logarithms = .000 OOlX-434 29. § 329. (23) Note. If we make M=l, the logarithms become Naperian, and this principle becomes identical with the preceding. d.) Reasoning as in § 334, we shall find, that, if any number whatever receive an increment very small in com- parison with itself, the corresponding increment of the log- arithm is approximately ecptal to the modulus into the fo- ment of the number, divided by the number. Thus, l(ynJ r D)-ly" = l(lJ r ^ ; )=M^. [See §§337. a; 332. (28), (29)]. Thus (§334), log 10 001— log 10 000 = .434X ro J o o = - 000 043 4 ' So, log 9865 = 3.994 097, log 9806 = 3.994141. And log 9866— log 9865 =.434 294x^5 = - 000 ° 44 - §336. e.) If a number exceed unity by a very small quantity, its logarithm exceeds zero by a very small quan- tity. Now, §332. (29), M= -^pU Therefore, the modulus approximately expresses or mea- sures the ratio (§ 230) of the infinitesimal excess of the logarithm above zero, to the corresponding infinitesimal ex- cess of the number above unity, § 337.] COMPUTATION. 257 m , .000 000 434 29 . 6jor Thus (§ 335. b), M= -^qoqqoI — = * 434 29 * /.) Or (§335. d), M= .X_/ , ' • Thatis, The modulus approximately expresses the ratio of an Infinitesimal increment of any logarithm, to the correspond- inc increment of the corresponding number, divided by tli> number itself. Thus (§ 335. d) M= ' 00 ° 1 044 = .434 294. § 337. Putting, in (27) of § 332, l+y instead of y, and of course y in place of y — 1, we have log(i+y) = ^(y-^ 2 '+^ 3 -|2/ 4 + &c.). («) Putting — y for y in («), we have l0g(l-y) = J/(-y-I 2 /2_^3_^4_ &c . )> (/)) Subtracting (5) from (a), 1-4-w E «(l-^)-'(l-jr)='^=2^(jr+i»?-+4e.)- W ••• l0g(l+y)=l0g(l-3/)+2iJ/^+^3 +i3/ 5 H _ &c . ) . (31) This series converges with tolerable rapidity, when y is a small fraction. Thus if y = .1, we have . 1-f.l 11 log n-iog 9 = 2ir(i 5 + g( L ) - +^-+ & c). , log 11 = log 9+2^(^ + 3^- +^- 5+& e.). Here, if no more than seven places of decimals ai*e re- quired, the fourth term of the series may be neglected. Now, § 329. (23), M— 0.434 294 48 ; and, §§ 316. 3 ; 322, log 9 = 2 log 3 = .954 242 5. *22 258 LOGARITHMS. [§338, 339. log 11 = .954 242 5-J-.868 588 96(.l+.000 333 3+&c). log 11 = 1.041392 8. log § 338. Making in (c) 1 1+y *4-l and = "*^i+SKKF+ H ' 8(2*fl)« ' 5(22+1)5 + &c -)' log (=+1) = log HJJr^ + p^s +&c.).(32) Thus, if * = 10, we have z-\-l = 11 ; and log 11 = log 10+.868 588 96(^- + -^- 3 +&c.). This series may be summed thus ; 2*4-1= 21.868 588 96 =23f (2*-f-l) 2 = 441 (2*4-1) 2 = 441 041 361 38-r-l --.041 361 38 93 794-3 = 31 26 214-5 = 4 .041 392 68. log 11 = log 10 4- .041 392 68 = 1.041 392 68. This series converges much more rapidly than the preceding. Many still more rapidly converging series have been devised. We shall give, however, but a single example. u ! — 1, and (32) be- §339. Let*4-l = ?o 2 . Then ; comes log .. =l„g («._l) + 2Jf ( s -^. + _ —+&*■)■ Or, as log w 2 = 2 log u, and log («- — 1) = log («4-l)4~log (w— 1), we have log (?t -j- 1) = 2 log u — log (« — 1) — 1 Uf (i5r=r+ 3(2« 2 — l) 3 ' 5(2u 2 — 1) 1 _Ti^+ & 4 (38) § 340.] EXPONENTIAL THEOREM. 259 Thus, if u = 12, we have log 13 = 21og 12-Iog n-2^(^ + 37 4 ? +&c.). Now 2 log 12 = 2 (log 34-log 4) = 2 (log 3+2 log 2), = 2(.477 121 25+.602 060) = 2.158 364 ; and (§ 336) log 11 = 1.041 392 68. The series may be summed thus ; 2« 2 — 1 = 287 .868 088 96 = 2,1/ (2**2— l) 2 = 62 369 .003 026 44-KL = .003 066 44 5-^3 = 2 .003 026 45. log 13 = 2.158 362 5—1.041 392 68— .003 026 45. = 1.113 943. For all lai-ger numbers, the first term of this series will give the value correctly to eight places. Note. The logarithms of the prime numbers only need be com- puted by such processes ; the logarithms of all other numbers being found by the proper combination of the logarithms of primes. Thus, log 4 = 2 log 2 ; log 6 = log 2+log 3 ; &c. The logarithms of 2 and 3 may be found from formula (32), and the logarithms of 5 and 7 from (32) or (33). EXPONENTIAL THEOREM. § 340. In the equation a x = y, we have found x in terms of y ; i. e. a logarithm in terms of the corresponding num- ber. We shall now find y in terms of x ; i. e. a number in terms of its logarithm. Put y[= a* = (l+o-l)*] = [(l+«-l )»]""; n being any number whatever, and of which the value of y is entirely independent (§ 276 ; 277 ; 280. e). Developing, (1+ ^l ) » = l +n ( _l ) _L.^d)( a _l)2 +&c .. 260 • LOGARITHMS. [§34l, or, (1+a—iy = 1+An+B?i*+Cn 3 + &c. ; A, B, C, &c. being functions of a ; and, evidently, A = a— 1— J-(«— 1) 2 -B(«— 1 ) 5 ~ & c- = La. § 326. £. Then we have y = [(i+o-i)*]» ;= [i+(^+^ 2 -Hbc.)]s: ... y = l+^(An+B^+&c.)+ ^=J } (^n+i^+&c.) 2 r^-rc) (*-2» )_ ( j w+jR|fl+&c , } 3+&c . . $ 295. *. or ?/ = l+a:(J+JBH-&c.)+ ^^\-4+JBh+&c.) 2 f x(x — n)(x — 2n) 1.2.3 -(^4+^+&c.) 3 +c&c. A 2 x 2 A % x z ■.(§277) # = «'=l+^+— +— ~ 3 +&c. (34) , (Za) 2 a: 2 , (Za)^ 3 , Or af=l+Za.aH- i Y72~+ i i-^73-+&c (3o) § 341. a.) We know the value of A from §§ 325. (12) ; 328. e. But, if we did not know it, we might find it from the equation (34) itself. Thus, Let x = -j. Then (§330) ^= 1 +I+l^ + i4.-3 + IT2 L 3-.T+ &C - = e - < 36) .-. (§52. N.) a — e A . (37) That is, -j- is the logarithm of e to the base a (§ 328. d, /') ; and A is the logarithm of a to the base e (i. e. the Na- perian logarithm of a [§ 328. e]). 5.) Or thus ; taking the logarithms of both members of (37), log a = A log e. §316.3. §342-344.] EXPONENTIAL EQUATION. 201 log a 1 1 , . A = r- 2 — = Lo°- a = -. = ^rp, a being the base. log e log e M That is, A is the reciprocal of the modulus of the sys- tem of logarithms whose base is a. Note. The logarithms may be taken in any system, provided both be taken in the same; the logarithms of two numbers having the same ratio in one system as in another (§ 328. b). §342. c.) We have found (§341) the value of A, in terms of a. We may, if Ave prefer, assign a value to A, and find the corresponding value of a. Thus, let A = l. Then, «' = 1+*+ ~ + —^ +&c. Making x = 1 , a — 1+1+ —% + OT3+ &C " = e ' § 343. d.) This is called the exponential theorem ; and y, in the equation, y = a x , is called an exponential func- tion of x. On the other hand, x is called a logarithmic function ofy ; being the logarithm of^ to the base a. e.) These two classes of functions ai*e also called teans- cendentai/ functions ; as transcending the elementary op- erations of Algebra. EXPONENTIAL EQUATION. § 344. The equation, a x = b, is called an exponential equation. If a is the base of a system of logarithms, we have simply x — log b. But if a is not the base of a system of logarithms, take the logarithms of both sides of the equation, in any system. The common system is usually most convenient. Then, log (a r ) = log b ; ov x log a = log b. § 316. 3. (d) Lat. transcendo, to exceed, surpass. 262, LOGARITHMS. [§ 345 _log b log « 1. Given 12* = 20, to find x. x log 12 = log 20 ; i. e. 1.079 181 25a: = 1.301 030. log 20 1.301030 x = f^T^ = i n~oiQio* = L205 57 &c - log 12 1.0/9 181 25 2. Given 60 3 ' = 7, to find x. Am. x = .475 273. 3. Given 125 x = 25, to find x. Am. x = §. § 345. c?.) If the equation be of the form, x x = b, we have x log a; == log b. This equation, can be most conveniently solved by the method of trial. For this purpose, find by trial two ap- proximate values of x. Substitute these values successive- ly in the equation, x log x =. log b, and note the error in each result. Then Diff. of the results : Diff. of the assumed numbers = the • rror in either result : the correction to be applied to the cor- responding assumed number. This correction, being applied, will give a nearer ap- proximation to the true value of x. This new value may now be taken as one of the assumed numbers, and a still closer approximation obtained ; and so on. 1 . Given x' : =z 100, to find x. Here we have x log x '== log 100 = 2. Also, since 3 3 = 27, and 4 4 z=. 25 G, the value of x lies between 3 and 4. Substituting 3 and 4 successively, we have 3 log 3 = 3 X 0.477 121 25 = 1.431 363 75 ; whence 2 — 1.431 363 75 = .568 636 25, error; § 34:5.] INTEREST. — POPULATION. 263 also 4 log 4 = 4 x 0.602 059 99 == 2.408 239 96 ; whence 2 — 2.408 239 96 = — .408 239 96, error; and 0.976 876 21 = difference of results. .967 876 : 1 =n— .408 239 96 : —.418, correction. 4 — .418 = 3.582 = x nearly. Again, we find x> 3.582, and <3.6. Therefore, substi- tute these values, and repeat the operation. Thus, 3.582 log 3.582 = 3.582 X 0.554 126 5 = 1.984 881. 2 — 1.984 881 = .015 119, error. 3.6 log 3.6 = 3.6 X 0.556 301 9 = 2.002 689. 2 — 2.002 689 = — .002 689, error. Also, .017 811 = difference of results. .-.. 017 811 : .018=— .002 689 : —.002 717, correction, 3.6 — 0.002 717 = 3.597 283 = x nearly. That is, 3.597 283 3 -597283 — ioo. 2. Given x' '— 5, to find x. Ans. x = 2.1293. 3. Given x* = 2000, to find x. Ans. x — 4.827 822 63. 4. Given a, m and I of an equimultiple series, to find n <^ 64 - J) - A , B= !2|lri21? +1 . I02; m 5. Let a =■ 2, I = 162, and m = 3 ; and find n. Ans. n — 5. 6s In how many years will p dollars amount to A dol- lars, at r per cent, compound interest ? We have (§ 258. 5) A —p{l-{-r) t . . low A — log p Ans.t— ° n ,\ J . log (1+r) 7. In what time will $100 amount to 8200 (i. e. in what time will a sum of money double itself )$ at 6 per cent compound interest? Ans. 11.89 years Here p = 100, A = 200, and 1+r = 1.06. 204 LOGARITHMS. [§845. Notes. (1.) The solution of most questions relating to com- pound interest may be greatly facilitated by the use of logarithms. (2.) The formula? of compound interest apply also to the increase of population in a country. 8. Find r fro'm the formula, A =p(l-\-ry. See § 258. 5? 9- . , ... , x log A — loo; p Ans. log (1+r) == — 5Ll . 9. The population of the United States in 1830 was 12 866 000, and in 1840, 17 068 000. What was the year- ly rate of increase ? Here A = ll 068 000, p '=. 12 866 000, and t — 10. . „ , . log 17 068 000— log 12 866 000 .-. lug(l+r)=-2 , 10 7.232183—7.110118 = .012 206 5. 10 1 -f r = 1.0285 ; and r = .0285 = 2J$ per cent. 10. At the same rate, what will be the population in 1850? Here p = 17 068 000; r=.0285, and < = 10. A[=p{l-\-ry~] = 17 068 000 (1.0285) } °. log A = log 17 068 000+10 log 1.0285. Ans. A = 22 654 000. 11. In how many years will the population amount to 50 000 000 ? Ans. In 38.24 years from 1830. 12. If the number of slaves in the United States in 1830 was 2 009 000, and in 1840, 2 487 000, what was the yearly rate of increase? Ans. 2% per cent. 13. At the same rate, what will be the number in 1850 ? in 1860 ? Ans. 3 078 700, in 1850 ; 3 811 000, in 1860. 14. The population of Virginia in 1830 was 1211 400. and in 1840, 1 239 700 ; that of New York in 1830 was 1 918 600, and in 1840, 2 428 900. What was the yearly rate of increase in each state ? Ans. In Virginia, .0023, or less than \ of 1 per cent; in New York, .0238, or more than 2^ per cent. CHAPTER XVI. THEORY OF EQUATIONS. § 346. We shall confine ourselves here to the considera- tion of equations containing but one unknown quantity. 1. If the exponents of the unknown quantity in such an equation be all integral, or if their differences be all inte- gral, the degree of the equation is correctly expressed by the difference beticeen the greatest and the least of those expon- ents (§§40. a;51.b). 2. But, if the difference between any two of the expon- ents be fractional, this difference between the greatest and least, obviously, may not express the degree of the equa- tion. Thus, evidently, x x' 2 -J- ax -\- bx^ -\- c — is not of the second degree. Reducing, however, the exponents to a common denom- inator, we have e 3 i x^ -\- ax 3 -f- foe 3 ~f- c = 0, which may be said to be of the § degree. In. fact, if we make 3 */x — y, we shall have y 6 + «y 3 + 5y + c = j an equation of the sixth degree in respect to y (i. e. in res- i pect to x 3 ). Hence, when the difference between any two of the ex- ALG. 23 ii66 THEORY OF EQUATIONS. [§ 347, 348. juts is fractional, the degree of the equation is the dif- ference between the greatest and least exponents, expressed in terms of the least common denominator of all the expon- mts. Thus, 1- 3 2 .> - -\- ax' J -\- b (=: x G -\- ax 5 -\- b) = is of the § degree. i i i x 3 -f- ax 1 -4- bx° -J- c = is of the f% degree. 347. It is evident that equations of this kind can be expressed in integral degrees, by reducing their exponents to a common denominator, m, and substituting a new un- known quantity for the mth root of x (i. e. by putting y=. y ). Hence, we shall need to consider equations of integral legrees only, and shall suppose them reduced to the fol- lowing form, viz. x n + A 1 x n - 3l ^-A z x n ~ s . . . -f A^^+Al—O. (1) We shall also assume, that every equation has at least >ne root. Note. A single symbol, as X, or f{x), is sometimes put for the il member of an equation. Thus, X=0, or /(a:) =0. DIVISIBILITY. — ROOTS. 18. Let a be a root (§ 39) of equation (1). Then, J-i^/'-'+ioa"- 2 • • • .+A n . 1 a-\-A n — 0. .-. A, r =~a n — A l a n - 1 —A 2 a n - 2 .... —A n . x a. Substituting this value of A n in (1), we have .--— a n )-\-A l (x , '~ l — a n ~ l ) . . . +A H - 1 (x — a) = 0. Now this expression is divisible by x— a (§§ 81, 9G). Hence (compare § 213. 5), If a be a root of the equation, rJ-i/" 1 i-A^ v c + A n = 0, Hie first member of the equation is divisible by x—a. $348.] DIVISIBILITY. — ROOTS. '-''''. a.) This principle may be demonstrated otherwise ; thus. If v» T e actually divide the first member of (1) by x—a. we shall have, representing the quotient by Q, and the re- mainder by i?, f(x) = x" -f- A x x"~ i . . . + A , = (x — a> of a, that the given polynomial is of x ; i. e. it will be what the given polynomial becomes, when a is substituted for x. ■See §211. 1. Notes. (1.) The remainder is independent of or. For, if it con- tained x, the division might be continued farther. R, therefore, since it does not contain x, will have the same relation to a, whatever value is given to x. (2.) It is evident also from this principle, that. if a is a root of the equation, the remainder will be zero, and the di- vision perfect. 1. Divide x 3 -}-^4 1 x 2 -|- A 2 x-]-A 3 by x — a. Bern. a$ -j- A x a} -f- A a q + A 9 > 2, Divide x? — 8x 2 + 1 Ix + 20 by x — a. Rem. a3_^8a 2 -f-llrt-f-2M, 268 THEORY OP EQUATIONS. [j 349, 85& § 349. c.) Conversely, if the first member of (I) be di- visible by x — a, then 7? = a" + ^ 1 a n -i + J 2 a n ~ 2 +J n = 0; i. e. the substitution of a for x satisfies the equation (§ 39) ; and, therefore, a is a root of the equation. d.) Hence, to determine whether a is a root of the equa- tion, x" + A x x n -i -\-A n =:(), we have only to divide the first member by x — a. And, (1.) If the division is perfect (§ 82. g), a is a root; (2.) if it is not perfect, the remainder is the value of the first member, with a substituted for x (§ 348. b). § 350. 1. Find whether 3 is a root of the equation. a;» -a;* — 25a; 3 -4- 85a: 2 — 96a; + 36=0, Divide by x — 3 ; thus (§ 86), 1 _ i __ 25 + 85 — 96 + 36 + 3 + 6 — 57 + 84 — 36 1 + 3 1 + 2 — 19 + 28 — 12, 0, the remainder, Hence, the remainder being zero, 3 is a root. 2. Find whether 4 is a root of the same equation. In performing these divisions, the first coefficient, being 1, need not be written. Thus, 1 — 1 — 25 + 85— 96 + 36 4 + 4 + 12 — 52 + 132 + 144 1 + 3 — 13 + 33+ 36, + 180, the remainder. Consequently, 4 is not a root ; and the substitution of 4 for x reduces the first member to 180. 3, What does the first member of the equation, x * _ 7^3 _ 20a: 2 + 30a; — 48 r= 0, become, when 7 is substituted for x ? Ans. — 818. This may, of course, be determined by the actual substitution of 7 for x. But we arrive at the same result much more conveniently bj dividing by x — 7 S as in the preceding examples. 5 351.] NUMBER OF ROOTS. 4. What does the first member of the equation, a* — 20x- T -96 = 0, become, when 7 and 9 are successively substituted for .<■ r Ans. 5, and — 3. NUMBER OF ROOTS. § 351. Let a x be a root of the equation, X = x*+ A x x n ~ 1 + &c. = 0. 1 Then if we divide by x — a v we shall have, evidently, an equation which will be satisfied, if either of its factors be equal to zero. Making x"~ l + B 1 x n ~ 3 + Sep. = 0, and supposing a 2 to be one of its roots, the primitive equa- tion will take the form (§ 348), X— (x — a x )(x — a 2 )(x"~ 2 -{- C^~- . . «. + C„_ 2 ) = <*. AVe may, obviously, proceed in this way, diminishing the degree of the polynomial by unity at each division, till w have taken out n factors of the form x — a. X=x n + A 1 x n -* .... -\-A. n _ lX -\-A u = (x — a l )(x — a 2 )(x — a 3 ) (x — a n )==0. (2) Now this equation will be satisfied, if any one of its n factors be equal to zero ; i. e. if x be equal to any one of the n quantities, « x , a 2 . . a a . Therefore, 1. Every equation of the form, X=x n + A 1 x''~i +A = 0, can be resolved into n binomial factors, of the form, x— (2.) Every equation has as many roots as there are mite in its degree. See § 213. 1, 2. Thus (§ 348. a), the equation, £3 _ 8x 2 -|- 1 \x +20 = (x — i)(x — 5) (x + 1) - 0, has the three roots, 4, 5, — 1. *23 270 THEORY OF EQUATIONS. [§352,353. § 352. Suppose b, a quantity different from any of the roots a 1 , a 2 , a 3 , &c, to be a root of equation (2). Then we have (5.— ai )(b — a 2 )(b — a 3 ) . . (b — a n ) = 0, an evident absurdity ; because, by hypothesis, b being not equal to any of the quantities, a x , a 2 , &c., no one of the factors, b — a v b — a 2 , &c. can be equal to zero. Hence, The number of roots of an equation cannot be greater than the number of units in its degree. Hence (§§ 351, 352) § 053. The number of roots of an equation is al- ways equal to the number of units in Us degree. a.) These roots may be all real ; or part or all of them may be imaginary (§ 216). b.) Again, they are not always different from one anoth- er. Any part, or all of them may be equal (§ 205). An equation will, of course, contain equal roots, when it-: first member contains equal factors. Thus, the equation, x 3 — 3x 2 + 3x — 1 = {x — l)(x — l)(x — 1) = 0, has three roots, each equal to 1. c.) If we know a part of the roots of an equation, wc may find, by dividing by the corresponding factors, the equation of a lower degree, which contains the remaining roots (§351). 1. One root of the equation, x* — 9x 3 -f 19x 2 + 9x — 20 = 0, is 1. Find the equation which contains the remaining roots. Ans. x 3 — 8x2 _j_ \\ x + 20 = 0. * * $ 2. Another root of the same equation is 4. Find the equation containing the other two roots. Ans. x 2 — 4x — 5 = 0. 3. Find the remaining two roots by § 207 or 208. $354,355.] NUMBER Off ROOTS. — COEFFICIENTS. 271 4. One root of the equation, x 3 — 1 = 0, i. e. x 3 = l, is, obviously, 1. What are the other roots (§ 207) ? Arts. i(- 1 + (— 3)*), and \ (- 1 - (- 3) *). d.) Either of the roots of the last equation, being cubed, will produce 1. Thus, every number has three cube roots ; one, real; and two, imaginary. In like manner, every number has four fourth roots ; and, in general, n nth roots. § 054. e.) The principle of § 353 may be applied to equations of fractional degrees (§ 346. 2). Thus, the number of the roots of the equation, x' 1 — 7ic Tj " -j- 6 = 0, may be said to be _, . For we find »* = 1, 2, or -3 ; and, consequently, x = 1, 4, or 9. Now these three values of x correspond to six values of x' 2 , only three of which satisfy the equation ; as will be seen, if we take x = — 1, —2, or +3. The values of x, therefore, i. e. the roots, may properly be said to be half roots (§12). i i So, the equation, x 5 — 2 = 0, i. e. X s = 2, obviously gives x~=. 8, or x — 8 = 0. But x- 8 = (J- 2)(*M- 1 -V-3)(a£+- 1 -fV-0), only one of which partial or component factors (§ 12), with the corresponding partial root, is found in the given equa- tion. The equation may, therefore, be said to contain only one third of a root. See § 221. 2, 3. COEFFICIENTS. § 355. Let a v a 2 . . . • a* be the roots of an equation. Then we shall have 2 72 THEORY OF EQUATIONS. [§ 355. a?-\- A-^x"- 1 . . -\- A n = (x — a x )(x — a 2 ) . . (x — a„) = 0. Multiplying (§283), v*+\>A x &~i . . . . + A n — x n — a x — a 2 — a. — a, x n ~- — a,a a, j. ^ o — a x a 2 a^ a 2 a 3 a 4 &c. + o. 2 a n &c. Hence (§ 277), A ± = — 0>\ — « 2 — a 3 • ■ ^4 2 — a l a 2 -\-a 1 a 3 . . -}-«!«„ x n ~ 3 . ,±a 1 a 2 . . a„ — ct„ + « 2 « rt + &( -'' J 3 == — a x a 2 a 2 — a 1 « 2 « 4 — a x a d a a Sec. ^ 4 = a 1 « 2 a 3 a 4 -(~ a i a 2 a 3 a 5 "T~& c ' A r . ± o 1 a 2 a a a 4 « 5 . . . o a . That is. (1.) The coefficient of the second term is equal to the sum of the roots loith their signs changed (§ 213. 3). (2.) The coefficient of the third term is equal to the sum of the products of the roots taken two and two (§ 213. 4); (3.) that of the fourth term, to the sum of their products taken three and three ; and so on, the signs of the roots be- ing changed in every case. (4.) The absolute term (i. e. the coefficient of x° [§ 208]) is the product of the roots taken cdl together, with their signs changed. a.) It is evident, that, in the third, fifth, seventh, &e. terms, the number of factors being, even, the result will be the same, whether the signs of the roots be changed or not (§213.4). b.) The last term will be positive or negative, according as the number of positive roots is even or odd (§215. 1, 2). c.) If the roots be all negative, the factors will be of the § 356.J FORM OF THE ROOTS. — NOT FRACTIONAL. 273 form x-\-a 1 , x-\-a 2 , &c, and the terms will, evidently, all be positive (§ 215. 1, 3) ; if the roots be all positive, the terms will be alternately positive and negative (§ 215. 1, 3). d>) If the coefficient of the second term be zero, the sum of the positive roots is numerically equal to the sum of the negative roots (§ 214. 1). e.) Every root of the equation is a divisor of the last term; and, hence, if the last term be zero, one of the roots must be zero (§ 214. 2); or rather, in this case, the equa- tion becomes of the (n — l)th degree (§ 203). 1. Form the equation, whose roots are 2, 3, and — 4, Ans. (x—2)(x — 3)(x4-4 : ) = x^—x n —lix-\-2i=zO. 2. Form the equation, whose roots are 1, 1, 2, and 3. Ans. x* — 7x 3 + 17x 2 — 17 -f- 6 = 0. 3. Given the roots, 2,-1 +y— 3,-1 — ./— 3 ; to find the equation. Ans. x 3 — 8 = 0. FORM OF THE ROOTS. §356. Let the equation, st*-\- A^x"- 1 . . -\-A n = 0, have its coefficients all integral (the coefficient of the first term being unity) ; it is required to determine whether it can haye a fractional root. If possible, let -r, a fraction in its lowest term?, be a root of the equation. Then we shall have a n a n ~ 1 Multiplying by b"~i, and transposing, ^=z-,A 1 a n -^—A 2 a n ~^b . . — AJT~*. Now all the terms of the second member of this equation, are whole numbers, while the first member is an irreduci- ble fraction. That is, we have an irreducible fraction equal to a whole number; which, evidently, is impossible. Ilence, 274 THEORY OF EQUATIONS. [§35?. If the coefficient of the first term he unity, and the other orfficients all integral,the equation cannot have a fraction- al root. a.) It is not, therefore, to be inferred, that all the root? are integral. They may be either integral, irrational (§§ 153, 175), or imaginary (§ 23./. 2). § 357. Let the coefficients of the equation, X= 0, be all real ; and let a -\- bj 1 — 1 be a root of the equation. The quantity bj— 1 can have resulted only from the extraction of an even root, which must have given, at the same time, — bj— 1 (§ 23./. 1). Consequently, a —■ bj—l must be a root of the equation. Otherwise ; the sum and product of the roots (§ 355. 1, 4) must both be real. Therefore, if one root be a + fl«/— 1, another must be a— bj— 1, so that their product (§ 186) and sum may both be free from imaginary expressions. Hence, If the coefficients of an equation be all real, the number of its imaginary roots must be even (§ 217. I.). 0.) Thus, there may either be no (§ 63. N.) imaginary roots, or there may be two, four, &c. Hence, b.) Cor. r. Every equation of an odd degree has at least one real root, with a sign (see c. below,) different from that of the last term (i. e. of the coefficient of x°), c.) We have [§§186, 162] (a + 6y— l)(a— bj— 1) z=a 2 -i-& , .i positive quantity (§ 11. N.). Hence, Ccr. ii. If all the roots of an equation are imaginary, the last term must be positive (§ 216). Hence, Cor. in. Every equation of an even degree, whose last term is negative, has at least tivo real roots ; one positive, and the other negative (§§ 68. a; 215. 2). 1. Giver, the roots, 5, 3 -\-*/ — 4, 3 — «/ — 4 ; to form the equation. Ans. a: 3 — 1 lx 2 + 13.x* — 65 ■—. 0. § 358, 359.] signs of the roots. 275 2. Form the equation, whose roots are — 6 -{- 5«/ — 1, — C — 5y— 1, 1 -fV— 4, and 1 — y— 4. Ans. x* + 10x 3 — 42a: 2 — 62a: + 305 = 0. 3. Form the equation, whose roots are 2, —2, 1-fV— 37 and 1— y— 3. .4ws. a: 4 — 2x 3 -j- 8a: — 16 = 0. § 358. d.) Again (§ 218. h), (x-a-5y-l)(x-fl-fiy-l) = (a: — a) 2 -H 2 ; a result necessarily positive for every real value of x. Con- sequently, Cor. iv. (1.) The product of all the imaginary factors is positive for every real value of x. Hence, (2.) The sign of the j£rs£ member, for any ?-caZ value of x, depends on the real factors. And, (3.) If all the roots are imaginary, the first member will be positive for every real value of x. e.) The product, (a; _ a — y— 5) (a: — a -+V— b) — x- — 2ax -f a- + 6 2 , of the factors corresponding to each pair of imaginary roots, or conjugate? roots, as they are sometimes called, is real. Hence, Cor. v. Every equation may be resolved into real fac- tors ; of the first degree, corresponding to the real roots ; and of the second degree, corresponding to each pair of imaginary roots. SIGNS OF THE ROOTS. § 359. Let a be a root of the equation, x' l + A 1 x n - lJ r A 2 x n ~ 2 . . . . -f A n ^ 1 x-\-A n = 6. [{I) Changing the signs of the alternate terms, we have x n — A jK"- 1J r A 2 x n -z— A^x n - 3 -\- &c. — ; (2) or (§ 44. a), changing all the signs of (2), — x n +A 1 x n - 1 —A 2 x n ---\-A 3 x'*- 3 —&c. = 0. (3) (e) Lat. conjugo, to join together. 27G THEORY 0? EQUATIONS. [§ 360. The equations (2) and (3) are, obviously, the same; as will be seen by transposing, in each, all the negative terms to the other side. Now, if -[-a be substituted for x in (1), and — a, in (2) when n is an even number, or in (3) when n is odd, the re- sults will be precisely alike. But the substitution of -f- a in (1) reduces the first member to 0. Consequently, the substitution of — a in (2) or (3) reduces the first member to 0, and therefore — a is a root of the equations (2) and (3). Hence, If the signs of the alternate terms in an equation be changed, the signs of all the roots will be changed. Form the equations, whose roots are 1, 2, and 3 ; and — 1, — 2, and — 3. § 3G0. A permanence* o£ signs occurs when two succes- sive terms are affected each by the same sign ; a variation, when their signs are different. Thus, x -f- a = -exhibits a permanence, and x — a = 0, a variation ; the first corres- ponding to a negative, and the second, to a positive root. I. Let the signs of the terms in their order, in any com- plete equation be -\- -| ( , and let a new factor, x — a =. 0, corresponding to a new positive root, be intro- duced. The signs will be as follows, viz, + + — - + - + ~ + + + ~ Now, in this result, it is manifest, that each permanence is changed into an ambiguity ; and that, whether there be one, or any greater number, of double signs, the single signs immediately preceding and following are always un- like. Hence the number of permanences may be dimin- ished, but cannot be increased. « Hence, the number of signs being one greater than be- (/) Lat. permaneo, to continue. §361.] 'SIGNS OF THE ROOTS. 277 fore, the number of variations also must be at least one greater. Now the equation, x — a = 0, containing one positive root, has one variation. Consequently, as every additional pos- itive root introduces, at least, one additional variation, The member of variations can never be less than the number of positive roots. II. (1.) By like reasoning it can be shown, that the in- troduction of a negative root (i. e. of the factor x-\-a) will introduce at least one permanence ; and that, therefore, The number of permanences cannot be less than the number of negative roots. (2.) Or, if we change the signs of the alternate terms, the variations will evidently become permanences, and the per- manences, variations ; and the negative roots will, at the same time, become positive (§ 359). But the number of variations in this equation cannot be less than the number of its positive roots. Therefore, the number of permanences in the primitive equation cannot be less than the number of its negative roots. Hence, universally, in a complete equation, § 361. The number of positive roots cannot be greater than the number of variations of sign ; nor the number of negative roots, greater than-the number of perman- ences. Note. A complete equation of the rath degree, * n + A*"- 1 +A n _ 1 x + A n =zQ, must, obviously, contain n4-l consecutive powers of x; and, of course, n + 1 terms (§§195, 196). 1. How many permanences and variations in the equa- tion, whose roots are 2, 2, and — 5 ? Ans. The equation is (x — 2)(x — 2)(x-\-5)=:x3-\-x 2 — l6x + 20 = 0; showing one permanence, and two variations, as we have seen there must be. alg. - 21 -' fl THEORY OF EQUATIONS. [§362. 2. IIow many permanences and variations in the equa- on, whose roots are 1, 2, 4, and — 4 ? a.) The whole number of variations and permanences must, evidently, be equal to the degree of the equation e equation being complete, or, if not complete, being rendered so by the introduction of cyphers, as in § 362). Therefore, Cor. i. If the roots of an equation be all real, the num- ber of positive roots must be equal to the number of varia- tions ; and the number of negative roots, to the number of permanences. See § 218. 1, 2, 3. § 362. b.) If any term of the equation be wanting, a cy- pher may be put in its place; and, obviously, either sign may be given to it without affecting the I'oots of the equa- tion. Thus, the equation, x 2 -f 25 = 0, may be written x 2 ± -f- 25 = 0. Now, in this equation, if the upper sign be taken with ;ie middle term, there will be no variations ; and, of < ourse, the equation has no positive root. But, if the low- er sign be taken, there will be no permanences ; and, there- fore, the equation has no negative root. Consequently, the roots of the equation are imaginary (§ 353). So, in the equation, x 3 ±0 + 4^ + 7 = 0, be upper sign be taken with the second term, there will be no variation, and no positive root ; and, if the lower si +A n ^x J ' + A n ) Or, putting B 1 , B. 2 , &c., for the coefficients of y i_1 , y n ~ 2 , &c. X^yn-t-BsjO-i + B^-s ...+B n _ l y + B n = U.(:; Or, again, X — (x—x / ) n -\-B l (x—x'y- 1 .. +B n - , (x— x J )-\-B n = ; (4 where x — x 1 may be regarded as the unknown quantity. Now, obviously, the roots of (2), (3) and (4) are the val- ues of y(— x — x/); and are, therefore, less by x' than the roots of the given equation (i. e. the values of x). Hence, the transformation required is effected by the substitution of y-\-x' (i. e. of \_x — x'~\-\-x') for x in the given equation. Thus, Find an equation, whose roots shall be less by 2 than those of the equation, x 2 — 9cc + 20z=0. Substitute y-\-2 for x. Then (y + 2) = -% + 2) + 20 = 0, or # 2 + 4 — 9 ■5y+6 = 0, y+ 2- — 9X2^=0, ovy' + 20 is the equation required, whose roots will be found to be less by 2 than those of the given equation. § 366. a.) The labor of effecting this substitution may be greatly abridged, especially in the higher equations. *24 282 THEORY OF EQUATIONS. [§ 366. For B n , i. e. the coefficient of y° in the transformed equation (2), is simply what the first member of the given equation becomes, when x / is substituted for x. That is, B n =f(x'). B n _ 1 is formed by multiplying each term of B n by the ex- ponent of x 1 in that term, and diminishing the exponent by unity. B n ^ 2 is formed by multiplying each term of B n _ x by its exponent of x f , diminishing the exponent by unity, and di- viding by 2 ; and so on. b.) In other words, each term of B n ^ x is the first de- rived function (§ 292. N. 3) of the corresponding term of B n ; i. e. of /(a/). Each term of B n _ 2 is half the first derivative of the cor- responding term of B H ^ 1 ; i. e. half the second (§ 292. N. 4) derivative of the corresponding term of B n . So, each term of -S„_ 3 is one third of the first derivative of the corresponding term of B n - 2 ; i. e. one sixth of the third (§ 292. N. 4) derivative of the term of B n . c.) Hence, B n ^ 1 is called the first derived polynomial of B n , or of the given equation ; and may be expressed by -S},orby/'«>. I? n _3 is half the second derived polynomial of the equa- B" f'ip^) tion, and may be expressed by —-, or by' — - — . bo, 2*„_ 3 — 2-3 — ~273 » ^«-4 — 2.3.4 2.3.4' ^-^-2.3.4.5-2.3.4.5' ^ C ' 1. Diminish by 2 the roots of the equation, x i _}_ 5 X -j- 6 = 0. The transformed equation will be of the form., ■ §367.] TRANSFORMATION. — COEFFICIENTS. 233 And we shall have B 2 =f(x f ) =zx f2 + 5x , + 6 = 2 2 -\-5x2 + G = 20; B 1 = B 2 '=f'(x f ) = 2x' + 5 = 2X2 + 5 — 9; B = *B 2 " = 1X2 = 1 {B denoting the coefficient of ,y 2 ). # 2 ~\~ 9y-f"20 = is the equation required. § 367. d.) A still more convenient method of finding these coefficients results from the form of equation (4). For, comparing (4) and (1), we have (x—x'y-\-Bf(x — x'y-i- +B n _^(x — x') + B K -x n -\-A x x n -^ -\-A n - x x-\-At. (5) Now every term of the first memher of this equation is divisible by x — x', except the last term, B n ; which will be the remainder. In like manner, every term of the quotient which results from this division is, evidently, divisible by x — x', except the last, B n _ x , which will be the remainder; and so on. But the second member being absolutely (§ 37. d) equal to the first, its successive divisions by a: — x 1 must result in the same quotients and remainders as the division of the first member. Hence, If we divide the given equation by x — x J , the remainder will be J?,', the coefficient of y° in the transformed equation. If we divide the resulting quotient by x — a/, the re- mainder will be B' n _ v the coefficient of y 1 ; and so on, each of the coefficients being formed by the successive division of the several quotients by a; — xf. e.) It is evident also from § 348. b, that the first remain- der will be B n [=f(x')~] ; i. e. what X becomes, when x' is substituted for x (§ 350. 2, 3, 4). 1. Transform the equation, x- -j- 9a: + 20 = 0, into another whose roots shall be less by 5 than those of thf> given equation. 284 THEORY OP EQUATIONS. [§ 367 :e' + 9.r + 20 x — 5 Or (§86), l+9-|-20 1 x* — bx x + 14 x — 5 + 5 + 70 5 14x x-\- 5 1 l + 14, + 90,2?„. \Ax — 70 19: = -B„-r + 5 90 = £„. l, + 19,£ n _ r y 2 ~h 1% + 90 = is the equation required. 2. Find an equation, whose roots shall be less by 3 than those of the equation, k 3 + 10x 2 — 15a + 30 = 0. Neither the first coefficient of the divisor (§ 350. 2), which is always 1, nor the first coefficients of the quotients, each of which is the same as the first coefficient of the div- idend, need be written. Thus, 1 + 10 — 15 + 30 (3 + 3 +39 + 72 -+13- + 2 4, + 102 = B n =B 3 . + 3+48 + 16, + 72 = B ft _ 1 = B a . 3 + 19 = B a _ 2 = B 1 . .-. y z + 19?/ 2 + 12y + 102 = is the equation required. 3. Find an equation, whose roots shall be less by — 2 (i. e. greater by 2), than those of the equation, x 3 + 8a: 2 — 20* + 25 = 0. "We must here, of course divide by x — ( — 2) ; i. e. by x + 2. Ans. y 3 + 2y 2 — M)y + 89 = 0. 4. Find the equation whose roots are less by 1 than those of the equation, x* — 2x--\-3x — 4 = 0. Ans. y 3 +y- + 2y — 2 = 0. 5. Find the equation, whose roots are less by 2 than those of the equation, X 5 + 2x 3 _ 6x 2 — 10* + 3 = 0. Ans. y 5 + l(ty* + 42y 3 + 86y 2 + 10y +12 = 0. § 367.J TRANSFORMATION. — CHANGE OF ROOTS. 285 6. Diminish by 2.8 the roots of the equation, x* — 12a; 2 + 12a; — 3 = 0. We may here either diminish the roots of the equation by 2, and then the roots of that equation by .8, or we may diminish the roots of the given equation at once by 2.8. The former method is generally the more convenient, Thus, 1+0— 12 + 12 — 3 (2 2+ 4 — 16 — 8 2— 8— 4,- 2+ 8 * -11 4 0,-4 2 +12 6, + 12 2 8+12—4 -11 (.8 .8 + 7.04 + 15.232 + 8.9856 8.8 +19.04 +11.232, — 2.0144 .8 + 7.G8 +21.370 9.6 + 26.72, + 32.608 .8 + 8.32 10.4, + 35.04 ■ 8 11.2 Diminishing the roots by 2, we find the equation, y 4 +83/ 3 + 12y 2 -5y-ll = 0. Diminishing the roots of this equation by .8, we have y* + \l.2y 3 + 35.04y 2 + 32.608y — 2.0144 — 5 the equation required. 7. Diminish by 1.3 the roots of the equation, x 3 — 7a; + 7 = 0. An*. x 3 + 3.9a; 2 — 1.93* + .097 = 0. 8. Diminish by 14 the roots of the equation, x 3 — 17a; 2 + 54a; — 350 = 0. Ans. x 3 + 25a; 3 + 166a; — 182 — 0, 286 THEORY OF EQUATIONS. [§ 368. § 368. If the coefficient of any power of y in equation (2) of §365 reduce to zero, that term will be wanting in the new equation. Thus the second term will disappear from the equation, if nx! + A z= ; i. e. if x' — . Hence. n To make the second term disappear, we must make x/ =: ; i. e. we must diminish the roots by ; or, which n n is the same thing, increase them by -\ . lb a.) This will be evident otherwise ; thus, The sum of the n roots of the primitive equation is — A A (<$ 355. 1). Now if each of the roots be increased by — , v y n their sum will be increased by A ; and will, of course, be equal to — A + A = 0. 1. Remove the second term from the equation, X i _ 4 X 3 _ 19 X 2 _|_ 106a; — 120. Here we have n = 4, and i = — 4. .*. x, — — — — x; n 4 and we must diminish the roots of the equation by 1. 1—4— 19 + 106 — 120 (1 1— 3 — 22 +84 _ 3 — 22 + SlT- 17 !^ = 5 4 . 1 — 2 — 24 — 2— 247+ 60 = # 3 . 1—1 — 1,-25 = J5 2 . 1_ y* — 25y - + GQy — 36 = is the equation required. Transform the following equations in like manner. 2. x* — 3x2 — 4 X + 12 = 0. Ans. x 3 — 7x + 6 = 0. § 369.] TRANSFORMATION. — REMOVAL OF TERMS. 287 3. a^-f. 14x4 + 12a; 3 — 20x 2 + 14x — 25 = 0. Am. y* — 78*/ 3 + 412?/ 2 — 757y-f-401 = 0. 4. x 2 -{-2^ + 5 2 = 0. Am. y*-\-(q*—p*) = 0. 6.) The last result leads to the common solution of the equation. For, by transposition, if- = p n ~ — q 2 ; and.%y=±Q>2 — q2)k But ^ — a?-|-^. i x+p = ±(j>*.- q *)\ x = — ^> ± (jp 2 — y*) ■ c.) If we would remove any other term from the equa- tion, we must make the coefficient of that term in (2) of § 365 equal to zero, and find the corresponding values of x 1 . By the substitution of a value so found for x 1 , that term will, of course, vanish. It is obvious, that, to remove the third term, we must solve an equation of the second degree ; for the fourth, one of the third degree, and so on. To remove the last term, we must solve an equation of the wth degree ; in fact, the given equation itself, with x 1 substituted for x. The values of x' found from this equa- tion will, therefore, evidently be the roots of the given equation. § 369. If, in the general equation, x n -\-A 1 x n - l + A 2 x n - 2 .... +A n _ l x-{-A n =0, we put y = rax (i. e. substitute — for x), we shall have m £+^S£ + A -^+ A "=°-- or (§ 46) y n -\-A l my n - x . . -\-A n - l m n - x y -\-A n m n — ; an equation whose roots are in times those of the primitive equation. Hence, An equation will be transformed into another, whose roots shall be equal to the roots of the first multiplied by any 288 THEORY OF EQUATIONS. [§ 370. number, as m, if we multiply the second term of the given equation by m, the third by ?n 2 , and so on. Hence, Cor. i. An equation having fractional coefficients may be changed into another with integral coefficients, by trans- forming it so that its roots shall be those of the given equa- tion multiplied by a common multiple of the denominators. Cor. II. If the coefficients of the second, third, &c. terms of an equation be respectively divisible by m, m 2 , &c, then the roots of the equation are of- the form mx, and conse- quently m is a common measure of them. 1. Transform the equation, 3x3-f4x 2 — 5x+6 — 0, into another whose roots shall be three times those of the given equation. Here m = 3. .*. y = Sx, and x = iy. Am. 3y3 + I2y 2 — 45y + 162 = ; or, y 3 + 4y 3 — 15y + 54 = 0. 2. Transform the equation, into an equation with integral coefficients. Am. x^ -f 8x 2 + 108* — 4320 = 0. § 370. If in the general equation, * l + A l a+-i+A a x*-* .... +A n _ 1 x + A n =0, we substitute - for x, we shall have y ^ + A~ + ^^ +A n _ 1 1 - + A n =0; or, clearing of fractions, and reversing the order of the terms, A^+A^^-i + ^ 2 y»+.4 1 y + l=0; an equation, whose roots are the reciprocals of the roots of the given equation. Hence, To transform an equation into another, whose roots shall §372.] RECURRING EQUATIONS. 289 be the reciprocals of the roots of the first, we have only to reverse the order of the coefficients. a.) Cor. We may also, evidently, transform an equation into another, whose roots shall be greater or less than the reciprocals of the roots of the given equation, or multiples of those reciprocals, by applying the processes of §$ 367, 369 to the coefficients taken in a reverse order. b.) It may happen, that the coefficients, when taken in the reverse order, shall be the same as when taken direct- ly. In such a case, the transformed will obviously be identical with the given equation ; and will have the same roots. Consequently, as the roots of the transformed are the reciprocals of those of the given equation, and, at the same time, are identical with them, one half of the roots of the given equation must be the reciprocals of the other half. Thus the roots will be a. - ; b, T '•> & c - a b c.) If the coefficients of corresponding terms are numer- ically equal, but have unlike signs, the same is true of the roots, in every equation of an odd degree ; and, in every one of an even degree, whose middle term is wanting. For, in both these cases, if all the signs of the transformed equa- tion be changed, (which will not affect the iralue of the roots,) the transformed will be identical with the primitive equation. § 371. d.) Such equations (§ 370. b, c), which remain the same, when - is substituted for x, are called recurring 9 x or reciprocal equations. e.) The general form of a recurring equation is, obvious- iy, ' x n -\-A 1 x n ~ l +A 2 x n - 2 + A ^-^A^x -+-1 = 0. Eecurring equations have certain peculiar properties, which will be considered hereafter. (g) Lat. recurro, to run back. ALG. 25 290 THEORY OF EQUATIONS. [§ 372, 873. LIMITS OF THE ROOTS. 72. In the equation, (x — « x )(^ — Q>$)(x — « 3 ) ... = 0, let a 1 , a 2 , a 3 , &c. be the real roots, taken in the order of {heir magnitude; i..e. a 1 >a 2 , a 2 >a 3 , &c. If now b lt > a v be substituted for x, we have (b 1 ~a l )(b 1 —a. 2 )(b l —a 3 ) . . . (b l —a n ), positive; all the real factors being positive (§§ 68. a; 358. 1, 2). If b. 2 , < a x and > a 2 , be substituted for x, we have (h — «i)( 5 2 — « 2 )(* 2 — « 3 ) • • • ( & 2 — «»)» negative- one of the real factors being negative (§ 68. a). So, if we substitute b 3 , a 3 , the product will be positive ; two of the real factors being negative, and the rest, positive. In like manner, the substitution of 5 4 , « 4 , will give a negative ; of & 5 , « 5 , a positive re- sult ; and so on. Hence, (1.) If a quantity, greater tlian the greatest real root of an equation, be substituted for x, the result will be positive : and, (2.) If quantities intermediate between the roots, begin- ning with the greatest, be successively substituted for x, the results will be alternately negative and positive. The roots of the equation, X 3 _ 5 X 2 _|_ 2x + 8 = 0, are 4, 2, and —1. Substitute 5, 3, 1, 0, and —2 fur x, and observe the signs of the results. § 373. a.) Hence, Cor. i. When two quantities are successively substituted for x, if the results have like signs, there is an even ; if un- like signs, an odd number of real roots between those quan- tities. Note. The even number may be (§63. N.). <^37-l, 375.] LIMITS OF THE ROOTS. 201 b.) From 1, and Cor. i., it is evident, that, Cor. ii. IF a number less than the least real root be sub- stituted for x, the result will be positive or negative, aecord- iug as the number of real roots is even or odd. c.) If the degree of the equation be odd, the substitution of -j- oo for x will render the first member ■positive ; and of — oo, negative. Hence (§ 373. a), Cor. in. (1.) Every equation of an odd degree must have at least one real root (§ 357. b) ; and (2.) the whole number of its real roots must be odd. d.) If the degree be even, and the last term negative, the substitution either of -\- oo or of — oo will render the first member positive ; and the substitution of will render it negative. Hence, Cor. iv. (1.) Every equation of an even degree has an even (§373. JST.) number of real roots; and (2.) every equa- tion of an even degree, whose last term is negative, has at least two real roots, one positive and the other negative ;§ 357. Cor. in.). §374. e.) If the substitution of p, and of every number greater than p, renders the result positive, then p is great- er than the greatest real root ; and is called a superior li?n- it of ^the roots. f.) So, if, the signs of the alternate terms being changed (§359), the substitution of q, and of every number greater than q, renders the result positive, then — q is less than the least real root (i. e. it is an inferior limit). §375. Let A h be the first, and A m , numerically the greatest, negative coefficient of any complete (§3G1. a) equation, x n + i/-i . . — A h x n ~ h . . —A m x"- m . . -f A n = Now, if all the coefficients after A h be negative, the sum of those terms will be numerically equal to the sum of the preceding, positive terms. 592 THEORY OF EQUATIONS. [§ 376. Consequently, any value of x, which renders the sum of the preceding positive terms numerically greater than the sum of the negative terms, is a superior limit of the roots. And, with still greater reason, any value of x, which renders x n numerically greater than the sum of the nega- tive terms, is a superior limit. The most unfavorable case possible is, evidently, when all the coefficients after A h are negative, and each of them is equal to A m , the greatest. Any value of x, then, which makes x»>J m (x' l -» + x n - h -i . . . + * + l), (1) or (§261) *>^« C^7 1 ). < 2 ) is a superior limit. Now (2) will certainly be ti-ue, if we have x'i-M-i x~<"- 1 } x" > A m ; or 1 > A, x _l ■ ~" x _i > or x -'<-i(. r _i) > J m . (3) But x— 1 A m . (5) Also, (4) and (5) give x— 1 = , or > (^„,)*; or x=z ,or>(J w )* + l. (6) That is, in a complete equation, § 376. 7f we increase by unity that root of the greatest negative coefficient, ivhose number is equal to the number of terms preceding the Jirst negative term, the result will be a superior limit of the roots. Find superior limits of the roots of the following equa- tions. 1. x* — ox 3 + 37x2 — 3x -f 39 = 0. § 377.] LIMITING EQUATION. 293 Here A m = 5, and h — 1. 1 i (A m ) h + 1 = 5 T + 1 = 6, the limit required. a.) If the second coefficient be negative, the limit found will be the greatest negative coefficient increased by unity. 2. aj«-f-7:K±--12a; 3 — 49fc*-f-52se — 13 = 0. Ans. (49)^ + 1 = 8. 3. a^-fll* 2 — 25* — 67 = 0. .4/w. (67)^+1 = 6. 6.) If the signs of the alternate terms be changed (§ 359), and a superior limit be found, that limit with its sign chang- ed will be an inferior limit ; or, as it is sometimes called, a superior limit of the negative roots. c.) A number, which is numerically less than the least positive or negative root is sometimes called an inferior limit of the positive, or of the negative roots. Let the equation be found, whose roots are the recipro- cals of the roots of the given equation ; and let the superi- or limits of the positive and negative roots of this new equation be found. Now those roots of the new equation, which are numer- ically the greatest, are the reciprocals of those of the given equation, which are numerically the least. Therefore, the reciprocals of the superior limits of the positive and negative roots of the new equation will be the inferior limits of the positive and negative roots of the given equation. LIMITING, OR SEPARATING EQUATION. § 377. Let a l , a„, a 3 , &c. be the real roots, taken in the order of their magnitude, of the equation, 3*+-4 1 ic B -i . . . + A n „ 1 x + J n =0. (1) Diminishing the roots of this equation by a/, we have (§365) *25 m 294 THEORY OF EQUATIONS. [§ 377. 9*+Ma*r* + B#— + B B _ 1 y-^B H =0; (2) in which ($§ 365 ; 366. c) J? n _ 1 -fix') = na"-i +(«- 1)4^-2 . . J- J„_ 1 .(3) Also, -5,,-!, is the sum of the products of the roots, with their signs changed, of equation (2) (i. e. of the products of x' — a v x 1 — a 2 , . . x 1 — a„), taken n — 1 at a time (§ 355). That is, B„_ 1 = (x'— a 2 ){x J — a 3 ){x>— a 4 ) . . . (x'—a n ) + (x'—a 1 )(x'—a 3 )(x'—a 4 ) . . . (x'—a n )-\- (x'—a l )(x ! —a 2 )(x'—a i ) . . . (x 1 — o„) + j • • • (^ «i)(*'' a 2)( X ' a 3) • • • (^ a n-l)jJ each term consisting of n — 1 factors ; and, of course, each factor being found in every term but one. If now, in this value of B n ^^ we make x* ' = a v a 2 , a 3 , &c, successively, we shall have (§ 68. a) B n . r — (a l —a 2 )(a 1 —a 3 )(a l —a 4 ) . . — +.+•+ . .= + ; B n . 1 = (a 2 — a 1 )(a 2 —a 3 )(a 2 — a 4 ) . . = — . + ."+..=— ; #«-, = (« 3 — « x )(« 3 — «2)(« 3 — a 4> • ■ — — • — ■ + •• = + ? &c. That is, if we substitute a 1 , a 2 , a 3 , &c. for x! in Ai-p the results are alternately positive and negative. Hence (§ 372), the real roots of B n _ l =. lie between a p a 2 , a 3 , &c. ; and therefore, putting x m place of x 1 , we have the equation, B ll ^ 1 ss na;"-! + (n— 1) ^a*- 2 + (n— 2)A 2 x n ~ 3 . . +A„^ 1 = ; whose real roots severally lie between those of the given equa- tion ; and which is thence called the separating or limiting equation. a.) B n - V we have seen (§ 366), is the first derived poly- nomial of the given equation. That is, f(x) = 0,or X' = is the limiting equation of f{x) — 0, or A' — 0. S 3 -\ § 378.] equal roots. 296 Hence, the separating or limiting equation is properly called the derived equation. b.) It is obvious, that iff(x) — have real roots (as as- sumed in the investigation), the greatest and least are res- pectively greater and less than the greatest and least real roots of/' (a?) =: 0. EQUAL ROOTS. § 378. c.) If the given equation have two roots equal, an a and -|- o> for x in the several functions. In this case, each function will have the sigi of its first term. (2.) Moreover, if we substitute for x, the number of variations lost from — oo to will give the number of neg- ative roots.; from to -f- oo, the number of positive root.-. It is obvious also, that the substitution of for x will re- duce each function to its last term, which is independent. of x. § 387. The theorem has been demonstrated on the hy- pothesis, that the equation contains no equal roots (§ 379). If, however, we have an equation containing equal roots, we shall find a common divisor of Xand X' ; and a remain- der, of course, equal to zero. If now we divide the functions, X, X', &c, by this great- alg. § 26 302 THEORY OV EQUATIONS. [c 888. est common divisor, we shall obtain a new series of func- tions, T, T', r 2 , &c. Now, it is evident, (1.) that Y= will contain no equal roots; and (2.) that the variations of sign in the series of new functions will be the same as in the primitive series. For, if the common divisor be positive, the signs will not be affected by the division (§§ 62. a; 80. b); and, if it be negative, all the signs will be changed. Hence, the theorem is applicable to equations having equal roots. § 388. 1. How many real roots has the equation, X = a; 3 — 7*4-6 = 0? Here X' = 3x s — 1 ; v — 7^ o •r X -4- X X' x 2 x 3 x = — oc gives — r -f" — ~f*> ^ variations. xz=zO " -\- — — -f~j 2 variations. x — -\- go " + + + +< variation. Hence, there are three real roots ; one negative, and 'two positive. We shall find, more nearly, the values of the roots by substituting different numbers for x. Thus, x =■ — 4 gives -+:-+; 3 variations ; x— — 3 " + - +; x=— 2 " + + - +. 2 variations ; *=— 1 " + +, 2 variations ; *= 1 " +; 5B== 1.6" - + + +, 1 variation ; k"= 2 « + + +; x '== 3 « + + + +> variation. have here found the three roots, - -3, 1, and 2 (the § 388.] stdrm's theorem. 303 values of x which reduce X to zero). We find also that a variation is lost in passing each of the roots. 2. Find the number and situation of the real roots of the equation, X=z x 2 -]-x — L = 0. Here X' = 2x + 1 ; X 2 =z+5. x — — co gives -f- — -f" , 2 variations ; x = " — 4~+j 1 variation ; a; = + co " +~h+5 variation. There are, therefore, two real roots ; one positive, and the other negative. Moreover, x = — 2 gives -J- — -|- , 2 variations ; x =z — 1" — — -f~ > 1 variation ; x ■=. -f- 1 " + + +> variation. There is, then, one root between — 2 and — 1 ; and one between and 1. The first figure of the negative root is — 1 ; and, by sub- stituting .1, .2, .3, .4, .5, .6, and .7, we find the first figure of the positive root to be .6. 3. How many real roots has the equation, X = x3 + llx 2 — 102x + 181 = ? Here X'= 3x 2 -\- 22x — 102; X 2 = 122* — 393; Hence, Ave find three real roots ; one negative, and two positive situated between 3 and 4. Now, diminishing the roots (§ 367) of the equations, X = 0, X' = 0, &c, by 3, we find T' = 3y 2 + 40?/ — 9 ; r 2 z=122y-27; 304 THEORY OF EQUATIONS. [§ 388. These functions show that the two positive roots of Y= lie between .2 and .3. Consequently, the two pos- itive roots of X = are between 3.2 and 3.3. Again, diminishing the roots of Y= 0, T' = 0, &c. by .2, we find Z — Z 3 + 20.6s 2 — .88z + .008 ; Z'= 3z* + 41.22 — .88; Z 2 = 122s — 2.6; Hence, the initial figures of the two positive roots of Z ~ are .01 and .02. Consequently, the first three fig- ures of the positive roots of X = are 3.21 and 3.22. Also, the sum of the roots (§ 355. 1) is — 11. — 11 — 3.21 — 3.22 = — 17.4, the negative root. 4. How many real roots has the equation, Xz=x 5 -\- 2x± + 3x 3 -f- 4x2 _j_ § x _ 20 — ? Here X' = 5x* + 8x 3 + 9x 2 + 8x + 5 ; X 2 = — 7x 3 — 21z 2 — 42a; •+- 255 ■> X 3 = — 13x + U; x = — co gives — -f- -f- -f- — , 2 variations. aj= + oo " + + — — — ? 1 variation. Hence the equation has one real, and four imaginary roots. The real root is, of course, positive (§ 357. b) ; and is found to be between 1 and 2. a.) "When we arrive at a function, as X m , such that the roots of X m = are all imaginary, we need not continue the divisions. For this function having the same sign for all values of x (§ 358. 1), can never conform to the signs of those be- yond it; and no changes of sign in those functions can af- fect the number of variations in the series (§ 384). The coefficients of an equation of the second degree, show at once, whether its roots are imaginary (§ 216). k 388.] STURM'S THEOREM. 305 In respect to equations of higher degrees, the question is not so easy of solution. It can, however, be determined by applying Sturm's theorem, as to an independent equa- tion. The roots of X' = 0, in the last example, are all imagi- nary ; and X and X' give the same result as the whole se- ries of functions. Note. X' =0 is a recurring equation (§371), and can be easi- ly solved by a process which will be explained hereafter. 5. How many real roots has the equation, X = x 3 -\-px -\-q z=: 0? Here X' = 3x 2 +p; X 2 zz: — 2px — 3q ; X 3 — — 4p 3 — 27q' 2 . b.) First, let p be positive. Then — 2p will be negative; and X 2 will be positive for x = — oo ; negative, for x = -f- oo. Also, — 4p 3 will be negative; and, as — 27q 2 is neces- sarily negative, X 3 will be negative. Thus, x = — co gives — + + — > too variations ; x = -|- co " + + — — , one variation. Hence, if p be positive, the equation has one real, and two imaginary roots. c.) Again, let p be negative. Then — 2p will be positive ; and X 2 will be negative for x = — co ; positive, for x = -J- oo. Also, — 4p 3 will be positive; and when — 4p 3 >27y*, i. e. when — 4p 3 — 27q 2 >0, or (§ 146. d) 4p 3 + 27«7 2 < 0, X, will be positive. If these conditions be fulfilled, we shall have, for x =. co, — -j- — -f- } three variations ; x =: -J- co, — I — — | — — J — — f — ^ no variation. ^ Hence, if both p, and 4p 3 + 27q 2 be negative, the equa- tion has three real roots. *26 306 THEORY OF EQUATIONS. [§ 389. d.) Again, suppose X 3 to be positive ; i. e. 4p3 + 27? 2 <0. Then 4p3 < — 27? 2 . § 144 N . •-.(§147) ^<-^or(|) 3 <-(|) 2 <0 . ,. P< 0. For — f|j is negative. Consequently, f||) is nega- tive ; which cannot be unless p is negative. Hence, if 4p 3 -j- 27g 2 < 0, the roots are all real. 6. How many real roots has the equation, X=x^-\-px + q — 0? Here .X'=2a:+^; X^^-4?. . First, if JF 2 be positive, x = — co gives -f- — -f- , two variations ; x = -\-cx> " -L._L.4-, no variation ; showing two leal roots. Again, if X 2 be negative, x = — co gives + — — > one variation ; a; __. -|- co " 4~ + — j one variation ; showing no real root. Consequently, the roots are real, or imaginary, accord- ing as p 2 — 4^ is positive or negative. Moreover, when X 2 is negative (i. e. p"* — 4^<0), we have p* (hp) 2 >0 (§ 216). NUMERICAL EQUATIONS. — I. INTEGRAL ROOTS. § 389. Let a be an integral root of the equation, X=zx n + A 1 x n - 1 +A 2 x n - 2 . . +^ M _ 1 a + ^ B = 0, (1) the coefficients being all integral. Then a n -\-A x a"-i . . -\-A n . 2 a^+A Jl . 1 a-\-A n = 0. (2) § 389.] INTEGRAL ROOTS. 307 Transposing, and dividing by a, we have £« = _„-! _^ ia »- 2 -A n _ 2 a-A^ 1} (3) a whole number. Hence, A n -±- a is a whole number; and a is an integral factor, or divisor (§ 80. d) of A „. Consequently, all the integral roots of an equation will be found among the divisors of the last term. They will also, of course, be contained between the superior and in- ferior limits (§ 374) of the roots. Therefore, we shall find all the integral roots of an equa- tion, by the method of §§ 349. d, 350., if we substitute for a, successively, the several factors of the last term, which are included between the limits of the roots. 1. Find the integral roots of the equation, Here, the limits, found by § 374, are 18 and — 5. It is evident, however, that there can be no integral root great- er than 6. Hence, the only numbers to be tried are 6, 3, 2, 1, — 1, — 2, and — 3. 1 — 7 + 17 — 17 + 6 (1 + 1— 6 + 11 — 6 1 — 6 + 11 — 6 (2 + 2 — 8 + 6 1 — 4+3 (3 + 3—3 1-1 (1 + 1 1. Consequently, the root3 are 1, 1, 2, and 3 (§ 355. e. 2). 2. Find the integral roots of the equation, X — x 3 + x 2 — 17a: + 15 =: 0. If X is divisible by x — a, it is, evidently, divisible by a — x -, the signs merely of the quotient being different, 308 THEORY OF EQUATIONS. [§ 389. Therefore, arranging the coefficients according to the as- cending powers of x (§ 33), and dividing by 3 — x, we have 15 — 17 + 1 + 1 3 _|_ 5_4_i _|_i — 12 5 — 4 — 1 + 5 + 1 5 + + 1 1 5 1 — 1. Hence, the roots are 3, 1, and — 5. a.) In this process, the root, evidently, must divide the first term of each remainder ; i. e. the sum of each term of the quotient and the succeeding coefficient. b.) In fact, transposing A n ^ l in (3), representing — " + A „_ 1 by B, and dividing again by a, we have a B a a n-2 — A x a n ~" -™-n—3 a -^n— 2> a whole number. In like manner, continuing to transpose the coefficient of a , and divide by a, each quotient will be a whole number ; and the last quotient will be the coefficient of x n with its sign changed. 3. Find the integral roots of the equation, x 4 — 21x- + 14k + 120 = 0. Here +7 and — 7 are limits. Moreover, only two of the roots can be negative, and two, positive (§361). Hence, having found two positive roots, we need try no more positive divisors. 6 + 1 120 + 14 — 27 + 20 + 34. + 1 34-^-6 not being a whole number, 6 is not a root. The roots are 4, 3, — 2, and — 5. § 390.] INCOMMENSURABLE ROOTS. 309 c.) If the equation is not in the common form (i. e. with integral coefficients, the first being unity), it should be re- duced (§ 369), and the method applied to the reduced equa- ion. 4. Find the roots of the equation, 3x 3 — x 2 — 3x + 9 = 0. Having found the values of y from the transformed equation, we shall have x = \y. II. INCOMMENSURABLE ROOTS. § 390. Find the roots of the equation, X=x- + 5x — 5 = 0. Applying Sturm's theorem, we find X = x 2 + 5x — 5 ; X' = 2x + 5 ; x 2 = +. Hence there is a positive root between .8 and .9, and a negative root, between — 5 and — 6. If, now, we diminish (§ 367) the roots of X — by .8, one root of the transformed equation, Z=y 2 + 6% -.36 = 0, will be between and .1. Applying Sturm's theorem again, we find Y=y 2 +G.6y — .36 = 0; r' = ^/ + 3.3; r 2 = + . Hence, there is a root between .05 and .06 ; and, conse- quently, the root of 1= is between .85 and .86. Again, diminishing the roots of T= by .05, one root of the transformed equation, Z — z 2 + Q.7z — .0275 = 0, will be between and .01 ; and will be found by the the- orem to be between .001 and .005, 310 THEORY OF EQUATIONS. [§391,392. Hence, the root of X = is between .854 and .855. Note. We might, in the same way, find any number of figures of the root. But the process would be tedious. The nature of the roots, however, of the equations, F=0 and Z-0, will suggest a more convenient method of determining the successive figures, as ap- pears in the following sections. § 391. We know that the root of the equation, ? 2 + 6%-.36 = 0, (1) is less than .1 ; i. e. we have y < .1, and, of course, y 2 < .01. Hence it is evident, that the equation, 6% + .36r=0, (2) will furnish a near approximation to the true value of y. or? \ In fact, we have, from (1), y = y+6.6' in which the first significant figure will be the same, wheth- er we take y := 0, or .09, as will be seen by dividing .36, successively, by G.6, and by 6.69. The same reasoning will apply, with still greater force, to the first figure of the root of Z= 0. Hence, we may find, in each instance, approximately, the next figure of the root by dividing the coefficient of y° and z° by the coefficient of y 1 and z 1 . The operation, then, will stand thus ; 1 _}- 5.8 —5 (.8541 .8 -f 4.64 — .36 - f .3325 — .0275 + .026816 — .000684 + .00067081 6.7082 — .00001319 § 392. To explain this method of solution in a more general form, let a root of the equation, 6.65 5 6.704 4 6.7081 1 §393.1 INCOMMENSURABLE KOOTS. ii X^^ + A^'-i + Asx"--* .... + A^ v c-{-A n ^0, be x = x'-\-y; x 1 being the part of the root already found, and y representing the remaining figures, and y being, of course, very small compared with x' (§ 174. N. 1). Then diminishing the roots of X —3 by x 1 , we have frr^+^f- 1 .... +B n _ y* + B n _ l y + £ n =n. But, y being very small, its powers above the first may, for the moment, be neglected ; and we shall have, nearly, B n - x y + B a =.0i or, also approximately, y — — ~ " . The correctness of the result will be verified by intro- ducing into the transformed equation the figure so found. Representing the figure so found by y', we shall have y = y~|- z ; and finding an equation, whose roots are less than those of Y= by y r , we shall, in like manner, find another figure of the root ; and so on. Hence, for finding a root of an equation of any degree whatever, we have the following RULE. § 393. 1. Find by Sturm's theorem, or by trial, the first figure, or the integral part, of the root. 2. Transform the equation into another, whose roots shall be less than those of the given equation by the part of the root already found. 3. With the last coefficient of the transformed equa- tion for a dividend, and the last but one for a trial divisor, find the next figure of the root ; and verify it by substitution in the transformed equation (§ 350). 4. Diminish the roots of the transformed equation by the figure just found, divide as before for the next figure ; and so on, as far as is necessary. 312 THEORY OF EQUATIONS. [§394. a.) The method is applicable to both positive and nega- tive roots ; each figure of a negative root being treated, in multiplying, as a negative quantity. h.) A negative root is, however, more conveniently found by changing the signs of the alternate terms, and finding the corresponding positive root (§359). 394. 1. Find the roots of the equation. X = x* + 10x° + ox — 260 = 0. Here, X' = 3x 2 -f 20a: + 5 ; X 2 = 17* + 239; x c= — oo gives two variations ; x=--\- ao, one. Hence, there is but one real root ; positive, of course (§ 357. b). We find, moreover, that the first figure of the positive root is 4. + 10 + 5 — 260 (4.1179 4 56 + 244 14 61 -16 4 72 '+ 13.521 18 ,133 ,— 2.479 4 2.21 1.376531 ,22.1 135.21 — 1.102469 1 2.22 .966221613 22.2 ,137.43 ,— .136247387 1 .2231 .124396356339 ,22.31 137.6531 ,— .011851030661 1 .2232 ,137.8763 22,32 1 .155359 ,22.337 138.031659 7 .156408 22.344 ,138.198067 7 .201167] L _ ,22.3519 138.21817371 § 394.] INCOMMENSURABLE ROOTS. 313 The coefficients of the successive transformed equations are marked with commas, the first coefficient in each being the same as in the primitive equation. Thus, we shall have r = y z + 22y 2 + 133y — 16 = 0; Z = z" + 22.3^2 4- 137.432 — 2.479 = ; and so on. Note. It will be observed, that the .1 added to 22, does not term a part of the coefficient of ?/ 2 , but was added to that coefficient in forming the next. A similar remark applies, of course, to the subse- quent coefficients; and to the example of § 391, where .8 is most con- veniently added to 5, by being written after it. a.) The coefficients of the two last terms (B n _ l , and B n , [§ 392]) in each of the transformed equations have unlike signs. This is as it should be, in finding a positive root. For, suppose that the least real root of X = is posi- tive ; and represent the part already found by x 1 . Then B n and -S„_ r are what Xand X become, when a/ is substituted for x. Therefore, x 1 being less than the least real root, B n and B n _ x (i. e. /(«') and /'{x 1 )) must have unlike signs (§§373. b; 385). b.) Similar reasoning will apply to any other positive root, provided xf differs from that root less than the next inferior root of X' = does (§ 385. a). See g, h, below. c.) In approximating to a negative root (§394. a), xf is greater than the root ; and, of course, if it is less than the next greater root of X' = 0, B n and B n _ x (i. e. /{xf) and /'(a/)), must have like signs. d.) If, having found the root, 4.1179, we divide X by x — 4.1179, we shall have an equation of the second degree, from which we may find the remaining roots (§ 353. c). e.) Otherwise ; we know that the coefficients of x 2 and x° in the given equation are respectively the sum, and pro- duct of the three roots with their signs changed. Also, the coefficients of x l and x° in the depressed equation will be the sum and product of the two remaining roots with their signs changed (§ 355. 1, 4). alg. 27 'jl4 theory of equations. [§394. Hence, if we diminish the coefficient of x 2 , and divide the coefficient of x°, in the given equation, by the root found taken with a contrary sign, we shall have the coeffic* ients of a; 1 and x° in the depressed equation. Thus, or x 2 + 14.1179 x + 63.1365 = 0, will give the remaining roots of the equation, which, are, evidently, imaginary (§ 388. 6). f.) "When the roots are all real, it is frequently quite as convenient to find a second root from the given equation, in the same manner as the first ; and then find the third by adding the two roots found to the coefficient of x~, and changing the sign of the result (§§ 355. 1 ; 388. 3). 3. Find the roots of the equation, X = x 3 — 7x + 7 = 0. Here X' = 3x* — 7; X 2 = 2x — 3 ; Hence, there are three real roots ; one between — 3 and — 4, and two between 1 and 2. Also, the first two figures of the roots are — 3. 0, 1.3, and 1.6. To find the greatest root, proceed thus. x — 7 + 7 (1.69202147 1 1 — 6 ~L — 6 ,+ 1 1 2 — 1.104 ~2 -4 ,— .104 1 2.16 .100809 ,3.6 — 1.84 ,— .003191 .6 2.52 4.2 ,+ .68 Find the other figures of the root. Also find the other root. g.) There are here two roots of X =. 0, and only one of X' = (viz. 1.528), greater than 1. § 394.] INCOMMENSURABLE ROOTS. 315 Consequently, the substitution of 1 for x renders X pos- itive and X' negative (§ 372) ; giving B n =f) =-.104, and B H _ l = f(x') = + .68. If we had substituted 1.5, B n _ 1 would have remained negative; because 1.5 is less than the greatest root of X' = 0. Hence, if the sign of B n changes, that of B n ^ 1 should change also. See a, above. h.) It may, however, not change at the same figure of the root, for that figure may be common to the next great- er root of X= and of X' = 0. This occurs in the great- est root of the following equation. See 4, below. i.) To find the negative root, we change the signs of the alternate terms (§ 393. b). 1—0 —7 —7 (3.048917 3 3 o O 2 18 ,20 _6 ,— 1 .814464 .3616 ,9.04 4 9.08 4 20.3616 .363 2 ,20.7248 .0730 ,9. 128 8 9. 136 8 ,9. 1449 9 T. 1458 20.7978 .0730 ,20.8709 82 24 24 88 12 3041 ,— .185536 .166382 ,— .019153 .018791 ,— .000362 208 592 408 228169 179831 873763 ,— .000153 146 ,— .000007 306068 211615 094453 20.879 8 14241 23122 20|8|8]737363 x — —3.048 917. 316 THEORY OP EQUATIONS. [§394, k) "We should evidently have obtained the same result, as far as we have carried the approximation, and with mnch less labor, if we had neglected all the figures on the right of the vertical lines in the several columns. 4. Find the roots of the equation, X = x 3 + 1 1x2 _. i02x + 180 = (§ 388. 3). The roots are 3.229 52, 3.213 127 7, and — 17.442 648 96. The greatest root of X' = is 3.2213. Consequently, 3.22 substi- tuted for x, will render both X and X' negative. But 3.229 will render X negative, and X positive. See h above. 5. Find the roots of the equation, 8x 3 — fry — 1 = 0. It is not necessary for the application of Sturm's theo- rem, or of this method of approximation, to reduce the equation as in § 389. c. We shall find, that there are three real roots ; one posi- tive, and two negative ; and that their initial figures are .9, — .1, and — .7. The equation may be put under this form, x & _ |aj— £ = x 3 — .75a: — .125 = 0. To find the negative root, proceed as follows. 1 0.7 —.75 +.125 (.76 &c. .7 _^49 —.18 2 IX =^26 ,— .057 .7 .98 .050976 ,2T6 ,+ 772 - .006024 .1296 .8490 The roots are — .760 04, —.1737, and .9397. 6. Find the real root of the equation a: 3 — 2 = ; i. e. find the cube root of 2. Ans. 1-259 921. 7. Find the roots of the equation a; 2 — 2 = ; i. e. find the square root of 2. Ans. ±1.414 213 6. Note. It will be observed, that the solution of the third and fourth examples i3 equivalent to the processes of §§ 174, 179. The § 394.] INCOMMENSURABLE ROOTS. 317 method is, obviously, equally applicable to the extraction of roots of large numbers. The trial divisor, however, approximates, of course, most closely to the complete divisor, when the part of the root not yet found is very small. 8. What is the cube root of 3 442 951 ? Ans. 151. 9. Find the roots of the equation, x* — 12x*-\-12x — 3 = 0. — 12 + 12 — 3 (2.8 &c 2 4 — 16 — 8 "2 — 8 — 4 -11 2 8 8.9856 4 ,-4 ,— 2.0144 2 12 15.232 6 ,12 11.232 2 7.04 ^8.8 19.04 Am. 2.858 083, .606 018, .443 276 9, and —3.907 378. Continue the operation, and find the other roots. The work may be Teatly abridged by rejecting all but one decimal figure in the col- umn of x3, two or three in the column of x 2 , three, four or five io that of x l , and four, five, six or seven in that of xo. 10. Find the roots of the equation, x 3 — 2x — 5 = 0. Ans. 2.094 55 ; the other roots are imaginary. 11. Find the roots of the equation, x± — x 2 -\- 2x — 1 = 0. Ans. 0.618, and — 1.618 ; the others, imaginary. 12. Find a root of the equation, X 5 _|_ 2a:* + 3x 3 + 4x 2 + 5x — 54321 = 0. Ans. x = 8.414 454 7. 13. What is the fifth root of 2 ? Ans. 1.148 699. Note. This method of finding the real roots of any equation, if incommensurable, approximately, if commensurable, exactly, is sometimes called Horner's method. *27 318 - THEORY OP EQUATIONS. [§ 395. RECURRING, OR RECIPROCAL EQUATIONS. § 395. The general form of a recurring, or reciprocal (§ 371) equation of an odd degree, is, obviously, x 2»+i+^ 1 a; 2n + A 2 x 2n ~i . . ± A 2 x 2 ± A x x ± 1 = ; (1) in which the like coefficients belong, one to an even, and the other to an odd power of x throughout. Now, if the corresponding coefficients have like signs, the substitution of — 1, and, if they have unlike signs, the substitution of -f- 1, for x, will render the corresponding terms numerically equal with contrary signs ; and will, therefore, reduce the first member to 0. Hence, One of the roots of a recurring equation of an odd de- gree is — 1, or -f- 1, according as the corresponding coeffi- cients have like or unlike signs. a.) Again, the equation may be written thus, (x*"+i±l)-\-A l z(x*»- 1 ±l)+A< i x*(x 2n -z±l) . . = 0;(2) in which x = — 1, if we take the upper signs, and x =z -L-l, if we take the lower signs, will render each of the quantities enclosed in parenthesis equal to zero. b.) Let 2n -\- 1 = 5. Then the equation becomes x sJ r A l x* + A 2 xZ±A 2 x2±A 1 x±l = 0; (3) or (x&±l) + A 1 x(x?&l)±A z x*(x±l) = 0. (4) Now, if we divide either the first member of (3) or each term of (4) by x ± 1 (§§ 98, 96), taking always the upper signs together, and the lower signs together, we shall have x 4 q: 1 x 2 qp 1 -Mi £-1-1=0; (5) x 3 + 1 + 4i evidently an equation of an even degree (the 2«tn), whose coefficients at equal distances from the extremes are equal (i. e. are numerically equal and have like signs). It is, therefore, a recurring equation (§ 370. b). The same reasoning will, obviously, apply to any similar equation as well as to that of the fifth degree. § 396-7.] RECURRING, OR RECIPROCAL EQUATIONS. 319 § 396. The general form of a recurring or reciprocal equation of an even degree, in which the like coefficients have unlike signs and the middle term is wanting (§ 370. c), is, obviously, x 2n+2_|_^ i;c 2.H-i # . -\-0x n — . . —A x x—l — 0. (6) Arranging (§ 34. c) according to the coefficients, we have (x 5 "+-— 1)+J 1 x(x 2n — l)+A 2 zi(x«- n - 2 —l) . . = ; (7) each term of which is, evidently, divisible by x 2 — 1 (§ 96), i. e. by (x+l)(x—l) [§ 93]. Hence, A recurring equation of an even degree, in which the middle term is wanting and the corresponding coefficients have unlike signs, has its first member divisible by x 2 — 1 ; and, of course, has the two roots, — 1 and -f- 1. a.) Let 2n-\-2 = 6. Then the equation becomes x*-{-A 1 x 5 -\-A 2 x' i — A 2 x 2 — A 1 x — l = 0; (8) or f (a;°- 1) + A x x{x± — l)+A 2 x°-(x* — 1) = 0. (9) Now if we divide either the first member of (8) or each term of (9) by x 2 — 1, we shall have x* + A x x^-\-A 2 x 2 +A l x + 1=0; (10) + 1 a recurring equation of an even degree, whose like coeffi- cients have like signs, as in § 395. b. b.) Otherwise ; the roots of the depressed equations, (5) and (10), are the remaining roots of the primitive equations (3) and (8) ; and one half of them are, therefore, the re- ciprocals of the other half. § 397. The general form of a recurring equation of an even degree, in which the like coefficients have like signs, is x 2n -\- A x x 2n ~^ . . +A n x n . . + .4 1 a; + l=:0. (11) Dividing by x n , we have x -\- A.^x ~ . . . -\-A. n ^]X -\- A. n -J- ^l u _ 1 — . . . x + ^i=i+=--^ ( 12 > or i -vt — A ■ 2* t 320 THEORY OF EQUATIONS. [§ 398. *«+^+A( xn - 1+ ^) ' • +A »-i( x+ l) +A » = ' (13) Now put x-\- - = z. (a) JO Then, squaring and transposing, x*+±; = z*-2. (b) JO Multiplying (b) by x -\- - = z, x 3 4-^--\-x-\-- — z^—2z ; orx3+-!- = zS—Sz. (c) X 3 X x d Or, in general, since we have, by transposing, ^'+^T=(^+p)(-+i)-(--'+-if)- (4 Thus, making to = 3, from (a), (3), and (c), x*+ i = (z^_ 3*)s - (*»- 2) = z*- 4*2+ 2. (/) Substituting these values of x-\-x~ 1 , x 2 -\-x~ 2 , &c. in (13), we shall have an equation of the nth degree in z; i. e. of half the degree of the primitive equation, (11). Hence, § 398. A recurring equation of an even degree, in which the like coefficients have like signs, can always be reduced to an equation of half that degree. a.) Hence (^§ 395, 396), Cor. A recurring equation of an odd degree (2» -f- 1), or one of an even degree (2n-\-2) whose middle term is ■wanting and whose like coefficients have unlike signs, can always be reduced to an equation of the «th degree. b.) The solution of the equation of the nth. degree gives § 398.] RECURRING, OR RECIPROCAL EQUATIONS. 321 the values of *; and the values of a? may be found from the equation, x-J-- = z; i. e. a? 2 — zx = — 1. x 1. Find the roots of the equation. X 5_ na;4_|_ I7a? 3 +I7a? 2 — \\x + 1 = 0. One root is —1 (§ 395). Therefore, dividing by * + 1 (§ 348, 350), we have x*— 12a? 3 -|-29a? 2 — 12a? +1 = 0. Dividing by a? 2 , (* 2 +^)- 12 (^ + ^)+ 29 = °- Substituting, z 2_ 2 — 12c + 29 = ; or s*— 12* _|_ 27 = 0. z = a? 4-- = 9, or 3. a? 3f2=»j a? 2 — 9a? = — 1, and x = i(9±y77) ; if 2 = 3, a? 3 — 3a? = — 1, and a? = £( Si^/5). Therefore, the five roots are 9 +y77 9 — y77 3 +y5 3 — y5 -1, g , g ' — 2"— ' and — Y ; or, rendering the numerators of the third and fifth roots ra- tional (§ 187), 9-K/77 2 3+^5 aQd _2__. _1 ' 2 ' 9+^77' 2 ' ana 3+V5' the third root being the reciprocal of the second, and the fifth, of the fourth (§ 120. d). 2. Find the roots of the equation, 4a?6— 24a?5-f 57a? 4 — 73a? 3 + 57a? 2 — 24a? -f 4 = 0. The reduced equation is 4z 3 — 24z 2 + 45s — 25 = 0, whose roots are 1, § and f. Hence, the roots of the given equation are 2, h 2, 1, 1 ~ i ^, ™* l =^F 1 - 322 THEORY OF EQUATIONS. [§ 399, 400. 3. Solve the equation, 5**4- 8z 3 + 9^2+ 8x + 5 = (§ 388. N.). l Ans. 5z 2 -f8z — 1 =0. x — .05825 ±^/(— .9966), x = —.85825 ±y(— 2634). 4. Solve the equation, a;«— 6|x 5 -f-llfx4— llfa; 2 +6|a;5— 1 (§398. a). The roots are 1, — 1, 2, £, 4, and J. BINOMIAL EQUATIONS. § 399. Equations of the form, y»±A=zQ, (1) containing but two terms, are called binomial equations. Suppose A n =. a, i. e. A = a". Then we have y n ±.a n = 0. Putting y = aa;, a n x"± a" = ; or x n ± 1 = 0. (2) § 400. I. Let n be an odd number, 2m-\-\. Then x 2m +i±l = 0, (3) being a recurring equation of an odd degree (§ 395), has one reed root equal to — 1, or -j- 1, according as the last term is positive or negative. 1. Let the equation be ar 2 "»+i — 1 = 0. (4) Then -j-1 is a root ; and dividing by x — 1, we have (§ 96) a; 2m_^ a ,2m-l_|_ x 2m-2 . . . -L. X * _L. a; -f- 1 = 0, (5) which can be reduced to an equation of the with degree (§ 398). Moreover, (4) has ?io o^er rea? root. For, if x be neg- ative, x*™+ 1 will be negative (§ 151. c) ; and, if a; be pos- itive and different from ljX 2 "^ 1 evidently cannot be equal to 1. § 401.] BINOMIAL EQUATIONS. 323 Consequently, all the roots of (5) are imaginary, a.) This is evident, also, from the number (2m) of con- secutive terms wanting in (4). See § 364. 2. 2. The equation, x 2ni + l + 1 = 0, (6) has (§ 395) one real root equal to — 1 ; and, reasoning as above, it is evident, that it can have no other real root. If we divide by x-\-l (§ 98), we shall have the equation containing the remaining roots, which can be reduced by §397. b.) Also, the roots of x- mJ r l -f- 1 = are the same as those of x 2m + l ] — 1 = 0, taken with contrary signs (§ 359). § 401. II. Again, let n be an even number, 2m. 1. Then x 2m — 1 = (8) has two real roots, -j- 1 and — 1 (§396). It has also no other real roots. For, if we divide by x 3 — 1, we have a . 2 m-2_J_ a .2m-4 _J_ X <1 _|_ 1 _ Q . (0) in which the powers of x being all even (§ 151. c), any real value of x, whether positive or negative, will render the first member positive (§ 358. 3), i. e. > 0. This equation can be reduced also to one of the (m — l)th degree (§ 398). a.) Moreover, we have x" m — 1 =(x m — l)(x m -f-l = 0. x m — 1 = 0, and x m + 1 = 0. 2. All the roots of the equation, x" m -\-l = 0, (10) i. e. x 2 "' r= — 1, are imaginary (§ 22. 2). This equation can be reduced to one of the mth degree (§398). b.) In each of the equations, (8) and (10), there is a de- ficiency of an odd number (2m — 1) of consecutive terms- 324 THEOBY OF EQUATIONS. [§ 402, 403. Consequently (10) must contain at least 2m imaginary roots; and (8), at least 2m — 2 (§ 364. 1). § 402. Let the real roots be suppressed from the equa- tion, x"^ 1 = 0; and let the equation in z, Z:= 0, be found (§397). Let also one of the imaginary values of x be a-\-by — 1. Then we shall have But (§§187, 162) 1 a — by— 1 a — by— 1 a + by— 1 (a + by— 1) (« — by— 1) a 2 +6 a Moreover, if a-\-by — 1 be a root of the equation, x n q: 1 = 0, a — b y — 1 must be a root also (§ 357). Hence we shall have ( a -{-by—l) n = ±l, and (a — by—l) n — ±1. .-. [(a + by-l){a — by-l)Y = (a*-+b 2 ) n z=l. And since a 2 -\-b- is a positive quantity, we have — -r-j-- — ; —a— b\y— 1 ; a-\-by — 1 and z = a -f- b y— l-\-a — b y— 1 = 2a. Hence, all the roots of the equation, Z — 0, are real. § 403. Let a be one of the imaginary roots of the equa- tion, x n — 1 = 0. Then we have a n = 1 ; a n - n = 1 ; a 3n = 1 ; &c. also «-" = 1 ; a~ 2n = 1 ; ar* n = 1 ; &c. Hence, If « be an imaginary root of the equation x n — 1 =0, then will any integral power of a be a root also. a.) As the equation can have but n roots, many of these powers of a must be equal to one another. Thus, the imaginary roots of a; 4 — 1 — are + */ — * and — y— 1. Now we have ($ 162) § 404, 405.] BINOMIAL EQUATIONS. 325 -(■v/-l) 2 = -l; (V-iy = -V-l; (v>-l)* = l; (y_l)5 = y_ 1; (y_i) 6= _ 1; (y_l)7 = _y_ 1; 6.) It must be understood, however, that these are only different ways of expressing the same roots. The equa- tion, x n ^:l = 0,-has no equal roots ; since its derived equa- tion nx n ~ x — has no common measure with it (§ 378./). § 404. Let a be an imaginary root of the equation, afrf-1 = 0. Then we have o tt = — 1; (a B ) 3 ==(a s ) B — — 1; (a") 5 = (a 5 )" = — 1; also (a")- 3 = («- 3 )' , = — 1; (a 2 " 1 -^)" = — 1. Hence, If a be an imaginary root of the equation, x n -f- 1 == 0, then will any odd integral power of a be a root also. Thus, the imaginary roots of x~-\- 1 = are -\-»/ — 1 and — «/ — 1 ; and all the odd integral powers of either of these roots are also roots (§ 403. a). § 405. Find the roots of the following equations ; 1. £3—1 — 0. Ans. 1, - , and . 2. x*— 1 = 0. Ans. 1, — 1, y— 1, and — y— 1. 3. *6_i — 0; i. e . (x3—l)(x 3 -\-l) = 0. . i±y— 3 _ liy— 3 Ans. 1, — 1, , and 2 ' 2 Note. The roots of the equations, a; 2 — l = 0, a: 3 — 1 =0, &c, are sometimes called the roots of unity. It is evident (§§151. a; 152. a), that the roots of any other number, of any degree, may ba found by multiplying one of them, most conveniently, the arithmeti- cal root, by the several roots of unity of the same degree. ALG. 28 CHAPTER XVII. CONTINUED FRACTIONS. § 406. A continued fraction is one whose nu- merator is a whole number, and whose denominator is a whole number plus a fraction, which also has a ivhole number for its numerator, ana for its denom- inator a whole number plus a fraction; and so on. We shall consider only those, in which each of the nu- merators is unity, and the partial denominators (a, below) are all positive. Thus, 1 (1) 1 (2) 1' a i 1 4 + &c. J ' « 3 +&c. are continued fractions. a.) The integral parts of the denominators are some- times called partial denominators, or partial quotients; and the fractions, \, -^, &c, — , — , &c, are called partial. or integral fractions. § 407. If, in (1) above, we neglect all but the first par- tial fraction, the denominator 2 will be less than the true denominator; and, of course, |is greater than the true val- ue of the continued fraction. Again, suppose we neglect all but two partial fractions. Then, the partial denominator, 3, being too small, the par- § 408.] APPROXIMATE VALUES, OR CONVERGENTS. 327 tial fraction, £, is too great ; and, consequently, 2^ being greater than the true denominator, the fraction, 1 _1_3 will be less than the true value of the continued fraction. Similar reasoning will, evidently, hold in respect to any number of terms ; and will apply equally to the general form (2), as to the particular example we have considered. Hence, If we include in the reduction an odd number of partial fractions, the result will be too great ; if an even number, the result will be too small. 11 1 . a.) The fractions, — , -, -. &c, 2 a -4- — - are approximate values of the given fraction ; and are sometimes called approximating or converging fractions, or simply, convergents. b.) It is evident, that the true value of the continued fraction, lying between two successive approximate values, differs from either of them less than they differ from each other. § 408. We have — = — , 1st approx. value. a x a x 1 ' fl^aa + l' «i + — 2d « «2 a * + aZ a 2 + — 0203 + 1 3d (. a l a 9 + 1 ) a 3+ a l 3'2& CONTINUED FRACTIONS. [§ 4Q8. We shall, evidently, find the fourth approximate value, or convergent, by substituting, in the third, a 3 -\ for a 3 . «4 Thus, flu«_ -4- 1 \a . A- ci- VS, the fourth conver- (a 2 a 3 -|-l)a 4 -[-«2 gent. We find, obviously, the numerator and denominator of the third convergent, by multiplying those of the second by the third partial denominator, and adding those of the first convergent. We find, in like manner, the fourth convergent from the terms of the second and third. To show the generality of this law, let it be admitted to hold good as far as the rath convergent (i. e. the conver- gent corresponding to a n ). ' L M N , P . ^ Let also jy, —-, -^, and — be the convergents cor- responding to ot„_ 2 , a„-u #«i and a n + x . Then, since the nth convergent is formed according to the above law, we shall have -=-. = , /r . " , -j-.. (3\ N' M'a n -\-L' v / N 1 If now we substitute in — , a n -\ for a n , we shall, JS' « ;i -f x P obviously, find -^. Thus, p_ _ M ("" + ~^r) +L (jfc„+z) g „ +1 +ii/ P Na n +, + M or, from (3), y , = ^— j-y (4) Consequently, if the law holds good for n convergents, it will for n -f- 1. Hence, to find the numerator and denominator of any § 409.] CONVERGENTS. 329 convergent after the second, as the (n -f- l)th, we have the following RULE. § 409. Multiply the numerator and denominator of the nth convergent by the (n-\-\)th partial denominator, and add to the products, respectively, the numerator and denom- inator of the (n — l)th convergent. a.) The numerator and denominator of any convergent must be respectively greater than those of the preceding ; each numerator and each denominator being at least equal to the sum of the two next preceding. b.) Moreover, each convergent is found by substituting, in the preceding, for the last partial denominator, an ex- pression known to approach more nearly to the true de- nominator. Hence, evidently, each convergent approximates more closely than the preceding to the true value of the con- tinued fraction. See § 410. a, b. 1. Find the successive convergents of the continued fraction, 1 2 + 1 l +Wr Ans. \, £, §, t* t , and §|£. c.) The first four convergents are approximate values of the continued fraction ; the last, %%^, is the true value. d.) A continued fraction is sometimes mixed (§112)? i. e. made up of a whole number and a fraction. Thus, 3 + 1— («) 2 + r 5 + &c. *28 330 CONTINUED FRACTIONS. [§ 410. In such cases, the integral part may be reserved and ad- ded to the convergents ; or it may be taken, with 1 as a denominator, for the first convergent. Thus, in the above example, Ave shall have the conver- gents, 8fe 3|, 3if ; or f, |, V, lg$. 1 (6) e.) This form, a -f- «i + a 2 -j-«fec, is sometimes assumed as the general form of a continued fraction ; the place of the integral part, when it is wanting, being filled with 0. In that case, the jz>st convergent is, evidently, too small ; the second, too great ; and so on, those of an odd order be- ing too small, and those of an even order, too great. See §407. Note. If the integral part be zero, the first convergent will of course be o § 410. If the second convergent of §408 be subtracted from the first, the remainder is unity divided by the product of the denominators. If the third be subtracted from the second, the remainder is minus unity divided by the product of the denominators. Suppose it has been proved, that this law extends to n — 1 convergents ; i. e. that L. - — LM' —L'M _ ±1 Z 7 M'~ z 17M<~ ~T7W' M _N M Ma n + L M' N' ~ M> M'a n + L ' _ L'M—LM' _ LM'—L'M the numerator of which is the same as that of (7), with a contrary sign. Hence, the principle proved in regard to the first three convergents, applies equally to the whole series. That is, If each convergent be subtracted from that ivhich next § 411.] CONVERGENT. — ERROR. — LOWEST TERMS. 331 precedes, the numerator of the difference will be ± 1 ; and the denominator will be the product of the denominators of the two convergents. a.) Again, the true value of the continued fraction lie? between any two successive convergents, and differs from either of them less than they differ from each other (§ 407}. M That is, the convergent — f , differs from the true value of the continued fraction by less than M'N< But (§ 409. a) M> < N> ; and .-. M'~ < M'N'. WW' < M™' That is ' Cor. i. The error, in taking any convergent whatever for the true value of the continued fraction, is numerically less than unity divided by the square of the denominator of that convergent. b.) The denominator of each convergent is greater than the next preceding by some whole number (§ 409. a). Hence, if the fraction be infinite, we may find a conver- gent whose denominator shall be greater than any given quantity ; and, consequently, Cor. ir. "We may find a convergent, which shall differ from the true value of the continued fraction by less than any given quantity. c.) Suppose that M and M' have a common divisor, D. Then D will, of course, divide L' M and LM', multiples of M and M 1 ; and, consequently (§ 102. Note c), the differ- ence of those multiples, LM' — L'M= ± 1. Therefore D must divide ± 1, which has no integral di- visor but unity. D = \. That is, Cor. in. Every convergent is in its lowest terms. § 411. One of the most obvious uses of continued frac- tions is, to express approximately, in small numbers, frac- tions whose terms are large. Thus, 332 CONTINUED FRACTIONS. [§ 412. i. £= 17 1 1 1 1 59~ w T 17 3 +17 Ct) 3 •+i Here we first divide both numerator and denominator (§113. 3) of \l by 17. We then reduce ff to a mixed number 3 T S T .; and, again, divide both terms of T 8 T by 8, and reduce to a mixed number; and so on. Evidently, these operations produce no change in the value of the given fraction. a.) Now the several convergents of the continued frac- found, are ^, f , and J|. 1 191 AVe find - = -~, too great ; 2 16 fi - = — -, too small, but differing from / oy the true value by only ; { 3 . 2. If the fraction proposed had been \ f , we should have found 59 _q I 8 -q_L l --J. 1 __3+-_3+-_o+— T ; ¥ 2 + 8 and the convergents, 3, §, and -Jf . § 409. d. b.) This reduction of a common, to a continued fraction, is, evidently, effected by applying to the terms of the given fraction the process of finding the greatest common divisor ; the several quotients forming the successive partial denom- inators. § 412. If it be required to transform any quantity what- ever, x, into a continued fraction, the nature of continued fractions will sufficiently indicate the following RULE. 1. Find the greatest integer contained in x, and denote § 413.] REDUCING TO A CONTINUED FRACTION. ooo it by a ; and denote the fractional excess of x above a by 1 1 1 ^ , — . Then x — a-\ . .•.:*:., = > 1. x 1 x x x — a 2. Find the greatest integer contained in x lt and denote it by a x : -and denote the fractional excess of x x above a x by — . Then x x = a x -{-■ — . 3. Apply the same process to x 2 , and so on. Thus, x=za + — = a-\ =a-\ — - «x+ — a x -] 1 2 a 2 -) . &.c. a.) If x < 1, we shall have a = 0. h.) We shall always have x x , Xc,, &c. > 1. For if x x = , or < 1, we have — = or > 1 ; and a is x x not the greatest integer contained in x. c.) Whenever we find a denominator, x n , equal to a whole number, we shall have x n =. a n ; and the continued fraction will terminate. This will happen, if the cpiantity, x, can be exactly ex- pressed by a common fraction. d.) If the quantity is not equal to a common fraction (i. e. if it is incommensurable), the continued fraction Avill extend to infinity. § 413. 1. Given n = 3.14159 (§ 247. N. q), employing only five decimal places. Reduce n to a continued frac- tion, and find approximate values. „ , 1 Ans. it=3-\ 15 + * Convergents (§ 400. e), 3, £, fff, Hlb &c - 334: CONTINUED FRACTIONS. [§ 414. Note. The second approximate value, 22 was found by Ar- chimedes; the fouith, 54^, by Adrian Metius. 2. The common, or tropical year consists of 365.242 241 mean solar days. Find approximate values for this time. Ans. 365i, 365^, 3653 8 3 , 365 T W, &c. Note. The third approximation shows an excess of the solar year above 365 days, of JL of a day. To preserve the coincidence between the solar and civil year, therefore, eight years in thirty-three must contain 366 days each. That is, a day must be added to every fourth year seven times in succession, and, the eighth time, to the fifth year. 3. The sidereal month (i. e. the time of the moon's side- real revolution) consists of 27.321 661 days; or, the moon revolves 1 000 000 times in 27 321 661 days. Find approx- imate values of this ratio. Ans. 27, 8 /, 7 ^, 3 T \° 3 ', &c. Note. These ratios show that the moon revolves about 3 times in 82 days; 28 times in 765 days; or, more exactly, 143 times in 3907 days. § 414. Continued fractions are also employed in finding the roots of equations, and in extracting the roots of num- bers. 1. Extract the square root of 3 ; i. e. find a root of the equation, x 2 — 3 = 0. (1) Here x ■= 1 -I . x x Diminishing the roots of (1) by 1 (§ 367), we have y* + 2y-2 = 0, (2) an equation, whose roots are equal to — . Transforming (2) by § 370, we find 2x? — 2x r — 1 = 0. (3) This gives x Y = 1 -| . •Ms n Transforming (3) in the same manner as (1), we have 1 x„ 2 — 2x 2 — 2 = 0; (4) and x 2 = 2 -\ . X 3 § 414.] ROOTS OF EQUATIONS. 33") We find, in like manner, 2aJ 3 2 — 2x z — 1 =0, (oi which being the same as (3), will have the same roots, and will give rise to transformed equations like (4) and (5). Hence, we shall have a repetition of the equations (3) and (4), and of their roots of which 1 and 2 are the inte- gral parts, in endless succession. x — 1 +- - = 1.732 &c. 1 + 2 + - l + i&c. The convergents are & f, f, f, \\, ff, ||, ||. a.) A continued fraction of this kind, in which any num- ber of the partial denominators are continually repeated in the same order, is called periodic. b.) It will be found, that every incommensurable root of an equation of the second degree may be expressed by a periodic continued fraction. Of course, when the first period is found, such a fraction may be developed to any extent, by simply repeating the period. 2. Extract the square root of 2. Convergents, \, §, £, \l, §$, f§, &c. ERRATA. \ Page 14, first line, for " 16 " read " 11." " 60, line 27, for " + b'," read " — ft/." " 62, line 19, for " 58," read " 28." " 69, last line, for " + b " read " +ab." " 83, first line, for " + 63" read "_&3." " 91, line 15, for " a =6" read "6 = a;" and for "a" — 6"" read "a" — a' 1 ." " 92, " 20, for «+ab " read "— ab." " 93, " 27, after " them " insert " taken as a divisor. , ' > " 96, " 26, for "5aZ>3" read " 5«36." " 99, " 30, and page 100, line 26, for " 12 " read " 11." " 106, " 17, for " dividing " read " multiplying." " 119, " 5, for " 13_y read " 23^." " 128, " 25, for "(2.32)" rea d "(2.32):?." " 148, " 11, for " a—b " read " a^—b." "163, " 4, for "^(^2^2)" r ead "y(p2_ ?2 ) - » "192, " 3, for "10" read "11." "198, " 3, for "(1)" read "(3)." " 292, " 14, after " have," insert "(x being > 1)." " 310, " 11, for "+" read "-." r it It, - jiliiliili