JINN I', iiil QJntnsU Hniowaitg Htbrarg ■'' BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 arV19298 College algebra Cornell University Library ■g- ■{924 031 226 503 olin,anx Cornell University Library The original of tliis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031226503 COLLEGE ALGEBRA A SERIES OF MATHEMATICAL TEXTS EDITED BT EARLE RAYMOND HEDRICK THE CALCULUS By Ellery Williams Davis and William Chaeles Brenke. - ANALYTIC GEOMETRY AND ALGEBRA By Alexander Ziwet and Louis Allen Hopkins. ELEMENTS OF ANALYTIC GEOMETRY By Alexander Ziwet and Louis Allen Hopkins. PLANE AND SPHERICAL TRIGONOMETRY WITH COM- PLETE TABLES By Alfred Monroe Kenyon and Lonis Ingold. PLANE AND SPHERICAL TRIGONOMETRY WITH BRIEF TABLES i By Alfred Monroe Kenyon and Louis Inqold. ELEMENTARY MATHEMATICAL ANALYSIS By John Wesley Young and Frank Merritt Morgan. COLLEGE ALGEBRA By Ernest Bhown Skinner. MATHEMATICS FOR AGRICULTURE AND GENERAL SCIENCE By Alfred Monroe Kenyon and William Vernon Lovitt. PLANE TRIGONOMETRY FOR SCHOOLS AND COLLEGES By Alfred Monroe Kenyon and Louis Ingold. THE MACMILLAN TABLES Prepared under the direction of Earle Raymond Hedrick. PLANE GEOMETRY By Walter Burton Ford and Charles Ammerman. PLANE AND SOLID GEOMETRY By Walter Burton Ford and Charles Ammerman. SOLID GEOMETRY By Walter Burton Ford and Charles Ammerman. CONSTRUCTIVE GEOMETRY Prepared under the direction of Earle Raymond Hedrick. JUNIOR HIGH SCHOOL MATHEMATICS By William Ledley Vosbukqh and Frederick William Gentleman. COLLEGE ALGEBRA BY ERNEST BROWN SKINNER ASSOCIATE PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF WISCONSIN ¥eJD gorfe THE MACMILLAN COMPANY 1917 All rights reserved Copyright, 1917, bt the macmillan company. Set up and electrotyped. Published September, 1917. J. S. Gushing Co. —Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE The shortening of the time given to algebra in the secondary school, together with the great extension of the elective system . and the consequent placing of mathematics in competition with a score of new subjects, has made some modification of the. traditional college algebra absolutely necessary. The first important change has been to put the college alge- bra upon a more elementary basis. Whether we like it or not, we are bound to recognize the fact that a large number of freshmen come to us with a single year of algebra, and that year so far in the past that it is of little help to the college teacher. The second change is one which the writer believes will in the end prove to be of great advantage, not only to algebra, but to mathematics in general. The college teacher has been obliged to exert every effort to make his work of interest to his students. To this end the subject has been made more concrete, applications to the affairs of everyday life have been emphasized, processes have been made more direct, and the road to the mathematics of the sophomore year has been shortened. In the following pages I have tried to introduce enough elementary material to meet the needs of students who have had the minimum training of the secondary schools, and at the same time I have tried to recognize the growth in mental capacity that normally takes place between the first year of the high school and the freshman year of the college. I have tried to emphasize the immediately practical side of algebra by drawing freely upon geometry, physics, the theory of in- vestment, and other branches of pure and applied science for illustrative examples. An amount of space greater than usual VI PREFACE has been devoted to the study of the functions which occur most frequently in practical work. At the same time I have endeavored to arrange and classify this material in such man- ner that the student may acquire some skill in manipulation — an accomplishment in which the average freshman is sadly lacking. Geometrical interpretations have been introduced at the outset and have been emphasized wherever possible. Long experience has convinced me that there is no better method to make the student thiuk of algebra in a concrete way. I have not hesitated to omit a number of topics that are ordinarily included in textbooks on college algebra. The alge- braic solution of special types of equations belongs to the theory of equations rather than to general algebra ; partial fractions can be taken up in connection with the iutegral calculus. I have followed the same policy with respect to proofs of important theorems where the proofs seemed to me to be be- yond the average freshman. Many theorems of this class can be taken up to better advantage at a later period. It gives me great pleasure to acknowledge my indebtedness to my colleagues, Professors L. W. Dowling, Arnold Dresden, and H. T. Burgess, Dr. Florence Allen, Dr. T. M. Simpson, and Dr. Gr. E. Clements, now of the U. S. Naval Academy, for valuable suggestions made while the book was in manuscript. I am also under obligations to Professor E. E. Maurer and to Professor L. E. Ingersoll for data for problems from their respective fields. ERNEST B. SPaNNEK. The University of Wisconsin, July 1, 1917. CONTENTS I. Introduction . 1 II. Algebraic Identities 12 III. Powers and Roots 22 IV. Logarithms 37 V. Functions of a Variable — Graphical Repre- sentation 49 VI. Quadratic Equations with One Unknown . . 79 VII. Systems of Linkar Equations — ^Determinants . 91 VIII. Systems Containing Non-linear Equations . 105 IX. Inequalities 120 X. Complex Numbers . . ... 125 XI. Polynomials — Equations of Any Degree . . 131 XII. Determinants and Linear Equations . . 158 XIII. The Binomial Theorem .... . 179 XIV. Progressions — Compound Interest . . . 185 XV. Permutations and Combinations .... 206 XVI. Probability . ^ 212 XVII. Sequences and Limits 220 XVIII. Infinite Series 235 Tables 251 Index 259 COLLEGE ALGEBRA CHAPTER I INTRODUCTION 1. Positive Integers. Algebra, like arithmetic, deals with numbers. The positive integers, or natural numbers, arise through the operation of counting. They are denoted by the symbols 1, 2, 3, 4, 5, 6, ... of arithmetic. When considered in their usual order they form the so-called natural scale.* The numbers of the natural scale may be represented geo- metrically by points equally spaced on a straight line. Begin- ? f f f t f Fig. 1. ning with 1, which is placed at a convenient distance from an arbitrary starting point, the numbers are attached to the equally spaced points proceeding toward the right as in Fig. 1. 2. The Number Zero. To the arbitrary starting point the symbol 0, called zero, is attached. Zero is a new number to be added to the scale. When numbers are used to denote * Tlie natural scale has three fundamental properties : (1) it hegins with a definite first symhol ; (2) it has uo last symbol ; (3) every symbol is followed by a definite next symbol. The natural scale does not depend upon any particular mode of geometric representation. See Fine's College Algebra, Chapter 1. 2 COLLEGE ALGEBRA [L § 2 quantity, as 3 pounds, or 7 feet, this new number denotes the absence of quantity. As a number of the scale it is simply the symbol that immediately precedes the symbol 1. 3. The Four Fundamental Operations. Upon the positive integers the four fundamental operations, addition, subtraction, multiplication, and division, may be performed. Of these opera- tions two, namely, addition and multiplication, may be per- formed without introducing new numbers. Subtraction and division may not always be performed, as will soon appear. 4. Subtraction and Negative Numbers. The subtraction of 6 from 8 may be considered as the operation of finding the fifth number before 8 in the scale. Without the addition of new numbers, 5 cannot be subtracted from 3, for there is no .fifth number before 3 in the natural scale. Neither could we express a temperature of 2 degrees below zero. To make sub- traction of integers possible in all cases, negative integers with symbols -1,-2,-3,-4,... are introduced. These negative integers are so arranged that — 1 goes before 0,-2 before -1,-3 before — 2, and so on. When the positive and the negative integers, and zero are ar- ranged as indicated above and in § 1, they form the so-called complete scale* The numbers of the complete scale are represented geometri- - or < between the two numbers, in such a way that the opening is toward the greater number. Thus, the inequalities 7>3, -1<3, -6<-3 mean, respectively, 7 is greater than 3, — 1 is less than 3,-5 is less than — 3. 6. Division and Fractions. With nothing but the numbers of the complete scale available, division would be possible only in case the divisor is contained an exact number of times in the dividend. The difficulty is obviated by the introduction of fractions. Eor example, the quotient of 5 by 3 is defined as that number whose product by 3 is 5, and is written 5/3 or |. Thus, we have by definition |x3 = 5. To represent 5/3 geometrically, each o,f the lengths 01, 12, 2 3, — of Pig. 1 is divided into three equal parts, so that the division points are three times as numerous as before. To the ? f f f f t f . , f , , f , , f , , f Fig. .3. first point to the right the symbol 1/3 is attached, to the second 2/3, and so on. The fifth point represents 5/3. Similarly, points to the left represent successively — 1/3, — 2/3, and so on. Figure 3 shows how this representation is accomplished. 4 COLLEGE ALGEBRA [I, § 7 7. Reduction to a Common Denominator. Each interval in the complete scale may be divided into any given number of equal subintervals. If each interval be divided into 3 subia- tervals, and in the same geometric representation the divisions obtained by dividing each interval into 12 subintervals be marked, it -will be geometrically evident that 5^20 3 12' The second fraction is obtained from the first by multiplying numerator and denominator by 4, and the unit is now one twelfth instead of one third. Similarly, the fraction 7/4 which means 7 units, each one fourth as long as the unit of the complete scale, may be ex- pressed as 21/12. Two fractions are said to be reduced to a corriTnon denomi- nator when they are expressed in terms of the same unit. The common denominator must be a multiple of the denominators of both fractions. Any multiple will answer, but it is usually better to use the least common multiple. Fractions having a common denominator are compared by comparing their numerators. For example, we have 7^5 . 21 ^ 20 - >-, smce — > — . 4 3' 12 12 8. The System of Rational Numbers. The totality of all quotients that can be formed by means of the numbers of the complete scale (division by zero excepted)* constitutes the so- called system of rational numbers. In other words, the system of rational numbers consists of all positive and negative in- tegers, all positive and negative fractions, and zero. Any number of the rational system bears a definite relation to any second number of the system in the sense that it is either greater than or less than the second. Geometrically, * The reason for excluding divisioii by zero will be explained later (§ 14) . I, § 9] INTRODUCTION 5 the first is represented by a point farther to the right or farther to the left than the point which represents the second.* In the rational system, the four fundamental operations, di- vision by zero excepted, are always possible. 9. The Introduction of Literal S3rmbols. Formulas. Or- dinary arithmetic deals principally with numbers which are expressed in terms of the nine digits and zero. Algebra, on the other hand, deals with numbers which are expressed, for the most part, by means of letters. Numbers represented by letters are combined by means of the same operations and under the same laws as numbers of the rational system. In this way algebraic formulas in which ordinary numbers may be substi- tuted at any time for the letters, are obtained. For example, a body moving with uniform velocity during time t moves through the distance d given by the formula d = vt. EXERCISES ' 1 . The formula for simple interest is ■ I=Prt, where P is the principal, r the rate, and t the time expressed in years. Find the interest when P = ^ 125, r = .06, and t = 2j\. ' 2. The distance traversed by a hody moving freely from rest under a uniform acceleration, is given by the formula 2 where s is the distance in feet, t is the time in seconds, and a is the accel- eration in feet per second per second, tind s vfhen J = 6, and a = 32.2. 3. If a freely moving body has an initial velocity of vo, the formula for space moved over is s ^-afi + Vot. Find « when t = 5, a =-32.2, and vo = 20. • * The rational system differs in two important particulars from the com- plete scale: (1) a given number is not followed by a definite next number; (2) between any two numbers of the rational scale there is one, and conse- quently, an indefinite number of other numbers. 6 COLLEGE ALGEBRA [I, § 9 4. The formula for uniform acceleration is u, — ) t2 — tl where vi aud v^ are the velocities at times ti and ti. Find the accelera- tion when the velocities at the end of 3 seconds and 7 seconds are 96.6 ft./sec. and 225.4 ft./sec, respectively. 5. The formula for compound amount is (7 = P(l + i)", where P is the principal, i the rate, and n the time in years. Find the compound amount of $500 for 3 years at 6 ^fc. 6. The formula giving the sum to which an annual deposit of P dollars drawing interest at rate i for n years will amount, is given by ,l_p Cl + »)"-! i Find the value of a man's savings if at the end of each year for five years he deposits 1 100 in a savings bank paying 4 fa. 7. According to the Newtonian law of gravitation, the formula for the force exerted upon a unit mass by a mass m at distance r, is / = km/r^. What is the force exerted by the earth upon a unit mass at the surface, if the mass of the earth be taken to be 614 x 10^^ grams, the distance to the center 637 x 10^ cm., and k-- 1543 X 104 10. The Three Fundamental Laws. The four fundamental operations are carried out in accordance with three fundamental laws, which are stated as follows : I. The Commutative Law. Tlie order of tJie terms of a sum, or of the factors of a product, may be changed without affecting the result. In symbols, we may write, for a sum and for a product, respectively, (1) a-\-b=b + a; (2) ab = ba. I, § 10] INTRODUCTION 7 II.. The Associative Law. The terms of a sum, or the factors of a product, may be grouped in any way we choose without chang- ing the result. In symbols, wq may write, for a sum and for a product, respectively, (3) a + (6 + c)=(a+6)+c; (4) a(hc) = (ab)c. III. The Distributive Law. The 'product of one factor by another which is the sum of tivo or more terms is the same as the sum of the terms obtained by taking the products of the first factor into each term of the second. In symbols, (5) a{b + c)=ab + ac. These laws, which are perfectly obvious for integers, and even for fractions, apply without exception to all numbers with which ordinary algebra deals. The four fundamental operations together with the three fundamental laws constitute the rules of reckoning for algebra. Many of the simpler problems in factoring are solved by direct application of the distributive law in the form ab + aG = a{b + c). For example, mnx — mny + mnz = m,n{x — y + z). Again, am — an + bm — bn =a{m — n)+ b{m — n) = {a + 6)(m — n). In this case the law is applied twice. The distributive law does not hold for all the operations em- ployed in algebra. For example, (a + b)^is not a^ + ¥, but is a^-i-2ab + b^. One of the commonest mistakes is to apply the distributive law in extraction of roots. The expression Va^ + b^ is not equal to a + &, as may be seen easily by substi- tuting numbers for a and b. 8 COLLEGE ALGEBRA [I, § 11 11. The Generalized Distributive Law. Applied twice, the distributive law gives (a+ b)(c + d) =(a + b)c+(a + 6)d,= ac + 6c+ ad+bd. In a similar manner, it is easy to show that (a+ 6)(c+d)(e+/)= ace +bce+ade+bde+acf+bcf+adf+bdf. The last product is formed in accordance with the generalized distributive law which applies to any number of parentheses and which is stated as follows. The product of any number of parentheses is equal to the sum of all possible products that can be formed by taking one and only one factor from each parenthesis. By means of this law products of several factors may be written down by inspection. EXERCISES 1. Write out each of the foUowmg products. '(a) {a + b)(x + y + z). ^(d) (Va +Vb){G + d + e). *(6) (x + a){x + b)(x+c). -(e) (a + 3)(& + c + 2)(5 + S). ■ (c) (mn+i)3)(rs + ««). ' (f) (a + b}(,x^ - xy + y'^). 2. Tactor each of the following expressions, noting the applications of the distributive law. - (a) ax — bay+ az. "(c) 3 a;^ — 7 a:^ + 8 a;. (6) OK* + 2bx' + C3?. (d) axJrbx + ay-\-by + az + bz. - (e) x^ ~ ax — bx+ ab. • (/) ax + ay + az + bx + by + bz + ex + cy + cz. 3. Columns of figures may he added by beginning either at the top or at the bottom. What law justifies this ? 4. Can you add a column consisting of twenty numbers by adding the sum of the first eight numbers, the sum of the last six, and the sum of the remaining six ? What laws give the authority ? 5. Illustrate the associative law for multiplication by considering the volume of a rectangular parallelepiped whose edges are 5, 7, and 10. I, § 12] INTRODUCTION 9 6. Show how the distributive law may be interpreted geometrically by considering the total area of two rectangles liaving a common altitude c and with bases a and b. 7. Show how the distributive law is used in finding the area of a trape- zoid with altitude a and parallel sides b and c. 8. How is the distributive law used in finding the area of a regular polygon ? 9. Verify the results obtained in Exs. 1-2 by substituting numbers for the letters. 12. Extension of Meaning of the Fundamental Operations. In arithmetic the first definitions of addition, subtraction, mul- tiplication, and division are given for positive iategers. These definitions have no meaning either for fractional or for nega- tive numbers. For example, the product of 4 by 3 is 4 -|- 4 -|- 4 where the multiplicand 4 is used 3 times. But the product of 4 by 2/3 cannot be interpreted in the same way. It becomes necessary, therefore, to make new definitions whenever new numbers are introduced. When the numbers to be combiued are fractions, the new definitions are given by the following formulas. (6) For addition : a^cad + hc^ (7) For subtraction : b d bd a _ c_ ad — be b d bd * (8) For multiplication : f X J = :"" (9) For division : b d bd ■ « ■ c _ ad b ' d~ be These formulas are merely the expressions in symbols of the rules given in arithmetic for combiniag fractions. They are assumed to hold for every kind of number that is used in ordinary algebra. 10 COLLEGE ALGEBRA [L § 13 13. Operations with Negative Numbers and Zero. Let a and b be any rational' numbers, either integers or fractions. The fundamental operations with negative numbers and zero are defined by the following equations. (10) a + (- 6) = a - &. (13) a x (- 6)= - ab. (11) a-{-b)=a + b. (14) (- a) x (- 6)=a&. (12)' 0+0 = 0-0 = a. (15) 0X0 = 0. (16) a-(-6) = (-a)^6=-|. (17) (_a)^(-6)=^. (18) 0^0 = 0. b EXERCISES 1. Perform the indicated operations in eacli of the following exercises, and check each result by substituting numbers for letters. -(«) i-(-i)- (6) ix(-^)- .(c) (-«)x(-A)- ■w (-I)^A- («) (f-. 5) (-1.25). ' (/) b e a ■ (?) ^ + ^ ■ a + b a — b w x+y y . / , x + y (.i) a+6 6 2 a^-b^ U) a + b ^,a-b c + d' c — d (k^ a2 + 2 a + 1 . a? - 1 \z X yl \a b c] (0) (?) (r) c2 _ d^ 6 - a 24 a; . 8 a; 16a;2-l 4a;-l 27a:8 . 9 a; a;2 _ 9 • 3 - X ' 1 — a + 6 1+- a— b Zxy ■ 9xy ^±^-a X a 2. Translate the formulas 6-18 of §§ 12 and 13 into ordinary language. I, § 16] INTRODUCTION 11 14. Division by Zero not Defined. According to the defi- nition of § 6 the division of one number a by another number 6 is defined by the equation (19) ^xh = a. If the quotient a/b be denoted by q, the defining equation takes the form a = qb. If the divisor 6 is zero, the equation can have no meaning unless a is zero also. In this case q may have any value what- ever. In either case we shall have to say for the present, at least, that the quotient has no meaning when the divisor is zero. 15. Parentheses. In algebraic reductions it is frequently necessary to operate with long and complicated expressions. The removal of parentheses inclosing such expressions is accomplished . by means of the distributive law together with the first four definitions in § 13. The parentheses may be removed one at a time. EXERCISES 1. Simplify each of the following expressions by removing aU parentheses, '(a) a-[-(6-c)]. '(c) 10 — 7[- 8 - 2(16 + 7)]. (6) - X - 3[y - 2Cx + y)-i. (d) (a + b - c)(x -y + z). (e) (X -(- 1) - 2 {(X + 2) + 3[(x + 5)- 4(a; H- 4)]}. ~Cf) b^~b{b + cla{b-c)+ 6(c - a)-|-c(a- 6)]}. '(g) l-{l-[l-(l -»)]} + »;. " 2. Inclose the last thtee terms oix^— Ty+iz — Swiua. parenthesis preceded by the negative sign. 3. Inclose the second, third, and fifth terms of 3 a6c - 4 62c -I- 7 a^c -1- 13 6c2 - 8 ac2 in a parenthesis preceded by the negative sign. 4. Fill out the parenthesis on the right side of the equality x^+2aW-b^ = x^-i ). CHAPTER II ALGEBRAIC IDENTITIES 16. Definitions. An algebraic identity is an equality be- tween two algebraic expressions such that one member may be transformed into the other by means of the rules of reckon- ing (§ 10, Chap. I). For example, the equality is an identity since the left member may be changed into the form of the right member by simple multiplication. When numbers are substituted for the letters in an algebraic expression the result is a numerical value of the expression. The most important property of an algebraic identity is given by the following principle, the truth of which is as- sumed without proof. If two expressions are identically equal, they are numerically equal for every set of values of the letters for which the expres- sions are both defined. Thus, when x = the identity given above reduces to 6 = 6; vehen a; = 2, or X = 3, it reduofes to = 0. A conditional equality, or a conditional equation, is an equality such that the two members are not numerically equal for every set of values of the letters for which they are both defined. For example, the conditional equality x^ — 5 a; -H 6 =0 is true for a; = 2 and for x = Z, but not for a; = 0, or for a; = 1. 12 [I, § 16] ALGEBRAIC IDENTITIES 13 Identities are frequently written with the ordinary sign of equality. If, however, it is desired to emphasize the difference between the identity and the conditional equation, the sign =, consisting of three bars, is used for the identity. Illusteative Examples. The equality x^-6x+6 _ x-S x^ — 2 X X is an identity, since the left member takes the same form as the right member when both numerator and denominator are divided by x — 2. The two members of the equality have the same numerical value for every value of x, except x = and x = 2, the two values for which they are not both defined. The equality x'' — 5x + 6 _Q x2 - 2 X ~ Is not an identity, but a conditional equation, since the numerical values of the two numbers are not equal for any value of x except x = 3. The equaUty 3x + 4j/ = 7 is satisfied by an infinite number of sets of values of x and y. It is, nevertheless, a conditional equality, since it is not satisfied by the system of values x = 2 and y = 1. The equaUties (x-2)(x- 3)=x2-5x + 6, a + b = b + a, x(,y + e)=xy + XZ, ■ a . c _ ad + bc b d~ bd ' are all identities. The distinction between the identity and the conditional equality is a fundamental one in mathematics. In the simpler cases the test may be made either by means of the definition or by the principle given on p. 12. 14 COLLEGE ALGEBRA [II, § 16 All changes in the forms of algebraic expressions are effected by means of identities; all statements of problems which require the finding of definite values for an unknown are expressed in terms of conditional equations. The product of a coefficient and positive integral powers of one or more variables may be called a simple term. For example, Sx'y'z, Baaf'y^ and iax*, are simple terms. In the seccind and third examples a may be thought of either as one of the variables, or a part of the coefficient. The sum of several simple terms is called a polynomial. Thus, 3xY-5 a^yz" + 14 o-V*^ is a polynomial. The degree of a term of a polynomial is the sum of the ex- ponents of the variables. In the example just given, the degrees of the terms are 6, 6, and 7, respectively. The degree of a polynomial is the degree of the term, or terms, of highest degree. The degree of a polynomial depends upon the particular variables under consideration. Thus, the degree of the polynomial given above is 7 if it is looked upon as a polynomial in x, y, and s, vrhile the degree in x and y is6; in X and z is 5; my and z is 4. Finally, the degree in a; is 3, in ?/ is 4, and in z is 2. A rational fraction is the quotient of two polynomials. The degree of a rational fraction is defined to be the degree of the numerator diminished by the degree of the denominator. For example, the degrees of the expressions x^+ 5x + 6 x^ + 5x + 6 , x — 2 — < —r-^ — -. 1 and , x + 1 x^ + lx + i x^ + lx+l are 1, 0, and — 1, respectively. II, § 16] ALGEBRAIC IDENTITIES 15 EXERCISES 1. Tell which of th^ following equalities are identities and which are conditional equations. - (6) -^— = l + x+ ""^ 1-x "^ '' x-S x-3 - (d) ^"-7^+1^ = 0. ^ ^ x-S ^ a;-3 ^■''^ a;-3 !c-3 2. Tell the degrees in x of each of the following expressions. (a) 5a:8-6a;2 + 14 a; -16. (6; a^K^ - 5 X + 17 . (c) - + X-2. X , ,, jc — 6 a;8_15 (e) aa;8 + fta; + c. (/) x8 - 2/8. a'- 15 (?) (A) a-15 ftoK" + 6ix»-i + ■■■ + K 3. Give the degrees in x, in y, in z, and in the combinations of x, y, and «, of the following expressions. (a) 6x«-3x^ + 5y^ + z. (c) 2x8 + Sj/' + 4«3 - lOxyz. 4. What is the degree in x and y of the expression 16 COLLEGE ALGEBRA [IL § 17 17. Some Important Identities. The following identities, which are used very frequently in making algebraic transfor- mations, should be memorized. The square of a first degree expression is expressed by the formula (1) {x + a)2 = ajs + 2 ax + a^. The product of the sum and the difference of tioo numbers is expressed by the formula (2) (as + y)(x-y)=x^- y\ The product of two first degree expressions is expressed by the formula (3) (x + a){x + b)=ac^ + (a + b)x + ab, or, more generally, (3 a) (mac + a) (nx + 6) = mnxi^ + {mb + na)x + ab. The product (4) (£c — y){x^ + xy + y^ = as' — y', . is important, as is also the product (5) {x + y){x^—xy + y^)=x^ + tr'. These identities are easily verified by performing the multipli- cations indicated in the left members. The formulas (2) and (4) are special cases of a more general identity that may be written in the form (6J (x - 2/)(i»«-i + x"-^y + x^-^y^ H h y"'^) ^x"- y". The fifth is a special case of th6 formula (7) (x+ y){x"-^-x'^^y+x'^^y^ -x"-*y^ + - -|-(-l)"-ij/"-i) = 05" + y". The proof of (6) and (7) will be postponed to a later section. (See Chap. XIV, § 129, Exs. 25, 26, and 27.) II, § 18] ALGEBRAIC IDENTITIES 17 18. Rational Factors. The process of breaking up an ex- pression into its factors depends essentially upon the recogni- tion of the appropriate identity. Many of the most important problems in factoVing may be solved by direct reference to the identities in § 17. EXERCISES 1. Factor each of the following expressions. - (a) 25 a2 - 64. (/) S^" -|- 7 x 3» -f 6. (fc) a^' - b": -(b) a'b^c^-^. (g) {l+xy-Sx\' ■ (l) 64a;S"r-7292/»+%6. (c) x2"+2-4. (A) 8a;6-729j/6. (m) 1000 a^ - 8 fii^c". (_d) x^ + ip-q)x - pq. (i) 27 a^ + 1. (n) 9 x^'^y^ - 4 z^™. (e) ay^ - 7 a^y + e a', (j) a^-l. (o) 25a*6B-16. (p) a:i2 -I- y 12. [Hint. a:i2 = (x^y and j/i2 = (j,4)8.] r(q) X* -)- a;22/2 -(- y^. [Hint. Add and subtract x^^.'] 00 (m -|-n)(m2_ j;2)_(m-ic)(m2- »2). (s) o3-63-2a26 + 2a62. (t) x^-S-6x^ + nx. ■ (a) x^ + 3 x^ — 2. [Hint. jCtomplete the cube of a; -|- 1.] 2. Express the following as the products of first degree factors, (a) bc{b — c)+ ca{c— a)+ ab{a—b). (fi) a-^{b-c)+b\c-a)+c\a-b)- (c) (a - 6)3 +{h- c)8 + (c - a)8. (d) 4a2c2-(a2_62 + c2)2. («) o3(6 - c) -)- 63(c - a)-!- c'Co - 6). 3. Prove that 62-fc2-a2 _ (6 + c-|-a)(6 +c-a) ■2bc ~ 2 be 1+- 4. Prove that ^ _ 52 + c2-a2 _ (a-6-|-e)(a-h 6 - c) _ 2 6c T 2 6c 6. Prove that &!5.Vi + 6i±^!:^^Ul - 6i + ^^A^U.(s - a) (. - 6) (s - c), 4 V 2 6c l\ 2bc I ^ where s =(a + b + c)/2. c 18 COLLEGE ALGEBRA [II, § 19 19. Reduction of a Quadratic Expression to the Sum or Difference of two Squares. Expressions of the form a!^ +px + q, and ax^ + hx + c, are called quadratic expressions. Whether or not such an expression can be resolved into the product of rational factors may always he determined by reducing it to the sum or dif- ference of two squares. The sum x'+px differs from the square of a; + p/2 by the term p^/i. If, therefore, p'/i be added to, and subtracted from, the first quadratic expression, we have (8) x^ + px + q = x^+px + ^-f-^-q\ If p^/4: > q, p^/i — q is a positive number ; hence it can be written in the form 4 It follows that £_,.(V|-,j. (9) 3:'+px + q^(^x+^J-(^yj^-qJ. If, on the other hand, p^/i < q, q — p'/i is positive, and the identity may be written in the form (10) x'^ + px + q^fx + lj+fq-^y or (11) ' x^+px + q=(x+A'+ ^yjq - ^J. To sum up the results, we may state the following theorem. The quadratic expression x^+px + q may be expressed as the sum or the difference of the squares of two real quantities accord- ing as p'^/i < q or p^/i > q. II, § 20] ALGEBRAIC IDENTITIES 19 It follows directly that a quadratic expression can be re- solved into real first degree factors only when jf/A: > q. EXERCISES 1. Reduce each of the following expressions to the sum or difference of two squares, and factor if possible. (a) a;2-6a;-|-8. (c) a;^ + 6a; + 7. (e) a^ - 7x+ 23. (&) a;2 - 6a; + 6. (d) x^ + Qx + 11. (/) 3?^- ax + b. 2. Factor the expression Sx^ + 5x + 6. [Hint. Sx^ - 5a; + 6 = 3(x2 - |x + 2),] 20. Highest Common Factor. A common factor of two numbers a and 6 is a factor that is contained in both numbers. The largest common factor contained in both numbers is called the highest common factor {H. C. F.) of the two numbers. Thus, 2, 3, and 6 are common factors of 18 and 24, while 6- is the H. C. r. Two numbers having no common factor except unity are said to be prime to each other. Similarly, two polynomials that have common factors, have a common factor of highest degree. If we ignore mere numerical factors, two polynomials have but one common factor of highest degree. This common factor is called the highest common factor. If the degree of the H. C. T. is zero, i.e. if the H. C. F. is a mere numerical factor, the two polyno- mials are said to be prime to ed,ch other. The polynomials 6a;3-80a;2-|-36a; and 16 a;^ _ io6 a;' + 150 a; have as H.C.F. x{x — 2), while the polynomials % a? — ZQ x"^ -{■ Z& x and 2x^ — 12 a; -(- 10 are prime to each other. The highest co'mmon factor of two polynomials is the product of all their prime factors (numerical factors excluded), each factor being raised to the lowest power in which it occurs in either polynomial. It is possible, therefore, to find by inspection the H. C. F. of 20 COLLEGE ALGEBRA [II, § 20 two numbers or of two polynomials as soon as their prime factors are known. The foregoing considerations apply with- out change to three or more numbers, or to three or more polynomials. EXERCISES 1. Find the H. C. F. of the following. •■ (a) ix + y)\x-u), (x-y)\x + y). (6) xs - 1, a;2 _ 7 a; + 6, (x- 1)8. (c) (a - by, a< - 2 a^b^ + b*, a' - a% - aft^ + 6=. 2. Reduce the following fractions to lowest terms. C , 2g'^5a: + 3 , , \ + bx + Qx^ ,. g* + 0^6== + 6* ^ ' bx^ + nx + l' ^^l + 6a; + 8x2' ^' ofi - b^ (h-) x^-(y-s)^ ,,-. x^ + Sx^ + Sx + 2 ,^ a-b *■ ■' (a; + j/)2_g2" '^ J x^-2x^-2x-3' ■" ai's - fti'^ 21. Lowest Common Multiple. A number which is divisible by each of several given numbers is called a common multiple of those numbers: If it is the least of such numbers, it is called the lowest common multiple {L. C. M.). Similarly, the L. C. M. of two or more polynomials is that polynomial of lowest degree among all the polynomials which are divisible by the given polynomials. The L. C. M. of several polynomials is the product of aU the dif- ferent prime factors (numerical factors included) of all the poly- nomials, each factor raised to the highest power in which it occurs in any one of the polynomials.* We are thus enabled to find the L. C. M. by inspection when the numbers or the polynomials are broken up into their prime factors. EXERCISES 1. Find the L. C. M. of the expressions x-1, (x-2)(x-iy, (K -l)2(x-2). Solution. The prime factors occurring are- x — 1 and r — 2, and the highest power to which each occurs is the second, consequently the L. C. M. i8(a;-l)2(a;-2)2. II, § 21] ALGEBRAIC IDENTITIES 21 2. In each of the following cases, find the L. C. M. of the given expressions. (a) 4x-l, 16a;2-l, Idz'^-lGx + l. (6) (x + y)(,x'-y«),(x-y)(x^ + y'). (c) X^ + X^ + X, X" — X^, X* — x'. (d) XS + 2,8^ x3 - J/8, X< + XV + 2/4. 3. The earliest application of the lowest common multiple is found in the reduction of fractions to equivalent fractions with the lowest common denominator. Perform the operations indicated on the following fractions. , , 1 1 6o * 3a-56 3a + 56 9a:^-25l W 71 r, -. + 7 ^ rr+: (b-c)(c — a) (c — a)(a-b) {a — b){b-c) 4. Prove that the L. C. M. of two numbers is equal to their product divided by their H. C. F. MISCELLANEOUS EXERCISES • 1. Add the fractions J, f, f, |, first by using the least common de- nominator, and then by using the common denominator 144. Compare the form of the results. » 2. Add the fractions b a c ax + ab' x^ — 6^ ' f)x — ab ,3. Add _^, -JL^, and i+^. y{x — y) m{y — x) my ' 4. Reduce ^^ - ^ a: ♦!- 8 ^ .rjO. ^^ -^^ lowest terms. x2 + 2x4-1 a;-4 • 5. Reduce x^ - 6 x + 8 ^ x^ - 5x + 6 ^ x^ - 7 x +12 ^ .^^ j^^^^, x2_9 x2-9x + 20 (x-2)2 terms. 6. If ^ and N are two polynomials having a common factor, and the degree of JV is less than that of M, prove that the remainder after dividing Jf by iV is divisible by the common factor. ■ [Hint. M/]\f= Q + H/N gives M=Nq + JJ.] CHAPTER III , POWERS AND ROOTS 22. Integral Powers. The expression a" is called the nth power of a. When si is a "positive integer, a" is defined, to mean the product of n factors each equal to a. Thus, a' is a short way of writing the product a- a- a. The number a is called a base and the number n an exponent. For the present, powers whose exponents are negative numbers, fractions, or zero, have no meaning. For powers with positive, integral exponents, five theorems, called the fundamental laws for exponents, may be demon- strated. I. a«^a"^ = a'"+». This law is known as the index law. II. o"» -:- a" = a™-", (m > n). III. («»»)« = a"™. IV. (a6)» = a»6«. [bj 6»' V. EXERCISES 1. Demonstrate each of the five fundamental laws. 2. Translate each of the above formulas Into words. 3. Prove that ((a")")' = a"""'. i. Prove that (a6c)" = o''6"c''. 22 Ill, § 24] POWERS AND ROOTS 23 5. Perforin the indicated operations in eacli of the following exercises, and put results in their simplest forms. (a) (-2a:)6. ^^ ^^a^'h^o^' + -a''h^A^ia%^c. •(6) -(6ccVs)8. 3V4 5 / ^"H"365" j (A) (a; + 2/ - 2)3 X (s + 2/ - z)2. /■;■) (»'')* -(8/^)* , (jN (a: - y)'° (a: + y)\ ^ ^ (a;2)2+ (2,2)2 (a:+2/)'' (x - 2/)» (e) (2»3»)8. ■ (j) (a2_3a36)x(aS + 2o62). 23. Zero Exponents. To fiad a meaning for a", we assume that the index law holds for all exponents. Consequently, a'^a" = a''+'' = a". But the only multiplier that leaves a number unchanged is 1. Therefore (1) «« = 1 (a ^ 0). 24. Negative Exponents. Again, assuming the index law to hold whatever a" may mean, a''ar'' = o° = 1. Consequently, division of both sides by a" gives (2) a-» = A. Two numbers whose product is unity are called reciprocals. According to this definition a~" is the reciprocal of a" and vice versa. On account of (2) a number which is a factor of one term of a fraction may be removed to the other side of the line if the sign of its exponent be changed. Thus, 5-"- 6 ^ 5-2 . 6^ _ 1^ . 6^ _ 6 17 17 52 17 52 • 17 Similarly, a2(a;^ ^ y''^ _ x^ + y^ &3 - a-263 24 COLLEGE ALGEBRA [III, § 25 25. Fractional Exponents. The meaning of a power with a fractional exponent is Setermined by means similar to those employed in finding meanings for powers with zero and nega- tive exponents. Assuming fractional powers to be deiined in such manner that the iudex law holds, we would have, for example, 2 2 2 Zj.2 . 2 a^ X a^ X a' = a'^ ' = a?. From this it is seen that a^^' is one of three equal factors of a?. But one of three equal factors of a number is the cube root of the number. Therefore, For the general case, " - i. f 4. !■+!!+... ton term, an . a» — to n factors = o» » = a" ; that is, (3) «»»/" = Vm™- Meanings * have been found for powers with zero, negative, and fractional exponents by assuming that they follow a single one of the five fundamental laws for positive integral expo- nents, namely, the index law. If only positive bases are con- sidered, it is possible to prove that all powers with rational exponents follow the remaining four laws. However, in order to save space, we shall assume without proof the following fundamental principle. Powers of positive f hases with zero, negative, and fractional exponents obey the Jive fundamental laws for powers with positive integral exponents. * The student should be careful to note that the meanings for zero, nega- tive, and fractional exponents are essentially definitions since they are based upon the unproved assumption that the index law holds. t All the laws except IV hold for negative, as well as for positive, bases. For the present it is sufficient to note that [(—2) (—3)]'^" is not equal to (— 2)V2(— 3)V2. This exceptional case will be considered In § 33. Ill, § 25] POWERS AND ROOTS 25 Theorem. Tlie nth root of the mth power of a positive num- ber is numerically equal to the mth power of the nth root of the number. Proof. It is necessary to prove that -y/a" = (Va)". Since all powers of positive numbers with rational exponents obey the same laws, ml 1^ _ a " = a"""* = (a")™ = ("v/a)", if a > 0. But by definition, a"*/" = Va" ; therefore, if a > 0, since both are equal to a"'". In simplifying expressions containing exponents, it is prac- tically always simpler to apply the laws first, and to insert the meanings in terms of radicals afterwards. EXERCISES 1. Free each of the following expressions from zero and negative ex- ponents. («) ^,- •(*) 3-2aiV. ' (c) 5a:-2y-4. 2. Perform the following indicated multiplications, and express each answer in terms of powers with positive exponents. (a) a36-2 X aV36-Vs. (j) _ |a;-5/62,2/3 x 2a:V6j,i/2. (c) S-'x-^ x 6-h^. (d) (oV2-|-a;-2)(aV2_a;-2). (e) (a:V2 + a;-V2)(a;V2-x-i/2). - (/) {a-" + a-8 + a-i) (a-2 - ao). (gr) (3^y-S/i + x2|/-l/2 + a;j,-l/4 ^ y. y2 = a''; whence 2/2 In the language of logarithms the last equation gives log„ ^ = Ki - »2, 2/2 IV, § 36] LOGARITHMS 39 or (6) lOga ^ = loga VI - lOga V^, 2/2 whicli was to be proven. Theokbm 5. The logarithm of apower is equal to the logarithm of the base of the power multiplied by the exponent of the power. For, if be the number, and m the exponent, then by III, § 22, From the last equation, log, 2/'" = ma;, or (7) \oSa y»^ = ni loga y. CoEOLLART. If m = 1/n, where n is an integer, equation (7) takes the form (8) loga Vy = ^losay. Translated into words, equation (8) says that the logarithm of the root of a number is equal to the logarithm of the number divided by the index of the root. 36. Systems of Logarithms. The logarithms of all real positive numbers with a given number as a base form a so- called system of logarithms. On account of its great conven- ience, the system whose base is 10 is used in all numerical computation. The system whose base is 10 is called the com- mon or Briggsian system. Unless otherwise stated, common logarithms will be used and all reference to the base will be omitted. For example, we shall write log 25, not logio 25. 40 COLLEGE ALGEBRA [IV, § 37 37. Characteristic and Mantissa. A table of powers of 10 may be written in exponential and logarithmic forms, as follows. Exponential Form LOGARITHMIO FOEM 10-6 = .000001 log .000001 = - -6 10-5 ^ .00001 log .00001 =: - -5 10-* = .0001 log .0001 = - -4 10-3 = .001 log .001 = - -3 10-2 = .01 log .01 = - -2 10-1 = .1 log.l = - -1 10" =1. logl = 101 =10 log 10 = 1 102 ==100 log 100 = 2 103 = 1000 log 1000 = 3 10* =10,000 log 10,000 = 4 105 = 100,000 log 100,000 = 5 106 = 1,000,000 log 1,000,000 1 = 6 The logarithm of an integral power of 10 is an integer, either positive or negative. The logarithm of a number which is not an integral power of 10 lies between two integers and must be expressed, either exactly or approximately, as an integer plus a decimal part. For example, the preceding table of powers shows that the integral part of log 258 is 2 because 258 lies between 100 and 1000, and its logarithm must therefore lie between 2 and 3. Prom the table of logarithms (Table, p. 262) the remaining part is found to be approximately .4116. We may then write log 258 = 2.4116. The integral part of a logarithm is called the characteristic, and the decimal part is called the mantissa. Even when the logarithm is negative it may be so written IV, § 38] LOGARITHMS 41 that the mantissa is positive. For example, it is known that approximately, log .025 = - 1.6021. When 10 is added and subtracted, this equality may be written in the form log. 025 = 10 - 1.6021 - 10= 8.3979 - 10. In the last form the mantissa is the positive number .3979, and the characteristic is the negative number 8 — 10, or — 2. In what follows it will always be assumed that the logarithm is written in such a form that the mantissa is positive. Theorem 6. — Tlie mantissas of the common logarithms of all numbers having the same sequence of figures are equal regardless of the position of the decimal point. Pboof. Moving the decimal point is equivalent to multiply- ing or dividing the number by a power of 10. The logarithm would be changed by adding or subtracting the logarithm of this power of 10. But the logarithm of a power of 10 is an integer, and adding or subtracting an integer cannot change the mantissa. Example. Log 2.58 = log ?^ = log 258 - log 100 = 2.4116 - 2 = 0.4116 38. Determination of the Characteristic. — From the table in § 37, it is clear that the characteristic of the logarithm of 1, or of any number between 1 and 10, is zero. Moving the dec- imal point one place to the right is equivalent to multiplying the number by 10, and therefore, by Theorem 3, to increasing its logarithm by 1. On the other hand, moving the decimal point one place to the left is equivalent to dividing the number 42 COLLEGE ALGEBRA [IV, § 38 by 10, aud, by Theorem 4, to diminishing the characteristic by 1. It follows that the characteristic of the logarithm of any number may be found by comparing the position of its decimal point with that of another number having the same sequence of figures and lying between 1 and 10. Thus to find the characteristic of log 2580 it is sufficient to note that 2580 = 10' x 2.58. The characteristic is therefore 3. Similarly the characteristic of log .000258. is — 4, since .000258 = 10-^ X 2.58, In general, the characteristic is equal to the exponent of that power of 10 which must be used as a multiplier to produce the given number from a number homing the same sequence of figures and lying between 1 and 10. An easy way to determine the correct characteristic is first to imagine the decimal point placed after the first significant digit (not zero) and then to count forward or backward to the actual decimal point. The number of digits passed over is the characteristic, counted positive if the motion was to the right and counted negative if the motion was to the left. A negative characteristic is written on the right of the man- tissa, or, it is broken up into two parts, one of which is a mul- tiple of — 10, and this multiple of — 10 is written on the right of the mantissa. Thus, log .000258 = .4116 - 4 = 6.4116 - 10, since the characteristic is — 4. EXERCISES 1. Using the table, find the logarithms of eact of the following numbers. (o) 2.34. (d) .359. (?) (6) .234. (e) .0293. (A) 28.4 8S.6 .0438' Cc) .000234, (/) (.0346).=. (i) g??)'. IV, § 40] LOGARITHMS 43 39. Interpolation. Logarithms, like square and cube roots, can be expressed only approximately in the ordinary notation. The tables in common use give the mantissas to four, five, or six decimal places, leaving the computer to supply the char- acteristic according to the rule of § 38. By means of a four-place table the logarithm of a three-figure number may be read off directly, but the logarithm of a four- figure number must be found by interpolating a logarithm between two logarithms of the table. Por example, suppose the logarithm of 258.3 is wanted. Clearly log 258.3 lies be- tween log 258 and log 259. Prom the table log 258 = 2.4116, and log 259 = 2.4133. The difference between these two mantissas is .0017. The difference between 258 and 258.3 is 3 tenths of the difference between 258 and 259. The difference between log 258 and log 258.3 is therefore 3 tenths of .0017, or .0005 ; and we write log 258.3 = log 258 + .0005 = 2.4121. 40. Multiplication and Division of Logarithms. A loga- rithm whose characteristic is positive is multiplied and divided in the same manner as any positive number is multiplied or divided. If the characteristic is negative the operation of di- vision requires some care. For example, to divide .4126 — 4 by 7, the logarithm should be written so that the negative part is exactly divisible by 7. Thus .4126-4=3.4126-7=66.4126 —70. Consequently, one seventh of .4126 — 4 is .4875 — 1, or 9.4876 — 10. Either result is in convenient form. EXERCISES Find the logarithms of each of the following numbers. 1. 534.2. 6. (.3826)5. 9. (.03562)3/2. 2. 534.2617. 6. (3.826)5. 10. 8J. 3. 6.278. 7. V:3427. 11. %\. 4. .05342. 8. v':02796. 12. /5ii^V. 128.16/ 44 COLLEGE ALGEBRA [IV, § 41 41. Antilogarithms. In order to complete a computation by logarithms it is necessary to find from the table the antiloga- rithm, that is, the number corresponding to a given logarithm. To illustrate, suppose log X = 2.5432 is given, and the number x is required. The nearest mantissa given by the table is .5428 which corresponds to the number 349. The difference between this mantissa and the given man- tissa is .0004, while the difference between the mantissa .5428 and the next mantissa ia the table is .0013. The difference in number is therefore ^ of 1, or, to the nearest tenth, .3. The first four digits in the required number are then 3493. From the relation between decimal point and characteristic as given in § 38, the point must lie between the 9 and the 3 so that X = 349.3. If log X = 8.54325 - 10, then x = .03493. 42. Cologarithms. The direct subtraction of logarithms may be avoided by the use of cologarithms. The eologarithm of a number is the negative of the logarithm, i.e. the logarithm of the reciprocal of the number. It is written in the form (8) colog m = log - = — log n = 10 — log w — 10, n where 10 has been added to and subtracted from — log n. For example, colog 258 = log ^ = 10 - 2.4116 - 10 = 7.5884 - 10. Again, colog .0258 = log -J— = 10-^(8.4116 - 10)- 10 = 1.5884. In finding the eologarithm from the logarithm, expert com- puters begin the subtraction on the left, taking every figure up IV, § 42] LOGARITHMS 45 to, but not including, the last significant figure from 9. For example, Log 34.58 X .01237 2.159 X .1673 = log 34.58 + log .01237 + colog 2.159 + colog .1673 = 1.5388 + 8.0924 - 10 + 9.6658 - 10 + 0.7765 = 0.0735. MISCELLANEOUS EXERCISES 1. Compute the value of each of the following expressions by means of logarithms. (a) 259 X 342 X. 0621. 8394 X 6728 (6) (c) id) 9342 3782 X v'ssig ; 9234 2594 X .8431 ■v/324i (e) (/) .8432 X -v^ .0231 5 .03421 7//'5294Ys ■V 1,6382 j ■ (3) if \/.03428 V.03795 (ft) 5.34-021. (0 631-". Compute the value of C-5347)'=(-y- 59.32) (- 76843)8 [Hint. Logarithms will give only the absolute value of the result. The sign must be determined independently.] 3. Find the area of a circle whose radius is 36.59. 4. Find the volume of a sphere whose radius is 2.634. 5. Find the volume of a right circular cone whose altitude is 5.374 and the radius of whose base is 2.634. 6. Find the simple interest on 1 239.26 at 5.75% for 2 years and 9 months. 46 COLLEGE ALGEBRA [IV, § 42 7. Find the compound amount of 1 1225.76 for 4 years with interest at 6 % compounded semiannually, given that the formula for the compound amount is where P is principal, r rate, n the time expressed in years, and p the num- ber of times the interest is compounded each year. Would the result obtained by means of four-place tables be acceptable in commercial practice ? 8. A man invested $ 5000 in business and at the end of 5 years drew ' out $ 7439.62. What rate at compound interest payable annually did he receive on the investment ? 9. The weight of a cubic foot of water is 1000 ounces. What is the weight in tons of a ship whose " displacement " is 59,200 cubic feet ? 10. The time in seconds of one complete oscillation of a pendulum is given by the formula r = 27rJi, where I is measured in feet and g = 32.2. What is the time of a complete oscillation of a pendulum 27 inches long ? 11. One meter is equivalent to 89.37 inches. How many inches are there in 1276 centimeters ? 12. Beduce 485 square centimeters to square inches. 13. Reduce 579 sqfiare inches to square centimeters. 14. Ordinary interest is figured on the basis of 360 days and exact interest on the basis of 365 days to the year. The relation between the two is given by the formula exact interest = ordinary interest x ||. Find by a single computation the exact interest on 1 532.75 for 139 days at 6%. 15. When air expands without gain or loss of heat the pressure is approximately given by the formula p = cpi ^1 where p is density and c is a constant. • Find the value of jp when c= 566.9 and p = .0026. IV, § 43] LOGARITHMS 47 43. The Logarithmic Scale and the Slide Rule. The loga- rithms of the integers 1 to 10 inclusive are, if we use two deci- mal places only, .00, .30, .48, .60, .70, .78, .85, .90, .95, 1.00. These numbers may be plotted easily upon any line segment one unit in length and divided into hundredths. If we attach to each point, not its logarithm, but the corresponding num- ber we obtain a so-called Gunter's or logarithmic scale. (See Kg. 4.) X ' 3 3 4 B 6 1 8 10 L I 1 \ I I I I L_l Fig. 4. A slide rule consists essentially of two logarithmic scales A and B which may be moved over each other as in Fig. 5. ± 7 8 ip III Fig. 5. To multiply two numbers, as 2 and 4, by means of a slide rule, we set the 1 of scale B upon the 2 of scale A and count forward to 4 on scale B. We have in this way that point on scale A which gives the sum of log 2 and log 4. The point is of course 8. Similarly, if we, wish to divide 8 by 4 we would set 4 of scale B upon 8 of A then count back on A to 1 of B. This point de- notes the difference between log 8 and log 4. Clearly the logarithmic scale gives only the mantissas of the logarithms. The determination of the characteristics presents no difficulty, at least in the simpler cases. The slide rule in actual use is considerably more elaborate than the pair of logarithmic scales in Fig. 5. The illustration 48 COLLEGE ALGEBRA [IV, § 43 o < CO o = = "^ OJ- = E CO - - = ~- ra = — OS i= = - ^ = ^ — - 11 i i CD : i = z: CO -: = = - CO ; ;| ■ =. = ■ " = - --■ ■ = — = " Il-O) a> - - — — _ CO = =: °° — =E - || .o in - - — \ . - - E = = "^ = ; E = 03 - : = : = E :e --- E ~ ■z = — ■ = E- .i E' -E Z~ ' E- — _ Z - Z . , ~ - — — — < ffi o Q \- -4 V (Fig. 6), which is a reproduction from a photograph of an actual slide rule, shows two pairs of scales. The upper pair consists of two logarithmic scales placed end to end. By means of a ten-inch rule prod- ucts and quotients, true to two or three figures, may be found with surprising rapidity. The longer the rule the greater the accuracy. o EXERCISES 1. Perform each of the following multi- plications by means of a slide rule. '(a) 3x2. Ce) 5.5 X 4.5. e (5) 3.5 X 2.5. (/) 6.6 X 7.3. R- (c) 2.7 X 3.4. (9-) 1.45 X 23. ^ (d) 5x4. (A) 14.5 X 2.3 2. Perform each of the following divi- sions by means of/a slide rule. (a) 6 -J- 3. (6) 6.5-^3.5. (c) 30 -:- 5. (d) 32.5 -=-4.6. (e) 21.4-4-63.5. (/) 4.52-^0.125. 3. I'erform each of the following indi- cated operations by means of a slide rule. (a) 3 X 4 X 5. (6) 3.7 X 4.5 X 5.4. (c) 6.9 X 11 -h 4.5. (d) (45)2 -=-63. (e) ir(4.52)2. (/) 4 ,r (68.2)3. CHAPTER V FUNCTIONS OF A VARIABLE- GRAPHICAL REPRESENTATION 44. Definitions. If a man walks at the uniform rate of 3 miles per hour the distance coTered in x hours is 3 a; miles. If y denotes the distance, the relation between distance and time is given by the equation y = dx. In this example x may represent any arbitrary number. More- over to every value of x there corresponds a definite value of y. N"ximbers like x and y in this example, which assume many values in the same discussion, are called variables. The im- portant fact to be noted in the example is that when the value of one variable x is given the value of the other is known. If two variables, x and y, are so related that when the value of X is given the value of y is k^iown, then y is said to be a function ofx. The notion of dependence of one variable upon the other is fundamental. The variable which is assumed to take arbitrary values is called the independent variable. The variable whose values depend upon the values of the independent variable is called the dependent variable. In the example just given it has been assumed that the values of x are arbitrary. There- fore X is the independent, and y the dependent variable. The notion of a function is one of the most important ideas with which mathematics has to do. The following examples will serve to illustrate this statement. E 49 50 COLLEGE ALGEBRA [V, § 44 Example 1. If 1 100 be loaned at 6 9i the interest is a function of the time, which is expressed hy /=|100 X .06 X «. Example 2. The distance through which a body falling freely from rest will travel in tinie {, is a function of the time given by where gisa. constant. Example 3. The circumference C and the area Aoia, circle are func- tions of the length of the radius given by = 2 Trr, and A = wr^, respectively. Example 4. If x denote any number and y its logarithm, the func- tional relation y = \ogx expresses the relation between the numbers of the number column and the corresponding logarithms of a table of logarithms. Example 5. If y denote the angle which a stairway with 12-inch treads makes with the floor, and if h denote the height of the riser, the steepness of the stqirway depends upon the height of the riser and is ex- pressed as the ratio of riser to tread. In trigonometry this ratio is called the tangent of the angle y, so that by definition, tan 2, = A. It should be carefully noted that in the examples just given, /, s, C, A, y, and tan y are merely symbols for the functions 100x.06x<, 2^, 2,rr, Trr^, logos, and A, respectively. All these functions are given by definite mathe- V, §44] FUNCTIONS — GRAPHS 51 matical expressions, a thing which is by no means necessary to make one variable a function of another. When it is desired to express the fact that a variable y is a function of x without stating the exact mode of dependence, it is customary to write (1) J/= /(«!), which is read " y equals a function of x," or " y is a function of X." EXERCISES , i. One leg of a right triangle is x and the other is 5. Express the length of the hypotenuse as a function of x. •> 2. Let X denote the side of an equilateral triangle. Express the area as a function of x. • 3. Let a denote the radius of a circle and let x denote the distance from the center of the circle to a chord. Express the length of the chord as a^function of x. •^i. Express the side of a square as a function of the area. 5. Express the area of an equilateral triangle as a function of the length of a side. 6. Express the side of an equilateral triangle as a function of the area. 7. Express the simple interest on one dollar as a function of the tintie t when the rate is 5 9fc ; as a function of the rate r, when the time is 10 years. 8. Express the amount of one dollar at compound interest as a func- tion of the time iwhen the rate is r. 9. At a given'time a man 12 miles from a given place begins to travel away from the place at the uniform rate of 12 miles per hour. "What function expresses his distance from the place at the end of t hours ? 10. In Ex. 9, .suppose the man begins to travel toward the place. What is the function ? 11. A man starts from a point 3 miles east of a certain place and travels eaat at the rate of 4 miles per hour. What function expresses his distance from the place at the end of x hours ? J" (2,4) X' )V=4 -\ — h 52 . COLLEGE ALGEBRA [V, § 45 45. Coordinates. The independent variable may take an unlimited number of values. To a given value of the inde- pendent variable corresponds one value (or several) of the function. This pair of values, one for the independent vari- able and one for the function, may be used to determine the position of a point in a plane in the following manner. Let the function be x + 2. To the value x — 2 corresponds the value y = 4. The value x = 2 can be represented by a line OA two units long laid off to the right of the arbitrary point in the arbitrary lineXX'(Fig.7). Similarly the value 2/ = 4 can be rep- _ resented by a line 4 units long laid off upward from A, parallel to a line drawn through 0, and perpendic- ular to the line XX'- j-iQ Y The arbitrary line XX', in which the values of x are measured, is called the x-axis, or the axis of the variable, while the line TT', parallel to which the values of the function are drawn, is called the y-axis, or the function-axis. The pair of values x = 2 and «/ = 4 determine a point which is found by measuring 2 units along the av-axis and then 4 units upward parallel to the y-axis. (See Fig. 7.) If either a; or y is negative the corresponding line is drawn in the opposite direction. Thus x= —2 and y = 4 determine a point two units to the left of the y-asis and four units above the !B-axis. The distance measured along the avaxis is called the abscissa and the distance measured along the y-a,xis is called the ordi- nate. The two together are called the coordinates of the point V, § 47] FUNCTIONS — GRAPHS 53 which they determine. The point in Fig. 7 is indicated by the symbol (2, 4). In general, we write {x, y) where x is the ab- scissa and y the ordinate. 46. The Graph of a Function. By giving arbitrary values to X, any number of values of the function may be found. The relation between the variable and the function may be exhibited as in the following table, which has been constructed for the function x-\-2. t -4 -3 -2 - 1 1 2 3 4 3 5^ 4 x + 2 -2 -1 1 2 -3 -,l I I "^„ I -^„ -3-2-10 2 3 4 ■4— f- If a pair of axes be drawn, each value of x with the cor- responding value of the function determines a point which is easily located by the method of § 45. The lines 1 Pi, 2 P-i, 3 Pi, ••• represent the values of the function a;-f-2 for the correspond- ing values 1, 2, 3, — of x. (See Fig. 8.) If the points on the «-axis be indefinitely near together, the points P will be indefinitely near together and will lie in a line, either curved or straight. The line thlis determined is called the graph of the function. The graph of a function is constructed by locating a sufficient number of its points and then drawing a smooth curve through these points. 47. Coordinate Paper. By means of paper ruled in small squares, all necessity for measuring instruments to determine -1 --2 Fig, 54 COLLEGE ALGEBRA [V, § 47 the lengths of coordinates is avoided, since the spaces on the paper serve as a measuring scale. Paper so ruled is called coordinate paper, or squared paper. Coordinate paper with 8 or 10 rulings to the iach, or 6 rulings to the centimeter, is most convenient. EXERCISES 1. Mark on coSrdinate paper the points (1, 1), (2, 3), (— 1, 3), (-1,0), (0,1), (-3, -2), (2, -3), (0,0). 2. Construct the graphs of the following functions. (a) 2x. (e) J-x. (i) ix + 2. (6) Zx. (/) Ix. U) i,^-2- (c) X. (g) x-2. (k) Sx + 1. (d) Ix. (ft) 2 a: - 2. (Z) Zx- 1. 3. What is your conjecture concerning the geometric character of the graphs in Exs. 2 (a) - 2 Q) ? at + So5 where v„ is the initial velocity and Sg is the initial space. 62 COLLEGE ALGEBRA [V, § 51 EXERCISES 1. Construct the graph of each of the following functions. (a) a;2. (d) x^-2x. (g) x^-2x-4:. (6) x^ + i. (e) x^ + 2x. (h) a;2 + 4a;. (c) x^-i. (/) a:2-2a; + 4. (i) -a;2-4a; + 3.' 2. Construct the graphs of the functions a;^ + 4a; + 1, x^ + ix + 2, CB^ + 4 a; + 3 with the same axes and the same units, and determine as nearly as you can what effect changing the absolute term has upon the graph. 3. Construct with the same axes and the same units the graphs of the functions a" — 2x + l, a;^ — 4a;+l, and a;- — 6 a; + 1 and giye your opin- ion as to the influence of the coefficient of x upon the position of the graph. 4. Construct the graph of a?/2 and compare it with graph of x^ con- structed upon the same axes, and with the same units. 6. By means of a table of logarithms construct the graph of the equation 2/ = logioa:. 6. By the same means construct the graph of the equation y = W. [Hint. Let the unit for ordinates be 10 times the unit for abscissas.] 62. Maximum and Minimum Values of Quadratic Func- tions. In § 51 it was stated that the graph of the function a?— 4:X + 5 has a lowest point, and that the graph of the func- tion — x^ + 4:X + 2 has a highest point. The number express- ing the length of the ordinate drawn to the lowest point of such a graph is called the minimum value or least value, of the function. Similarly, the number expressing the length of the ordinate drawn to the highest point of a graph, like that in Fig. 11, is called the maximum value or greatest value of the function. A minimum value is less, and a maximum value greater, than any other value in the immediate neighborhood. V, § 52] FUNCTIONS — GRAPHS 63 Maximum and minimum values of quadratic functions are easily found by transf ormiag the function into the sum or the difference of two squares by the method of § 19. Tor example, the function a;'- 4a; + 5' may be written in the form (k _ 2)2 + 1. In this new form the first term (x — Tf alone is variable, and, siuce it is a square, it is either a positive number, or zero, for any real value of x. The least value of the function is, there- fore, the value that corresponds to the least value of {x — 2y. Now, the least value of (x — 2)* is zero. When (x — 2)^ is zero, X = 2, and the corresponding value of the function is 1. (See Fig. 10.) Similarly, the function — x^ + ix + 2 may be written in the form In this case the first term is always negative, or zero, and the greatest value of the function corresponds to (« — 2)^ = ; that is, the maximum value of the function is 6. This value is found by makiug x = 2. (See Tig. 11.) If the coefficient of x^ is not unity, the function must first be put in the form a(x^ +-X +- \^ a. a then the factor within the parentheses may be transformed as before. Thus we may write 9 , I, , r/ , & \^ (b^-4:ac)l (See Ex. 2, p. 19.) 64 COLLEGE ALGEBRA [V, § 62 EXERCISES 1. Find the maximum, or minimum, as the case may be, of each of the following functions, and give the corresponding value of x. ,(a) x^-2z+T. (d) 2x^-4:X + 9. (6) a;2-3a; + ll. ' (e) -2x2+4x + 9. (c) -x^ + 2x+T. U) -3x^+2x-7. 2. The motion of a body thrown vertically upward in an unresisting medium is given by the equation s=—igfi + vot, where g = 32.2, and vo is the velocity with which it is thrown. Find the maximum height to which a ball thrown upward with a velocity of 100 feet per second, will rise. 3. Solve the problem in Ex. 2 in general terms, i.e. without substitut- ing values for Vq and g. i. How long will the ball in Ex. 2 rise if the initial velocity is 100 feet per second ? [Hint. What value of t corresponds to the maximum value of s ?] 5. What are the area and the dimensions of the largest rectangle that can be inscribed in a right triangle whose base is 4 feet and altitude 6 feet ? [Hint. Let the area be given by a = xy, then by" means of similar tri- angles y can be expressed in terms of x and so a will be expressed as a quadratic function of cc.] 6. Find the area and the dimensions of a rectangular ventilating window, of maximum opening, that can be put in a window that is an equilateral triangle whose side is 6 feet. 53. The Power Function. A function of the form (11) ace", which is the product of a number a and a power of x, is called a power function. The great importance of the power function lies in the fact that by means of it many fundamental formu- las of geometry and physics may be expressed. The following examples will give some indication of the r61e that it plays. (a) The circumference of a circle with radius r is the power function 2 irr, and the area is the power function vr^ ; (6) the volume of a sphere is the power function f irr' ; V, § 53] FUNCTIONS — GRAPHS 65 (c) the law for a body falling freely in a vacuum is given by the power function J ffJ^ ; (d) tbe time required for a body to fall freely through a space s is the power function v'2/5f • s'^" ; (e) the Newtonian law of gravitation is given by the equation (/) in the so-called adiabatio expansion of air the pressure p expressed in terms of the density p is the power function p = c- pi-^os. The simplest power function is x". For n = l the graph is a straight line, as vre already know, and for n = 2 the graph is a parabola whose equation is y = x\ rigure 12 shows those portions of the graphs of x^'^, x, and a;^ which lie above the a;-axis and to the right of the y-susis. The graphs of all power functions of the form x", where n is posi- tive, pass through the two points (0, 0) and (1, 1). Why ? 66 COLLEGE ALGEBRA [V, §54 54. Power Functions with Negative Exponents. Infinity. The simplest example of a power function with a negative exponent is a;"', which may be written 1/a;. Brief considerar tion of this function will serve to bring out a property not found in any of the functions that have been studied hereto- fore. The graph of the function is the graph of an equation which may be written ia any one of the three forms y = x y=-, yx = l. X Clearly .the graph passes through the poin^ (1, 1), but it does not pass through the point (0, 0), since the substitution of for X would give y = 1/0, which has no meaning. (§ 14.) Nevertheless, if we construct a table of values for the func- tion y = i/x corresponding to the values x = l, x= .1, .01 we shall see how y behaves as x becomes smaller and smaller. X 1 .1 .01 .001 .0001 .00001 y = yx 1 10 100 1000 10,000 100,000 From this table, it is perfectly clear that as x continues to decrease y, or 1/x, will continue to increase and as x becomes indefinitely small y will become indefinitely large. We say that y becomes infinitely great, or simply that it becomes infinite, and for the word infinity we use the symbol " oo ." The symbol oo cannot be looked upon as a symbol for a number like 1, or 2, or 5/6, or 0, or V2, since it does not obey the ordinary rules for reckoning by the fundamental operations. For a negative value of x, the function y = 1/x is negative, taking a value equal in magnitude, but opposite in sign, to the V, § 54] FUNCTIONS — GRAPHS 67 value for the corresponding positive value of x. Thus for X = .01, y = 100, and for x = -.01, y =- 100. ' The graph consists of two apparently distinct parts, as in Fig. 13. The curve is called an equilateral hyperbola. Fig. 13. The graphs of the functions x~", where — n is a negative integer, are somewhat similar in shape to the equilateral hyperbola. EXERCISES 1. Oonstruct the graph of each of the following power functions, using i inches, or 10 centimeters, as unit. (a) x8. (e) 3a;2. (i) xr-K (6) ^. (/) 3a;i/2. (J) x-v\ (C) xl/2. (9) ^"^. (k) 2x-^. (d) a;3/2. Qi) 2a;-i. (l) 3x-^\ 2. Give two points through which all the curves whose equations are of the form y = ax", pass, if m > 0. 3. Give one point through which all the curves whose equations are of the form y = axr", pass, if n>0. 68 COLLEGE ALGEBRA [V, § 55 55. Variation. We say that one number y varies as, or is proportional to, another number x, if y is equal to a constant times a;, that is if y = ex. If one number y is related to another number x according to the relation y = cx-\ we say y varies inversely as x. In general, if (12) y = can", where n is any rational number, positive or negative, integral or fractional, we say that y varies as the wth power of x. The connection between variation and proportion is easily established. Tor, if x^ and ajj be two values of x, and y^ and 2/2 the -corresponding values of y, according to the relation (12), then 2/1 = ex" and 2/2 = ca;^. When one of these equations is divided by the other the resulting equation may be written as the proportion (12 a) yi-yi = x\: x% Conversely, if equation (12 a) holds for every pair of values of X and y, then ^ y-i — ~^i- But if x^ is fixed, y^ is fixed, and 2/2/3:2 is a constant which may be denoted by c. Then y^ = ex". The last equation is identical in form with equation (12). Consequently, the equa- tion (12) is identical in meaning with the proportion (12 a). In practical problems the constant c is usually determined by experi- ment. For example, if a body travels with uniform velocity, the distance traversed is proportional to the time, and we have s = ct. If we meas- ure the distance s corresponding to a definite time t, c is known immedi- ately. It is, of course, the velocity. \ V, § 55] FUNCTIONS — GRAPHS 69 Again, under the Newtonian law of gravitation, the force by which two unit masses are attracted to each other varies inversely as the distance between them. The formula for the force is therefore /=ci. To find c experimentally, it would be necessary to take two bodies of unit mass, measure the distance between them, and then measure the force/. The determination of the constant c in variation problems depends upon the units employed in measuring the numbers that are proportional to each other. In the well-known formula for the motion of a, body falling from rest, s = \ gfl, the constant g is approximately 32.2 only when t is measured in seconds and y in feet ; if i is measured in seconds and s in centimeters, g is approximately 981. When in equations like s = ct^, t denotes time and s space, the number 2 c is called the acceleration. EXERCISES ^ 1. The weight w that can be raised by a pull of p lb. by means of a pulley block varies as p, that is, w = kp. Find fe if w = 250 when p = 100. > 2. The height ft of an object varies as the length I of its shadow at any given place and time. If the shadow of a pole 10 feet high is 8 feet long, find the relation between h and I. Hence find the height of a building whose shadow is 65 feet long. 3. The amount of extension e of an iron rod under tension of p lb. varies asp. For a certain rod e = 0.009 in. when p = 1200 lb. Express e in terms of ^. 4. The space traversed by a ball rolling down a rough inclined plane varies as the square of the time. If the ball rolls 23 feet in 2 seconds, what is the acceleration ? 5. There are 2.54 centimeters in 1 inch. If in the formula s = ^ gt^ s is measured in centimeters and t in seconds, what is the acceleration ? 6. The ordinate of a moving point varies as the square of the abscissa, and when the abscissa is 4 the ordinate is 64. What is the equation be- tween X andy? 7. We know that the circumference of a circle varies as the radius, i.e. if c and r denote circumference and radius, respectively, c = kr. Meas- ure the circumference of a circle with known radius and thus determine k. 70 COLLEGE ALGEBRA [V, §55 8. Boyle's law states that the volume u of a given quantity of gas, at constant temperature, varies inversely as the pressure p, measured in pounds per square inch. If, for a certain gas b = 2 cu. ft. whenp = 91 lb., find the relation between^ and v. Hence find v whenp-= 200 lb. 56. Discontinuous Functions. In the previous section ■graphs were discovered which contained a break or gap. Tor example, in the graph of ar^ it would not be possible to trace out with a pencil point the curve between x = — 1 and x = + 1 without removing the pencil from the paper. We say that the function is discontinuous at a; = 0. Other examples of discontinuous functions are numerous. For example, the postage on written rnatter is " 2 cents per ounce or each fraction." The postage to be paid on x ounces is 2 X cents provided x is an integer. The dotted straight line of Pig. 14 does not, however, rep- resent the situation, for the post- age is the same for all weights between zero and one ounce, then between one ounce and two ounces; and so on. The real situation is indicated by the series of unit segments of straight lines all parallel to the as-axis, and each having its right-hand extremity in the line y = 2x, as in Fig. 14. The function is discontinuous, at a; = 0, a; = 1, a; = 2, •-, constant for fractional, and nonexistent for negative values of x. 67. Statistical Graphs. One of the most common, and at the same time most useful, applications of graphical methods is the representation of tables of statistics by graphs. Such a graph is called a statistical graph. Statistical graphs are employed extensively in the study, of political economy and kindred sub- / ,'0 1- Jt Fig. 14. V, § 57] FUNCTIONS — GRAPHS 71 jects, and, indeed, in the study of all subjects which involve the collection and comparison of numerous data. For example, suppose it be required! to represent graphically the data of the following table giving the population of the United States for each decade from 1850 to 1910 inclusive. TEAit POPTTLATION 1850 23,000,000 1860 31,000,000 1870 39,000,000 1880 50,000,000 1890 63,000,000 1900 76,000,000 1910 92,000,000 90 / P, SO / 70 / (' SO / < SO / •2 / Pi so 1 / P, 20 / 10 ism 1860 1870 1S80 ism 1900 ISIO Taking 10,000,000 as the unit of measurement on the population-axis and 10 years as the unit on the time-axis, it is not difficult to see that the known points of the graph are the points Pi, Pj, Pa, P4, Ps, Pe, and P^, of Fig. 15. It is usual to as- sume that for the first approximation the change is uniform between two con- secutive known points so that the portion of the graph be- tween two such points is a segment of a straight line. The graph is then a broken line which may change its direction abruptly from time to time. If desired the graph may be "smoothed out" by drawing a smooth curve through the known points. Fig. 15. 72 COLLEGE ALGEBRA [V. §57 Another common form of the statis- tical graph is ob- tained by drawing a' rectangle with a unit base and alti- tude equal to the length of the ordi- nate. The value of the function corre- sponding to a given value of the inde- pendent variable is then represented by an area instead of by a length. In the example just given the graph would he obtained by con- structing a series of rec- tangles each with unit base and having for their altitudes the ordinates of lengths 2.3, 3.1, 3.9, and so on. The graph is shown in Kg. 16. Another form of statistical graph is obtained by using the number of individuals reaching a certain standard as the ordi- nate and the standard itself as abscissa. Suppose a class is marked according to the following table : 90 SO 70 60 S '" S *• ao 20 10 • laso ism 1870 isao isso 1900 1910 Fig. 16. Number of Students 10 20 35 50 75 78 50 20 Grade 50 60 70 75 80 85 90 95 The graph obtained by locating the points (10, 60), (20, 60), (35, 70), and so on is called the curve of distribution of the marks. V, § 57] FUNCTIONS — GRAPHS 73 It is important to note that for a statistical graph, no formula for the function need he known. Indeed in most cases occurring in statistics, no function formula can he known. EXERCISES 1. What does the graph in Fig. 15 show ahout the period of most rapid growth in population 1 2. The high school attendance in the United States since 1876 is given by the following table : Year 1876 1880 1890 1900 1910 Attendance . 22,986 26,609 202,963 519,251 915,061 Compare graphically the increase in high school attendance with the in- crease in population, taking the data for population from the table given above. Use both forms of graph. 3. The following table gives the average weights of men of different heights and ages. Plot graphs of each of the rows and compare them. Height Age 15-24 Age 25-29 Age 80-34 Age 35-89 Age 40-44 Age 45^9 Age 50-84 ■Age 55-59 Age 60-64 Age 66-69 Lb. Lb. Lb. Lb. Lb. Lb. Lb.. Lb. Lb. Lb. 5 feet 120 125 128 131 133 134 134 134 131 — 5 feet 1 inch . . . 122 126 129 131 134 136- 136 136 134 — 5 feet 2 inches . . 124 128 131 133 136 138 138 138 137 5 feet 3 Inches . . 127 131 134 136 139 141 141 141 140 140 5 feet 4 inches . . 131 135 138 140 143 144 145 145 144 143 5 feet 5 inches . 134 138 141 143 146 147 149 149 148 147 5 feet 6 inches . . 138 142 145 147 150 151 153 153 153 151 5 feet 7 inches . . 142 147 150 152 155 156 158 158 158 156 5 feet 8 inches . . 146 151 154 157 160 161 163 163 163 162 5 feet 9 inches . . 150 155 159 162 165 166 167 168 168 168 5 feet 10 inches . . 154 159 164 167 170 171 172 173 174 174 5 feet 11 inches . . 159 164 169 173 175 177 177 178 180 180 6 feet 165 170 175 179 180 183 182 183 185 185 6 feet 1 inch . . . 170 177 181 185 186 189 188 189 189 189 74 COLLEGE ALGEBRA [V, §57 4. The following table gives certain statistics of tiie Post Office during tlie years 1902-1915. Draw graplis of all the columns and compare them. United States PosT-OrFiOE Statistics Number OF Post Offices Extent OF Post Routes IN Miles Eetentte of TUB Depart- ment Expenditure OP THE Department Amount Paid fob PiecAi Tearb Compensation to Postmasters Transporta- . tioD of the Mail lfi07 1908 1909 1910 1911 1912 1913 1914 1915 62,659 61,158 60,144 59,580 59,237 58,729 58,020 56,810 56,380 463,406 450,738 448,618 447,998 435,388 436,469 436,293 435,597 433,334 1183,585,005 191,478,663 203,562,383 224,128,657 237,879,823 246,744,015 266,619,525 287,934,565 287,248,165 1190,238,288 208,-351,886 221,004,102 229,977,224 237,648,926 248,525,450 262,067,541 283,543,769 298,546,026 ^24,575,696 25,599,397 26,669,892 27,521,013 28,284,964 28,467,726 29,126,662 29,968,515 30,400,145 .$81,090,849 81,381,421 84,052,596 85,259,102 88,058,922 89,154,811 92,278,517 98,002,421 104,701,200 5. Plot a graph for the following table of velocities of water with a given "head." Theoretical Velocity of Watek in Feet per Second Head, Feet Yelocitv, Feet per Second Head, Feet Velocity, Feet per Second Head, Feet Velocity, Feet per Second Head, Feet Velocity, Feet pee Second 10 12 15 18 20 22 25.4 27.8 31.1 • 34.0 35.9 37.6 25 30 35 40 45 50 40.1 43.9 47.4 50.7 53.8 56.7 55 60 65 70 75 80 59.5 62.1 64.7 67.1 69.5 71.8 85 90 95 100 125 150 74.0 76.1 78.2 80.3 89.7 98.3 58. Interpolation of .Values of Functions. The construc- tion of a graph is usually accomplished by finding the values of the function for a few isolated values of th6 independent variable, and then drawing a smooth curve through the ex- V, § 58] FUNCTIONS — GRAPHS 75 tremities of tlie ordinates so determined. Frequently the values of the function for intermediate values of the independ- ent variable are not less important than the values for which the independent variable is given. The process of iinding these intermediate values without reference to the function itself is called interpolation. We have already made use of this process in connection with the use of logarithmic tables. If the graph of the function whose values are to be inter- polated is a straight line the intermediate values may be Fig. 17. found exactly. To find the correction to be added to y sup- pose T/i and ^2 are known values of the function corresponding to the values Xi and ajj of the independent variable. Let OMi = Xi, OMi = a^, Jf iPi = yi, M^P^ = 2/2) and let MP = 2/ be the unknown value to be interpolated between yi and ^2- (See Fig. 17.) Draw P1N2 parallel to the ovaxis. Then z = NP is the correction that must be added to 2/1 in order to obtain y = MN + NP. For, from the similar triangles NPiP and N^PiPz, NP-.N^Pi-. -.P^N^-.P^N^, 70 COLLEGE ALGEBRA [V, § 58 But NP = z, iVaPj = 2/2 — yi, -Pi-^ =:x— Xi, and P1N2 = a^a — "'ij so that the proportion may be written z:y2 — yi = x—Xi:X2 — Xi. Consequently, (13) « = ^^^ (2/2- 2/0- X2 — X1 In this expression x — Xi, x^ — Xi, and 3/2 — 2/i S're all known, so that the correction z is readily found. If the function in- creases with x the correction must be added to yi, if it de- creases with X the correction must be subtracted from yi- If the graph of the function is not the straight line P1P2 but the curve P1CP.2, the true correction would be JVC and not NP. However, for elementary work it is sufficiently accu- rate to assume that the graph is a straight line between two near-by points Pi and P,. The correction is therefore found in all cases by means of formula (13). Example. Suppose we have given (5.56)2 = 30.9136 and (5.57)2 ='31.0249 and wish to find (5.5642)2 to the same number of decimal places. The problem is therefore to interpolate a value of the function x^ between the values for 5.56 and for 5.57. The difference Vi — yi is .1113. Also we have X X2 — Xi_ -Xi .0042 .0100 42 100 The correction z is, therefore, 100 1113 = .0467, and we have approximately, (5.5642)2 = 30.9603. This result could have been found easily by other means, but there are many cases where interpolation is practically the only method available. V, § 59] FUNCTIONS — GRAPHS 77 EXERCISES 1. ^556 = 23.5797, and V667 = 23.6008. Find V556.34. 2. -v-^eo = 7.7194, and v-iel = 7 .7250. Find 4^46056. 3. The areas of two circles with radii 6.52 and 6.53, are 33.3878 and 33.4901, respectively. Find the area of a circle with radius 6.5231. 4. Log 2513 = 3.40019 and log 2514 = 3.40037. Find log 2513.7. 5. The population of Chicago was 1,698,575 in 1900 and 2,185,283 in 1910. What was it in 1906 ? 59. Zeros and Infinities of a Function. A zero, or a root, of a function is a value of tlie variable x for which the value of the function is zero. In symbols Xi is a zero of f(x) if f{xi) = 0. An infinity of a function is a value of x for which the function becomes infinite. Thus, 2 is a zero of the func- tion 2 a; — 4. Again, 2 is an infinity of the function l/(x — 2), or of the function (x'' — 7x + ll)/(a; — 2). The real zeros of a function are represented graphically by the abscissas of points where the graph crosses the avaxis. EXERCISES 1. Point out zeros and infinities of the following functions, as far as possible. (a)2x + 4. (h) {x+3)(2x+l). ,. (a;-l)(a:-2) (b)ix + l. {i) (5-x)(,3-x). ^ {X-3XX-4) (c) ia;-2. (J-) _J^. (d)2x. ^-^ W^±^- ' (e) Vx - 3. W -;j== (/) V4^. ^^^ a _ ^^^ ^_2 {g) {x - 1) (a; - 2). y/a^ - x^ Vk^ - 4 2. The graph of a certain continuous function f{x) is negative for x = a and positive for x = b. Show by graphical representation that the function has at least one zero between x = a and x = b. Can it have an even number of distinct zeros between x= a and x = 6 ? 78 COLLEGE ALGEBRA [V, § 60 60. Implicit Functions. If an equation exists between two variables, it is frequently possible to solve the equation for one variable in terms of the other. The first variable is then expressed as a function of the second. For example, if a;2 + ?/2 = 4, we may transpose the term a;^ and take the square root of both sides of the resulting equation. The iinal result is given by the equation y = V4 — x"^, which is implied in the equation k" + 2/2 = 4. A function which is implied by a mathematical relation in the form of an equation is called an implicit function. The function exists even though it may be impossible to find its form, as in the example just given. The graph of an implicit function may always be constructed if the form of the function can be found ; otherwise the con- struction is usually impossible by elementary means. As in the example given above, the process of finding the form of the function frequently leads to a function having more than one value for each value of x. The function 2/=V4 — a;2 found from the equation x'^ + y^ = i,, has two values for every value of x, one positive, and the other negative. EXERCISES 1. Construct the graph of the function of x determined by each of the following equations. (a) a;2 + 2/2=,4. (d) xy = -2. (6) a;2 - 5^2 = 4. (c) x-'^y = 2. (c) 2a; + 33; + 7 = 0. (f) x'y = 7. CHAPTER VI QUADRATIC EQUATIONS WITH ONE UNKNOWN 61. Definitions. A quadratic equation with, one unknown is an equation which has, or may be reduced to, the standard form (1) adcP- + 6a5 + c = 0, where a, &, and c are constants, and x is the unknown number. The equations' 3a; ■ are all quadratic. The numbers a, h, and c are called the coefficients of the equation. The number c is called the constant or the absolute term. A number is said to satisfy an equation if, when it is sub- stituted for the unknown number, the two members of the equation become numerically, or identically, equal. For example, 2 satisfies the equation a!2-5a; + 6 = 0, since 22-6-2 + 6 is numerically equal to zero. Again, b/a satisfies the linear equation ax=: b, since a x{b/a)=b. A root of an equation with one unknown is a number which satisfies the equation. 79 80 COLLEGE ALGEBRA [VI, § 62 62. Solution of the Quadratic Equation. If the equation (1) ax^+bx + c = be diyided through by a, and the term c/a in the resulting equation be transposed, it is reduced to the form x^ +-x = a a Completing the square of the left member by adding b'^/i a" to both sides, we obtain the equation- x^ + bx-\ = ; or. 2aJ 4 a2 Extracting the square root of both sides of this equation, we have , b , V62-4ac 2a 2a and finally, by transposing the known term -— , (2) ^^_±:i:i^ZA^. Equation (2) gives two values for x in terms of the coeffi- cients of equation (1), one for the positive and the other for the negative sign of the radical part. That both these values satisfy equation (1) and are, therefore, roots of it, may be seen by actually substituting the values for x and showing that the resulting equality is an identity. Equation (2) may be looked upon as a formula for the solu- tion of arty quadratic equation which has been reduced to the form (1). It is much better, however, to master the process and to solve each quadratic equation by the method used above. VI, § 63] , QUADRATIC EQUATIONS 81 Thus, to solve the equation 3x2-5a;-6 = 0, it is better to- go through the successive steps : a;2 - I K - 2 = 0, 6 6. 6 6 EXERCISES 1. Solve each of the following equations. (a) a;2 — 7 X — 11 = 0. (/) mx^ —nx = p. (6) 2a;2_7a;_i3 = o. (3) x*- 5x2 + 6 = 0. (c) 2x= 12-11 a;2. (A) x^ - Sx^ + 6 = 0. (d) 3 X - 7 = 4 xA (i) (X - 0)2+ 3Cx - o) - 14 = 0. (e) 0x2-14 = 6x2. (j) 6(m+ »)2 + 5(m + n)-4 = 0. 2. Find t from the equation '=i9'«2 + »i« + W3. 3. The difference between two numbers is 4 and their product is 21. What are the numbers ? Explain the double result. 63. Solution by Factoring. By the method of § 19 every quadratic expression may be reduced to the difference of two squares. For, clearly, ax^ + bx + c = al x^ + -X + -] ■ \ a aj If b^/i a be added to and subtracted from the expression in the parenthesis, the quadratic equation takes the form ' a 4a2 4a2 ' ^J-(^f ^2^ ft V /'./&2-4acV" 2 a) V^ 4a2 = 0. 82 COLLEGE ALGEBRA [VI, § 63 By the method of factoring the difference of two squares', the equation is reduced to the form ^ ' \ 2a 2a ^V 2a 2a J By hypothesis, a is not zero ; consequently, either since a product cannot be zero unless at least one of its factors is zero. The two equations (4) are both linear. If Vi and r^ denote their roots, then (^) '^^ — 2^+ 2a ' '=-~2^ 2^ These solutions are identical with (2) of § 62. Ti! & V6^ — 4 ac J 6 , V6^ — 4 ac ■ . • /os i. If — and - — h - — m equation (3) be ^ a u Qi Z a Z a replaced by their values — r^ and — r^, we may write (6) aa?-\-hx-\- c = a{ic — r-^{x, — ri}, where r^ and r^ are the roots of the quadratic equation. From the identity (6) it follows that a quadratic equation may be solved by factoring the left member, after the equation has been reduced to the standard form. EXERCISES 1. Solve each of the following equations by factoring. T (a) a!2_5a; + 6=0. \ (c) a;^- 5a; + 7 = 0. - (6) a;2- 7a; + 10 = 0. {d) 9x^-12x + i = 0. 2. Solve each of the foUovcing equations by factoring. First reduce the right-hand side to zero. VI, § 64] QUADRATIC EQUATIONS 83 (o) a;2 = 10a; + 21. 'fc) (x- 3)(a; - 4) = ^. (6) 123:2= 17x-6. (d) (x-2){x-5)=i. 3. Solve the equation x^ = 64. [Hint. Transpose the constant term and factor.] 4. Solve the equation X* = 81. 5. By means of the identity ax^ + bx + c = ax'^ + bx + c — (ar'^ + br+ c), where r is a root of the equation ox" + 6x + c = 0, prove that x — r is a factor of the quadratic function ax^ + bx-\- c. 64. The Formation of a Quadratic Equation having Known Roots. By means of the identity (6), it is possible to build up a quadratic equation having any two numbers, real or imaginary, as its roots. If, for example, the roots of a quadratic equation are 3 and 5, the equation must be (x-3)(x- 5)=0, or x2-8x + 15 = 0. EXERCISES 1. Construct the equations whose roots are as follows, (a) 2,4. (d) 2+\/3, 2- Vs. (g) 1, VS. •(6)3,-7. («)p+V^,p-Vg. W-| + ^^«,-|-^^. (c), -4,-3. (/) -V3, VS. (J) 0,2. ■ 2. What is the simplest quadratic equation whose roots are both zero ? 3. Prove that if the roots of a quadratic equation are conjugate radicals A+VB and A— VB, where A and B are rational integers, the coefficients of the quadratic are rational integers. Is the converse true ? 4. One root of a quadratic equation with rational coefBcients is 3 + VS. What is the equation ? 5. Prove that the expression ax^ + bxy + cy^ may be resolved into two linear factors that are rational in x and y, though not necessarily rational in the coefficients of x and y. 6. In view of what has gone before, can you factor x^ + y^? 84 COLLEGE ALGEBRA [VI, § 65 65. Character of the Roots. The roots of a quadratic equation ax^ + 6a! + c = 0, may be either real or imaginary. (See § 33.) Their character in this respect is determiaed by the sign of the number ¥ — i ac, which occurs under the radical signs of the roots b . V62 — 4ac ., 6 y &a — 4 ac 2a 2a 2a 2a Clearly, if 6^ — 4 ac is positive, the roots are both real ; and if &^ — 4 ac is negative, the roots are both imaginary. Moreover, if 6^ _ 4 dc is zero, the roots both reduce to — 6/2 a, and are therefore equal. These results may be stated in the following theorem : The roots of the quadratic equation ax^ + 6a! + c = 0, are (a) real and unequal, if 6^ _ 4 etc > ; (6) real and equal, if 6^ — 4 ac = ; (c) imaginary, if 6^ — 4 ac < 0. To say that the roots of a quadratic equation are imaginary does not mean that they do not exist, but rather that the facts given by the equa- tion are incapable of geometrical or physical interpretation in the ordinary sense. For example, if the hypotenuse of a right-angled triangle be 20 and one leg 25, the third side is given by the equation x^ + 252 = 202, or, a;2 + 225 = 0. The roots of this equation are imaginary and, as we know, the triangle is impossible. Another characterization of the roots is fully as important as the preceding one. VI, § 65] QUADRATIC EQUATIONS 85 The roots of a quadratic equation with rational coefficients are (a) rational, if b'^ — i ac is zero or is a perfect square ; (6) irrational, if b^ — 4:ac is neither zero nor a perfect square. The number 6^ — 4 «c is called the discriminant of the quad- ratic equation. EXERCISES 1. Find the discriminant of eacli of tlie following equations, (a) a;2- 5x+ 6 = 0. (c) 4x^=Ux + 9. (6) 3 k2 - 7 a: + 8 = 0. (d) S>x = ^^:^. X 2. Determine the character of the roots in each of the following equa^ tions without solving. (a) a;2-10x-|-40=0. (&) .3x2+14 a; _16 = 0. (c) 10 x2_i7 a; + 1000=0. 3. What value of k will make the roots of the equation equal? Sx^-6x+2k^0 4. Suppose the equation 5x^ + bx + 20 = is subjected to the con- dition that its roots must be equal. What is the value of 6 ? 5. For what values of b will the equation in problem 4 have imaginary- roots ? 6. The law of motion for a body thrown vertically upward is given by the equation s=—16.1t^ + vt where v is the velocity with which it is thrown, s the space, and t the time. If d=100 feet per second, in what time will the body be at a distance of 100 feet above the earth ? Inter- pret the results. 7. Under the conditions of problem 6, at what time will it be at a height of 26,000/161 feet ? 8. Under the conditions of problem 6, at what time will the body be at an altitude of 200 feet ? Interpret the result. 9. What must be the initial velocity in order that a body thrown vertically upward shall rise exactly 100 feet ? 10. Prove that all quadratic equations for which the signs of the coeffi- cient of X and the absolute term are unlike have real roots. 86 COLLEGE ALGEBRA [VI, §66 66. Geometric Interpretation. From the definition of the roots of a quadratic equation ax'^ + bx + c = 0, it is clear that they are the zeros (§ 59) of the quadratic function ax^ + bx + c. The roots are, therefore, represented graphically by the dis- tances from the origin to the points where the graph of the Fig. 18. function ax'^ + bx + c crosses the avaxis. If the discriminant ¥ — iac of the quadratic equation is positive, the graph crosses , the a>axis in two distinct points as in I (Fig. 18). If 6" — 4ac VI, § 67] QUADRATIC EQUATIONS 87 is equal to zero, it touches the a-axis, which is sometimes ex- pressed by saying that the graph crosses the axis in two coincident points. If 6^ _ 4 cic is negative, the graph does not cut the a!-axis, but lies wholly on one side" of it. In the last case we may say that the graph cuts the a^axis in two imaginary points. It follows, therefore,, that when 6 — 4 ac ^ 0, the sign of ax''' + hx -\- cis the same for every -value of x except when the function is zer6. If 6'' — 4 ac > 0, however, ax^ + 6a; + c is positive for some values of x and negative for others. For this reason the function ax^ + bx + c is said to be definite when 6^ — 4 ac^O and indefinite when &'' — 4 ac > 0. Note. If the quadratic expression is definite, a single value of x will suffice to determine whether the sign is always positive or always negative. EXERCISES 1. The graphs I, II, and III in Fig. 17 are the graphs of the functions !c2 _ 5 3; + 7.25, x^— 6x + 5.25, and x^— bx + 6.25. To which function does each graph belong ? Prove your statements. 2. Tell without constructing the graphs of the following functions which graphs cross the K-axis, which touch it, and which do not cross it : (a) 3 a;2 + 7 a; - 6. (J>) Sx^ - 7 x + 'd. (c) 9 a;2 _ 24 a; + 16. 3. Sketch roughly the graph of a quadratic function belonging to the quadratic equation whose roots are 4 and 5. Is there more than one such graph ? 67. The Sum and the Product of the Roots. When the right member of the identity (6) of § 63 is multiplied out it takes the form (7) ax^ +bx + c = ax^ — a{ri + r^x + ariv^. Then (8) bx-\-c = - a(ri + ri)x + ar-ir^. , 88 COLLEGE ALGEBRA [VI, § 67 Since an identity is true for eveij value of x for which both members are defined (§ 16), the value x = gives ariVi = c, or, (9) »-i»-^ = |- Also on account of (9), the identity (7) is further reduced- to the form bx = — o(ri + r^x, from which it follows that (10) ri + r2=-|. EXERCISES 1. Write down the sum and the product of the roots of each of the fol- lowing equations without solving the equation. (o) 3 x2 - 5 a: + 16 = 0. (c) ix^-ll =ix. (6) - a;2 + 7 X + 6 = 0. {d)x + §A±i = o. 2. rind ri + ra from formulas (5), § 63, by direct addition. 3. Find rir^ from formulas (5), § 63, by direct multiplication. 4. Find the value of the symmetric function r? + rf of the roots of the equation ax^ + 6x + c = in terms of the coefficients. ' [Hint. Square both sides of the relation j-j + j-j =— 6/a.] 6. Find the value of ri — I'a for the equation ax"^ + 6a; + c = without using formulas for n and ri. [Hint. From (n + r^y subtract 4 rirj. J 6. If xi and x^ are roots of ax'^ + 6a; + c = 0, find the equation whose roots are 7^ and xl [Hint. What are the-coefficient of x and the absolute term in the new equation ?] 7. If Xi and Xj are roots of ax^ + 6x + c = 0, find the equation whose roots are Xj/xz and xf/xi. VI, § 68] QUADRATIC EQUATIONS 89 68. Equations containing Radicals. The unknown num- ber in an equation frequently occurs under one or more radi- cal signs. In such cases both sides of the equation must be raised to powers corresponding to the indices of the radical signs as often as need be to remove the radicals. For example, the equation V6 X - 11 = x-S must be squared once. The resulting equation is easily reduced to the equivalent equation a;2 _ 12 a; + 20 = 0, whose roots are Ki = 2 and Xi = 10. In solving such equations great care must be exercised since raising both sides of an equation is liable to introduce extra- neous roots, that is to say, roots which do not satisfy the original equation. Any such extraneous root must be dis- carded, as it is not, properly speaking, a root. For example, the equation a;-2 = 3 has a single root, x = b. But if both sides are squared, the resulting equation is a quadratic equation a;2 _ 4 x + 4 = 9, which has the two roots, a; = 5, and x=— 1. The root a; = — 1 does not satisfy the original equation and is therefore extraneous. If it be stipulated that in the equation V6 a; — 11 = x-3 the radical shall have the positive sign only, the root a; = 2 obtained above will not satisfy it, while if it be stipulated that the radical shall have the negative sign, the root a; = 10 is extraneous. In solving equations containing radicals it is therefore nec- essary to test every root found by substituting it in the original equation. Unless stated explicitly to the contrary, it is under- 90 COLLEGE ALGEBRA [VI, § 68 stood that every radical expression in. the given equation is to be taken with the positive sign. EXERCISES 1. Find the values of the unknowns in each of the following equations. , _ C2 (a) Va; + 9 — Vk = 1. (c) x + V-c" —ax = (6) v'kT7+Vk = 7. Vc^ — ax (d) a;2 + 11 + Ve^ + 11 = 42. (e) V2a; + 9-v'a;-4 = Va; + l. (/) v'(a;-l)(a;-2)+V(x-3)(x-4)=V2. (si) yJa^-x + V'fe'-' + a; = a + 6. ^.x Vk + Va; — 5 2a;— 5 Vk — Va; — 5 ^ [Hint. Rationalize the denominator.] (i) 6V2J:V^^=2a:-6. v^ — -n/k — 6 2. Solve the equation + ^Jx + 1=0 and verify the solution. What is the cause of the difficulty ? CHAPTER VII SYSTEMS OF LINEAR EQUATIONS, DETERMINANTS 69. Two Linear Equations with Two Unknowns. In many problems it is required to find several unknown num- bers which satisfy several equations at the same time. Such a set of equations is called a system of simultaneous equa- tions, or, briefly, a system of equations. The simplest system of equations consists of two linear equations, that is, two equations of first degree with two unknowns. Such a system may be written in the form f UiX + b^y = Ci, ■ a2X + b^v = Cj. (1) i "' ■ By the well-known method of multiplying the first of these equations by 62; a^nd. the second by 61, then subtracting the two equations thus found, it may be shown that (2) ^^cA-C26i. Orfii — ajbi Similarly, usiag a^ and Oj as multipliers, we find /Q\ „ _ ctiCa — «2Ci Wifta — ttibi If each of the expressions on the right of (2) and (3) has a meaning, that is, if the common denominator is not zero, the pair of values of x and y is called a solution of the system, aeeordirig to the following definition : A solution of a system of equations is a set of values, Qne for each unknown, which satisfies every equation of the system. 91 92 COLLEGE ALGEBRA [VII, § 69 The final test of a solution is that when the values of the set are substituted for the unknowns, all the equations of the system reduce to numerical equalities or to identities. EXERCISES 1. Solve each of the following systems of equations and verify the solutions found. (a) 8x -iy = :10, = 20. (6) 1x -iy- -14 = = 0, 8x + Sy- 12 = :0. (c) w , 4 a; , 10 \ -iy = 3x. = 3 2/ + 14, 2x. r 5a; 18 = 70. Determinants. The expression 0162 —«2&i>'w^liich occurs in the denominators of both x and y, is called a determinant of second order, and when it has reference to the system of equations (1) , it is called the determinant of the system. The determinant 0162 — 02^1 is usually written as a square array in the form ffli h 02 62 (4) The numbers ai, 0.2, &i, b^, are called the elements of the determinant. The terms 0162 and 0261 are formed by taking one element, and only one, from each row and each column of ' the square array. Clearly, Ciftj — C2&1 and aiC^ — aiCi are also determinants which may be obtained from the. determinant 0162 — «26i by replacing, first, the a's by the c's and, second, the &'s by the c's. Hence the solution of the system (1), § 69, may be written in the form (5) Cl 61 «! Cl C2 &2 y = «2 <'2 «! 6] «! 61 Wo 62 a2 62 VII, § 70] LINEAR SYSTEMS — DETERMINANTS 93 The equations (5) may therefore be translated into the fol- lowing rule, called Cramer's rule, for writing down mechani- cally the solution of any system of linear equations in which the number of equations is equal to the number of unknowns. For the denominators, write the determinant of the system. For the numerator of each unknown, write the determinant obtained from the determinant of the system by replacing the coefficients of the unknown by the absolute terms. EXERCISES 1. Solve the following simultaneous systems. 7x + lly = 55, !2x + 3y = 6, ^ ' \Sx-6y = 7. w ix-3y = 21. (c) I X + 4:y = n. r 5 X - 4 y + 14 = 0, 6x + 5y + n = 0. ! mx + ny +p = 0, [rx (/) rx + sy + t = 0. 4 5 (?) W 5 + 2/ 12 + X 2 X + 5 2/ = 35. 3x + iy + 3 _ 2X + 7-2/ ^ g y -8 10 15 5 ' 9j/ + 6x-8 x + y _ 7x + 6 12 4 ~ 11 ■ 5 7 = 14, X y 1 + 8 = 42. X y (0 — \- -= a, X y X y Hint. Solve for land 1.1 X y J 2. Pind two multipliers ki and k^ such that when the equations of the system f 3 X + 4 2/ = 20, [5x + 82/ = 17, are multiplied by &i and fe respectively, and added, the coefficients of x and y in the new equation will be 4 and 0. 94 ^ COLLEGE ALGEBRA [VII, § 70 3. Two children weighing 25 and 40 pounds, respectively, are riding on a "teeter board" 14 feet long. How far should the support of the board be from the lighter child in order that they should just balance ? [Hint. The mechanical principle of the lever that is balanced is given by widi = W2d2, where loi and w^ are the weights and d^d^, their respective distances fpm the fulcrum.] i. A grocer has two brands of coffee worth 25 and 40 cents a pound. How many pounds of each must he take to make a mixture of 100 pounds worth 35 cents a pound ? 5. From Ex. 4 find the relative amounts of each brand regardless of the amount of the mixture. 6. Coffee worth a cents a pound is mixed with cofiee worth 6 cents a pound to make a mixture worth c cents a pound. Give the rule for mak- ing the mixture. 7. A goldsmith wishes to mix 10 carat gold with 18 carat gold to make 20 ounces of 14 carat gold. How many ounces of each must be taken ? [Pure gold is 24 carat.] 71. Graphic Solutions. In §§ 48 and 49 it was shovro that the graph of a linear equation in two variables is a straight line. The system (1), [aix + biy = Ci, . aa^ + 622/ = C2, is therefore represented geometrically by two straight lines, one for each equation. In general, the two straight lines will intersect in a point and the coordinates of this poiat will satisfy both equations. The coordinates of the intersection, therefore, constitute a solution of the system. Consequently, for equations in two unknowns, the definition of a solution of a system given in § 69 may be translated into geometric language by substituting for the words pair of values, one for each unknown, the single word point, and for the words satisfies all the equations of the system the words lies on all the graphs of the system. For greater clearness the two definitions are given in parallel columns. VII, §71] LINEAR SYSTEMS — DETERMINANTS 95 The graphic solution of a system is a point which lies on all the graphs of the system. An algebraic solution of a system with two unknowns is a pair of values, one for each unknown, which sat- isfies all the equations of the system. The approximate values of x and y are readily found by con- structing the two graphs on coordinate paper and then reading from the paper the coordinates of the intersection. For example, if it be required to solve graphically the system, 3x — 7y = 4:, the graphs may be constructed as in Fig. 19. tt""""^-^w""1t i: ^^ ^s ^^3 T* *• '^xT '~ ~ " -~ ~- - - s^f^ , , s. C3. / f.9 ) . S. -"^ N- -- : -.1 - ^''1^ ^ - - - - ._HiisA' - -^^ ,r,l^^ - -=. ^^ V - ^ :: p Z' '^' 1^ 1 Fig. 19. From the figure the coordinates of the point of intersection are seen to be approximately, x = 3.6, y = 1. With coordinate paper ruled to tenths the solutions may be found accurately enough for most purposes. 96 COLLEGE ALGEBRA [VII, § 71 EXERCISES 1. Solve graphically each of the following systems of equations, giving the results to the nearest tenth. '^"^ l5x + 22/ = 15. W \^ + Sy=-12. HA f2x + 5y = 19, |x-2/ = 0, ^ ^ \6x-4.y = 13. '''[3x + 4y = 12. nx + Sy = 18, |x-3 = 0, ^ ■' \x + ey = 10. *■•' M 3 a; - 4 3/ = 12. 72. Consistent, Inconsistent, and Dependent Systems. Any attempt to solve the system of equations 6x + 9y=7, by the ordinary methods of elimination will lead to a con- tradiction. For example, it is possible to eliminate x by multiplying the first equation by 3 and subtracting the second. But y disappears along with x and nothing remains but the untrue statement, 15 — 7 = 0. Systems of equations which, like the system just given, lead to a contradiction, are said to be inconsistent. If the determinant solution of the system be examined, it will be seen that the determinant of the system, 2 3 6 9 ' is zero. Since division by zero has no meaning (§ 14), it is rea&onable to expect diflB^culty in such cases. The real nature of the difficulty is most easily discovered by going back to the general system (1) r ajx + biy = Ci, a^x + h<2y = C2, whose solution is given in (2) and (3) in the form y. _ C162 - C2&1 ^ aiC; - aaCi 0162 — «2^1 '^J^'i ~ '^J^l VII, § 72] LINEAR SYSTEMS — DETERMINANTS 97 If, now, ai62 — Ca^i = Oj division by — bfii shows that h 62 But if bi and 62 are different from zero, equations (1) may be written in the form (7) a, , Ci 62 h Since — ai/bi and — aj/ftj are the slopes of the lines (§ 49), which represent equations (1), it follows from (6) and (7) that if the determinant a-fi^ — aj)i is zero, the graphs of the system (1) are parallel, and consequently do not meet in any finite point. A pair of inconsistent linear equations in two unknowns is represented geometrically by two parallel Ivies. If the numerators, as well as the denominators, of (2) and (3) are zero, the equations are said to be dependent. Por de- pendent equations not only are the graphs parallel, but (8) C1&2 — C2&1 = 0, and aiC, — a^Ci = 0. The equations (8) are easily reduced to the forms (9) ^1 = ^, and ^i = £2, which show that the intercepts of the two lines are equal. But if the slopes and the intercepts of two lines are respec- tively equal, the lines are coincident. A pair of dependent equations in two unknowns is represented geometrically by a pair of coincident straight lines, that is, by a single straight line. A system of equations which is neither inconsistent nor dependent is said to be consistent. Consistent systems have always a common solution. 98 COLLEGE ALGEBRA [VII, § 72 EXERCISES 1. Betermine in each of the following whether the equations are con- sistent, inconsistent, or dependent, and construct the graphs for each system. ,„. f2a:+7y = 5, f3x-6y = 4, ^ ' \ix+Uy=-5. ^ ' 1 4 a; -8 2/ = 4. f3a;+15j/ = 12, ^'^ U+5, = 4. «*) x-2y = 3, 2x + y = b, x + Sy = l. 2. Prove that when the equations (1) are dependent it is possible to find a constant k, which, used as a multiplier on one of the equations, will produce the other. Find k. 3. Can you recognize a system of two inconsistent or dependent equa- tions in two unknowns without applying the determinant test ? If so, how? 4. Prove that it in the system (1) we have 0162 — 0261 = and Oibi — C261 = 0, then also aica — azCi = 0. 6. Prove that the lines determined by the three equations Ax + By+ = 0, -x+-y+ — = 0, MAx + MBy + MG = 0, are identical, and from the conclusion deduce the proposition that multi- plication or division of a linear equation by a constant does not aSect the solutions of the equation, or the solution of any system of which it is a part. 6. Shew by drawing a figure that a system of three equations in two unknowns is, in general, inconsistent. 7. Given a system of three linear equations in two unknowns, aix + biy -I- Ci = 0, 023; + bay -I- C2 = 0, a^x + bsy -|- ca = 0, find the condition that the system sl^all be consistent by solving two of the equations for x and y and substituting the results in the third. VII, § 73] LINEAR SYSTEMS — DETERMINANTS 99 73. Linear Equations with Three Unknowns. The general form of a system of three linear equations in three unknowns is OiX + h-dj + Ci» = di, (10) a^x + &22/ + CiZ = di, ■ ajS! + % + C3» = dj. If the first equation be multiplied by h^c^ — h^Ci, the second by 63C1 — 61C3, and the third by 61C2 — 62C1, and the resulting equa- tions be added, the coefficients of y and z in the new equation will be zero. With both y and z eliminated we can find as the value of X (IV) X = <^i&2C3 — difegCg + d.J)sOi — djbiCs + djiic^ — d^b^Ci _ ai&uCg — 016362 + QibiCi — UibiCs + ajbic^ — O362C1 The denominator of this fraction is called a determinant of third order. It contains nine elements which may be arranged in the square array (12). (12) / X X A A = D By inspection of the terms in the denominator of x, it will be seen that the three positive terms are the products of elements lying on the arrows that start downward and to the right, while the three negative terms lie on the arrows ranging downward and to the left. D is called the determinant of the system. Moreover, it is clear that the numerator of x differs from the denominator only in the substitution of dj, d^, d^ for a^, a^, a^, 100 COLLEGE ALGEBRA [VII, § 73 that is in the substitution of the absolute terms for the coeffi- cients of X. Hence x can be found by Cramer's rule (§ 70). It can be shown in a similar way that y and z may be found by the same rule, so that the value of the three unknowns may be written down as follows : (13) di 6i Cl «! dl Ci «1 61 dj <*2 62 02 az dj C2 as 62 y = as ds -C3 , « = «3 63 ^3 «i 61 Cl «! 61 Ci «i 61 Ci «2 62 C2 ao 62 C-2 «2 62 C2 as 63 C3 CK3 63 C3 as 63 C3 EXERCISES 1. Pind the values of the unknowns in the following systems. (a) 3x + 4j/-5z = 20, a;+2y-33 = 8, a; -3-1/ + 72 = 30, (c) a; - 2/ = 4, 5x + 4!/ + z = 40. y + z = 6. ix +3y-2e-10 = = 0, X y z 3 a: — 2 J/ + 43— 20 = 5a; + 2/- 2 +30 = 0. 0, (d) i-? + ^=6, a; J/ 3 ?-l + ? = 5. a; 2/ (6) 2. Show that the multipliers das — CaSa, caoi — cias, and ciOa — CzSi will enable us to find y asx was found at the beginning of this section. 3. Prove that the values of x, y, and given in (13) are correct. 4. If x, y, z satisfy the system a\x + 61?/ + C\z = 0, a-ix + 62J/ + c-iz = 0, prove that (14) x 1/ S 61 Ci Cl «1 «! 61 &2 "2 C2 W2 a2 &2 [Hint. Divide through by a, then solve for the ratios x]z and y/z.] VII, § 74] LINEAR SYSTEMS — DETERMINANTS 101 74. Applications. (1) Determination of equations of given form belonging to loci which pass through given points. For example, the equation of a straight line may be written in the form y = ax + b, but we know nothing about the line until the numbers a and b are known. From geometry we know that a line is determined by two points. If, therefore, any two points on the line are given, the numbers a and b may be found. Suppose the line passes through the points (1, 2) and (2, 3). The fact that it passes through the point (1, 2) is expressed by the equation 2 = a . 1 + 6. Similarly, the fact that it passes through (2, 3) is expressed by the equation 3 = 0-2 + 6. From these two equations we readily find a = 1, 6 = 1. Consequently, the equation of the line passing through the two points (1, 2) and (2, 3) is '2/ = a; + 1, or x — y -{-1=0. (2) Functions with unknown coefficients. The law of motion of a body projected upward in a vacuum and subject to gravity is given by a function of the time having the forms s = -^gf^ + vt, where s is the altitude, g the acceleration due to gravitation, and V the velocity of projection. An observer with a stop watch and a measuring line, notes that at the expiration of 1 second the body is at an altitude of 83.9 feet, and at the expiration of 6 seconds, at an altitude of 20.4 feet. 102 COLLEGE ALGEBRA [VII, § 74 To find g and v we have the two equations i83.9 = -^g + v, \20A = -18g+6v. From these equations g = 32.2, and v = 100 ft. per second. (3) Alloys and chemical compounds, rrom two lots of brass, one containing 2 parts copper to 1 part zinc, and the other con- taining 3 parts copper to 2 parts zinc, it is desired to make 60 pounds of brass that shall contain 5 parts copper to 3 of zinc. How much must be taken from each lot ? If X and y denote the number of pounds required of each kind X + y = 60. Moreover, 2/3 of the first lot, 3/5 of the second, and 5/8 of the mixture, are copper. Hence, 2,3 5 ^ c„ 75 --iatercepts are found by putting y equal to zero and solving for x, and the iy-intercepts are found hy making x equal to zero and solving for y. When the equation of the conic is not in one of the standard forms (4), (6), (6), (7), (8), or (9), the curve may be constructed by solving the equation for y and then making a table of values as in § 46. It is important to note that for each value of x there will be, in general, two values of y on account of the radical sign that enters. EXERCISES 1. Find the distances between each of the following pairs of points. (a) (0, 0) and (3, 4). (6) (1, 1) and (2, 4). (c) (1, 0) and (5, 6). (d) (-1,7) and (1,-1). (e) (k, y) and (c, 0). if) (K, 2;)and(-c, 0). 2. Find the equations of the circles with centers and radii as follows. (a) Center (5, 6), radius 7. (6) Center (5, — 7), radius 4. (c) Center (5, — 7), radius 4. (d) Center (— 2, — 3), radius 6. (e) Center (— 5, 4), radius 3. (/) Center (1, 2) , radius 0. (g) Center (0, 0), radius 0. 3. Locate the center and find the radius of each of the following circles. (o) s? + 2^2 -Ok + 8 3^-24 = 0. (6) x^ + y^ + 6x + 8y-24:=0. (c) x^ + y^-6x + 4y-S0 = 0. (d) ifl + t/^-ax+l>y—c = 0. (e) x^ + y^ + 4:X + 6y + 20}=0. VIII, §76] NON-LINEAR SYSTEMS 111 4. Construct the graphs of the following equations, giving the intercepts and Interpreting any imaginary results that occur, (a) a;2 + 2/2 = 36. (6) 5 a;2 + 9 j,2 = 45. (c) 5a;?- 92/2 = 45. (d) a;2 + 2/2 - 2 X + 3 2/ + 1 = 0. (e) x^ + y^-Zx-iy- 10-= 0. (/) xy = 6. 77. The Graphical Solution of the Linear-Quadratic System. The graph of the linear equation is a straight line and that of the quadratic is a conic. It is a fundamental property of a conic that it may be cut by a straight line in two real points, or in no point. If there are two real inter- sections, the system has two solutions ; if the lines do not intersect, there are two imaginary solutions. EXERCISES 1. Solve each of the following systems, and interpret the results by constructing the graphs. (a) /^ +y =^' (c) /3a; + 42, + 7 = 0, \x^ + y^ = 16. \ y^-'7x = 0. (b) /^ +y =*' (d) I 2a; + 32/ = 6, I x2 - 2/2 = 16. \ 16 x2 + 25 2/=' = 400. 2. One point moves in suph a way that Its distance from the 2/-axls is 3 times Its distance from the x-axis, and another moves along the circle whose radius is 5 and whose center is at the origin. Where do the two paths cross ? 3. Find the value of k such that the straight line y =:2x + k will just touch the circle x^ + y^ = 25. What does the double sign for the radical in your answer mean ? 78. Systems Linear in x^ and yK Such a system is neces- sarily of the form From this system the values of x^ and y"^ may be found as in 112 COLLEGE ALOEBRA [VIII, § 78 systems of linear equations. TJie values of x and y are then found by a simple root extraction. In this case there are always four solutions, which are obtained by using all possible combinations of signs for x and y. The system is represented graphically by two conies which Fig. 24. may intersect in four points, or in two poiats, or not at all (Fig. 24). What is the character of the solutions in each of the three cases ? EXERCISES 1. Solve each of the foUawing systems of equations. («) ^ 4 a;2 _ 6 j/2 = 9. (c) 8j,2_4a;2-ll=0, 1 a;2 + j/2 = 9. ,j, r63c2+142,2_7=0, ,^, f8a;2-6j,2 = 15, ,., f2a;S + 3s(» = 100, l3a;2 + 8!/2_9=0. \5x^ + Sy^ = l5. j 3a;8 - 4yS = 75. 2. If the two conies 4 a;2 + 9 j/2 = * and »2 - 4 2/2 = 4 just touch each other, what is the value of k, and in what points do the conies touch ? VIII, §79] NON-LINEAR SYSTEMS 113 79. Systems of Symmetric Equations. An equation in two unknowns is symmetric if it remains unaltered when the un- knowns are interchanged. The most general form of a system of symmetric quadratic equations is I a;2 + j/2 4. a^xy + 62(0; + y) — c«. A symmetric system may be solved by first making the sub- stitution (12) a; = i{ + u, y =.u — V. The original equations then become aS) I ^ ^ "'^"' '^^~ ""^"^"^ + 2 61M = Ci, I (2 + a,:)u^ + (2 - a^v" + 2b2U = Cj. When «2 is eliminated from equations (13), a single quadratic equation in u is obtained. This equation can be solved for u, and either root substituted in one of the equations (13) will give another quadratic in v from which v may be found. When u and v are both known, x and y may be found from equations (12). There enefour solutions. EXERCISES 1. Solve each of the following systems of equations. ,j. I2(x^+y^) + 3xy-5{x+y) = 19, , \3{x^+y^)-2xy + 4(x + y) = il. ,. \-5xy + 6(x + y)=15i, l2(a;2 + 2/2)-|-5(x-|-j/) = 127. , „ I 3(a;2 -f y^) -6xy + \2(x + y)= 135, \4ix^ + y^)+7xy + 13{x + y)=5m. \xy = ib, \3{x'^ + y^)-exy-^ll{x + y)=Vl. ... fa;2 + 2/2 = 25, 1 2 a;2 -F 6 Ki/ -H 2 !/ - 7(a: -h !/) = 2^. 114 COLLEGE ALGEBRA [VIII, § 79 2. rind multipliers kt and fe for equations (11) such that when the multiplied equations are added, the result will be a single equation of the form x'^ +1^ +2xy + (&i6i + kzb2){x + y) =kiCi + feca. 3. Solve the system ix^ + y^ + ixy + 6(,x + y)= 31, \ x^ + y^ + S xy - 4(^x + y)= 9, by finding an equation similar to that found in Ex. 2, and then finding from this equation two linear equations. 80. Systems with all Terms of Degree 2 or 0. Such a system has the form To solve this system, let y = vx, and find («! + biv'^ + Civ)x'^ = di, (a2 + byv^ + C2v)x^ = di. The elimination of «" leads to an equation in v from which two values, Vi and Vi, are found for v. Either one of the equations y = VjX ot y = v^ together with one of the original equations forms a linear quadratic system from which x and y are readily found. There are four solu- tions. Why ? EXERCISES 1. Solve each of the following systems of equations. (16) x^ + by'i + ixy = il, J3a:2-7»/2 + 8xy = 73, |2a;-2-6y2 + 14a:j/ = 30, f 62/2 - 5a:y = 66, ^ ' l3a;2+7j/2-14a;!/ = -9. ^^' j 3x2 + 4a:i/ = 228. . |3a;2_62/!!+5a;2,=_12, | 2 a;^ - 7 xj/ = 18, 2. In either of the equations (14) — x and — y may be substituted for + X and + y respectively without altering the equation. What is the geometric significanpe of this fact ? VIII, §81] NON-LINEAR SYSTEMS 115 81. Systems solved by Various Devices. No solution of the most general system of simultaneotis quadratic equations in two unknowns can be given at the present time since such a solution involves the solution of an equation of the fourth degree. It is possible, however, to find solutions for many special systems of quadratics as well as systems involving equations of higher degrees. EXERCISES 1. Solve each of the following system of equations. x^ + y2 + z + y = 330, ^ ^ { ^/x + y + x + y = 12, («) { .,, lx^ + y^ + x + y=18, f (*) [xy = 6. (^^ U-y = 2. , ^ f xV = 180 - Ky. , , I k3 + 2/= = 407, (") U + 3, = ll. ^'^ U + J/ = ll. x^-y^ + x-y = 150. ^ ' (x^ + y^ = il. r> — y« = 98, .,, I xy + xy^ = 12, Ix^ + i 'xy + xy^ = 12, ^^^ { a:^ + j/S = 152, ■ xy + y^ = 19- 2. After appropriate substitution of new letters, solve each of the following systems of equations. f a;i/3 + 2,1 3 = 5, I ^ + 2, = 4^ ^"-' \x + y = 36. ^"^ U + 2/4^82. ra;V2 + 3,V2 = 4, I a;V4 + 2,1/3 = 5, '■''•' [ a:3 2 _,_ 2,3/2 ==28. ^ M xy' + y^'^ = 13. 3. Could you have foretold the symmetric form of the results in problems 2(a) - 2(c) ? 4. Prove that the elimination of y from the system f a;2 + 2,2 _ 2 X = 4 y - 20 = 0, 1 a:2 + 252/2 = 25, leads to an equation with rational coefficients of degree 4 in x. What is the geometric meaning of the result ? 6. Prove by means of two equations each having the form x'' + y^ + ax + by + c = 0, that two circles cannot intersect in more. than two real points. [Hint. Subtract one equation from the other. The intersections will lie on the graph of the resulting equation.] 116 COLLEGE ALGEBRA [VIII, § 82 82. The Graphic Solution of Quadratic Equations. Tlie accurate construetion of the graph of a quadratic funption in itself constitutes a graphic solution of the correspondiag quadratic equation when the roots are real. Another solution more in harmony with the methods for the solution of equa- tions of the third and fourth degrees is given here. Let the quadratic equation be (16) ax^ + ba} + c = 0, and let The equation (16) is then equiv- ^ alent to the system (17) 2^ = 052, ay + 6aj + c = 0. When the roots are real, their ^ approximate values may he read off directly from the intersec- tions of the parabola y = x^ and the straight line ay + bx ^'°- 25- + c = 0. If the roots are im- aginary the fact is apparent at once from the figure. Example. Let it be required to solve graphically the quadratic equation 3x2-2a;-3 = 0. The solution is found from the intersections of the parabola y = x'' and the straight line Sy — 2x — S = 0. The approximate values as read off from Fig. 25 are xi = 1.48 and Xj = — .83. The parabola must, of course, be drawn with care. This can be done by plotting the points for fractional values of x. With a good copy of the parabola cut out of cardboard, and a ruler, graphic solutions of quad- ratio equations may be found rapidly. VIII, § 83] I^ON-LINEAR SYSTEMS U7 EXERCISES 1. Solve graphically each of the following equations. (a) x^-4:X + l = 0. (e) a;2 - 6 a; + 8.9 : 0. (6) a;^-4a;-3 = 0. (c) 4a;2-8a;-3 = 0. (d) 9 x2 - 36 + 23 = 0. (/) a:2- 10a; + 25 = 0. (g) a;2-4a;-97 = 0. (A) 2 x2 - 8 a; - 431 = 0. 83. Graphic Solution of the Incomplete Cubic. Any cubic equation is easily reduced, by a process that will be explained in § 100, to the so-called incomplete form, (18) 3-l-2>a3 + g'=0. If 2/ = a^, this equation is equivalent to the system (19) ry + px + g = 0. The graph of the first equation is the graph of the power function a? and the graph of the second is a straight liae. The approximate values of the roots are easily read off from the intersections of the two lines. Example. Solve graphically the equation 3^3-5 3; + 1 = 0. This equation is equivalent to the system f y = x\ \_Zy -5x-\-l=0. From Fig. 26 the approximate values of X are read off to be asi = — 1.3, Xi = .2, Xs = 1.2. Fig. -JH. 118 COLLEGE ALGEBRA [VIII, § 83 EXERCISES 1. Solve graphically each of the following equations. (a) 3a;8-5a; + 2 = 0. (6) 3a;8 + 5a; + 2 = 0. (c) 3a;'-5a; + 5 = 0. id) 60a;8 + 115a; -56 = 0. 84. Graphic Solution of the Incomplete Biquadratic. An equation of the fourth degree is called a biquadratic. In a later section (§ 100) it will be shown that every biquadratic can be reduced to the incomplete form, (20) 05* + puB^ + qx + r = 0^ or, by a slight rearrangement of the terms of equation (20), to the form (21) x^ + x*+{p— l)x^ + px + r = 0. If y be chosen so that y = a?, the equation becomes equiva- lent to the system {p-l)y + r = 0. The graph of the first equation is a parabola and that of the -second is always a circle. The real roots will be read off from the intersections of the graphs. Example. Solve graphically the equation K*-5!c2-4a;-3 = 0. The term — 5 x^ may be written in Fia. 27. the form - 6 a;2 + x^. VIII, § 84] NON-LINEAR SYSTEMS 119 Then if ^ = x^, the equation becomes equivalent to the system. 2/ = a;2, x2 + 2/2 - 6 !/ - 4 a; - 3 = 0. When the squares of the terms in z and of the terms in y are completed the second equation takes the form (x - 2)2 + (!/ - 3)2 = 16. In this form the graph is readily seen to be a circle of radius 4, and with its center at the point (2, 3). From Fig. 26, the real values of x are seen to be Xi = — 2, and xi = 2.6. Two roots are imaginary. EXERCISES 1. Solve graphically the following equations. (a) x*-5x2-4x-7 = 0. (c) x* + 7x2- 2x - 38 = 0. (6) x* - 5 x2 - 2 X - 27 = 0. (d) 2x* + G x2 - 8 x - 41 = 0. CHAPTER IX INEQUALITIES 85. Definitions and Theorems. An inequality is a state- ment "which af&rms that one number, or one quantity, is greater than another. If a is greater than h the fact is expressed hy writing a> b, or 6 < a. The opening of the inequality sign is always turned toward the greater number. Two inequalities such as a > 6 and a;^ > 5 a; — 6, which have the inequality sign turned in the same direction, are said to exist in the same sense. It is everywhere assumed that the numbers in the inequalities arfe real. The rules of operation for inequalities are, with two important exceptions, the same as the rules of operation for identities, as the follow- ing theorems show. Theoeem I. An inequality with constant terms remains true in the same sense if any number, either positive or negative, is added to both sides. In symbols, the theorem states that if a > 6, then a ±m>b ± m. li a > b, a = b + d, where d is the difference between a and 6. Therefore I a + m = b + m + d. If the positive number d be dropped from the right member the equation becomes the inequality a + m > b + m. 120 IX, § 85] INEQUALITIES 121 CoBOLLAET. Any number- may be transposed from one side of an inequality to the other if, at the same time, its sign is changed. Theorem II. An inequality remains true in the same sense if both sides are multiplied or divided by the same positive constant. In symbols, the theorem states that if a > 6, then ma > mb., where m is a positive number. As in Theorem I, a = b +d where d is a positive number ; then ma = mb + md. If the positive number md be dropped, the equation ma = mb + md becomes the inequality ma > mb. That the theorem is true for division is clear if we note that the theorem is true for m = 1/n, where n is positive. Theoeem III. The sense of an inequality is changed if both sides be multiplied or divided by a negative number. For, if a>6, then — ma = — mb — md, as before. Discard- ing the negative number — md from the right member of the equation increases that member and the result is the inequal- ity — ma < — mb. If both sides of an inequality are positive, they may be, raised to any positive integral power and ,the result will be a new inequality having the same sense. Every numerical inequality can be changed in form by transposition so that both sides will be positive numbers. The solution of an inequality with one unknown is the pro- cess of finding all the values of the unknown for which the inequality is true. EXERCISES 1. Prove that a^ ^ j2 j> 2 a6 for every real value of a and 6, unless a = 6. [Hint, (o- 6)^ > 0, unless a - 6 = 0.] 2. Prove that when a and ft have the same sign a/h + h/a > 2, unless a = 6. 3. Prove that a'' -f- ft^ -f- c^ > 2 06 + 2 6c -I- 2 ac for real values of a, 6, and c. 122 COLLEGE ALGEBRA [IX, § 85 4. Prove the following inequalities. ■ (a) VIO > v'2 + V3. (d) V2 + \/5 > V3 + Vi. (6) VTI < -i/S + \/4. (e) V12 - V3 > VlT- v^. (c) VTi > 2 + VS. (/) vn - Ve + VB < Vio. 5. Of the following pairs of numbers which is greater, (a) V3or2 + V3? (c) VS - Vf or VS - V2 ? (b) V2I + ^/W or a/17 + vl3? (d) v^ - vT? + V3 or 2.2? 6. For what values of x is the inequality 5a; — 3>3a;— 7 true ? Algebraic Solution. By means of the theorems on inequalities this inequality is easily reduced to the form a; + 2 > 0, and from this form ' we see that x must be greater than — 2. Gkaphical Solution. A graphical so- lution is easily obtained by drawing the graph of the function x + 2. The graph is a straight line crossing the x-axis at x = — 2 X as shown in Eig. 28. The graph lies above the X-axis for all values of x that are greater than — 2. In other words the function X + 2 is greater than zero for x > — 2. Fig. 28 7. Solve the inequality x2<5x-6. Solution. The inequality is easily reduced to the form x2 - 5 X + 6 < 0, or (x - 2) (x - 3) < 0. The graph of the right member of the last form is a parabola which crosses the x-axis at x=2 and x=3. The figure (Fig. 29) shows that the function is less than zero, that is, the original inequality is satisfied for values of x lying between 2 and 3. The solution may be written in the form 20. id)x^ + x+l<0. (b) -a;2>6-5K. (e) (a; - l)(x - 2)(!i;-3)>0. (c) x^ + x>l. (/) (a;-l)(a:-2)(x-3)<0. 9. Prove that the inequality ax^ + 6a; + c > is true for all values of a; if ft'' — 4 ac < and a is positive. 86. Inequalities with Two Unknowns. The solution of an inequality with two unknowns consists in finding all pairs of values of the unknowns for which the inequality is true. For example, to solve the inequality 2x + 3y-5>0, note that the line whose equation is 2x + 3y-5 = 0, separates all the points of the plane into three classes : (1) Points whose coordinates satisfy the equation 2x + 3y-5 = 0; (2) points whose coordinates satisfy the inequality 2x + 3y-5>0; (3) points whose coordinates satisfy the inequality 2x + 3y-5<0. The points in the first class lie on the line. If the equation be written in the form 5-2a! it becomes clear that the points of the second class lie above the line, for if x is unchanged and y increased from y to y', we have 2/'>-3— Similarly, it may be shown that points of the third class lie below the line. In the same way, if we consider the line whose equation is 3x-iy + 5 = 0, 124 COLLEGE ALGEBRA [IX, § 86 it may be shown that points whose coordinates satisfy the inequality 3x — iy + 5>0 lie below the line, while points whose coordinates reverse the sense of the inequality lie above the line. Finally, the points which satisfy the two simultaneous inequalities f2a; + 32/-5>0, \3x-4:y + 5>.0, s Y s ^■> II <^y / \\: 5o „ ,/ III y < I y / \ \ y y IV s \ X y \ Fig. 30. ifiust lie above the first line and below the second, therefore lie in the region I of Fig. 30. They must EXERCISES 1. Solve graphically each of the following systems of inequalities. 2x + Sy-7>0, -x + 2y + 3>0. 2x + 3y-1>0, (a) <2x + i ^^' \-x + 2y + S<0. f2a; + 3j;-7>0, - \-x + 2y + 3>Q. (d) a;2+2/a-16>0. (e) (9) (ft) 2x + Sy-1>0, x-y-l>0, a; - 3 > 0. (/) x2 + j,2_i6<0. •x2 + j^2-16<0, x + y-2>0. !x^-y^-16>0. ,2» + 3/-l>0. CHAPTER X COMPLEX NUMBERS 87. Definitions. The solution of the quadratic equation a;2 + 1 = ■ is given by a; = ± V — 1, or, as it is usually expressed, x = ±i, where ^ is a symbol used to denote V — 1. The number de- noted by i cannot be real, for the square of every real number, either positive or negative, is positive, while P = -l. Numbers of the form bi, where 6 is a real number, are called pure imaginaries (§ 33). Assuming that i obeys all the laws of ordinary algebra, in- cluding the Index Law, the powers of i are given by the following table: f = i, i2 = - 1, i* _ ,:2,;2 — + 1, Moreover, (6i)2 = bi-bi = bH^ = - 6^ K'umbers having the form a + bi, where a and b are real, are called complex numbers. Since bi is not real, it follows that if a -(- bi = 0, then a = 0, and 6 = 0, since, otherwise, bi = — a, that is, a pure imaginary is equal to a real number, which is impossible. From this it follows that two complex numbers % -I- bii and a^ + b^i are 125 126 COLLEGE ALGEBRA [X, § 87 equal, when, , and only when, the coefficients of the real and of the imaginary parts are separately equal. For, if we have tti + bji = aa + hi, then ai — ch+ih— h^i = 0, and consequently, a^ = a-j and bi = 62. 88. Geometric Representation. The complex number a+&s, or a-l+h-i, contains two fundamentally different units, 1 and i. In the geometric representation of complex numbers this difference is recognized by measuring the real part along one axis called the axis of reals, and the imaginary part along another axis perpendicular to it and called the axis of imagi- naries. It is agreed that the axis of reals shall coincide with the a>-axis and the axis of imaginaries shall coincide with the y-axia. On the basis of this agreement, the complex number a + bi is represented, either by a broken line meas- ured a units along the axis of reals, then 6 units along the axis of imaginaries, or by the point P reached by the process indicated. Figure 31 shows, the geometric representa- 3 ^ p 3+St 1 1 ^_ L 1 y^ axin of reals ' A Fig. 31. tion of 3 + 2 i. The numbers 3 + 2i, - 3 + 2 1, - 3 - 2 i, and 3 — 2 1 are represented by numbers lying in the tirst, second, ■third, and fourth quadrants respectively. X, § 89] COMPLEX NUMBERS 127 The positive number Vcs^ + 6^ which measures the distance OP is called the absolute value or the modulus of the complex number a + hi. The absolute value of a complex number is denoted by | a + &j | or by Tnod (a + hi). The angle measured counter-clockwise from the axis of reals to the line OP, is called the angle or the amplitude of the complex number. The complex number a— hi is called the conjugate of the complex number a + hi. A complex number and its conjugate have the same absolute value. If a complex number is zero, its absolute value is zero, and conversely. EXERCISE Prove that the ahsolute value of a real number is identical with its numerical value. 89. The Fundamental Operations on Complex Numbers. Thbokbm. The sum, the difference, the product, and the quotient of two complex nuTnhers are complex numbers expressible in the standard form a -f- hi. Let tti +' 6i« and a^ + 62* be two complex numberp. Their sum is «! + hii + a^ + b^i = Oi + aa + (&i + 62)*'- Their difference is fli + 61 j — (Oj + 62O = «! — a2 + (pi — hi)i. Their product is (a. + &ii)(ffl2 + 62«) = («ia2 — &A)+ {o-A + a^h^i. Their quotient is g^ + 6^^^ _ (a^ + hii){ai — h^i) Bj + b^i (02 + b2i){ai — 62*") _ (a^a^ + bA) + («2&i — a Ay al + &i al + bl ai + bl 128 COLLEGE ALGEBRA [X, § 89 The final result in each case is a complex number of the standard form, and the theorem is therefore proved. That the apparent exceptions where the result is either real or pure imaginary are not really exceptions is shown by the fact that a real number may be written iji the form a-\-0 ■ i, while a pure imaginary takes the form + bi. The geometric representation of a complex number is most easily found by first reducing the number to the standard form. EXERCISES 1. Give the geometric representation and determine the absolute value of each of the following numbers. (a) 2 + Si. (e) -2. (i) ''-r"\ (6) 2-3i. (/) -i- U) (1 + 0'- (c) 4(2 + 3 i). <" h W (l-i)"- (d) 2. (ft) (3-4i)2. W (fi + \iY- 2. Prove that the sum and the product of a complex number and its conjugate are real. What of their difference ? 3. Construct the graph of z^ + 2 z + 3, when z = 1 + 2 i. 4. Construct the graph of z" + 2 z + 3, when z = 1 — 2 i. 5. Find the real and imaginary parts of /(o + bi) and /(o — hi) if f{z)=aos* + aiz^ + a^z'^ + aaz + a*, and the a's are all real. What difference in the results do you note ? 6. Prove that if /(z) is an integral quadratic function of z with real coefficients, that /(o + bi) has the form A + Bi and /(a — bi) has the form A — Bi. Is the result true for an integral function of any degree ? 7. By means of the absolute value, prove that if a + bi = 0, then a = and 6 = 0, and conversely. 90. Geometric or Vector Addition. In § 88 it was shown that the complex number a + bi may be represented geomet- rically either by the broken line OAP, or by the point P X, § 90] COMPLEX NUMBERS 129 (Fig. 31). Still another view is possible. We may say that a + M is represented geometrically by the directed line OP. Such a directed line is called a vector. From this point of view the geometric representation of a complex number is comparable to a force, or to a velocity, which is given when its magnitude, its direction, and its point of application are given. Let OPi and OP^ represent the complex numbers a^ + bii and 02 + ftji'i (Fig. 32). Construct the parallelogram OP^PPi, aj+aj+(Jj+4j)i FiQ. 32. with diagonal OP. Draw the ordinates M^P^, M^P^, and MP, and draw P^N parallel to the axis of reals. From the equal tri- angles OilfjPj and PyNP, M^P^ = NP=\ and OM^ = PN= a^. Consequently, 0M= OM, + M,M = Oifi + Pi-ZV = ai + a^, and MP = MN +NP = M,Pi + M^P^ = b,+ b^. The coordinates of P are, therefore, a^ + a^ and b^ + &2, so that the point P, or the directed line OP represents the number «i + 02 + {b\ + h)i which is the sum of the numbers % + &ii 130 COLLEGE ALGEBRA [X, § 90 and Oj + h^i. Therefore, if two complex numbers he represented by vectors, their sum is the diagonal of the parallelogram con- structed upon the two vectors as sides. It follows that geometric addition of complex numbers is precisely like the composition of forces or of velocities ia physics. EXERCISES 1. Construct each of the following sums by constructing the directed line for each complex number. (a) (l + i) + (2-i). (c) (3-5i) + (2 + 7i). (6) (3-2i) + (-4 + 3i). id) (5 + 60 + (2 + 3i). 2. Frame the rule for geometric addition in such a way that it will apply to the simpler case of addition of segments of a line. 3. Show geometrically that I (ai + bii) + (a2 + 62O | < | ai + 6ii | + J 02 + W |, and state the theorem expressed by the formula in words. CHAPTER XI POLYNOMIALS— EQUATIONS OF ANY DEGREE 91. Definitions. An algebraic expression of the form (1) fix) = «„«;" + ajx"-i + ag*"-^ + ... + a„, where n is a positive integer, and the coefficients a„, aj, — a„, are constants, is called a rational integral function, or, briefly^ a polynomial in x. A zero of the polynomial (1), that is, a number which when substituted for the variable x, causes the polynomial to vanish, is a root of the algebraic equation (2) «„!»" + a^a?"-! + a^*'""^ + •••+«„ = 0. , The coefficients a^, %, Oj, •.• a„, may be assumed to be in- tegers without loss of generality. 92. The Fundamental Theorem of Algebra. One of the prime objects of investigation in the study of polynomials relates to the number, the values, and the character of the zeros. To this end the most important step is a fundamental theorem, which may be stated as follows. Fundamental Theorem of Algebea. Every algebraic equation has at least one root.* This theorem, whose proof presents many diffi-culties, is here assumed to be true. * The first demonstration of this theorem was published in 1799 by the German mathematician Gauss. Proofs will be found in such books as Dickson, Theory of Equations, Bvrnside and Panton, Theory of Eqnor- tions, or Webeb, Lehrbuch der Algebra. 131 132 ' COLLEGE ALGEBRA [XI, § 93 93. The Factor Theorem. If r is a root of the equation f(x) = 0, X — r is a factor of the polynomial f(x), and conversely. By the fundamental theorem, the equation f(x) = has at least one root r. Consequently, /(?•)= 0, either numerically, or identically, so that /(«)=/ (a;)— /(r). But f(x)-f(r)= a,x- + aix»-i + ... + a„- (a„r» + air»-i+ ... + a„) = a„(x'' — r") + %(»"-' — r"-i) H +a,.-i(a! — r). Every one of the terms a„(x'' — r"), ai(x''~^ — ''""')> '"> ^n-iQ" —'>') is divisible by a; — r. Consequently, f(x) is divisible by a; — r-. To prove that the converse is true, note that if a; — ?■ is a factor of /(a;), then (3) f{x)~{x-r)f,(x), . where /i(a!) is a polynomial of degree one less than that of /(a;). Consequently, f(f) = (r-r)f(r) = 0. The theorem is, therefore, proved. CoEOLLAEY. A polynomial f(x) = a^x" + aia!»-i + OiX"'^ -\ f- «„ of degree n is the product of n linear factors and may he written in the form (4) f{x) = a(,(as — r{)ix - r^) ... (as - »•„), where r-i, r^, ••• ?•„ are the roots of the equation fix) = 0. The proof follows immediately, if we note that the polyno- mial /i (x) in equation (3) is divisible by x — ?-2, where r.^ is a root of the equation f(x) = 0. Then by the factor theorem, /i(a!)=(a;-r2)/2(a;), where /2(a!) is of degree one less than that of fi{x), or two less than f(x). The process can be continued until the quo- tient becomes a constant. Moreover, for every division the first term of the quotient has a„ as its coefficient. XI, §95] POLYNOMIALS 133 94. The Number of Roots. Theorem. An algebraic equa- tion of degree n has n, and only n, roots. In § 93 it was shown that f{x)=a„(x - ri)(a; - r^) - (x - r„), where Ti, r^, —, r„ are the zeros of /(»). Clearly, /(a;) vanishes when X is put equal to any one of the r's. If k of the linear factors X— r are equal, we say that the equation has k equal roots. With this understanding there are at least n zeros of the function ; that is, at least n roots of the equation f(x) = 0. Moreover, if r' be any number different from every one of the r's, then /(r')= a„(/ - n)(r' - r,) - (r' - r„) ^ 0, since no one of the factors can be zero. Therefore f(x) has only n zeros ; that is to say, f{x)=0 has only n roots. EXERCISES 1. Is 2 a zero oiifi- 6x^ + nx — 6? oiz^ + 6x^ + nx — 6? ' 2. Assuming that 3 is a zero of a;^ — 6 x!^ + 11 a; — 6, find all the linear factors. 3. One root of the equation x3 _ 6 a;2 + 6 X -f 8 = 0, is 4. Find the other two. 4. Prove by the factor theorem that a" — 1 is always divisible by a; — 1 , and by a:^ — 1 when n is even. 5. Prove that x" + 1 is divisible by x + 1 when n is odd, but not when X is even. 6. Prove that yz(y — z) + zx(z — x) + xy{x — y) is divisible Toy x — y, y — z, and z — x. 95. The Graph of /(oj). The graph of fix) may be found by constructing a table of values of the function exactly as was done in Chapter V. Owing, however, to the greater number of turns in the graph of a polynomial of degree above 2 the student 134 COLLEGE ALGEBRA [XI, § 95 will find it advisable to use shorter intervals between values of the independent variable. Thus, if it be required to construct a graph of the function the table of values to the nearest tenth, from x=0 to a: = + 6, where the value of /(k) is found for intervals of |, is as follows : X i 1 H 2 H 3 Si 4 H 5 5i 6 /w -5 3.4 8 9.6 9 6.9 + 4 1,1 -1 -1.6 4.6 13 When a smooth curve is drawn through the points given by the table, it is seen to have the form indicated in Fig. 33. Not only does the graph in Fig. 33 give visual evidence of the fact that this cubic equation has three roots, but it tells us approximately what they are. From the figure the curve is seen to cross the a^axis in the vicinity of the points x = .3,x = 3.7, and x= 5. Moreover, the position and values of the maximum and minimum or- dinates are approximately known. When the roots are all real and are all known, an approximate idea _ X 'of the form of the graph is easily ob- tained. FiQ. 33. For example, suppose it is required to construct the graph of a function whose zeros are -2, -1, 1, 2. By the factor theorem the function is /(a) = (k + 2)(a; + l)(a; - 1) (a; - 2) =a!4 - 5a ' + 4. For values of x less than — 2, all four factors are negative. Therefore, /(as) is positive, and consequently, the graph crosses from the positive to XI, § 95] POLYNOMIALS 135 the negative side of the a>axis at a; = — 2. It crosses the x-axis again in the points ic = — 1, a = 1, and x = 2. The graph has the form indicated in Fig. 34. EXERCISES 1. Construct the graphs of each of the following functions, and locate approximately the real roots. (a) x^-dx^ + Sx + S. (6) xS-5a;2 + 4a; + 6. (c) x^-5x^+5x + 3. 2. Construct roughly the graphs of each of the following expressions. (a) (a;-l)(c«; + l)(a;-2). (c) (a;^ - 5a; + 6)(x2 - 12a; + 35). (6) (x-2)(a;-3)(a;-4)(si;-7). (S) (a;- 2)2 (a;- 5) (a; - 7). 3. Prove that if ri, r2, ■•• r„ are all real and ao is positive, the graph of ao(x — ri)(x — Ti) ■•■ (x — r„) is above the x-axis, for values of x which are less than the smallest r when n is even, and below the x-axis, when n is odd. 4. An equation of odd degree has at least one real root. 136 COLLEGE ALGEBRA [XI, § 96 96. The Approximate Values of the Real Roots. No sim- ple method for determining the approximate values of the real roots exists.* If there is a siagle real root (or an odd number) between two given values of x, the fact is disclosed by means of the following theorem, the truth of which is almost self- evident. If fix) he a rational integral function, and if for any two real values of X, x = a, and x = b, f(a) and f(h) have opposite signs, at least one real root of the" equation f{x)= lies between a and h. For clearness suppose that /(a) is negative ahd /(6) is posi- tive. Then, since /(x) is continuous, as x passes from x = a to x = b, f(x) passes through all real values between /(a) and/(6). One of these values is zero. The corresponding value of x is therefore a root of the equation f(x) = 0. From the geometric point of view the theorem states that if the graph is below the x-axis at a; = a, and above it at a; = b, it must cross the axis once, or an odd number of times, between x = a and x=b. If the real roots of an equation do not lie too close together, this theorem furnishes a convenient means of locating them between two consecutive integers. For example, if /(x) = x* - 6 x'+ 5 0:2 -I- 10 a; -I- 2, the following table of values is easily computed : X - 1 1 2 3 4 5 /w 4 2 12 10 -4 -6 52 By the theorem, one root lies between 2 and 3, and another between 4 * A complete solution of the problem of determining the number of real roots between any two values of x was published in 1835 by J C. F. Sturm . For the demonstration of Sturm's theorem see Dickson, Theory of Equations, or BuRNSiDB AND Panton, Theory of Equations. XI, § 97] POLYNOMIALS 137 and 6. Closer examination shows that /(— .5) = — .4375. Consequently, one of the two remaining roots lies between x = — I and x = — .5, and the other between x— — .6 and a; = 0. EXERCISES 1. Locate approximately the real roots of the following equations, (a) x*-x^-2x^-3x-l = 0. (6) a;* - 12 a;3 + 36 3;2 - 12 X - 5 = 0. (c) x*-5x» + ix^-Sx-l=0. (d) 2 x5 _ 13 X* + 13 a;3 - x2 + 14 K - 1 = 0. 97. The Remainder Theorem. The formation of the table for a polynomial is greatly facilitated by an important theorem which may be stated as follows. Remainder Theorem. If a rational integral function of x be divided by x — a, the remainder is f{d). Proof. Whatever value of /(a) may have, f(x)^f(x)-f(a)+f(a). But f(x) -f{a) = a„(a;» - a") + ai(a;"-' - a""') + ■ • • a„-i(a; - a), or, f{x) = ao(a;" - a") + ai(a;"-i - a"" ') + ••■ + a„_i(a! - a) + /(a). Every term on the right of the last identity, except /(a), is divisible by cc — a. Therefore, 5) JA^=Q(a;)+-—-, , x — a f{a) where Q{x), which is the quotient of f{x) —f{a) by a; — a, is a rational integral function of x. Prom (5) it follows that the remainder is /(a). If both sides of (5) be multiplied by a; — a, the identity takes the form (6) / (05) = (05 - a) Q(a5) +/(«). 138 COLLEGE ALGEBRA [XI, § 97 By means of the remainder theorem the value of /(a) may be found as the remainder after dividing f(x) by a; — o. For example, if the function /(a;) = X* - 3 a:8 + 2 x2 - K + 13 be divided by x — 2, the quotient is q{x) = xs - x2 _ 1, and the remainder is 11. Therefore /(2) = 11. EXERCISES 1. Find the value of / (3) when / (x) = x* — 7 x^ + 8 x^ — 6 x + 14, and verify the result by actual substitution of 3 for x. 2. If /(x) =3x8-5x2 4- 4x+ 19, what is /(4)? Verify the result. 3. Prove the faQtor theorem by means of the remainder theorem. 98. Sjmthetic Division. The work of finding /(a) may be still further shortened by an abbreviated process for division called synthetic division. In order to explain this method of division, let /(a;) be of degree 4 ; then f(x) = a„x* + OjO^ 4- a^x^ + a^x + 04. When fix) is divided by a; — a, the quotient is of degree three with a constant remainder R. Therefore equation (6) of § 97 has the form (7) a^'^+aia?+a^x'^+a i^t takes the form (10) (Dia; - N,){D^ - N,)(D,x - N,) - (D^x - N,) = 0. From the form (10), (11) «o = ^1^2^53 - ^n, and «„ = ( - ir^i^^ - i^^„. It is clear from equations (11) that the denominator of any rational root is a divisor of a,,, and the numerator is a divisor of a„. It follows that the rational roots may be found by trial, since the number of integral divisors of Oo and a„ is finite. EXERCISES 1. Find the roots of the equation 4a;« + 4a;3-5a;2-9a;-9 = 0. Solution. Since the denominator of any rational root must be a di- visor of 4 and the numerator of 9, the rational roots are to be found in the set of numbers, ± 1, ± 3, + 9, ± J, ± }, ± I, ± J, ± §, ± f. Actual trial shows that | and — | are roots. The remaining roots may be found from the equation, 4a;4 + 4a:8-5x^-9x-9 .^^, ^^0_ 4 a;2 - 9 2. Find the roots of each of the following equations. (a) a;8-a;2-4x-6 = 0. (c) 6x* - 35a;8 + 68a;2 - 40a; -|- 7 = 0. (6) 2a;8-a=-14a;-t- 10 = 0. {d) 12^4- 65a;8 + 86 a;"- 41 a;+ 6=0. XI, § 100 POLYNOMIALS 141 100. The Transformation of Equations. The operation of changing one equation into another whose roots bear a definite relation to the roots of the first, is called a transformation. The most important transformation is one that changes the equation into another whose roots are greater or less by a fixed number than the roots- of the original equation. Problem I. To diminish the roots of an equation by a fixed number. Suppose it be required to diminish the roots of the equation fix) = a„(x - ri){x - r,)(x — r^) - (x - r„) = by the fixed number h. If x be replaced everywhere by x'+h, the new equation is (12) f(x' + h) = a,{x' + h- r,){x' + h - r,) - (x' + h- r„) =0. The roots of the equation (12), obtained by setting the sepa- rate linear factors equal to zero, are (13) x'l = ri — h, x\ = rj — h, ■■■, x'„ = r„ — h. That is to say, each root of the new equation is equal to the corresponding root of the old equation diminished by h. The required transformation is therefore accomplished by writing everywhere x' + h for x, or, as it is ordinarily expressed, by the substitution, (14) x = x' + h. If A is a negative number, the transformation (14) will in- crease all the roots by h. Geometrically, this transformation changes the position of the graph of f(x) by moving it bodily from right to left, or from left to right, according as h is positive or negative. It is often used to transform an equation into another which lacks a specified term. (See Ex. 4 below.) 142 COLLEGE ALGEBRA [XI, § 100 Pkoblem II; To change the signs of the roots of an equation.. Another important transformation is one which changes the equation into another whose roots are the negatives of the roots of the origiaal equation. It is accomplished by the substitution x= — x'. The proof is left to the student. EXERCISES 1. Transform the equation /(a;)=a;8_ 3 a;2 -f- 2 a; — 3 = into another whose roots are less by 2. SoLDTioN. The required transformation is a; = a;' + 2 and, conse- quently, the required equation is f(ix' + 2) = (a;' + 2)3 - 3(a;' + 2)2 -f- 2(a:' + 2) - 3 = 0. When the operations indicated are carried out and the accents dropped, the equation reduces to a;^ + 3 a;^ + 2 x — 3 = 0. 2. Transform the equation ^ — is?-\-U:x—l = (i into another whose roots are greater by 2. 3. Transform tlie equation a:* — 9 a;^ + 9 a;^ + 8 a; + 18 = into another having a root between and 1. [Hint. By the theorem of § 96 it is found that one root of the given equation lies between 2 and 3.] 4. Transform the equation x^ — Zx^ + 14 a; — 1 = into another in which the coefficient of x'^ is zero. [Hint. Let x = xi + h; then /(a;' + A) = (a;' + A)' - 3(a;' + ft)^ + 14(a;' + A) - 1 = a;'" + 3 hx'^ - 3 a;'2 + terms of lower degree. The co- efficient of x'2 in the new equation is 3 A — 3 and this coefficient will be zero f or A = 1.] 6. Transform the equation x* — 9 x^ -|- 9 x^ -f- 8 x -(- 18 = into another which lacks the term in x'. 6. Find a method by which an equation may be transformed into another whose roots are the roots of the original equation each multiplied by a given number m. 7. The roots of the equation 12 x' — 40 x^ -|- 41 x — 12 = are rational, but not integral. Transform the equation into another whose roots are integers. [Hint. The coefficient of x' is a multiple of the denominators of all the roots. Use the result of Ex. 6.] XI, § 101] POLYNOMIALS 143 8. Transform the equation x^ + 4 a;^ *|- 3 a; — 14 = into another which lacks the term in i?. 9. Prove that any equation of degree « can he transformed into another which lacks the term of degree n — 1. Give the substitution when the equation is written in the form ao^;" + aix""! + ■•• + a„_ia; + o„ = 0. 10. Find the condition that the cubic equation aox' + a\^ + ajx + flSa = can be transformed into another which lacks the term of first degree, by a real rational trdnsformation. 11. Solve the quadratic equation oqx^ + aox + 02 = 0, by first trans- forming it to the form Ax^ + B = 0. 12. Pind the approximate values of the roots of the following equations by first removing the second term and then using the graphical methods of §§ 83 and 84. (a) a;8 _ 3 J.2 + 14 a; _ 1 _ 0. (6) X* - 9 a;3 + 9 a;2 + 8 a; ;f 18 = 0. (c) ar* — 6 a;8 + 8 x2 + 4 a; - 4 = 0. 101. Horner's Method for the Approximate Determination of Real Roots. When tlie roots of an equation of degree greater than 2 are irrational, it is usually impossible to find usable values for the real roots except by methods of approxi- mation.* The most useful elementary method of approximation is known as Horner's method. Horner's method consists essentially in locating a root be- tween two integers, and then diminishing this root by successive transforma,tions until an equation is reached whose root differs from zero by an amount as small as we please. The details of the method will be made clear by an example. * Equations of the third and fourth degrees may be 'solved algebraically by well-known methods, though the forms of the solutions are such that they are not easily available for practical purposes. These solutions may be found in works on advanced algebra. It was proven by Abel in 1826 that the algebraic solution of the general equation of degree greater than four is impossible. 144 COLLEGE ALGEBRA [XI.HOl FiKsr Transformation. * Let it be required to find the approximate value of a root of the equation (15) /(a;)=cB« + a!'-Ha;2 + 12a;-10 = 0. By theorem of § 96, it is easily shown that one root lies be- tween x = 2 and a; = 3. If, therefore, this equation be trans- formed by the substitution a/ = x + 2, the corresponding root of the new equation will lie between and 1. Since the forih of the new equation is known, we may write /(»' + 2)= («;' -f 2y + (x' + 2)3 - ll(a;' + 2)2 + 12(a;' + 2)- 10 = A^x'* + A^x'^ + A.^^ + A3X + Ai. But since x = xf + 2, x' = x — 2^ Therefore if x' be replaced by a; — 2, the resulting function will be identical with the original /(a;). That is, /(a!)= a;« -f a^ - 11 a;' + 12 a; - 10 = A„{x - 2y + A^{x - 2y + A^(x - 2)2 -t- A^ix - 2) -f- A.^. Clearly, the remainder obtained by dividing the right member of this identity by a; — 2 is At. But the remainder after dividing the left member of the identity by a; — 2 is — 6. Therefore, ^4 = — 6. The integral part of the quotient obtained by dividing the left member is Qi(a;)=a^ + 3a;2-6a;-|-2. Consequently, Qi(a;)=a,-3 + 3a;2-5a; + 2 = A,ix - 2)3 + A,{x- 2y + A^(x -2)+ A,. The coefficient A3 may be found in exactly the same way that Ai was found, namely, by dividing both sides of the new identity by a; — 2. The process can be continued in this way until all the coefficients are found. XI, § 101] POLYNOMIALS 145 The successive quotients and remaiuders may be found by synthetic division and the work may be arranged as follows : 1 1 -11 12-10 (a; -2 2 6 -10 4 1 3 2 -5 10 2 10 -6 = --A,., , Qi{x)=afl + 3a^-5x + 2. 1 6 2 5 14 12 = = ^3, Q2(x)=x'-+5x + 5. 1 7 2 19 = ■^2i Q,(x)=x + 7. 1 9 = --A„ Q4(«^)=l. The new equation is, therefore, (16) x^ + 9 a;3 + 19 x^ + 12 a; - 6 = 0. Second Teansfokmation. Since a root of equation (15) lies between 2 and 3, the corresponding root of (16) must lie be- tween and 1. Closer examination shows that it lies between .3 and .4. If the roots of (16) be diminished by .3, the corre- sponding root of the new equation will lie between and .1. The work is arranged as follows : 1 9 19 12 - 6 (a; - .3 ■3 2.79 6.537 5.5611 1 9.3 21.79 18.537 - .4389 = ^'4, Q'i(a;)=a^+9.3a;'' + 21.79a; .3 2.88 7.401 -1-18.537. 1 9.6 24.67 25.938 = A'3, Q'i{x) = a;2-|-9.6 x + 24.67. .3 2.97 1 9.9 27.64 = A'2, Q'-i(x) ^x + 9.9. ■3 110.2=J.^, Q'iix)=i- The new equation is (17) X* + 10.2 x> + 27 M x" + 25.938 x - .4389 = 0. 146 COLLEGE ALGEBRA [XI, § 101 Thied Tkansfokmation. Since the root of equation (17) lies between and .1, the sum of the first three terms will be very small, and the root will be found approximately by neglecting the higher powers of x and using the linear equation 26.938 X - .4389 = 0. Clearly, the root of this equation lies between .01 and .02, so that the third transformation must be arranged to diminish the root by .01." By using x — .01 as a divisor, equation (17) is transformed into (18) a^ + 10.24 a^ + 27.9466 x^ + 26.493864 x - 0.17674679 = 0. Using the last two terms of equation (18), we find the root is approximately .006. By transforming (18) we should find a fifth equation, one of whose roots lies between and .001. In this way the work may be carried on until an equation is obtained with one root as near to zero as we please. Collecting results obtained thus far, we find that the root which originally lay between 2 and 3, has been diminished by 2.31, and that it now lies between and .001. It is evident, therefore, that 2.31 is the approximate value to the second decimal place. The process of finding the next figure of the root by solving the linear equation obtained by discarding all terms of degree higher than 2, is frequently called the method of trial divisors. It is evident that the accuracy of the trial divisor found in this way, increases as the work proceeds. Indeed, the next two figures of the root could have been obtained from the equation 25.938 a; -.4389 = 0. The approximate value of the root of the equation (16) to the fourth decimal place is 2.3166. XI, § 101] POLYNOMIALS 147 EXERCISES 1. Find the real roots to the third decimal place. (a) K* - 12 scs + 36 a;2 - 12 a; - 5 = 0. (6) K* - 10 a;2 + 1 = 0. (c) a;* - 5 a;3 + 4 a;2 - 3 a; - 1 = 0. {d) 2x^- 13 a;4 + 13 zs _ a;2 + 14 3; _ 12 = 0. 2. Find the positive roots of the equation xs + 12 a;2 - 6008 x + 179200 = 0. [Hint. To locate the roots, try 10, 20, 30, ■••.] 3. Find the positive roots of the equation kS - 1882 a;2 + 10695 x - 201474 = 0. 4. When one end of a beam of length I, and carrying a uniform load, is " built in " to a wall and the other rests upon a fulcrum, the distance x from the fulcrum to the point of maximum deflection is given by the equation Sx' - 9lx'' + l^ = 0. Find X for a beam 10 feet long. [Mekriam, Textbook on Mechanics of Materials, p. 1-53.] 8. Find the 5th root of 349 to 4 places of decimals by applying Horner's method to the solution of the equation x? — 349 = 0. 6. If the slope made by a line with the x-axis is a, the slope a; of a line whose angle with the a;-axis is one third as great is given by the equation x' — 3 ax^ — 3 x + a = 0. What is the smaller slope when the greater is 1 ? When the greater is 2 ? 7. A man invests $ 100 a year for 5 years in a savings association and at the end of the period his stock is worth § 560.25. The formula for computing the amount saved in n years is i where P Is the amount paid in each year and i is the rate. What rate of interest does he receive ? 8. The volume » of a spherical segment of one base is ^^^ (r-xy(x + 2r) 3 where r is the radius of the sphere, and x the distance of the base from the center. What is x when the segment is one half a hemisphere and the radius is 3 ? 148 COLLEGE ALGEBRA [XI, § 101 9. A rectangular beam 10 inches wide is used to brace a rectangular -12'- FlG. 35. frame whose inner dimensions are 10 by 12 feet. (See Fig. 35.) What is the length of the beam exclusive of the tenon ? 10. Bryan's equation for lateral stability of an aeroplane in horizontal steady motion, with coefficients determined by Bairstow for a machine of given dimensions, is M + 9.31 XS + 9.81 >? + 10.15 X - 0.161 = 0. Knd one positive and one negative root to 4 decimal places. [Technical Repoet, Bkitish Advisokt Committee foe AisKONAUTics.] 11. Bryan and Bairstow 's equation for lateral stability of an aBroplane in downward flight 1 — 6 with propeller cut off, is X* + 9.31 xs + 9.81 X2 + 10.25 X + 0.467 = 0? Find two real roots to 4 decimal places. [/6irf.] 12. The theory of symmetrically reinforced concrete beams depends upon the equation ^.3_3(|-|)^. + 12».l^=64i^2gy]. Find K to the second decimal place when c/ft = 0.6, a/A = .04, 'p =. .01, n = 15. [TuBNEAUBE AND Maubbe, PrincipUs of Beinforced Concrete Con- struction, 1912, p. 105 ] 13. If, in using Horner's method, a trial divisor should be too large, how could the fact be detected ? XI, § 102] POLYNOMIALS 149 102. Relations between the Roots and the Coefficients. 'To find the fundamental relations between the roots and the coefficients, the equation f{x) = may first be reduced to the form (19) f{x) = aj» + jji »"-! + P^x"-^ + ...+p„ = 0, by dividing both sides by Oq and then writing (20) ^=p„^=p„...,^=p^. For clearness let f(x) be of degree 4. By the factor theorem, it is evident that X* + pio? + p^x^ + p-iX +p^=(x- ri) (a; — r^ {x — r,) {x — r^. When the indicated multiplications are performed, the last identity takes the form, (21) !B* -frpio? +PiX^ +Ps?ii +Pi = x'^— (ri+ri+rs+Ti)!!^ + (Vz + Va + nr4 + nrs + r^n + r^r^x^ — {nriri + nr^r^ + Tir^r^ + r.^r^r^x -\-{rxr.{r^r^. An identity similar to (21) holds for every equation which has the form (19), whatever its degree may be. In general, pi is the sum of all possible products that can be found by taking the roots i at a time with the sign factor (— l)'. We may write therefore (22) P\ = - (n + ^-a H h »*n)= Sn, i>8 = — (n^an + n^an + •■• + nt-2»'»t-i»'.i) = — S^irgj-g The expressions Sri, 2>-i9-2 • • ■ which are abbreviations for the sums of the products, one at a time, two at a time, and so on, are defined by the equations in which they occur. 150 COLLEGE ALGEBRA [XI, § 102 EXERCISES 1. Give the sums of the products of the roots, one at a time, two at a time, and so on for each of the following equations. (a) a;S-5a;4 + 6a;8 + 7x2-3a; + 14 = 0. (c) aji" - 8 a;6 +' 17 =0. (6) 3x* + 7a;8-18a;2+ 14x-19 = 0. (d) x" -1=0. 2. Write out without multiplying the linear factors x— r, the equationo whose roots are as follows. (a) 1, 2, 3, 4. (c) 2, 2 + V3, 2 - \/3. (6) 3, - 2, 4, - 5. (d) 2 + 3 i, 2 - Si, 4 + i, 4 - i. 103. The Character of the Roots. Theokem. The imagi- nary roots of an equation with real coefficients occur in pairs of conjugates. The theorem asserts that if a + 6i is a root, a — Mia also a root. Consequently, [x—{a + bi)']{x — (a - hi)'] = {x - ay + 6^ must be a factor of /(a;). Pboof. To prove this fact, we may examine the remainder after dividing f(x) by (a; — o)^ + 6^. Clearly, the remainder cannot be of degree higher than 1. If it. be denoted by Sx +, T, (23) f{x) = l{x - a)^ + 6^] q{x) + Sx+T, ■where Qix) denotes the integral part of the quotient, and S and T are real numbers. By hypothesis, /(a + 6i) = 0, that is, [(a + bi - ay + &"] Q(a + bi) +S(a + bi) + T = 0. Since (a + 6i — ay + 6" = 0, we must have S(a + bi)+ T=Sa +T+Sbi = 0. But by § 87 if a complex number is equal to zero, the real and the imaginary parts are separately equal to zero. That is to say, Sa+T=0, and S =0. But ii S = 0, T= also, so that the remainder Sx+ T is zero, and the division is exact. The theorem is therefore proved. XI, 104] POLYNOMIALS 151 CoKOLLAET I. An equation of odd degree with real coeffi- cients has at least one real root. CoBOLLAKY II. Any rational integral function with real coefficients may be broken up into real factors, all of wfiich are of the first, or of the second, degree. 104. The Graph of f{a>) when Some Factors are Imaginary. Since imaginary roots occur in pairs of conjugates, f(x) will have a quadratic factor {x — ay + b^ corresponding to every such pair. Moreover, according to § 66, the sign of this quadratic factor never changes. Every such quadratic factor there- fore diminishes the number of Later- sections with the a;-axis by two. Usually, but not always, the graph has an elbow corresponding to each pair of imaginary roots, as in Kg. 36. Geometrically, these considerations become almost self-evident if we imagine the a>-axis to be lowered (or the curve raised) in Fig. 33 until the point B lies above the axis. The curve ia Fig. 36 is identical with that of Fig. 33, but each of its points are three units higher than the corresponding point in Fig. 33. Fio. 36. EXERCISES Sketch roughly the graph of each of the following functions, (a) f(x) = (x-l){x-2){x^ + x+l). (6) f(x) = (_x- 3){x^ - 4:x + 13). (c) f(x) = (s - 3)2(a;2 -ix+ 13). (d) fdx) = (a;2 -I- X -I- l)(a;2 - a; + 1). (e) /(a:) = (a;2-Ha;-|-l)(a;2-x + l)(a:2-4a;-|-13). 152 COLLEGE ALGEBRA [XI, § 105 105. The Number of Real Roots. Descartes's Rule of Signs. Sturm's theorem, to which teferenee was made in § 96, is the only method known for the determination of the num- ber of real roots of a numerical equation. There is, however, a simpler method due to Descartes which, in every case, gives an upper bound to the number. Any change in sign between two consecutive terms of f{x) is called a variation. Thus x^ + 3 a? — 2 a:? — x + 13 contains two variations, one from + to — and the other from — to +. Descaetes's Rule. An equation f(x) = cannot Jiave more real positive roots than there are variations of sign in f{x), nor more real negative roots than there are variations of sign in /( —x). Suppose the product of all quadratic factors corresponding to pairs of conjugate roots is a function for which the sequence of signs is + + - + H + + +. To iatroduce another root r, which is real and positive, it is necessary to multiply the function by a; — r, for which the sequence of signs is H . The multiplication of signs may be carried out as follows : + + - - + H + + + + - + + - - + + + + + - + + + +±-+±-±+±±- In the product the double sign indicates that the sign is in doubt. It is not difftcult to see that the sign ia the last line which stands directly underneath the second sign of any variation, will be the same both in the multiplicand and the product. It follows, therefore, that if we consider only those signs up to and including the second sign of the last variation, the number of variations in that part of the product standing XI, § 105] POLYNOMIALS 153 directly underneath will be at least as great. But one varia- tion is iatroduced in the remaining terms, because the last sign of the product is different from the last sign of the multiplicand. Therefore, at least one variation has been introduced. The conclusion just reached holds whatever the sequence of signs in the multiplicand may be. Since the introduction of each real positive root increases the number of variations by at least one, it follows that the equation cannot have more real positive roots than it has variations of sign. To prove the second part of the rule it is only necessary to note that the roots of /(—») = are the negatives of the roots of/(a;) = 0. (§100.) Example. The equation f{x) =3^ — Sx^-5a?-x + l=Q cannot have more than two positive roots ; and since it cannot have any negative root. It must, therefore, have at least four imaginary roots. EXERCISES 1. Prove that the equation a;^ — 1 = has only one real root. 2. Prove that the equation a;* — 3 x^ + 1 = has at least two complex roots. 3. Prove that the equation a;» — 1 = has n — 1 complex roots when n is odd and m — 2 when n is even. 4. Prove that a;» + 1 = has no real root when n is even, and one, when n is odd. 5. What is the number of real roots of the equation a;6 + x* + a;3 + a;2 4. j; + 1 = ? [Hint. Increase the degree by introducing the root 1.] 154 COLLEGE ALGEBRA [XI, § 105 6. What is the number of real roots of the equation xo + x^ + x* + x« + x^ + x + l = ? 7. Generalize the results in Exs. (5) and (6). 8. The formula for the amount of an annuity is i and when S, a, a;nd n are known the formula becomes an equation for the determination of the rate of interest. By substituting a for 1 + i, prove that the equation has only one root that can have any significance in financial problems. 9. Prove that if the coefficients ot f(x) are alternately positive and negative, the equation /(k) = has no negative root. 106. The Highest Common Factor of Two Polynomials. Let us consider two polynomials f{x) = aoaj" + aia!"-! + a^^af^' H 1- a„, (x) = b^x" + bix"-' + bj^x"-' + ••• + 6„, whose degrees in x are m and n, respectively. Let us suppose that m ^n. Only in exceptional cases can these polynomials be factored. The method for finding the highest common factor given in § 20, therefore, fails, and some other must be sought. If f{x) be divided by <^{x), the quotient will be made up of two parts, one an integral part Qiix), which is a rational inte- gral function of x, and the other, a fractional part whose numerator Bi{x) is a rational integral function whose degree is an integer less than the degree of the divisor (x). The re- sult may be expressed in either of the two forms (24) Z^ = Q^(a,)+:?iM, or f{x}= Q,{x).l>(x) + Ry{x). XI, § 106] POLYNOMIALS 155 Similarly, if (x) be divided by Bi(x), the remaiader iZaC:") "'^iH be of degree less than that of Bi{x). This process may be carried on until a remainder is reached which is either zero or a constant. The result? may be expressed in the following identities, which taken together are called the Euclidean algorithm : f f(x) = q,{x){x) = Q^{x)R,{x) + R^{x), E,{x) = Qi{x)Ri{x) + R^x), (25) [ Ri_^{x) = Q,ix)R,_i(x) + R,(x). From the first identity of the Euclidean algorithm, it is clear that any divisor of f(x) and <^(a;) is a divisor of Ri(x) and from the second, that this same divisor is a divisor of RJix), and so on. It is clear, therefore, that when Ri{x) = 0, Ri_i{x) is the highest common factor, and that when Ri(x) =^ 0, no high- est factor, apart from a constant, exists. It is important to notice that Euclid's process for finding the highest common factor involves only operations leading to ratimial results. Furthermore, the result will not be affected, if at any stage of the work the dividend be multiplied or divided by any constant. EXERCISES 1. Find the highest common factor for each of the following pairs ,of polynomials. (o) K* - 3 a;8 + a;2 + 4 and 4 a;^ - 9 a:^ + 2 a;. (6) a;8 - 1 and aj8 - 6 a; + 11 X - 6. (c) xs - 5 a;2 + 17 a: - 13 = and x* — 3 x^ + 10 x^ + 9 x + 13. (d) X* - 16 and x* + a^ + 5a;2-(-4x + 4. (e) X* — 16 and x' — 6 x^ + 11 x — 6. (/) x«-16 and x8-8x2 + 17x- 12. 156 COLLEGE ALGEBRA [XI, § 107 107. Roots Common to Two Equations. Multiple Roots. Let fix) =0 and <^{x) = be two equations of degrees m and n. Then, f{x) = a„(x - r,)(x - rj) - (x- rj, 4,{x) = \(x - r\){x - r\) - {x - r'„), where Vi, r^, •••, r„ are the roots oif(x) = 0, and r\, r'2, — '"'„ are the roots of (x) = 0. If a certain number of roots are common to the two equar tions, the corresponding linear factors are common to f(x) and 4>(x). It follows, therefore, that the highest common factor of f(x) and (x) will contain those, and only those, linear factors which correspond to common roots. If g{x) be the highest common factor, the common roots may be found by solving the equation g(x) = 0. An important application of the foregoing paragraph is the , determination of multiple roots of an eqp.ation, that is, roots which correspond to linear factors that are repeated. In the calculus, it is shown that a multiple root of an equar tion/(x) = is common to the two equations (26) f(x) = 'a^x" + aix"-' + a^x'^-' + • • • + «„-ia; + a„ = 0, (27) /'(») = na„a;"-' + (m-l)oia;"-=4 (n-2)a2a;"-^H |-a„-i= 0. The function/'(a;) is called the derivative of /(»). It is formed from f(x) by multiplying the coeificient of each term of f{x) by the exponent of x and then diminishing that exponent by 1. Example. If f(x)=afi-6x^ + 8x-4, then f'{x =3x^-Wx + S. The highest common factor of /(x) and/'(x) is x — 2. Therefore x = 2 is-a multiple root of /(x) = 0. Since (x — 2)^ is a factor of /(x) , the re- maining linear factor is easily found to be x — 1. The roots of the equst- tion are therefdre 2, 2 and 1. XI, § 107] POLYNOMIALS '157 EXERCISES 1. rind the roots common to the following pairs of equations, (a) x^-Sx^ + x'' + i = and ix^ -dx'^ + 2x = 0. (6) k" - 5 x2 + 17 K - 13 = and a^ - 3 xS + 10 a;2 + 9 a; + 13 = 0. (c) a:<— 16 = and x^ + x^ + bx^ + ix + i = 0. 2. Examine the following equations for multiple roots, (a) a;3 _ 9 j;2 ^. 24 a; - 16 = 0. (6) X* - 2 x8 - 11 x2 + 24 x - 12 = 0. (c) kS - a^ - 24 xs + 24 x2 + 144 x - 144 = 0. {d) x6 _ 5 a;5 + 8 a;4 - 5 x3 + 5 x2 - 8 X + 4 = 0. 3. Prove that an equation of the form x" ± a = cannot have a mul- tiple root. CHAPTER XII DETERMINANTS AND LINEAR EQUATIONS 108. Definitions. Determinants of orders two and three were defined in §§ 70 and 73. A determinant of order n is a square array of n^ elements, which is interpreted to mean the algebraic sum of all possible products that can be found by taking one, and only one, element from each row and each column, with the signs of the terms determined according to the rule of signs explained below. A determinant of fourth order is written in the form, «! 6i Ci di a2 bi C2 di as 63 Cs ds Ui bi Ci di (1) or, briefly, {aJiiCadi). Any term of this determinant will be made up of factors a, b, c, d, with subscripts all different, since, if two letters were alike, the product would contain two elements from the same column, and if two subscripts were alike, two elements from the same row. When tjie letters are written in the normal order, that is, the order in which they occur in the alphabet, the subscripts will occur in every possible order. An inversion from the order of the natural scale occurs whenever a digit stands before another that, in the natural scale, precedes it. Thus, in the term 016403^2, 4. stands before 3 and before 2, and 3 stands before 2. There are then three inversions in the term. 158 XII, § 108] DETERMINANTS 159 The E.ULE of Signs. Wlien the letters are arranged in the normal order, the sign of a term is plus or minus according as the number of inversions of subscripts is even or odd. By the rule, the sign of the term ajbfiid^ is minus, while the sign of the term a^^c^d^ is plus. The algebraic sum of the terms written out according to the definition, is called the expansion of the determinant. Since the subscripts will occur in every possible order, the number of terms is equal to the number of possible arrange- ments or permutations, of the subscripts. In a later section it will be shown that this number is n\, where n ! is defined to be the product of the n integers 1, 2, 3, — , n. Thus, the num- ber of terms in the expansion of a determinant of order 3, is 6 ; of order 4, is 24 ; of order 5, is 120. EXERCISES 1. Write out the expansion of the determinant ai 6i Cl di aa 62 2+ diDi + dj)i. Similar statements are true for a determinant of any order. The numbers Ai, A^, A^, A^, Bi, -Bj ••• a^re called cof actors of the corresponding elements a^, a^, a,, a^, 61, 62 — ■ A closer examination of the cofactors yields valuable in- formation. The cof actor Ai of Oi is made\ip of products of the letters b, c, d, with the subscripts taken in every possible order, and with the sign determined by the order of the sub- scripts 2, 3, 4, since the subscript 1 is always first. But this combination of terms is exactly the determinant h Ci d^ 63 C3 d. h Ci d. which is pbtained by striking out the first row and the first column of A. To find the meaning of A^, the first and second rows of A may be interchanged in order to bring a^ to the left-hand upper corner of the determinant. The terms which are multi- plied by Oj 8're then seen to be exactly the terms in the ex- pansion of the determinant, h Cx d. h C3 d. 64 C4 d. But considered as belonging to the original determinant A every sign is changed since two rows have been interchanged. Consequently A^ is the negative of the determinant just written. 6i Cl di 61 Cl di a., C2 cf2 62 C2 d, ,^4 = - 62 C2 <^2 ,5i = - fflj C3 da &4 C4 d, (/■) Co ttg a^ C4 (^4 XII, § 110] DETERMINANTS 165 In a similar way it is shown that (7) ^3=4- A determinant obtained by striking out a row and a column of a determinant is called a complementary minor, or simply a minor, of the element that stands at the intersection of the row and the column stricken out. A minor is denoted by the letter A„ with the element to which it is complementary written as a subscript. Thus, A„ , A„ , ••• are the miuors of ai, a^, •■•. From the foregoing it is clear that the cofactor is equal numerically to the minor, but that as we proceed from the left- hand upper corner, along either rows or columns, the alternate minors, beginning with the second, differ in sign from the cor- responding cofactors. The rule is that the minor corresponding to the element in the ith row and the Mh column is equal to the cofactor, or to the cofactor with its sign changed, according as i + k is even or odd. The reason for the rule is easily seen to be the fact that the sign of the origiual determiaant is unchanged or not, by bring- iag the element ia the ith row and A;th column to the upper left-hand corner, according as the number of interchanges I -)- & is even or odd. Usiag the minors instead of the cofactors we have for the determinant (1) of fourth order (8) A = aAa, - a2^a, + ai\^ - a^a, = - bi\ + b~i\ - hs\ -f bi\, = CiA,^ - C2\ + C3\ - c,\ = - di\ + d2\ - dsA,, + dAic 166 COLLEGE ALGEBRA [XII, § 110 In this way we have 2 n different expressions for a determinant in terms of its minors, or of its cofactors. Almost as important as the foregoing expressions for a deter- minant, are the identities that exist between its elements. Thboeem 6. The sum of the products of the elements of a row {or column) each multiplied by the cofactor of the corresponding element of another row {or column) is identically zero. For example, siace A = a^Ai + a^Ai + a^A^ + 04,44, the expression &i^i + 62^2 + 63^3 + M4) is a determinant obtained from A by replacing Ox, Oj, ... by 61, 62, ..., and, consequently, has two columns identical. Therefore (9) 61^, + 62^2 + 63-43 + 64-44 = 0. Many other identities of this type may be written out. There are altogether 24 of them for a determinant of order 4. 111. Computation of Determinants. The computation of a determinant whose elements are known numbers is best illustrated by concrete examples. Example 1. Compute the determinant 9 13 17 4 18 28 33 8 30 40 54 13 24 37 46 11 If from the first, second, and third columns, twice, three times, and four times the last column be subtracted, according to Theorem 5, Cor. I, the result will be XII, § 111] DETERMINANTS 167 A = 1 1 4 1 1 2 4 2 1 1 1 1 2 6 2 3 The last determinant is obtained by subtracting the sum of the first three columns from the fourth. Subtracting the third column from the other three in turn, and then noting that the sum of the products of the elements in the first row by the corresponding minor contains a single term, we have A = 1 3 2 -1 2 1 3 2-1 2 The last determinant is reduced to a determinant of order 2 by sub- tracting twice the first row from the second, and then striking out the first row and the first column. We find 13 A = 2 0-7 4 2 1 Example 2. Compute the determinant 5-10 11 = -15. A = 4 2 -6 - 10 - 11 12 11 12 - 11 4 2 Adding the sum of the second and third rows to the fourth, and sub- tracting twice the third from the second, we find 5 -10 11 -32 -35 34 11 12 -11 2 1 5 3 Adding the first row to the second, then subtracting 8 times the first from the third, we find A = = -2 5 -10 11 -82 —35 34 = -10 1 5 3 5 -2 1 -32 -7 -1 118 A=+ 10 27 -39 -2 1 3 1 9 = 90 -39 17 17 = 8100. 168 COLLEGE ALGEBRA [XII, § ni In the foregoing, the orders of the determinants have been reduced by bringing them to such a form that all elements but one in a row or a column ^.re zero. This process may always be accomplished by multi- plying .the elements of the columns, say, by such factors that the elements in a row will be replaced by a common multiple. All factors introduced in this way must be taken out again by division. For example. 2 3 1 ^ 6 6 6 1 3.2 5 7 8 6 3 4 ■_ 1 ^3-2.6 15 14 48 18 6 24 1 1 _ 1 3.2 1 12 1 14 34 6 18 = 1 1 3.2 12 15 14 48 18 6 24 341 18 34 1, 3 EXERCISES 1. Compute the values of the following determinants. 6 12 25 1 5 3 4 2 (a) 8 20 25 (0 5 4 2 3 10 20 30 3 6 7 3 4 5 6 6 3 (b) 4 5 6 3 5 6 3 4 6 3 4 5 (/) 10 10 10 1 2 5 7 8 12 3 4 3 4 6 3 (c) 5 6 7 8 9 10 11 12 13 14 15 16' (g) a h g h b f- 3 5 7 9 g f c (d) 3 6 9 7 a — X h g 4 17 8 w h b-x f 3 8 7 4 g f C— X 2. Express the determinant aS 68 cs (a + X)8 (6 + X)8 (C + X)8 (2a + X)8 (2 6 + X)a (2 c + X)8 • as a po ynomial in \. • [Salmon, I ligher Alg ebra.'\ XII, § 111] DETERMINANTS 169 3. Prove that ai 61 ci 11 1 02 62 C2 = O261C1 biCiOl C2O161 fl!3 63 C3 ^ ^ '■ O361C1 fesCifli CsOifti Show how the result can be utilized in the computation of determi- nants whose elements are known numbers. 4. Prove by the factor theorem that x — a and x — p are factors of 1 1 1 X a p x^ a? ^' 5. Prove that 111 a /3 7 as ^8 ^8 = (^ - t)C7 - a)(« - i3)[i.a + MP + JV7]. [HiHT. Use the method of Problem 4, then note degree of determinant and the coefficients of three terms suitably chosen.] 6. Find the expression for the determinant 1111 a jS 7 S «2 ^ 72 32 «» i8s ^ as 7. Find the expression for the determinant 1 1 1 a p y a* ^* 7* [Hint. The result must be of degree five and can change sign only when any two of the numbers a, /3, 7, are interchanged.] . 8. The area of a triangle whose vertices are the points (xi, 2/1) , {Xi, 2/2), (xs, 2/3), is given by the formula xi 2/1 1 Area = \ x^ y^ 1 • Ks ys 1 Find the area of the triangle whose vertices are at the points (3, 2), (5, 6), (4, - 7). 170 COLLEGE ALGEBRA [XII, § 111 9. The volume of a tetrahedron whose yertioes are at the points (ki, J/i, zi), (a;2, !/2, za), (%, J/s, zs) , and (xi, yt, zt), is given by Volume = \ a;i, Vu zi, 1 X2, 2/2, Z2, 1 Ks, 2/3, Zs, 1 X4, 2/4, Z4, 1 Find the volume of the tetrahedron whose vertices are at the points (0,0,0), (1,0,0), (0,1,0), (0,0,1). 112. Application to Linear Equations. Let a^oc + b^y + c^z + d2W = ejj Wja; +632/ + CsS + e^jW = 63, . a4a; + 64 j/ + c^z +diW = 64, (10) be a system of four linear ecLuations in four unknowns. The determinant, is called the determinant of the system. If the first equation be multiplied by Ai, the second by A^, the third by A3, and the fourth by A^, where the A's are cofactors, and the resulting equations added, the coefficient of x will be exactly equal to A, by (6), § 110. On the other hand, the coefficients of y, z, and w will be zero, by (9), § 110. We have, therefore, A a; = e^Ai + 62-^2 + ^3^3 + e4^4- , The expression on the right is A with e's substituted for a's. Consequently (") ei 61 Ci di e-i 62 Cj d^ 03 63 C3 <^3 C4 64 C4 d. XII, § 112] LINEAR EQUATIONS 171 Similarly, (12) «1 «1 Cl d. «2 ^2 C2 a. «3 63 cs d. «4 ^4 ^4 d. «i 6i «! t?i a^ 62 ea da as 63 63 e?3 «4 64 C4 ^4 «i 61 Cl «! Wj 62 C2 ^2 as 63 O3 63 "4 64 C4 C4 AAA This solution of the system is simply an extensioh of Cramer's rule (§§ 70 and 73) to four equations. Clearly, the method applies to any number of linear equations, provided the number of unknowns is equal to the number of equations, and the determinant of the system is not zero. EXERCISES 1. Solve the following systems of equations. (a) (6) (c) (d) • !>; + 2/ + z = 6, x+y-2z + w=10, Zx-y + 2^ = 1, (e) 2x— y + z — vi = — 7, .4a; + 32/-s = 7. x + Sy + 2z-w = 0, 3a; + 2/ + 3s'+2M) = 19 3 a; + 4 2/ - 5 2 = 32, ■x + y = 4:, 4a;-5!/+3z = 18, (/) y + z = 6, .5a;-3?/-4z = 2. .z + x = 8. x + y = l, x-Qy + 30-lOto= -102, (9) y + z = 2, 2x-\-1y — z— w = 6\ z+w = S, Sx + y + 5z + 2w = 10, w + x = 2. Ax — 6y-2z-9w = -67. X y ■-x + y + z + w = a, (.h) Ui=6, x — y + z + w = b, y z x + y — z + w = c, i + Uc. ■ x + y + z-w = d. . Z X Find the coefficients A, S, O, and D, which will make the curve y = Ax« + Bx^ + Cx + D through the points (0, 1), (1, 1), (2, 0), (3, 1). 172 COLLEGE ALGEBRA [XII, § 113 (13) 113. Homogeneous Linear Equations. A system of linear eqiuations in which every absolute term is zero is called a homogeneous system. A system of four homogeneous linear equations may be written as follows : ' a]X + biy + Ci* + ditv = 0, UiX + hiV + CiZ + d^ = 0, ttiX + biV + C3» + d-iW — 0, . a^as + biV + C4« + d.iW = 0. Let A = (afi-fi^d^ be the determinant of the system ; then by Cramer's rule, (14) Ao; = 0, Aj/ = 0, ^z=^Q, Aw = 0. Therefore, if A ^ 0, we must have (15) a; = 0, 2/ = 0, «=0, w = 0. This solution, which might have been obtained by inspection of the system (13), is of little importance and is called the trivial solution. It is also clear from (14) that no solutions other than the trivial solution can exist unless A = 0. This demonstration applies to n equations in n unknowns. Therefore the necessary condition that n homogeneous linear equations in n unknowns have a solution other than the trivial solution, is that the determinant of the system shall he zero. The determinant Oi h Cl c? Oi h C2 d, as h C3 d, a-i 64 C4 d. is called the eliminant of the homogeneous system (13). 114. The use of Eliminants in Geometry. Suppose, for example, that it is required to find the equation of a straight line which passes through two given points (xi, y^,) and (ajj, 2/2). XII, § 114] LINEAR EQUATIONS 173 According to § 49 tlie equation of the line has the form (16) Ax + By+C:=0 where A, B, and C are undetermined coefficients. Since the coordinates of the points {x^, y^, and {x^, y^, satisfy equation (16) we have the system \Ax +By +0=0, (17) \Ax, + By,+ C=0, [Ax2 + By2 + C=0, from which to determine A, B, and C. The values of A and B in the terms of C might be determined from the last two equations of the system (17). These values could be inserted in the first equation, and the required equation would then be obtained by dividing by C. The same result may be obtained directly by looking upon the system (17) as a system of equations homogeneous in A, B, and G. According to § 113 if A, B, and C are not all zero, we must have (18) X y 1 Xi 2/1 1 X2 2/2 1 = 0. Equation (18) is the equation of a straight line since it is a linear equation in two variables. Moreover, it is satisfied by the coordinates of both the points (xi, y{) and {x2, 2/2), since, if the coordinates of either point be substituted for x and y, the determinant will vanish identically. Example. The equation of the line through the two points (2, 3) and (3, 5), Is X y I 2 3 1=0, 3 5 1 2a;-2/-l = 0. 174 COLLEGE ALGEBRA [XII, § 114 EXERCISES 1. Assuming the equation of a plane to be of the form Ax+ By+ Cz + D = 0, find the equation in determinant form of the plane through the three points (sci, yi, zi), {x^, y^, sOi »"4y=- «! &1 di «2 &2 d2 ag 63 da tt] di Cl ctj d2 Cj 6(3 da ■C3 * It is assumed in the proof that the determinant (oiftjCa) =^ 0, though this is not necessary. (See Bochbb, Higher Algebra, The Macmillan Company, p. 47.) XII, § 117] LINEAR EQUATIONS 177 But the determinants on the right become, after suitable inter- changes of columns, exactly A-i, £4, O^, so that (23) D^x = AiW, D^y = B^w, D^z = dw, or (23a) Ai'- s w 117. Two Equations having a Common Root. Let the two equations with one unknown be Multiply the first by a;^ and by x, and the second by x. In this way five equations are obtaiued, a^* + ai3? + a^x"^ = 0, a„a?' + a^x'^ + a^x = 0, , a„aj' + aiX + aj = 0, h^ + hiO? + hiX^ + h^x = 0, 6„a;3 + 6ia;2 + h^x + 63 = 0. If these five equations are looked upon as five homogeneous equations with the unknowns X^, sfi, x^, x, 1, the necessary con- dition that they have a common solution is, by § 113, (25) 61 «0 &1 aj ai Oi 63 62 &3 =0. The determinant in equation (25) is called the resultant, or the eliminant, of the equations (24). The resultant of two equa- tions of degrees m and n is a determinant of order m + n. Proofs of the sufficiency of the condition may be found in Fine, College Algebra, and in Dickson, Elementary Theory of Squations. 178 COLLEGE ALGEBRA [XII, § 117 EXERCISES 1. Show that the condition that the equations aox + ai = 0, and bax + 6i = 0, have a common root is ao6i — ai6o = 0. 2. Find the condition that two quadratic equations aax!' + aia; + 02 = and box!' + 6ia; + 62 = shall have a common root. CHAPTER XIII THE BINOMIAL THEOREM 118. Statement of the Theorem. The binomial theorem is a theorem which enables us to write out at once any power of a binomial. For a positive integral exponent n the theo- rem may be expressed by the following formula : (1) (a; + a)« = a:'«+7£c»-i«+^^^iff^a!"--^a2 Proof. For n = 2, (x + a)2 = a;2 + |a;a + ^^^^^a^ = a;2+2 m + a'. Similarly, for n = 3, (x + 0,)? = sfi + Sic^a + ^(^ ~ ^) a;a2 + ^^^~^)(^~ ^) a' "■ ' ^ 1.2 1.2-3 = iB3 + 3 a;2a + 3 a;a2 -f a^, which again agrees with the result of direct multiplication. In this way we are led to suspect the truth of the theorem as stated. Verification is obviously impossible for more than a few values of n. The proof for any positive integral exponent is obtained by a method called mathematical induction. By this method it is shown that if equation (1) is true for any particular value of n, it is true for the next larger value ; and, consequently, since it is known to be true for one or two small values, it is true for every positive iategral value of w. 179 180 COLLEGE ALGEBRA [XIII, § 118 Suppose the theorem is true for n = k. The hypothesis is then (2) (a; + ay= x''+ kx'^^a + ^^^^ i^'^a^ ^ ^ 1..2-3-(r-l) If both sides be multiplied hj x + a, the result is (x + a)'+i = «;'+' + Jcx'a + ^(^~-^) x''-'a^ + - 1 " ^ fc(fc-l)-(fc-r + 2)^,, ... , ^ 1 . 2 • 3 •■. (r - 1) 1.2-3 - (r -2) + - + a*+\ When like terms on the right are combined, it will be seen that the coefficient of a;*a is A; + 1 ; the coefficient of a;*-ia2 is ^(?'~-'^) + A; = (^ + 1)^ ; \ • Ji 1. * Ji the coefficient of a;'"+^a'~' is fe(fc-l) - (fc-r + 2) A;(fc-1) - (fc-r + 3) 1 . 2 . 3 - (r - 1) 1 . 2 . 3 ... (r - 2) ' which reduces to (^ + W-l) •;• (fe- r + 3). 1 . 2 . 3 ■■■ (r - 1) Moreover, k,k — l,k — 2, •■■ may be written in the forms A; + l — 1, fc + 1 — 2, fc + l — 3, ■••. Consequently, (a; + a)*+' = x"-^' + (fc + 1) x^+'-^a + (^ + 1) (^ + 1 - 1) ^i-n-2a2 1 ■ ^ + ... , (k+l)ik+l-l)(k+l-2) ... (k + l-r + 2) „„-,«.^r ■^ "^ I.2.3... (r-1) + - + a*+^- The last formula is exactly like (2) except that everywhere & in (2) is replaced by fc + 1. In other words, if the theorem holds for n = k, it holds also for m = fc + 1. This proves that XIII, § 119] THE BINOMIAL THEOREM 181 if the theorem is true for one value of n, it is true for the next larger value of n. But it is true for n = 3, then for n = 4, then for n = 6, and so on. It is therefore true for any positive inte- gral value of n, as was to be proven. The expansion of (x — a)" is found directly by writing it iu the form [a;+(— a)]". It follows that all the terms contain- ing odd powers of — a, that is, the first, third, fifth, powers and so on, are negative, while all other terms are positive. 119. The Number of Terms and the «.th Term. When n is a positive integer, the number of terms in the expansion o, (a; 4- a)" is n -|- 1, since the first term, which does not contain a, is followed by the terms contaiaiug the first n consecutive powers of a. The n -I- 1 numbers /q-, 1 n n(n — 1)- n(n — l)(n — 2) ^"^^ ^' T' 1-2 ' rT2T3 ' •••' n(n — l)(w — 2) ••• {n — r + 2) ,^. ^ l-2'3 — (»--l) ' * ' ' are called the binomial coefficients, and are frequently denoted by the symbols (*) nCof n^i, nCg, •••, nf^r> '"> tiCn,. The rth coefficient is „C,_x, and the rth term is (5) „(7,_ia;"--«a'-^ ^ n{n-l){n-2){n-Z) -{n-r + 2) ^„-r+i^r-i 1 . 2 ■ 3 - (r - 1) The product of the n consecutive integers beginning with 1 is called factorial n, and is written n !, or [w. By definition the denominator of the rth term is (r — 1) ! If both numerator and denominator be multiplied by the prod- uct of the factors (n — r + 1), n — r, n — r — 1, —, '6, 2, 1, 182 COLLEGE ALGEBRA [XIII, § 119 that is, by (n — r + l) I, the numerator becomes the product of the n integers from n down to 1 inclusive. It will then be n ! The coefficient of the j-th term may then be written in the form, n — Vil " "' (r-l)!(n-r + l)!' and the expansion itself takes the form, (, + «)» = «,» + _AL_ «,»-.« + _!^^.-v + ... + (.- 1) ! (I -.+1)! '^^"^""^" +•••+""• The most important property of factorial n is expressed by the identity nl = nx (n — 1) ! 120. Pascal's Triangle. The following triangular array of numbers formed by adding two adjacent nurabers in a line and writing the sum under the one on the right is called Pascal's Triangle: 1 1 1 12 1 13 3 1 14 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 The numbers in the first few horizontal lines of the triangle are seen at once to be the binomial coefficients of (x + a)", (x + a)', (x + of, and so on. That the numbers in any line are the co- efficients of a power of x + a is easily shown by adding two consecutive numbers in a line, say „Cr_i and „C„ and show- ing that the sum reduces to n+iO,.. The proof is left to the student. (e) (x^ - 2 ax + a2)*. V^y (r) (x2 - 2 ax + 2 a^)^: XIII, § 120] THE BINOMIAL THEOREM 183 EXERCISES I. Expand the following binomials. (a) (x + yy. (J-) (^a-v'6)6. (6) (x-y)^. (^-j (1.03)2. (c) (x-2 2/)6. (;(.) (t.025)2. W (a -3 6)5. (;) (1.01)4. («) fx+iy (») (53)^- ^ */ (n) (69)2. (7) (x-2Vi;)o. (o) (84.)2. (Sr) (a^ + Sftci)". (p) {x+[y+z})*. V V9/ 2. Write down the 11th term of (a + xy^. 3. "Write down the 12th term of (z — xy. i. Write the middle term of (x — 2!/)i8. 5. Prove that the coefficient of a^b'c' in the expansion ,of (a+ft + c)" is nl/(r!sU!). [Hint. The ({ + l)st term of ([a+6] + c)" is — (a + 6)"-2 = 1 + iOl + ia- 1) (.01)2 + JQ _ 1) (^ _ 2) (.01)8+ ... = 1 + .005 - .0000125 + .0000000625 + .- = 1.0049875625.... This result is correct to the 8th decimal place. EXERCISES 1. Compute the numerical values of each of the following quantities. (a) (1.02)V2. , (6) (1.02)1/4. (c) _1_. 2. Extract the 6th root of 70 by finding the value of (64 + 6)1 6 = 641/6(1 + ^^)V6_ 3. The formula A = A{1 + i)" for compound interest is assumed to hold for n less than one year. Find the compound interest on 1 1000 for one inonth at 4 per cent. Compare your result with the simple interest on the same principal for the same time at the same rate. 4. The expression p[(l + ty/p — 1] where i is rate of interest andp the number of times interest is compounded in a year, is important in the theory of investment. Find its value when i = .04 and p = 4. 6. "When the rate of interest Is i, the present value of a sum due n years hence without interest is .4(1 + i)-». Find the present value of 1 1000 due in 5 yr. without interest if money is worth 5 -'-) i'- ■?' and m 4, -2, i. -i- the progression is said to be decreasing. As in arithmetic progressions, there are five fundamental numbers to be considered, namely, the first term Ui, the ratio r, XIV, § 128] PROGRESSIONS 189 the number of terms n, the nth term a„, and the sum of n terms s„ . With this notation the progression %, a2, Ojz, «4, •••, may be written in the form ai, ajT, ttir", air',---, where the general, or type term, has the form a-if*. 127. The nth Term. By inspection, we observe that the wth term of the progression is given by the formula (6) an = ain^-^. That this formula is correct, may be proved rigorously by a mathematical induction similar to that employed in finding the formula for the nth term of an arithmetic progression. 128. The Sum of n Terms. If we multiply both sides of the equation s„ = tti + a^r + a^r' + — + fti?'""-' by the ratio r, we obtain a second geometric progression whose sum is rs„ = a^r + a^r'^ + — + a^r^-'^ + a^-r. When the first of these equations is subtracted from the second, all terms on the right disappear except a and ar". The result- ing equation is (r - l)s„ = aiJ-" - ai. Consequently, (6) Sn = -^ —^^ This formula may be put in a somewhat more convenient form by replacing ar^ by its value ra„, which is readily ob- tained by multiplying both sides of (5) by r. The value of s„ is then given by the formula {&a) sn = ——--' 190 COLLEGE ALGEBRA [XIV, § 129 129. The Solution of Problems. All problems arising in geometric progressions may be solved by meai^s of formulas (5) and (6), or (5) and (6 a), looked upon as a pair of simul- taneous equations in two unknowns. Problems requiring the number of terms lead to exponential equations. EXERCISES 1. Find the 10th term of the geometric progression 1, 2, 4, •••. 2. Find the 10th term of the geometric progression 1, — 2, 4, .-.. 3. Find the 12th term of the geometric progi'ession V3, -3, 3V3, •••. 4. Find the 7lh term of the geometric progression 2 - VJ; 2 V3 - 3, 6 - 3 V3, .... 6. Find the nth term of the geometric progression a, a(l + i), a(l + i)2, -. 6. Find the nth term of the geometric progression 1, J, J, —. 7. Find the sum of 7 terms of the geometric progression 1+2 + 4 + .... 8. Find the sum of 7 terms of the geometric progression 1+1 + U-. 2 4 9. Find the sum of n terms of the geometric progression 1+1 + 1+-. 2 4 10. Find tlie sum of n terms of the geometric progression a + a(l + 0+a(l + i)^+ -• 11. Find the sum of n terms of the geometric progression a + a(l + iy- + a(l +i )-■' + -. 12. Insert 5 geometric means between 2 and 6. 13. Insert 5 geometric means hetweens 2 and 3. 14. Given ai = 5, os = 78125, m = 6. Find r and se. 15. Given ai = 5, se = 97655, n = 6. Find r and ae- 16. Given a = J, r = |, n = n. Find o„. 17. Find the sum of n terms of the geometric progression 18. Given r, n, and s„. Find a„. XIV, § 130] PROGRESSIONS 191 , 19. Given r, n, and a„. Find s„. 20. Given,- ai, r, and a„. Find n. 21. Prove that if a set of numbers form a geometric progression, their kth. powers form a geometric progression whose ratio is «■'. 22. The sum of three numbers in geometric progression is 221 and the last exceeds the first by 136. What are the numbers V 23. Find the sum of the geometric progression ^r-iy + (r^-L^%{r^-r^\ ...|,n_lj^ [Bourlkt.] 24. Prove that if a set of numbers form a geometric progression, their logarithms form an arithmetic progression. 25. Find the sum of n terras of the geometric progression a;"-! _ x"-2y + a;»-S{;2 + •••• 26. Find the sum of n terms of the geometric progression 27. From Exs. 26 and 27 deduce the theorems for the divisibility of X" + y" and X" —y". 130. Infinite Geometric Progressions. If an air pump whose cylinder contains 1 liter be attached to a vessel contain- iag 10 liters, the first stroke of the piston will remove one tenth of the air, leaving nine tenths behind. The air left behind expands immediately to fill the whole space, so that the second stroke removes one tenth of the remainder, and so on. If the weight of the air at the beginning of the experi- ment be denoted by 1, the weights of air in the vessel at the beginning of each successive stroke will be 1, .9, .81, .729, .6661, -, and the weights removed will be .1, .09, .081, .0729, .06561, -. Clearly, both of these sets of numbers are geometric pro- gressions with ratios equal to .9. Theoretically, the total weight removed after the mth stroke will be the sum s„ = .1 -H .09 -I- .081 -I- .0729 + -, to n terms. 192 COLLEGE ALGEBRA [XIV, § 130 By (6), § 128, this sum is Suppose, now, that this process is carried on indefinitely. As n increases (.9)" becomes smaller and smaller, and by tak- ing 11 large enough it can be made smaller than any previously assigned number. When this is done, the sum 1 — (.9)" will differ from 1 by a number less than any previously assigned number. We say that s„, or 1 — (.9)", approaches the limit 1, and we write lims„ = lim[l-(.9)"]=l. ft=CO 71 — CO The physical interpretation of this process is, of course, that the weight of air in the vessel tends toward zero as the pump- ing is continued indefinitely. A geometric progression like the one just discussed, in which the ratio is less than unity, and the number of terms is per- mitted to increase indefinitely, is called an infinite geometric progression. The notion is a very important one in both pure and applied mathematics. In general, let K = «i + %'• + ai*"^ H f- ai*"""* be a geometric progression in which »• < 1. By formula (6), _ ttjr" — «! _ ay _ tti?'" r — 1 1 — 9- 1 — r Since r is a proper fraction, the powers of r approach zero as a limit. Consequently, air"/(l — r) approaches zero, and (7) lim sn = -^^^— . Formula (7) enables us to solve the air-pump problem directly ; for, we have Oj = .1 and r = .9, so that lim s„ becomes n=eo .1/(1 -. 9) =1. XIV, § 131] PROGRESSIONS 193 EXERCISES 1. Pind the limit of the sums of each of the following infinite geometric, progressions. Cc)l.|.|..... ((?) l+2-\/3 + 7-4V3+ ..-. 2. Find the common fraction which gives rise to the repeating decimal .555 •••, hy finding the limit of the sum of the series .5+ .05 + .555+ •■•. Frame a rule by which the common fraction may be written down at once. 3. rind the common fractions from which the repeating decimals may be derived. (a) .272727 -. (6) .109109 •••. (c) .347347 •-. (c«) .02323 -. (e) .52323 •■• = .5 + .02323 ■••. 4. Find the limit of the sum of the series 1 + x + x^ + x^ + ••■, when X <,1. , 5. If an elastic ball thrown upward to a height of 20 feet rebounds each time it strikes the ground to one third the height from which it fell, theoretically, how far wiU it travel ? 6. A farm yields $ 1000 net per year. If this rate of return can be kept up Indefinitely and money is worth 51fo, what is the farm worth ? [Hint. The first annual return due one year hence is worth now 1000/1.05; the second 1000/(1.05)2 . and so on.] 7. Suppose that instead of producing $ 1000 net each year the farm in Ex. 6 must lie fallow every other year, making a return in alternate years only. What would it be woi-th ? 131. Means and Averages. The arithmetic mean or simple average of several mimbers is the sum of the numbers divided by their number. In symbols, the arithmetic mean of the n ■ numbers ai, a^, •••,«„ is Q 194 COLLEGE ALGEBRA [XIV, § 131 The geometric mean of n numbers ai, On, — a„ is the nth root of their product and may be written (9) 6? =v'«i«a ••■«„. Closely connected with the arithmetic mean is the weighted average, which figures largely in problems in statistics. The weighted average of several numbers a,, Os, — , a„ is the 'sum of the products obtained by multiplying each number by a num- ber called a weight, divided by the sum of the weights. It is given by vjitt^ + w^a^ + ••• + «/•„«„ (10) W= w,-^w^-\- 1- w The weighted average has an important physical meaning also, namely, if Wj, w^, —, m„ be the weights of n masses lying in a line located at distances ax, a^, —, a„ respectively from a given point 0, then W is the distance from to the center of gravity of the masses whose weights are w^, w^, •■•, m„. When a set of numbers form an arithmetic progression, their reciprocals form an harmonic progression, and the reciprocal of the arithmetic mean of two numbers is the harmonic mean of the reciprocals of the numbers. In finding averages great care must be taken to use the proper mean. EXERCISES 1. Prove that any term of an arithmetic progression is the arithmetic mean of the two adjacent terms. 2. Prove that any term of a geometric progression is the geometric meain of the two adjacent terms. 3. Prove that the harmonic mean of a and 6 is 2ab (11) H. a + b i. Express the geometric mean of two numbers in ternis of their arithmetic and harmonic means. XIV, § 131] PROGRESSIONS 195 5. A line divided internally and externally so that the segments have the same ratio in both divisions, is said to be divided harmonically. Prove that when a line is divided harmonically, the whole Une is the harmonic mean of the two segments. [Hint. Let the line AB = b, and the segments AP = a and APi = c ; then express 6 in terms of a and c] 6. Prove that the roots of a quadratic equation ax^ + 2bx + c = Q are (a) real when a, b, and c are in arithmetic progression ; (b) equal when a, b, and c are in geometric progression ; (c) imaginary when a, 6, and c are in harmonic progression. 7. Prove that the logarithm of the geometric mean of two numbers is the arithmetic mean of the logarithms. 8. Three masses weighing 6, 7, and 11 pounds are located 2 feet to the right, 3 feet to the left, and 4 feet to the right, of a given point, respectively. ■ Where is the center of gravity with respect to the point ? 9. A student receives marks as follows : 85 in a 2-credit course, 90 in a 5-credit course, 93 in a 3-credit course, 91 in a 3-credit course, and 80 in a 3-credit course. What is his weighted average ? 10. A community has 1 citizen with income of 1 10,000, 10 with in- comes of 15000, 30 with incomes of $3000, 50 with incomes of 12000, 20 with incomes of 1 1000. Pind (a) the average income of citizens of the community ; (&) the income of the average citizen of the community. 11. A merchant increased his capital the first year by 15 %, the second by 20 %, and the third by 25 %. What was the average rate of increase for the three years 7 [Hint. If r be the average rate of Increase, the capital at the end of the three years would be the original capital multiplied by (1 -|- r)^.] 12. Find the formula for the average rate of increase, over a period of n years when the annual rates are ri, r^, rs, •■•, »•„. 13. Two successive miles are traveled, the first at the rate of a miles per hour, and the second at the rate of 6 miles per hour. What is the average rate ? What kind of a mean expresses the average rate ? 14. Three successive miles are traveled at the rates of 3, 5, and 7 miles per hour, respectively. What is the average rate ? 196 qOLLEGE ALGEBRA [XIV, § 132 132. Compound Interest. When a sum is placed at com- pound interest, the interest is added to the principal, or con- verted into principal, at stated intervals. The rate of interest is reckoned at a given per cent per year, though the conversion interval is frequently less than a year. The total amount due at the end of n years is called the compound amount. Problem. To find the compound amount on a principal A for n years at rate i. At the end of the first year the amount will be A-^Ai = A{1 + O- At the end of the second year it will be A-\-Ai +(J. + Ai)i= A{1 + i)\ Similarly, at the end of n years the compound amount will be (12) A< = AO. + i)" For convenience of computation formula (12) is supposed to be true for fractional as well as for integral values of n. The value of A' is easily found by logarithms when A, i, and n are given. Moreover, from formula (12) the value of any one of the numbers A, i, and n may be found where the other three numbers in the formula are known. 133. Present Value. The most important formula to be derived from (12) is that found by solving for A. We find or, if we replace A and A' by the symbols V and P, (13) F=P(1 +*)"". The quantity V is called the present value of the principal P due n years hence. It is the amount which, if placed at in- terest now, will amount to P in m years. In other words, it is the cash value of the obligation P due in n years without in- terest. The value of V like that of A' is easily found by XIV, § 133] COMPOUND INTEREST 197 logarithms. However, the numbers (1 + i)", the amoimt of unit principal, and (1 + 1)~", the present value of unit principal due in n years, occur so frequently that their values have been tabulated for a large number of rates, and for times up to 100 years or more. A table of compound amounts and present values is given in the tables at the back of this book (Tables D and E). EXERCISES 1. To what sum will ?> 1000 amount if put at compound interest at 4 fo for 10 yrs. ? 2. Find the amount of $1345.27 at 4tfo compound interest for 10 years and 6 months. [Hint. (IM)^''^ = (1.04)1" . (i.04)i.] 3. Find the compound amount of $ 1235.27 for 3 years at 6 %, com- pounded semiannually. [Hint. The multiplier will be the same as the multiplier for 6 years at 3 fo compounded annually.] 4. Find the compound amount of 12378.21 for 4 years at 4 fo, com- pounded quarterly. 6. What sum put at compound interest for 5 years at 4 ^i, com- pounded annually, will amount to 1 1000 ? 6. "What is the present value of $5000, due in 4 years without interest, when money is worth 6fo? 7. A man owes two sums, 1 2000 due without Interest in 2 years, and 13000 due in 4 years without interest. He wishes to arrange to discharge hoth of these obligations in 3 years. What will be the amount required if money is worth 6fo? 8. In what time will a sum of money double itself if placed at com- pound interest at 5 ^fj ? 9. At what rate must a sum of money be invested at compound in- terest in order that it will double itself in 15 years ? 10. Find the formula for the time when the amount, the principal, and the rate are unknown. 11. Find a formula for the rate when the amount, the principal, and the time are known. 198 ' COLLEGE ALGEBRA [XIV, § 133 12. In 1907 the consumption of ooal in the United States was 480,363,000 tons, which was 7.36 <^o in excess of the consumption in 1906. Assuming the rate of increase to continue, what will be the consumption in 1927 ? (Use logarithms and give result in millions of tons.) 13. A common rule for the time in which any sum of money will double itself at compound interest is, "Divide .69 by the rate and add one third of a year." Prove that the rule is approximately correct. [Hint. log. 2 = .69 and log„ (1 + i) = i - '_ + f ]. 14. At a certain university which had 4000 students in 1910, the in- crease is about 10 'Jo each year over the previous year's attendance. Assuming the rate of increase to continue, what would the attendance be in 1920 ? 15. Construct a graph to show the compound amount of 1 dollar at 4 % as the time varies. 16. If money is worth 6 %, what is thecash price of a farm for which a man pays $ 1000 cash, 1 1240 at the end of one year, $ 1180 at the end of two years, 11120 at the end of three years, and 11060 at the end of four years ? Compare result with the case of a man who buys a farm for $4000, agreeing to pay |1000 cash and 1 1000 pet year with interest at 6 9i) on all sums remaining unpaid, and draw your conclusion as to the necessity for compound interest. 134. Annuities. An annuity, or, to speak more accurately, an annuity certain, is a series of equal payments made at equal intervals of time during a fixed term of years. The first pay- ment is supposed to be made at the end of the first year. The elements that enter into an annuity are the amount of the periodic payment, the rate of interest, and the time, which we shall denote by a, i, and n, respectively. We shall speak of an annuity of a per annum when the sum of the payments made in one year amounts to a. The amount of an annuity is the sum that would be due at the end of the term, if no payments were made but all were put at compound interest as soon as due. XIV, § 134] COMPOUND INTEREST 199 The present value of an annuity of a per annum is the amount that one could afford to pay in cash for the privilege of receiving the payments in regular order. Pkoblbm I. To find the amount of an ahnuity of a per annum. The amount of annuity is the sum of the amounts of the annual payments, each accumulated to the end of the transaction. The first payment will be on interest for n — 1 years, the second for n — 2 years, and so on, the last payment being a cash payment. By the compound interest formula, the amounts of the payments are a(l + 0"-S 0(1 + 0""'^ -. a(l + i), a, and the amount of the annuity is the sum of these amounts. "Written in reverse order, the amounts form a geometric pro- gression with ratio 1 + i. Denoting the sum by S we have S = a + a{l + i} + a{l + i)-\ h a(l + i)""'. The value of this sum, as given by problem (10), § 129, is (14) s = a ^'' + fr-' . Tables for the amount of an annuity with a unit payment are given in textbooks on the mathematical theory of investment. The value of S is easy to find from the formula (14) if a table of compound interest is available. Peoblem II. To find the present value of an annuity of a per annum. . The present value of the annuity is the sum of the present values of the payments. The present value of the first pay- ment is a(l + iy^, of the second .a(l + 1)~^, and so on, to the last, whose present value is a(l + i)~". These numbers form a geometric progression whose ratio is (1 + i)"^- Denoting 200 COLLEGE ALGEBRA [XIV, § 134 the sum by V and making use of the results of Example 11, § 129, we have (16) F=«. l-^l + ^>"" . Like the values of S, the values of V for a unit payment have been tabulated, so that with a table at hand V is found by multiplying a by the appropriate number taken from the table. EXERCISES 1. A man saves 1 100 annually which he deposits with a savings bank which pays 4 fo. How much vfill he have saved at the end of 10 years ? 2. In 1907 the coal consumption of the United States was 480,363,000 tons and the rate of increase over the preceding year was about 7 fo. At that rate what would be the total consumption to the end of the year 1927, beginning at the first of the year 1908 ? 3. A man agrees to pay $ 1000 cash at the end of each year for five years for a house. If money is worth 6 fo , what is the cash value of the house ? 4. A mine will produce 1 10,000 net per year for 20 years and money ,is worth 6 9fc. Leaving aside extraordinary risks, what is the mine worth ? 135. Sinking Fund and Amortization Problems. When a debt S, payable at some future time, is contracted, provision for payment should be made by setting aside each year a sum large enough so that the total accumulation' will equal the debt when the latter falls due. The sums set aside each year constitute an annuity that will amount to S, and the fund into which the annual payments are made is called a sinking fund. A fund of this kind may be accumulated for the purpose of replacing buildings or equipment that are wearing out. In this case the fund is called a depreciation fund, and the annual payment is a charge for depreciation. The theory is identical for the two cases. XIV, § 135] COMPOUND INTEREST 201 Let As denote the annuity that will amount to S. The value of As is found by replacing a by As in equation (14), § 134, and then solving for^^. The required formula is (16) As = S (1 + i)" - 1 Another important problem arises when, the debtor agrees to pay principal and interest in equal annual installments. This form of payment is one form of amortization. Here again the annual payments constitute an annuity, which, in this case, has D the face value of the debt for its present value. The annual payment is spoken of as the annuity that D will purchase. Let Ajj denote the annuity that D will purchase. The value of Aj) is found by replacing F by Z) and a by A^ in equation (15) and then solving the equation for A^. The result is given by the equation (17) Ad = D 1- (l + i)-" The factor «/[(! + 1)" — 1] occurring on the right of equation (16) may be found directly from a table for "the annuity that will amount to 1," while the factor i/[l — (1 + *)""] ^^ (17) may be taken from a table for " the annuity that 1 will purchase.'' Formulas (12), (13), (14), (15), (16), and (17) are the basis of the mathematical theory of investment. EXERCISES 1. What sum must be set aside at the end of each year to repay a loan of f 100,000 due in 20 years, if the payments set aside can be accumulated aXifol 2. A locomotive costing 1 76,000 wears out in 25 years. If money can be accumulated at 5 %, what should be the annual charge for depreciation ? 202 COLLEGE ALGEBRA [XIV, § 135 3. A company borrows $ 100,000 for 20 years, agreeing to repay the loan, principal, and interest, at 6% in 20 equal annual installments. What is the yearly amortization payment ? 4. A man buys a house for 1 6000, agreeing to pay 1 1000 down and the balance, principal, and intei-est at 6 ^fc, in 6 equal annual payments. What is the amount of the annual payment ? 136. The Compound Interest Law. If a be any base, the consecutive integral powers of a may be arranged as in the following table. Exponent = x 1 2 3 4 5 6 7 Power = y 1 a a2 a> ai a^ a^ a^ The relation between an exponent x and its corresponding power y is given by the equation (18) y = a' It is clear that if the exponents form an arithmetic progres- sion the corresponding powers will form a geometric progres- sion, whether the exponents are integers or not. Since x is the logarithm of y to the base a, this fact may be stated as follows. If a set of numbers are in geometric progression, their logarithms are in arithmetic progression, and conversely. The formula for compound interest is a special case of (18). If interest be converted into principal m times a year, the compound amount A at rate i for a principal A at the end of t years is given by the formula (19) \ m Xmt Let i/m = 1/m, so that m = ui. With this notation IN (20) A' = A 1+- XIV, § 136] COMPOUND INTEREST 203 Suppose, now, that the conversion interval becomes shorter and shorter, so that in the limit interest is converted into principal instantaneously. In this way we arrive at the notion of instantaneous or continuous compound interest, a thing that is actually approximated in the business of large concerns which receive interest payments many times each day. However, the great value of the notion is diie to its application to physical rather than to financial problems. To find the formula for instantaneous compound interest it is necessary for m, and consequently for u, to increase indefi- nitely. In books on analysis it is shown that (21) limf 1 +iY = 2.718218 ••-. This limit is denoted by the letter e.* If y denote the limit toward which A' approaches, formula (20) becomes (22) y = A&t, (e = 2.718218 ■•■ ). With the value of e known approximately, the amount at instantaneous compound interest is easily found. To realize more clearly the meaning of equation (22), it is important to note that if A be the principal at the beginning of any conversion interval, the interest for the interval is iA/m, and the amount at the end of the interval is A+ iA/m. In other words, the interest, or increment per interval, is propor- tional to the principal. Equation (22) may therefore be looked upon as a formula giving the law of growth of any magnitude whose rate of increase at every instant is proportional to the quantity at the instant. This law, which is followed in the case of many natural phenomena, has been named by Lord Kelvin the Compound Interest Law. * The number e, which Is one of the most important in the whole field of analysis, is the base of the so-called Napierian system of logarithms. 204 COLLEGE ALGEBRA [XIV, § 136 If the rate i is negative, the change per interval in A, namely, iA/m, is a decrement, or, in financial matters, a discount. In all cases the constant A is the measure of the magnitude at the time t = 0. Why ? Among the many applications of the compound interest law the following may be mentioned : (1) By Kewton's law of cOoling the temperature at time t is given by the formula e = floe", where 6o is the initial temperature, and 6 is a constant rate to be determined for each substance. (2) The number of bacteria per cubic centimeter at the time t in the presence of unlimited food, and under proper temper- ature conditions, is given by N= Noe"', where N^ is the initial number and k is the rate of increase. (3) The difference in potential at time t of the two coatings of an electrical condenser of capacity k, discharging through resistance B, is where Vq is the potential at the beginning of the discharge. EXERCISES 1. Find the instantaneous rfompound interest on $1000 for 3 years at 6 % and compare the result with the ordinary compound interest convertible annually, for the same time and at the same rate. 2. Eadium is dissipated according to the law q — 906-", where k is the rate of dissipation, qo the initial quantity, and q the quantity at time t. Find k on the hypothesis that one half the quantity disappears in 1800 years. How much is left at the end of 100 years ? XIV, § 136] COMPOUND INTEREST 205 3. The amount of light that passes through incompletely transparent material of thickness t is L = ioe-*', where La is the intensity of the beam striking the material, and A is a constant rate depending upon the material. Find fc if 98 % of the light is transmitted through 1 cm. of glass. 4. In a certain culture 100 thousand bacteria were present at time f = and 900 thousand at time t = 10. Find the rate of increase k. Find an expression for the number present at any time t. CHAPTER XV PERMUTATIONS AND COMBINATIONS 137. Definitions and Fundamental Theorem. Consider a finite number of things which, for the moment, may be denoted by the letters %, a^, Oj, •• , a„. In general, any r of these n things may be arranged in several different orders. Each order,, or arrangement, of a set of r things taken from the set of w things is called a permutation of the r things. As a special case, r may be equal to n. Thus, the three letters a, b, c, give rise to the six permutations, abc, acb, bac, bca, cab, cba, when taken three at a time ; to the six permutations ab, ba, ac, ca, be, cb, when taken two at a time ; and to the three permutations, a, b, c, when taken one at a time. A set of r things considered without regard to order is called a combination. The three letters, a, b, c, give rise to one com- bination, abc, of three letters, three combinations db, be, ca, of two letters, and three combinations a, b, c, of one letter. 138. Fundamental Theorem. If a specified act can be per- formed in p ivays and a second independent specified act can be performed in q ways, the two acts can be performed in combiim- tion in pq ways. The theorem is nearly self-evident. To fix ideas, let us suppose that the first act is passing out of one room having p openings into a hall, and the second passing from the hall into another room, having q openings (Fig. 37). Using the first 206 XV, § 139] PERMUTATIONS AND COMBINATIONS 207 opening out of the first room, the passage can be made in q ways. This can be repeated for each of the p openings of the Fig. 37. first room. Therefore, there are pq ways of going out of the first room and into the second. 139. Formulas for Permutations. Pkoblem I. To find the number of permutations of n things taken all at a time. The number of permutations of n things taken all at a time is denoted by P„. ' It is easy to see that Pj = 1, P2 = 2 ■ 1, P3 = 3 • 2 • 1. Prom these simpler cases it is reasonable to guess that P„ = n{n — 1) ••• 3 • 2 • 1, or, with the briefer notation (1) I'n = nl The truth of this statement is easily proTen by mathematical induction. For, suppose that (1) is true for n = k. Then, by hypothesis Pj, = A; ! Suppose that a (fc + 1) st thiag be added so that there are fc + 1 things, any one of which may occupy the first place. In order to form all possible permutations, there are two acts to be performed ; first, the placing of one of the k + 1 things first, second, the arranging of the remain- ing k things. The first act may be performed in fc + 1 ways and the second in k ! ways. By the fundamental theorem the number of ways of performing the two acts is (fc + l)/i: ! But (A; + 1) X A; ! = (A; + 1)! Therefore, if (1) is true when n = k, it is true when w = A; + 1, and the induction is complete. 208 COLLEGE ALGEBRA [XV, § 139 Problem II. To find the number of permutations of n things taken r at a time. Let ^P^ denote tlie required number. For r = 2, the first thing may be arranged in n ways and the second in n — 1 ways. Therefore, by the fundamental theorem „P2 = n{n — 1), which is a special case of the more general formula (2) mPi. = »(»- l)(n- 2) •••(«.-»•- 1). To prove that (2) is true for any integral value of r, which is not greater than n, suppose it to be true for r = k. Then by hypothesis, „Pj = n{n- V){n _ 2) ■.. (w - W^l). If the things be taken fc + 1 at a time the required permutar tions may be formed by two acts. The first of these two acts is the arrangement of A; things and the second is the selection of the (k + l)st thing from the remaining n — k things. By hypothesis the first act may be performed in n{n - l)(w - 2) ••• (n - fc^^) ways. Since the second may be performed in n — k ways, the two acts may therefore be performed in n(n - l){n - 2) •■■ (n -k-1) x(n-k) ways. Consequently, „P^i = n{n- l)(n - 2) .•• (n - fc - !)(« -{k + 1) - 1). This formula is exactly formula (2) with j- replaced by A; + 1. If, therefore, formula (2) is true for r = fc, it is true for r = k+l, and by complete induction, for any value of r. If numerator and denominator of (2) be multiplied by (n — r)l the numerator becomes n !, since w — ?• is the next in- teger below n — r + 1. Consequently, (2) takes the useful form (3) ^Pr = —^. XV, § 140] PERMUTATIONS AND COMBINATIONS 209 EXERCISES 1. There are five routes by which a passenger may go by rail from Madison to Chicago, and six from Chicago to Saint Louis. In how many ways may tlie trip from Madison to Saint Louis by way of Chicago be made ? 2. How many five figure numbers can be written by means of 5 digits if no digit is repeated in any number ? 3. Every term of -a determinant of the 6th order is of the form a^fijCAe /„ where the subscripts x, y, z, u, v, w, are the numbers 1, 2, 3, 4, 5, 6, in some order. How many terms does the expansion of the de- terminant contain ? 4. A signal corps has 6 flags to be displayed in a horizontal row three at a time. How many signals can be made ? 5. How many numbers can be wi-itten with six digits if no number contains the same digit twice ? 6. How many signals are possible with six flags of different colors dis- played in a line ? 140. Formulas for Combinations. Problem III. To find the number of combinations of n things taken r at a time. Let the number required be denoted by „C,. Consider a single combination of r things. The things of this combiaa- tion may be arranged in r ! ways according to (1), § 139. Moreover, this statement is true of every one of .the „C, com- binations, and the totality of perniutations thus obtained is the number of permutations of n elements taken r at a time. Con- sequently, C X y'= P - '^- From this equation we obtain (4) „Cr= »*' rl{n —r)l 210 COLLEGE ALGEBRA [XV, § 140 EXERCISES 1. A society has 30 members. In how many ways may a committee of 3 members be chosen 1 2. How many different products, each containing four factors, may be constructed from 8 prime numbers ? 3. How many numbers between 100 and 200 may be written with the digits 1, 2, 8, 4, a,nd 5, if no digit is used twice in any number ? 141. The Binomial Theorem. The theory of combinations furnishes an elegant method for the proof of the binomial theorem when the exponent is a positive integer. The expression {x + a)" may be considered as the product of n factors a; + a,, x+ a-^, x + Us, -, x + a„, when the a's all become equal to a. By the generalized dis- tributive law (§ 11), (5) (x + ai){x + a^) - {x + a„) s a;" + (mi + as + - + «„)«»-' + (aia2 + OiMs + - + «»-]««)»:""'' +(aia2«3 H h a»_2a„-ia„)a!""^ -f- ••• + aia^a^ ••• a„, where the coefficients of a;"~^, x"'^, x"~^, ■•■• are the sums of all possible products of the a's taken one at a time, two at a time, and so on. If aj = 0.2 = as = ••• = a„ = a, (5) may be written in the form (6) (a; + ay = a;" + Ciax"-' + Cs,aV'^ + C,aV-' + - + a", where Gi, Oj, C3, ■•■ are integers which represent the number of terms in the coefficients of a;""', a;""^, — respectively. But a closer examination of (5) shows that Oi is the number of combinations of the a's taken one at a time, C^ the number of combinations taken two at a time, and so on ; that is, XV, § 141] PERMUTATIONS AND COMBINATIONS 211 Hence the binomial theorem may be written in the form (7) {x + a)" = a:" + „Ci aa;"-' + A a^a^""^ + nC^ a'a!"-^ + ... + „C,tt''af-''+ •■. +a". That this form for the binomial theorem is identical with that given in § 118, is easily seen by writing out the values of the binomial coefficients, (8) nCl, nCj, nCj, ..• nO, •". The coeifi-cient of the rth term is, clearly. (9) „CVi = (r - 1) ! (n - r + 1) ! MISCELLANEOUS EXERCISES 1. In a baseball nine one man can catch and one can pitch. In how many ways can the team be played ? 2. In how many ways can a committee composed of two Germans and three Englishmen, be chosen from 10 Germans and 15 Englishmen ? 3. How many lines can be drawn through 15 points, each line to pass through two points, if no three points lie on a line ? 4. If no three of a group of 21 points lie on a line and no four in a plane, how many planes are determined by the group ? 5. Prove by the theory of combinations that the binomial coefBcients equidistant from the ends 'are equal. 6. If n things lie on the circumference of a circle (or on any closed curve) a given arrangement is called a circular permutation. Prove that the number of circular permutations of n things is (n — 1) ! 7. In how many ways may the letters of the word algebra be arranged ? [Hint. Note the repetition of one letter.] 8. Find a formula for the number of permutations of n things taken all at a time when r of them are aUke. 9. Find a formula for the number of permutations of n letters taken all at a time, if r of the letters are a's, and s are 6's. 10. Write down the coefficient of a''6«c' in (a + 6 + c)". CHAPTER XVI PROBABILITY 142. Simple ProbabUiiy. If a bag contains 5 white and 3 black balls and a ball be drawn at random, any one of tbe 8 balls may be drawn. This fact may be expressed by saying that any one of the possibilities is equally likely to happen. Of the 8, any one of 5 would result in drawing a white ball, and any one of 3, in drawing a black ball. Common sense would seem to point out that 5 possibilities or 5 of the 8 cases are favorable to drawing a white ball and 3 unfavor- able. Upon this basis Laplace formulated the definition of simple probability as follows : Tlie probability that, among several equally likely events, an event will happen in a certain way, is the ratio of the number of favorable cases to the total number Of cases. In the example of the balls the probability of drawing a white ball is, according to the definition, 6/8 and the proba- bility of drawing a black ball is 3/8. In, the general case, if p represents the probability, a the number of favorable cases, and 6 the number of unfavorable cases, (1) The definition of probability, though it may be said to be based on common sense, is, like all definitions, purely arbi- trary. It does not mean that if the drawing were to be per- formed 8 times, 5 drawings would' bring white balls and 3 drawings would bring black balls, but, rather, that if a great number of drawings were to be made under exactly the same 212 XVI, § 142] PROBABILITY 213 conditions, the ratio of white balls drawn to black balls drawn, would be as 6 to 3. If the cases are all favorable, the event is certain and while if no case is favorable the event is impossible and (3) p = _A^ = 0. a -\-b Consequently, certainty is expressed by 1, and impossibility byO. Theorem. If p is the probability that an event will happen and q the probability that it will fail, (4) i> + g = 1. Tor, p = a/ (a + b) and q = 6/(a + 6). Therefore a , b ^ a + a + b EXERCISES 1. A single cubical die with faces numbered from 1 to 6 Is thrown once. What is the probability that the face numbered 4 will lie upper- most ? Solution. One case is favorable and 5 are unfavorable. The proba- bility required is therefore 1/6. 2. Two coins are thrown into the air simultaneously (or in succession) . What is the probability that both will fall " heads " ? [Hint. They may fall in only four possible ways ; namely, both heads, A heads and B tails, A tails and B heads, both tails. ] 3. A die is thrown once. What is the probability that the number of points uppermost shall be less than 4 ? 4. Ten balls numbered from 1 to 10 are placed in a bag and two drawn at random. What is the probability that the two are numbered 4 and 7? [Hint. How many ways are there of selecting 2 out of 10 things ?] 214 COLLEaB ALGEBRA [XVI, § 142 6. What is the probability that 12 coins tossed into the air at the same time will all fall heads 1 What is the probability that a single coin tossed into the air 12 times will fall heads every time ? [Hint. In how many ways may 12 coins fall ?] 6. Two balls are drawn simultaneously from a bag containing 4 white and 6 black balls. What is the probability that both balls will be white 1 That both will be black. That one will be black and one white ? 7. Statistics of the insurance companies show that of 100,000 children living at age 10, 740 will die within a year. What is the probability that a child 10 years of age will die within a year 1 That it will survive for one year ? 8. Of 100,000 persons living at age 10, 69,804 are supposed to be alive at age 50. What is the probability that a boy aged 10 will die before he is 50? 143. Partial and Total ProbabUity. An event may happen in any one of several series of mutually exclusive events. Th.e probability that it will happen in a given series is called partial probability. The probability that it will happen with- out regard to any series is called total probability. To illus- trate, suppose a bag contains 10 white balls of which 6 are marked with crosses and 4 not ; 8 black balls, 5 with crosses and 3 without ; and 7 yellow balls, 4 with crosses and 3 with- out. The probability of drawing a ball of any given color and marked with a cross is partial probability and the proba- bility of drawing a ball with a cross, but without regard to color, is total probability. In the example just given there are three partial proba- bilities, as follows : The probability of drawing a white ball with a cross is 6/25 ; that of drawing a black ball with a cross is 5/25 ; and that of drawing a yellow ball with a cross is 4/26. The total probability^ that is, the probability qf draw- ing a ball with a cross without regard to color, is 16/26, since 15 out of 26 balls have crosses. The important fact is that XVI, § 144] PROBABILITY 215 the sum of the partial probabilities is exactly equal to the total probability. Theorem. The total probability of an event is the sum of its partial probabilities. • Let m be the number of possible cases, and let Ui, a^, —, a„ be the number of favorable cases in each one of n series of events. Further, let p be the total probability, and^i, p2, •••,J5„ the partial probabilities. By the definition of total probability, ai 4- a, + ••• a„ a, , ao , , a„ m mm m But the fractions -J, -^, •••, J are the partial probabilities, mm m Consequently, (5) P=Pi+P'i+Pz+ — +Pn' as was to be proved. 144. Compound Probability. If the happening of either one of two events has no influence upon the happening of the other, the two events are said to be independent. The proba^ bility that two independent events shall happen simultaneously (or in succession) is called compound probability. For example, if two coins are thrown into the air, or if two throws are made with one coin, the probability that both falls will give heads is compound. Theorem. The compound probability of tivo events is the product of their simple probabilities. Let pi = tti/mi and p^ = ai/niz be the simple probability, and p the compound probability. The two events may happen through any one of the ai favorable cases for the first and any one of the a2 favorable cases for the second. By the fundamental theorem for permutations and combinations (§ 138), the number of favorable cases for the happening of miBia mi X^, ma V = 3>i Xj02, 216 COLLEGE ALGEBRA [XVI, § 144 the two events is a-fiii. In the same way the number of possible eases is wiiwia- Consequently, ■P-- or, (6) as was to be proven. CoKOLLABY I. The probability that several independent events whose simple probabilities are pi, p^, — , p„, will happen simultaneously (or in succession), is ' CoKOLLART II. The probability p that n events whose simple probabilities are pi, p^, ■•-, p„ shall ail fail is (8) i-p = (1-pi)(1-P2)-(1-k); the probability that the first r of them shall happen and the re- mainder fail is ' (9) V^^^ - A(1 - Ph-i) - 0- - P.)- EXERCISES 1. If the probability that A wiU win a race is | and that B will win it is J, what is the probability that either A or B will win it ? 2. What is the probability that a single throw of two dice will be less than 3 ? 8. The probability that A working. alone can solve a problem is J and that B working alone can solve it is J. What is the probability that the problem will be solved if both work at it, each alone ? [Hint. The problem will be solved if (o) A succeeds and B fails ; (6) A fails and B succeeds ; (c) both succeed,] 4. If pi represent the probability that a man will die within a year and P2 the probability that his vrife will die within a year, what are the probabilities (a) that the man will die and his wife survive ? (6) that the man will survive and the wife will die? (c) that both vdll die? (d) that both will live ? XVI, § 145] PROBABILITY 217 145. The Mortality Table. One of the most important applications of the theory of probabilities is the application to certain problems relating to the duration of human life. The problems confronting the life insurance company, or the employer, either public or private, who wishes data for a pen- sion system, or the judge who seeks to determine a life inter- est in an estate, all depend for their solution upon the theory of probability. The instrument actually employed hx the solution of all these problems is a mortality table. A mortality table in its simplest form is a record showing how many persons out of a large number, all of the same age, dife during each successive year until all are dead. Such a table is based upon observa- tion and not upon mathematical considerations. Mortality tables differ for different countries, for different classes living side by side in the same country, and for persons of the same class engaged in different occupations, etc. The table given in this book (Table F, p. 258) is called the American Experience Table and was compiled from the results of the experience of thirty American insurance companies to the end of the year 1874. This table is the basis upon which most of the life insurance business of this country is con- ducted. The methods actually emplpyed in constructing a mortality table from observed data need not concern us here. In the above table the column headed x gives the age, the column headed l^ the number living at age x out of the 100,000 alive at the age of 10 years, and the column headed d^ the number dyiag between the ages x and a; -|- 1. The probability that a person aged x years will live to age X + 1 years is assumed to be i+i 218 COLLEGE ALGEBRA [XVI, § 145 and the probability that he will die within the year is assumed to be '^^ or ^^+1 - ^-^ . EXERCISES 1. What is the prohability that a man aged 30 will be alive ten years later ? 2. A man is 45 years of age and his son is 15. What is the probabil- ity that both -will live 10 years ? 3. A man and his wife are 30 and 28 years old respectively when their first child is born. What is the probability that both will be alive on the child's twenty-first birthday ? That one will be alive on that date ? That only one will be alive ? 4. Find a formula for the probability that two persons of ages x and y will both live n years. 5. Find a formula for the probability that a man of age x will live n years and another of age y will die within n years. 6. From the Mortality Table plot the curve which shows the proba- bility of dying for each year from ages 10 to the end of the table. 146. Mathematical Expectation. The value of a mathe- matical expectation is the product of a sum of money to be paid upon the happening of a specified event by the proba- bility that the event will happen. For example, an insurance company agrees with a man 30 years of age to pay $ 1000 to his heirs in case of his death within one year. By the mor- tality, table the probability that a man of 30 will die within a year is .00843. Consequently the value of the mathemati- cal expectation is $ 8.43 at the end of the year, and if money is worth 3 % the present value of the expectation is $ 8.43 -;- 1.03 = $ 8.19. This is the simplest form in which a problem in life insurance can present itself. Tor example, a life insur- ance company could afford to insure the lives of 10,000 men of age 30 for one year for $ 81,900 cash plus the cost of con- ducting the business. XVI, § 146] PROBABILITY 219 EXERCISES 1. A stake of $ 10 is made contingent upon a throw of dice being less than 4. What is the mathematical expectation of the player ? 2. If 9997 ships out of 10,000 of a given class and in a given condition reach port safely what would be the cost of insuring a ship with its cargo worth $ 500,000 for a single voyage, if interest, expenses, and profits are neglected ? 3. What would be the cost of insuring a house worth $5000 for a year if two houses out of 1000 in its class burn down each year ? 4. On the basis of the mortality table what would be the minimum cost of insuring the life of a man of age 35 for one year, neglecting inter- est and expenses ? CHAPTER XVII SEQUENCES AND LIMITS 147. Definitions. A succession of numbers formed accord- ing to some definite law is called a sequence (§ 30). The set of numbers .3^ 33^ 333^ 3333 ,__^ which present themselves when we attempt to express the fraction 1/3 in the decimal notation is a sequence. If we attempt to find an expression for the square root of 2 we are led successively to the numbers of the sequence 1, 1.4, 1.41, 1.414, 1.4142,-. The law of formation in this case is the rule for extracting the numerical square root. The process of finding the ratio of the circumference to the diameter of a circle leads to the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ••.. If P^ denote the perimeter of a regular polygon with r sides inscribed in a circle with unit diameter, the sequence '3) -• 6) ■fl2> -^24) •■• is the sequence of numbers frequently used in finding the ratio of the circumference to the diameter. In all these sequences the process of finding additional terms can be carried on indefinitely. A variable may take for its valaes the numbers of a sequence. In such a case we say the variable runs through the numbers of the sequence. For example, let s„ be the variable which de- 220 XVII, § 148] SEQUENCES AND LIMITS ^ 221 notes the sum of n terras of the geometric progression 1 + 1 + 1 + 1+.... 2 4 8 16 Then s„ runs through the sequence 1 3 7 16 31 2' 4' 8' 16' 32''"' as n increases. 148. Limits. In every example that has been given the sequence has the property that as the number of terms is in- creased, the successive terms approach nearer and nearer to a fixed number. Thus, the numbers of the sequence 1 3 7 15 2' 4' 8' 16'"' approach nearer and nearer to 1, and if a suQicient number of terms are taken the difference between 1 and the terms of the sequence becomes indefinitely small. In such eases we say of both the sequence and the variable which runs through the terms of the sequence, that they approach a limit. The sequence just given and the variable s„ which runs through the values of the sequence approach the limit 1. The sequence of perime- ters approaches the circumference of the circumscribed circle as a limit. The complete definition of the limit of a variable correspond- ing to such a sequence, is as follows. If a variable x runs through the numbers Of a sequence Xi, X2, ajs, Xi, -, a;„, -, and if a constant number a exists, having the property that corre- sponding to any arbitrarily chosen positive number 8, a positive integer n may be found such that every one. of the absolute values (1) |a-»J, |a-a;„+i|, \a- x.+i\, - is less than 8, the constant number a is coiled the limit of the variable x. 222 COLLEGE ALGEBRA [XVII, § 148 The relation between the variable x and its limit a is ex- pressed by writing (2) or, simply lira «„ = a (3) lira x = a Formulas (2) and (3) are read the limit of a;„ (or of x), as n be- comes infinite, is equal to a. A sequence whose numbers approach a limit is said to be convergent. Any complete discussion of the theory of limits must be based upon a criterion for the existence of a limit. The f pllow- ing criterion, first given by Cauchy, has been called by Du Bois Eeymond, the General Principle of Convergence: The necessary and sufficient condition that a sequence of real numbers, Xi, X2, Xg, Xi, •■■, may have a limit, is that for any arbitrarily chosen positive number e, a value of n may be found such- that every one of the absolute values (4) |a;„-a;„+i|, |a;„-a;„+2|, | a;„ - a;„+3 1 , ••• shall be less than e. This general principle is frequently stated more briefly by saying that the sequence is convergent if a value of n can be formed such that (5) l«„-a;„+J<£ for all positive integral values of p. The general principle is here assumed to be true. Proofs may be found in textbooks on analysis. Example. Suppose s„ denotes the variable sum of n terms of the geometric progression 1-^1 + 1+... 2^4^8^ ' as n increases. Then s„ runs through the sequence 13 7 2' 4' 8''" XVII, § 149] SEQUENCES AND LIMITS 223 as n takes the values 1, 2, 3, •••. Moreover, according to § 128, s„ is given by the formula " 2" To show the existence of a limit it is necessary to consider the absolute value I 1 11 1 2"V 2^ 1 2" 2"+" It follows that this absolute value can be made less than a given number if n can be found to satisfy the inequality (6) |^(l-^.)<^'°^2">i(l-i- When logarithms are taken of both sides of the inequality written in the last form, we obtain the inequality ^^) -> ^ log2 • The number 1 —1/2" is less than unity for every admissible value of p, and its logarithm is negative. Consequently it will take a larger value of n to satisfy the inequality log- log e (S) n> , or n>-- — — , ^ ' log 2 .30103 than it takes to satisfy the original inequality. It is easy to find values of n to satisfy the inequality (8), whatever e may be. For example, if e = .0000001, logl/£ = 6 and the in- equality (8) is satisfied by n = 20 or any larger positive in- teger. The existence for a limit of the sequence is therefore proven by formula (8). In a similar manner it may be shown that the limit of the sequence, and consequently of the variable s„ is 1. 149. The Limiting Values of Functions. In applications of the theory of limits, the problem usually presents itself in 224 COLLEGE ALGEBRA [XVII, § 149 the form of an inquiry as to the value toward which a function tends as the independent variable approaches a definite limit, or increases indefinitely. The following examples will make the meaning clear. Example 1. In the problem of finding the sum of the geometric pro- gression 1 1 1 as the number of terms is increased indefinitely, the sum of n terms is given by 2 2" Here Sn is a function of n and the problem is reduced to finding lims„ = lim[l-iV Example 2. If a rectangle of base x be inscribed in a semicircle of radius r, the rectangle is , ^ A^xy^r'-'L (Fig. 38). In a later problem (Problem 4, § 157) it is shown that the maximum area of this rectangle is found to be the limit of .4 as a: approaches rV2. That is to say, ' maximum value ot A = lim x\r^ — — • In the second example we are not, properly speaking, deal- ing with a sequence, for x does not take a definitely determined sequence of values, but may take every value between two given values that lie between a; = and x = rV2. In such a case we say that the function f(x) approaches the limit A, as x XVII, § 151] SEQUENCES AND LIMITS 225 approaches a, if, and only if, Urn fix) = A for every sequence of values of x which approaches a. 150. The Limit of an Algebraic Function. An algebraic function of a; is a function which is formed, from x by means of a finite number of the operations addition, subtraction, multi- plication, division, and root extraction. In more extended treatments it is proven that for any two functions which approach limits, the limit of their sum, the limit of their product, the limit of their quotient, are equal respectively to the sum of their limits, the product of their limits, and the quotient of their limits, provided that in the case of the quotient, the limit of the denominator is not zero. It is also proven that the limit of a power of a variable is equal to the same power of the limit of the variable. These theorems may be summed up in a single easily re- membered statement, as follows. If fix) denote an algebraic function of a variahle x, then (9) liin/(a;)=/(a), for all values of a for which the limit of the denominator is dif- ferent from zero. Example. Let ■^^^^" 25 -x^ ' and suppose x approaches the constant a ; then ,. ., . ,. 2a;3-H3a;-5a!''2 2a3+3a-5a'" lim f(x) = lim '— = ~- , :,=« ^ '^ ^a 25 - a;2 25 - a^ provided \a\^5. 151. Infinity. Emphasis has been laid upon the fact that limits are finite and definite numbers. It frequently happens, however, that functions must be considered for which no finite limit exists, but for which the values increase beyond all 226 COLLEGE ALGEBRA [XVII, § 151 bounds as the independent variable approaches a particular value. (See § 64.) The simplest function of this sort is the function -• which increases indefinitely as a; approaches zero. In such cases it is customary to say that the function approaches the improper limit, infinity, or that the function becomes infinite. This circumstance is indicated by the notation (10) lim- = oo. asbO X The improper limit is said to be + oo or — oo according as the function increases numerically through positive or through negative values. For example, if x runs through the sequence of values 1, .1, .01, .001, .0001, -, the function 1/x runs through the sequence 1, 10, 100, 1000, 10,000, -, ■which approaches the improper limit + oo . For the same sequence of values for x the function — \/x runs through the sequence _^^ _^q^ _-^qq^ -1000, -10,000, -, which approaches the improper limit — oo. An important example of the limit of a non-algebraic func- tion is lim log X. When x runs through the sequence of values 1, .1, .01, .001, .0001, -, log X runs through the sequence of values 0, -1,-2, -3,-4,.... Clearly, as x approaches zero, log x approaches — oo. Note. The student should note that oo is not a number in the sense that it is an entity which obeys the ordinary rules of reckoning. For example, we may write oo -I- c = oo where c is any number, zero or other- wise. XVII, § 152] SEQUENCES AND LIMITS 227 152. The Indeterminate Form 0/0. A fractional function whose numerator and denominator are both zero for a given value of X is said to be indeterminate, or to take the indeter- minate form 0/0, for that particular value of x. Up to the present time it has been necessary to say that a fraction is not defined for values of x which make it indeter- minate. Nevertheless, the theory of limits furnishes a means of defining such functions for values for which they become indeterminate. , Example 1. The fraction .1 -z» /(«) = i 1-x becomes indeterminate for a; = 1, although it has a definite, finite value for every other value of x, no matter how little that value may differ from 1. Suppose that in the function given above, x runs through the sequence 0, .9, .99, .999, .9999, •■•, as it approaches 1. The function will then run through the sequence 1, 2.71, 2.9701, 2.997001, 2.99970001, ■■•, which apparently approaches the limit -S. That the sequence actually does approach a limit may be proven rigorously. It is convenient, therefore, to define the fraction (1 — a^)/(l — x), for the value a: = 1, as lim - — i-. 1=1 1 — a; In general, if the fraction n. 3. Prove that for sufficiently large negative values of x, the graph of AqX" + Uix"-'^ + •■• + BB,(ao > 0), ■will be found above the x-ax\s when n is even and below the a-axis when re is odd, and for sufficiently large posi- tive values, it will be found above the a^-axis, « even or n odd. 154. The Derivative of a Power Function. Let f(x) be any function of x. The limit (11) lim f(ae+h)-f(a>) h is called the derivative oif(x) and is usually denoted by /'(«). The derivative of a power function ex" (§ 53) is easily found by the preceding methods. Eor the power function the deriva- tive is h^ h JiO h The expression on the right is precisely the indeterminate form of Ex. 4, p. 228. To evaluate it when m is a positive inte- ger, note that, by (6), § 17, (as + A)" — a;" = [a; -J- ft —a;] [(a; + ft)»-i-|- a;(a; + 7i)"-2 + ... 4. a.»-i||. 230 COLLEGE ALGEBRA [XVII, § 154 The first factor on the right is h so that the derivative of ca;" reduces to lim [(a; + 7i)»-i + x{x + /i)"-^ -\ h a;"-^]. In the limit each one of the n terms within the bracket reduces to a;""' and consequently the derivative of ca;" is (12) ncx^-^. A slight modification of the reasoning leads to a result having exactly the same form when n is not a positive integer. Hence the rule for finding the derivative of the power func- tion is as follows. Diminish the exponent ofxbyl and multiply the result by the original exponent. 155. The Derivative of a Polynomial. Let us consider the polynomial f(x) = a^" + aix"-^ + a2a;"-2 -\ 1- a„. According to the definition, the derivative is /' (a;) = lirn ""^'" + ^^^"+ ""^^^ + ^^^""^ "' ^ ""~ ^"°"'" "* ^ "") - h^ h The limit on the right reduces directly to the sum J. j^ agjx + hy - apX" _^ j.^ ai(g + h)"-'- - ajX'-'- ^ ___ A=y) h hMI h 4. lim "n-iCai + ^O -«„-!« . ;i=M) h This sum is precisely the sum of the derivatives of the power functions a^", aia!""', a2X''~^ ■•• a^_iX. The derivative of the polynomial is therefore (13) /'(«) = na^ac"-^ + (n - l)a^x"-^ + ... + a«-i. XVII, § 156] SEQUENCES AND LIMITS 231 EXERCISES 1. Find the derivative of the function f{x) = 2 K* - 5 a;8 + 3 a;2 + 4 X + 6. Solution. By wliat precedes the derivative is the sum of the deriva- tives of the power functions 2 x*, — 5 x^, 3 x-, ix. By § 154 this sum is /(x) = 8 xs - 15 x2 + 6 X + 4. rind the derivatives of each of the following functions. 2. 6 x^ — 7 x^ + 14. 4. X" — a". 3. 8x*- 16x2+ 14x. 5. X" + a". 156. Geometric Interpretation of the Derivative. Slope of a Curve. Let / (x) be a polynomial, whieli, for definiteness, may be assumed to be of degree 3. Its graph will, therefore, be similar to the curve in Pig. 39. T/ Let P and Q be twp points on the curve ■whose abscissas are OM — a and ON = a + 7i. The ordinates are /(a) ~o and /(a + li) respec- tively. Draw the chord PQ and the line PR parallel to the as-axis meeting the ordinate NQ in R. Then RQ/PR is the slope of the chord (§ 49). But RQ=NQ-NR = Nq-MP= f(a + h) - f{a) and Pii; = MN = h. Therefore the slope of the chord is Fig. 39. (14) /(a + /i)-/(a) h If the point Q move along the curve toward P, the point R will approach P and h will approach zero. The limiting posi- tion of the chord will be the tangent line at P and the slope 232 COLLEGE ALGEBRA [XVII, § 156 of the chord becomes the slope of the tangent. The slope of the tangent is therefore given by the formula (15) Slope = llm /(« + fe)-/W =y>,(„). - - h It^O The slope of a curve at a poiat is defined as the slope of the tangent at the point. Therefore the geometric interpretation of the derivative of f(x) at a point x= a is that it is the slope of the curve y = f{x) at the point whose abscissa is a. EXERCISES 1. Draw a line having the same slope as the curve whose equation is y = x^ — 5x + 6 has at the point for which x = 3. 2. Find the points at which the tangent to the curve y=3?— 7x^+6— H is parallel to the K-axis. 157. Maxima and Minima. In § 52 the maximum value of a function was defined as a value larger than any other value determined by nearby values of the independent variable. In symbols, f{a) is a maximum if for values of h sufficiently small, /(a — h) and /(a + h) are both less than /(a). r Fig. 40. The definition of a minimum differs only in the substitution of the phrase less than for the phrase larger than. In the adjoining figure (Fig. 40) the ordinate at the point A represents a maximum, and the ordinate at the point B a minimum, value of the function. XVII, § 157] SEQUENCES AND LIMITS 233 Clearly, at either a maximum or a minimum point the slope is zero. But the slope is given by f\x). Therefore the maximum and minimum values of /(«) correspond to the values of x which satisfy the equation (16) nx)=o. The values of x corresponding to a maximum or a minimum Inay be found if the equation (16) can be solved. Example. Let f{x) =!i;8 - 9 a;2 + 23 a; - 15. /(a;)s3ij;2_ 18j; + 23. The roots of /'(a;)=0 are found to be, approximately, a; = 1.85 and X = 4.15. When these values are substituted for x in /(a:), the maximum and minimum values of the function are found to be approximately /(1.85)=3.1 and/(4.15)=-3.1. Brief consideration of the function shows that the first is a maximum and the second is a minimum, for the graph crosses the jz-axis a.ty = — 15. Y Then Fig. 41. The condition f'(x) = is necessary, but not sufficient, to insure a maximum or a minimum. A glance at Fig. 41 will show that a curve may have a point like the point A for which the slope is zero but the point is, nevertheless, neither a maxi- mum nor a minimum. 234 COLLEGE ALGEBRA [XVII, § 157 EXERCISES 1. Find the maximum and minimum values of each of the following functions. {a) x'-63fl + 9x-l. (c) 2a;s_i5a;2 + 36a;-14. (6) X* - 15. id) »» - 15. [Hint. The function a;' — 15 has neither a maximum nor a minimum. Verify the statement.] 2. What is the capacity of the largest box that can be made from a piece of pasteboard 24 by 36 Inches by cutting a square from each corner ? 3. Show that if an equation /(a;)= has a real double root it is geometrically evident that this root satisfies the two equations /(a;) =0 and/'(x)=0. 4. Find the rectangle of maximum area which may be inscribed in a semicircle with radios r. CHAPTER XVIII INFINITE SERIES 168. Series with Constant Terms. If (1) Ml, Ma, Ma, -, M„, - be an infinite sequence, the expression (2) ■ Ml + W2 + Ma H h M„ + - is called an infinite series. The term m„ is called the general term, or the type term. From the type term alone the series may be constructed. For example, if in the geometric pro- gression 1 + ^ + 1+.. .+k 2 + 4 + 8+ +2"' the number n be allowed to increase without bound, we obtain the infinite series 2+4 + 8+ +2"+ ' which has for its terms the terms of the infinite sequence 111 1^ 2' 4' 8' "■' 2"' '"' The terms correspond respectively to the values, n = 1, 2, 3, 4, ..... Again, when the binomial expansion for (l.Oiy/^ is found, the result is the infinite series 1 + .0025 - .000009375 + .0000000646875 -, ...-. For this series the general term is found by finding the nth term of the binomial expansion. 235 236 COLLEGE ALGEBRA [XVIII, § 158 In the case of the geometric series, it is possible to find the expression for the sum of n terms and by means of this sum to find the limit (if it exists) of the sum as n increases. In the general case it is not possible to find the sum of n /terms, much less the limit of the sum. We are concerned mainly with the question of the existence of a limit for the sum of n terms as n increases. Let (3) S„ = Ml 4- Ma 4- Ms + - 4- w„, or more briefly, (4) s„=2iMi, • 1 where the sign S denotes the sum of such terms as the sample term that fol-lows it. When the ferms of the series are con- stant, s„ is a variable which depends for its value upon n alone. If, now, n be increased indefinitely, one of the three following cases will necessarily occur. (1) The sum s„ may approach a finite number as a limit, as in the case of the series 1 + 1 + 1+1 + ...+!+.... 2 4 8 16 2» (2) The Sum may increase beyond all bounds, as in the case of the series i+J+i+i + •••+-+•••■ 2 3 4 n (3) The sum may take one of several values depending upon the form of n, as in the series 1+1-1-1 + 1 + 1-1-1+ ..., whose sum is 1, 2, or 0, according as n has the form 4 r ± 1, 4 r + 2, or 4 r. XVIII, § 159] INFINITE SERIES 237 In the first case the series is said to be convergent, in the second, divergent; in the third, oscillating. It is simpler to put divergent and oscillating series into a single class and call both of them divergent. A series is convergent if a finite limit exists for s„ as n in- creases; otherwise, it is divergent. If lim s„ = S, 'S is called the sum * of the series, and it is cus- tomary to write (5) S=Ui + U2+U3+ -. 159. Criterion for Convergence. The sums (6) Si, Sj, S3, -, s„, - of 1, 2, 3, — n — terms, form a sequence. The criterion for convergence of a series with real terms is therefore identical with the criterion for convergence of a sequence. When the General Principle of Convergence (§ 148) is applied to the se^ quence of sums *1> ^2) *3) ^4! •••) ^n) it takes the following form. The necessary and sufficient condition for the convergence of a series is, that corresponding to any given positive number e, a value for n may be found, such that all of the absolute values, (7) |s„+i-s„|, |s„+2 — s„|, |s„+3-s,„|, - |s„+j, — s„|, -, are less than e. This criterion may be expressed more briefly by writing (8) |s„+^-sJgti, U2>gt2, •", and or, U„>gT„. 240 COLLEGE ALGEBRA • [XVIII, § 160 But r„ increases without limit. Therefore U„ increases -with- out limit and the series Ml + Ma + "3 + •" is divergent. COEOLLARY. If he a convergent series with positive terms, and SMj ='Ui + U2 + U3-\ be a series such that u^ < «,-, for every value of i, then Sm^ is con- vergent ; if, on the other hand, the series S*,- is divergent and Uf > i;, then the series Sm; is divergent. The corollary is merely a restatement of the theorem for the cases, first where every Jc is less than 1, and second where every k is greater than 1. 161. Some Comparison Series. For the comparison test it is necessary to have at hand a number of series whose character is known.. The following are some of the commoner series used for this purpose : 1. Any geometriQ series (11) ar + ar^ + ar^ + — is convergent when r is less than 1, and divergent when r > 1. Tor, the sum of n terms is, by (6), § 128, (12) ^„=^n^. r — 1 This sum has a limit a/(l ^ r) for r < 1, but no limit for »• > 1. 2. The series (13) ^ + i + p + p+-+i+- is convergent for k> 1 and divergent for k'Sl. XVIII, § 161] INFINITE SERIES 241 The terms may be grouped as follpws : (14) -i + i + 1' 2«^ + : + -. ,+ ...+- + _(2» + l)' (2" + 2)* (2"+i)«^_ Every one of the 2" terms which go to make up the general terms of the series with the new grouping of the terms is less than 1/2"'. Consequently, the general term of the regrouped series is less than 2"(l/2"'), that is, less than l/2"('-». The terms of. the series in the new form are, after the second term, less than the corresponding terms of the series (15) ^ + 4 + ^+0-^+ • ^ "•" 2»(*-i) After the second term the series (16) is a geometric series with ratio 1/2'"^. This ratio is less than 1 for fc > 1. Therefore the series (14), or what is the same thing, the series (13), is convergent for Zs > 1. On the other hand, each of the 2" terms which make up the general term of the regrouped series is equal to or greater than 1 ' 2(»-i) * and the general term is therefore greater than 2". , or — • • 2(n+l)«:' 2' 2"<*~1* Each term of the regrouped series is, after the second term, greater than the terms of the series 1,1,11,1 1. ,1 1 1"' (16) 2* 2' 2'-i 2' 22<*-i> + + 1 ' 2* 2"<*~i^ + After the second term the series (16) is a geometric series with ratio l/2*~i. The ratio is equal to or greater than 1 when fc ^ 1. The series (13) is therefore divergent when fc < 1. 242 COLLEGE ALGEBRA [XVIII, § 161 EXERCISES 1. Determine the character of the serie? 11 2!^3! n\ SoLUTioji. If the first two terms of the series be set aside, the terms of the new series will all be less than the corresponding terms of the con- vergent geometric series 1+- + - + ... + — +■■■ 2 22 2" because the general term 1/n ! is less than the general term 1/2". The series is therefore convergent. 2. Determine the character of the series 1 2 22 23 12 3 4 SoLtTTiON. The general term is 2»-i/jj which is greater than the general term of the divergent series 1^2 3 n The series is therefore divergent. 3. Determine the character of the following series. ^ ■' 1.2-2 2.3.22 3-4.28 4.5-24 1 1 • (6) l + 4: + 4^+-- (c) -L + J_+J_ + J_+ .... ^ ' 1-2 2.3 3-4 4.5 162. D'Alembert's Test-ratio Test. The test-ratio of a series zti + Mj + ^3 + • ■ • is the ratio of the general term to the term preceding it- It may be written m„/m„_i, or what amounts to the same thing, m„+i/w„. Thbokbm. If in a series with real positive terms, lim (it„+i/M„)= t is less than 1, the series n=oo is convergent; if * > 1, the series is divergent. If t =1, the character of the- series is not determined by this test. XVIII, § 162] INFINITE SERIES 243 Pkoof fob convergent series. Let fc be a number lying between t and 1 (Fig. 42). If f = lim (m„+i/m„) is less than 1, there will be a value m of n such that m,„+i/m„ and all subsequent ratios will be less than h, that is 1 l-M — O tk 1 Fig. 42. (17) M„+i < Teu„, w„+2 < S;m„+i , m„+3 < fcM^^rj, • • •. When ■M„+i, M„+2> Wm+3, ••• are eliminated from the second, third, fourth, — , inequalities, the set (17) may be written in the form The inequalities (18) state that beginning with the (m + l)st term the terms of the series Sm,- are less than the correspond- ing terms of the geometric series (19) leu„ + khi„ + l^u„ + ..: When fc < 1 the series (19) is convergent, and consequently the series Sm^ is convergent. Pkoof foe divebgbnt series. By a process of reasoning similar to that used in the first part of the proof it may be shown that when < > fc > 1, and further, that It follows that from the with term onward, the terms of the series Smj are greater than the terms of the divergent geo- metric series ku^ + fc%„ + ¥u„ + -. Hence the series is divergent. When t=l the test fails, since t=l for some convergent and for some divergent series. For example, the series 244 COLLEGE ALGEBRA [XVIII, § 162 i + i4.1+... 13 ^ 23 33 for wMch t = l, is convergent, while the series 1+2+3+ ' for which * = 1, is divergent. (See § 161.) EXERCISES Determine the character of the following series. 1. J-.l + J- .i + J_.i+.... .. ' 1.2 22.3 22 3.4 28 2 JL 3 , J_ 3^, J_ 3f ■ 1.2 2 2-3 22 3-4 2' " " 3. J- + J- + J-+.... i. ^.l+-L.L + J^L+.... 1-3 3.5 5.7 1-3 2 3-522 5.728 5. 1 + 1 + 1 + 1+.... 6. J-l + LlllJ+l-2-3-^+.... 3 32 32 32 10002 10003 ^ 1000* 163. Series with Negative, or with Imaginary Terms. A series Ul + U2 + U3+ — composed of positive and negative terms, or of complex terms, is said to be absolutely convergent, if the series (20) 'ImjI+Im^I + ImjI + •-, formed of the absolute values of the terms of the original series, is convergent. If the original series is convergent but the series of absolute terms is divergent, the original series is said to be semi-convergent. Example 1. The series l_l + i_l + l... 2 4 8 16 is absolutely convergent since the series of absolute values 1+1+1+1+J_+, 24^816^ is convergent. XVIII, § 164] INFINITE SERIES 245 Example 2. The series 2 3 4 is semi-convergent since the series of absolute values 1 + 1+1 + 1+... 2^3 4 is divergent. Example 3. The series 1+^- + -^ + ?_+... 3 + 4i (3 + 40^ (3+4 i)s is absolutely convergent since the series of absolute values is the conver- gent geometric series 1+1 + 1 + 1+.... 5 52 58 164. Power Series. A series -whose terms are power func- tions with increasing exponents is called a power series. Such a series has the form (21) a„ + a^x + a^x^ -| 1- a„x^ -\ . A power series may be convergent for some values of x and divergent for others. The test-ratio test is the most useful means for determining the values of a; for which the series is convergent or divergent, For example, one of the simplest power series is the series The general term is nx" and the test-ratio is (n -f- l)a;"+i/(wa;"), which reduces to (n + V)x/n. The limit of the test-ratio is therefore x. Consequently, the series is convergent for 1^ ^ ^ Fig. 43. I X I < 1. The iaequality | a; |<1 is satisfied for every value of x which lies between -1- 1 and — 1, Fig. 43 ; or, as it is usually expressed, for values of x such that — 1 < a; < +1. 246 COLLEGE ALGEBRA [XVIII, § 164 The test-ratio for the general power series is a^^^x/a^. In order that the series may be convergent, it is necessary that or that ^ = Iim a' < lim-^ The last inequality may be written in the form (22) — lira < a;i=o n a a 1.2.3-(w-l) \a) In order that the series may be convergent, the absolute value of this limit must be less than 1, that is (26) < 1, or — a < a; < a, if a is positive. The series (24) represents the expansion of (1 + x/a)'" when, and only when, the condition (26) is satisfied. 166. The Exponential Series. In § 136 the expression lim (1 + x/ny occurred, and it was there denoted by e'. It can' be proven rigorously that It is easily shown that this power series is convergent for all values of x. (See Ex. 3, § 164.) ■ 248 COLLEGE ALGEBRA [XVIII, § 167 167. The Logarithmic Series. The computation of loga- rithms depends directly, or indirectly, upon the series (28) i„g,(i+«,)=^_|%|«_^+.... This power series converges (Ex. 7, § 164) for values of x such that - 1 < a; < + 1. The series (28) converges slowly, that is, it requires many terms to give a good approximation to the value of log (1 + x). A series which converges much more rapidly may be obtained by subtracting from (28) the series (29) log,(l-a;) = -%-|-|-J-..., which is obtained from (28) by changing the sign of x. The new series thus obtained is (30, ,„,.l±_- = .[. + | + |+...]. The series (30) converges rapidly for values of x lying between — 1 and + 1. If, for example, x = 1/10, three terms will give the Napierian logarithm of 11/9 to the sixth decimal place. We find log,ll/9 = 0.200671. To find the common logarithm of 11/9, the result just ob- tained must be multiplied by the modulus for reducing Napier- ian to common logarithms. This modulus is, to the sixth decimal place, (31) Jf= 0.434294. It follows that logio (11/9) = 0.087149. 168. Series as a Means of Computation. One of the most important uses to which series are put is that of finding the approximate numerical values of functions whose values other- wise can be found only with great difficulty, or perhaps not at XVIII, § 168] INFINITE SERIES 249 all. A few examples will make the matter clear. The use of series in the computation of logarithms was shown in § 167. Among the most important functions in. elementary mathe- matics are the sine and cosine of an angle. If the angle x be measured in radians, the sine of x is given by the power series ntO /nO rpl (32) sin. = .-|^ + |^-Hj + .... The sine of one radian, which is approximately 57°.29578, is given by the series sinl = l- — + — -- + -. 3! 5! 7! Tour terms give sin 1 = .8415, a result that is correct as far as it goes, since further terms would not affect the fourth decimal. To find sin .1 to the fourth decimal place only two terms of . the series, would be required. Another series of very great iniportanee in both pure and applied mathematics is the power series which represents the exponential function. This series is Consequently (33) e' = l+-+^ + — + -- K ) 1! 2! 3! Ten terms of the series (33) give to the nearest ten-millionth, 6 = 2.7182818. This result is correct to the seventh decimal place. Eoots of e, such as Ve = e-' and Ve = e-^, may be computed even more rapidly. • 250 COLLEGE ALGEBRA [XVIII, § 168 EXERCISES 1. Find approximate numerical values for each of the following func- tions given by power series. (a) cos sc = 1 - — H — — when x=l radian or 57°.29578. ^^ 1.2 1.2.3-4 ^ ' 11.2 1.2.3 (c) Ve = e-i. 2. Compute the value of (1 + .06)Vi2 to the fourth decimal place. 3. Extract the seventh root of 1.02 by the binomial series. 4. Extract the seventh root of .99 by the binomial series. 6. By means of the series (30) of § 167, and the modulus, 21f=0. 434294, find the Napierian and the common logarithms of |. 6. Find the Napierian and the common logarithms of 3. 7. In trigonometry it is shown that and that - = tan-i - + tan-i - + tan-i - , 4 2 5 8' /wS /i*5 rt"? tan-la; = a; -- + ±- _ ±-. + 3 5 7 From these data find the value of ir to four decimal places. TABLES Table A. Four- place -Looarithms op Numbers Table B. Powers and Roots .... Table C. Important Constants . Table I). Compound Interest Table Table E. Compound Discount Table Table F. American Experience Table of Mortality Paae 252 254 255 256 257 258 251 252 Table A — Four- Place Logarithms of Numbers N 1 2 3 4 5 6 •r 8 9 12 3 4 6 6 7 8 9 10 0000 0043 0086 0128 0170 ,0212 0253 0294 0334 0374 4 812 17 21 25 29 33 37 11 12 13 14 IS 16 17 18 19 0414 0792 1139 1461 1761 2041 2304 2553 2788 0453 0828 1173 1492 1790 2068 2330 2577 2810 0492 0864 1206 1523 1818 2095 2355 2601 2833 0531 0899 1239 1553 1847 2122 2380 2625 2856 0569 0934 1271 1584 1875 2148 2405 2648 2878 0607 0969 1303 1614 1903 2175 2430 2672 2900 0645 1004 1335 1644 1931 2201 2455 2095 2923 0682 1038 1367 1673 1959 2227 2480 2718 2945 0719 1072 1399 1703 1987 2253 2504 2742 2967 0755 1106 1430 1732 2014 2279 2529 2765 2989 4 811 3 710 3 610 3 6 9 3 6 8 3 5 8 2 5 7 2 5 7 2 4 7 15 19 23 14 17 21 13 16 19 12 15 18 11 14 17 11 13 16 10 12 16 9 1214 9 1113 26 30 34 24 28 31 23 26 29 2124 27 20 22 25 18 2124 17 20 22 16 19 21 1618 20 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 1113 15 17 19 21 22 23 24 25 26 27 28 29 3222 3424 3617 3802 3979 4150 4314 4472 4624 3243 3444 3636 3820 3997 4166 4330 4487 4639 3263 3464 3665 3838 4014 4183 4346 4502 4654 3284 3483 3674 3856 4031 4200 4362 4518 4669 3304 3502 3692 3874 4048 4216 4378 4533 4683 3324 3522 3711 3892 4065 4232 4393 4548 4698 3345 3541 3729 3909 4082 4249 4409 4564 4713 3365 3560 3747 3927 4099 4265 4425 4579 4728 3385 3579 3766 3946 4116 4281 4440 4594 4742 3404 3598 3784 3962 4133 4298 4456 4609 4757 2 4 6 2 4 6 2 4 6 2 4 6 2 4 5 2 3 5 2 3 5 2 3 5 13 4 8 1012 8 10 12 7 9 11 7 9 11 7 9 10 7 8 10 6 8 9 6 8 9 6 7 9 14 16 18 14 16 17 13 15 17 12 14 16 12 14 16 111315 11 12 14 11 12 14 10 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 13 4 6 7 9 10 11 13 31 32 33 34 35 36 37 38 39 4914 5051 5185 5315 5441 5563 5682 5798 5911 4928 5065 5198 5328 5453 5575 5694 5809 5922 4942 5079 5211 5340 5465 5587 6705 6821 5933 4955 6092 5224 5353 5478 5599 5717 6832 5944 4969 5105 5237 5366 5490 5611 5729 5843 5955 4983 5119 8250 5378 5502 6623 5740 5855 5966 4997 5132 6263 5391 5514 5635 6752 5866 6977 5011 5145 5276 5403 5527 5647 5763 5877 5988 5024 5159 5289 6416 5539 5658 5775 5888 6999 5038 6172 5302 5428 6551 5670 6786 6899 6010 13 4 13 4 13 4 12 4 12 4 12 4 12 4 12 3 1 2 3 6 7 8 5 7 8 6 7 8 5 6 8 5 6 7 5 6 7 5 6 7 5 6 7 4 ,5 7 10 11 12 91112 91112 910 11 91011 81011 8 911 8 910 8 910 40 6021 6031 6042 6053 6064 6076 6086 6096 6107 6117 12 3 4 6 6 8 910 41 42 43 44 45 46 47 48 49 6128 6232 6335 6435 6532 6628 6721 6812 6902 6138 6213 6345 6444 6542 6637 6730 6821 6911 6149 6253 6356 6454 6551 6646 6739 6830 6920 6160 6263 6365 6464 6561 6656 6749 6839 6928 6170 6274 6375 6474 6571 6665 6758 6848 6937 618S 6284 6385 6484 6580 6676 6767 6857 6946 6191 6294 6395 6493 6590 6684 6776 6866 6955 6201 6304 6405 6503 6599 6693 6785 6875 6964 6212 6314 6415 6613 6609 6702 6794 6884 6972 6222 6325 6425 6522 6618 6712 6803 6893 6981 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 4 6 6 4 5 6 4 6 6 4 5 6 4 6 6 4 5 6 4 5 6 4 6 6 4 4 6 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 7 8 7 7 8 7 7 ,8 6 7 8 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 12 3 3 4 6 6 7 8 51 52 63 54 7076 7160 7243 7324 7084 7168 7251 7332 7093 7177 7259 7340 7101 7185 7267 7348 7110 7193 7276 7356 7118 7202 7284 7364 7126 7210 7292 7372 7135 7218 7300 7380 7143 7226 7308 7388 7152 7235 7316 7396 12 3 12 3 12 2 1 2 2 3 4 6 3 4 6 3 4 6 3 4 6 6 7 8 6 7 7 6 6 7 6 6 7 N 1 2 3 4 s 6 7 8 9 12 2 4 5 6 7 8 9 The proportional parts are stated in full for every tenth at the right-hand side, The logarithm of any number of four significant figures can be read directly by add' Table A — Four-Place Logarithms of Numbers 253 N 1 2 3 4 6 6 7 8 9 12 3 4 6 6 7 8 9 55 56 57 58 59 7404 7482 7559 7634 7709 7412 7490 7566 7642 7716 7419 7497 7574 7649 7723 7427 7505 7582 7657 7731 7435 7513 7589 7664 7738 7443 7520 7597 7672 7745 7451 7528 7604 7679 7752 7459 7536 7612 7686 7760 7466 7643 7619 7694 7767 7474 7651 7627 7701 7774 12 2 12 2 1 1 2 112 1 1 2 3 4 6 3 4 5 3 4 5 3 4 4 3 4 4 5 6 7 5 6 7 5 6 7 5 6 7 5 6 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 112 3 4 4 6 6 6 61 62 63 64 65 66 67 68 69 7853 7924 7993 8062 8129 8195 8261 8325 8388 7860 7931 8000 8069 8136 8202 8267 8331 8395 7868 7938 8007 8075 8142 8209 8274 8338 8401 7875 7945 8014 8082 8149 8215 8280 8344 8407 7882 7952 8021 8089 8156 8222 8287 8351 8414 7889 7959 8028 8096 8162 8228 8293 8357 8420 7896 7966 8035 8102 8169 8235 8299 8363 8426 7903 7973 8041 8109 8176 8241 8306 8370 8432 7910 7980 8048 8116 8182 8248 8312 8376 8439 7917 7987 8055 8122 8189 8254 8319 8383 8445 1 1 2 1 1 2 112 112 1 1 2 112 112 112 112 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 5 6 6 5 5 6 5 5 6 6 5 6 6 5 6 6 6 6 5 5 6 4 5 6 4 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 112 3 3 4 4 5 6 71 72 73 74 75 76 77 78 79 8513 8573 8633 8692 8751 8808 8865 8921 8976 8519 8579 8639 8698 8756 8814 8871 8927 8982 8525 8585 8645 8704 8762 8820 8876 8932 8987 8531 8591 8651 8710 8768 8825 8882 8938 8993 8537 8S97 8657 8716 8774 8831 8887 8943 8998 8543 8603 8663 8722 8779 8837 8893 8949 9004 8549 8609 8669 8727 8785 8842 8899 8954 9009 8555 8615 8675 8733 8791 8848 8904 8960 9015 8561 8621 8681 8739 8797 8854 8910 8965 9020 8567 8627 8686 8745 8802 8859 8915 8971 9025 1 1 2 112 1 1 2 112 112 112 1 1 2 112 112 3 3 4 3 3 4 2 3 4 2 3 4 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 4 5 6 4 5 6 4 5 5 4 5 5 4 6 6 4 4 5 4 4 5 4 4 5 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 112 2 3 3 4 4 5 81 82 83 84 85 86 87 88 89 9085 9138 9191 9243 9294 9345 9395 9445 9494 9090 9143 9196 9248 9299 9350 9400 9450 9499 9096 9149 9201 9253 9304 9355 9405 9455 9504 9101 9154 9206 9258 9309 9360 9410 9460 9509 9106 9159 9212 9263 9315 9365 9415 9465 9513 9112 9165 ■9217 9269 9320 9370 9420 9469 9518 9117 9170 9222 9274 9325 9375 9425 9474 9523 9122 9175 9227 9279 9330 9380 9430 9479 9528 9128 9180 9232 9284 9335 9385 9435 9484 9533 9133 9186 9238 9289 9340 9390 9440 9489 9538 112 112 112 1 1 2 112 112 112 Oil 1 1 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 2 3 2 2 3 4 4 5 4 4 5 4 4 5 4 4 6 4 4 5 4 4 5 4 4 6 3 4 4 3 4 4 90 9542 9547 9552 mbi 9562 9566 9571 9576 9581 9586 Oil 2 2 3 3 4 4 91 92 93 94 95 96 97 98 99 9590 9638 9685 9731 9777 9823 9868 9912 9956 9595 9643 9689 9736 9782 9827 9872 9917 9961 9600 9647 9694 9741 9786 9832 9877 9921 9965 9605 9652 9699 9745 9791 9836 9881 9926 9909 9609 9657 9703 9750 9795 9841 9886 9930 9974 9614 9661 9708 9754 9800 9845 9890 9934 9978 9619 9666 9713 9759 9805 9850 9894 9939 9983 9624 9671 9717 9763 9809 9854 9899 9943 9987 9628 9675 9722 9768 9814 9859 9903 9948 9991 9633 9680 9727 9773 9818 9863 9908 9952 Oil Oil Oil Oil Oil 1 1 1 1 Oil Oil 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3- 2 2 3 2 2 3 2 2 3 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 t 4 3 3 4 3 3 4 N 1 2 3 4 5 6 ■ 7 8 9 12 3 4 5 7 8 9 ing the proportional part corresponding to the fourth figure to the tabular numher corresponding to the first three figures. There may he an error of 1 in the last place. 254 Table B — Powers and Boots Squares and Cubes Square Boots and Cube Roots No. S«TrARE CUBB SQrABE KoOT Cube EOOT No. Squase CrsE Sqfaee KooT Odbk Root ■ 1 1 1 1.000 1.000 51 2,601 132,661 7.141 3.708 2 4 8 1.414 1.260 52 2,704 140,608 7.211 3.733 3 9 27 1.732 1.442 53 2,809 148,877 7.280 3.756 4 16 64 2.000 1.587 54 2,916 157,464 7.348 3.780 5 25 125 2.236 1.710 55 3,025 166,375 7.416 3.803 6 36 216 2.449 1.817 56 3,136 175,610 7.483 3.826 7 49 343 2.646 1.913 57 3,249 185,193 7.550 3.849 8 61 512 2.828 2.000 58 3,364 196,112 7.616 3.871 9 ' 81 729 3.000 2.080 59 3,481 206,379 7.681 3.893 10 100 1,000 3.162 2.154 60 3,600 216,000 7.746 3.916 11 121 1,331 3.317 2.224 61 8,721 226,981 7.810 3.936 12 144 1,728 3.464 2.289 62 3,844 238,328 7.874 3.968 13 169 2,197 3.606 2.351 63 3,969 250,047 7.937 3.979 14 196 2,744 3.742 2.410 64 4,096 262,144 8.000 4.000 IS ~ 225 3,375 3.873 2.466 65 4,225 274,625 8.062 4.021 16 256 4,096 4.000 2.520 66 4,356 287,496 8.124 4.041 17 289 4,913 4.123 2.571 67 4,489 300,763 8.185 4.062 IS 324 5,832 4.243 2.621 68 4,624 314,432 8.246 4.082 19 361 6,859 4.359 2.668 69 4,761 328,609 8.307 4.102 20 400 8,000 4.472 2.714 70 4,900 343,000 8.367 4.121 21 441 9,261 4.583 2.759 71 5,041 357,911 8.426 4.141 22 484 10,648 4.690 2.802 72 5,184 373,248 8.485 4.160 23 529 12,167 4.796 2.844 73 5,329 389,017 8.544 4.179 24 576 13,824 4.899 2.884 74 6,476 406,224 8.602 4.198 25 625 15,625 5.000 2.924 75 5,625 421,876 8.660 4.217 26 676 17,576 5.099 2.962 76 5,776 438,976 8.718 4.236 27 729 19,683 5.196 3.000 77 5,929 466,533 8.775 4.254 28 784 21,952 5.292 3.037 78 6,084 474,552 8.832 4.273 29 841 24,389 5.385 3.072 79 6,241 493,039 8.888 4.291 30 900 27,000 5.477 3.107 80 6,400 512,000 8.944 4.309 31 961 29,791 5.568 , 3.141 81 6,561 631,441 9.000 4.327 32 1,024 32,768 5.657 3.175 82 6,724 651,368 9.055 4.344 33 1,089 35,937 5.745 ■3.208 83 6,889 571,787 9.110 4.362 34 1,156 39,304 5.831 3.240 84 7,056 692,704 9.165 4.380 35 1,225 42,875 5.916 3.271 85 7,225 614,125 9.220 4.397 36 1,296 46,656 6.000 3.302 86 7,396 636,056 9.274 4.414 37 1,369 60,653 6.083 3.332 87 7,569 668,503 9.327 4.431 38 1,444 54,872 6.164 3.362 88 7,744 681,472 9.381 4.448 39 1,521 59,319 6.245 3.391 89 7,921 704,969 9.434 4.465 40 1,600 64,000 6.325 3.420 90 8,100 729,000 9.487 4.481 41 1,681 68,921 6.403 3.448 91 8,281 763,571 9.639 4.498 42 1,764 74,088 6.481 3.476 92 8,464 778,688 9.592 4.514 43 1,849 79,507 6,557 3.503 93 8,649 804,357 9.644 4.531 44 1,936 85,184 6.633 3.530 94 8,836 830,584 9.695 4.547 45 2,025 91,125 6.708 3.557 95 9,025 857,375 9.747 4.563 46 2,116 97,336 6.782 3.583 96 9,216 884,736 9.798 4.579 47 2,209 103,823 6.856 3.609 97 9,409 912,673 9.849 4.595 48 2,304 110,592 6.928 3.634 98 9,604 941,192 9.899 4.610 49 2,401 117,649 7.000 3.659 99 9,801 970,299 9.950 4.626 SO 2,500 125,000 7.071 3.684 100 ^0,000 1,000,000 10.000 4.642 For a more complete table, see Thb Macmillan Tables, pp. 94-111. Table C — Important Constants Cbrtain Convenient Values for n = 1 to n 255 = 10 n 1/ft Vn VtT n\ l/« ! ' . LoGio n 1 1.000000 1.00000 1.00000 1 1.0000000 0.000000000 2 0.500000 1.41421 1.25992 2 0.5000000 0.301029996 3 0.333333 1.73205 1.44225 6 0.1666667 0.477121255 4 0.250000 2.00000 1.58740 24 0.0416667 0.602059991 S 0.200000 2.23607 1.70998 120 0.0083333 0.698970004 6 0.1()6667 2.44949 1.81712 ■720 0.0013889 0.778151250 7 0.142857 2.64575 1.91293 5040 0.0001984 0.845098040 8 0.125000 2.82843 2.00000 40320 0.0000248 0.903089987 9 0.111111 3.00000 2.08008 362880 0.0000028 0.954242509 10 0.100000 3.16228 2.15443 3628800 0.0000003 1.000000000 Logarithms of Important Constants 71. = NUiMBKR ' Value op n LoGio n IT 3.14159265 0.49714987 l-^JT 0.31830989 9.50285013 7r2 9.86960440 0.99429975 V? , 1.77245385 0.24857494 e = Napierian Base 2.71828183 0.43429448 .M= logio e 0.43429448 9.63778431 lH-ilf=logelO 2.30258509 0.36221569 180 -i- TT = degrees in 1 radian 57.2957795 1.75812262 TT -> 180 = radians in 1° 0.01745329 8.24187738 ir -4- 10800 = radians in V 0.0002908882 6.4H372613 T ^ 648000 = radians in 1" 0.000004848136811095 4.68557487 sin 1" 0.000004848136811076 4.68557487 tan 1" 0.000004848136811152 4.68557487 centimeters in 1 ft. 30.480 1.4840158 feet in 1 cm. 0.032808 8.5159842 inches in 1 m. 39.37 (exact legal value) 1.5951654 pounds in 1 kg. 2.20462 0.3433340 kilograms in 1 lb. 0.453593 9.65666(i0 g (average value) 32.16 ft./sec./sec. = 981 cm./sec./sec. '1.5073 2.9916690 weight of 1 cu. ft. of water 62.425 lb. (max. density) 1.7953586 weight of 1 cu. ft. of air 0.0807 lb. (at 32° F.) 8.907 cu. in. in 1 (U. S.) gallon 231 (exact legal value) 2.3636120 ft. lb. per sec. in 1 H. P. 550." (exact legal valu )) 2.7403627 kg. m. per' sec. in 1 H. P. 76.0404 1.8810445 watts in 1 H. P. 745.957 2.8727135 256 Table D — Compound Interest Table Amount of One Dollak Peincipal with Compound Intebebt at Vabious Rates Yeabb 2J% 3% 3i% 4% 4J% 5% 5J% 6% 6J% 7% 8% 1 S1.02S $1,030 $1,036 $1,040 $1,045 $1,050 $1,055 $1,060 $1,066 $1,070 $1,080 2 1.051 1.061 1.071 1.082 1.092 1.103 1.113 1.124 1.134 1.146 1.166 3 1.077 1.093 1.109 1.125 1.141 1.158 1.174 1.191 1.208 1;225 1.260 4 1.104 1.126 1.148 1.170 1.193 1.216 1.239 1.262 1.286 1.311 1.360 5 1.131 1.159 1.188 1.217 1.246 1.276 1.307 1.338 1.370 1.403 1.469 6 1.160 1.194 1.229 1.265 1.302 1.340 1.379 1.419 1.459 1.601 1.587 7 1.189 1.230 1.272 1.316 1.361 1.407 1.455 1.504 1.664 1.606 1.714 8 1.218 1.267 1.317 1.369 1.422 1.477 1.635 1.594 1.655 1.718 1.861 9 1.249 1.305 1.363 1.423 1.486 1.651 1.619 1.689 1.763 1.838 1.999 . 10 1.280 1.344 1.411 1.480 1.553 1.629 1.708 1.791 1.877 1.967 2.159 11 1.312 1.384 1.460 1.539 1.623 1.710 ■ 1.802 1.898 1.999 2.105 2.332 12 1.345 1.426 1.511 1.601 1.696 1.796 1.901 2.012 2.129 2.262 2.618 13 1.379 1.469 1.564 1.666 1.772 1.886 2.006 2.13S 2.267 2.410 2.720 14 1.413 1.613 1.619 1.732 1.852 1.980 2.116 2.261 2.415 2.579 2.937 15 1.448 1.558 1.676 1.801 1.935 2.079 2.232 2.397 2.572 2.759 3.172 16 1.485 1.605 1.734 1.873 2.022 2.183 2.366 2.640 2.739 2.952 3.426 ^ 17 1.522 1.653 1.795 1.948 2.113 2.292 2.485 2.693 2.917 3.169 3.700 i IS 1.560 1.702 1.867 2.026 2.208 2.407 2.621 2.864 3.107 3.380 3.996 19 1.599 1.764 1.923 2.107 2.308 2.527 2.766 3.026 3.309 3.617 4.316 20 1.639 1.806 1.990 2.191 2.412 2.663 2.918 3.207 3.624 3.870 4.661 21 1.680 1.860 2.059 2.279 2.520 2.786 3.078 3.400 3.753 4.141 6.034 22 1.722 1.916 2.132 2.370 2.634 2.925 3.248 3.604 3.997 4.430 6.437 23 1.765 1.974 2.206 2.466 2.752 3.072 3.426 3.820 4.256 4.741 5.871 24 1.809 2.033 2.283 2.563 2.876 3.225 3.615 4.049 4.633 5.072 6.341 25 1.854 2.094 2.363 2.666 3.006 3.386 3.813 4.292 4.828 5.427 6.848 26 1.900 2.157 2.446 2.772 3.141 3.556 4.023 4.649 6.142 5.807 7.396 27 1.948 2.221 2.532 2.883 3.282 3.733 4.244 4.822 6.476 6.214 7.988 28 1.996 2.288 2.620 2.999 3.430 3.920 4.478 6.112 6.832 6.649 8.627 29 2.046 2.357 2.712 3.119 3.584 4.116 4.724 5.418 6.211 7.114 9.317 30 2.098 2.427 2.807 3.243 3.745 4.322 4.984 6.743 6.614 7.612 10.063 31 2.150 2.500 2.905 3.373 3.^14 4.538 5.258 6.088 7.044 8.145 10.868 32 2.204 2.575 3.007 3.508 4.090 4.765 5.647 6.463 7.602 8.715 11.737 33 2.259 2.652 3.112 3.648 4.274 5.003 5.852 6.841 7.990 9.326 12.676 34 2.316 2.732 3.221 3.794 4.466 5.263 6.174 7.261 8.509 9.978 13.690 35 2.373 2.814 3.334 3.946 4.667 6.616 6.514 7.686 9.062 10.677 14.786 36 2.433 2.898 3.450 4.104 4.877 5.792 6.872 8.147 9.661 11.424 16.968 37 2.493 2.986 3.671 4.268 5.097 6.081 7.250 8.636 10.279 12.224 17.246 38 2.556 3.075 3.696 4.439 5.326 6.385 7.649 9.154 10.947 13.079 18.625 39 2.620 3.167 3.826 4.616 5.566 6.705 8.069 9.704 11.668 13.995 20.115 40 2.685 3.262 3.959 4.801 5.816 7.040 8.513 10.286 12.416 14.974 21.725 41 2.752 3.360 4.098 4.993 6.078 7.392 8.982 10.903 13.223 16.023 23.462 42 2.821 3.461 4.241 5.193 6.352 7.762 9.476 11..557 14.083 17.144 25.339 43 2.892 3.666 4.390 6.400 6.637 8.160 9.997 12.260 14.998 18.344 27.367 44 2.964 3.671 4.543 5.617 6.936 8.667 10.546 12.985 15.973 19.628 29.666 45 3.038 3.782 4.702 5.841 7.248 8.985 11.127 13.765 17.011 21.002 31.920 46 3.114 3.896 4.867 6.075 7.574 9.434 11.739 14.590 18.117 22.473 34.474 47 3.192 4.012 5.037 6.318 7.915 9.906 12.384 16.466 19.294 24.046 37.232 48 3.271 4.132 6.214 6.671 8.271 10.401 13.065 16.394 20.649 25.729 40.211 49 3.353 4.266 6.396 6.833 8.644 10.921 13.784 17.378 21.884 27.530 43.427 60 3.437 4.384 5.585 7.107 9.033 11.467 14.542 18.420 23.307 29.467 46.902 Table E — Compound Discount Table 267 Present VALtjE of One Dollar Dub in a Certain Number op Years Yeabs 2i% 3% 3i% 4% 45% 5% 55% 6% 6*% 7% 8% 1 $.9756 $.9709 $.9662 $.9615 $.9569 $.9524 $.9479 $.9434 $.9390 $.9346 $.9259 2 .9518 .9426 .9335 .9246 .9157 .9070 .8985 .8900 .8817 .8734 .8573 3 .9286 .9151 .9019 .8890 .S763 .8638 -.8516 .8396 .8278 .8163 .7938 4 .9060 .8885 .8714 .8548 .8386 .8227 .8072 .7921 .7773 .7629 .7350 6 .8839 .8626 .8420 .8219 .8025 .7835 .7651 .7473 .7299 .7130 .6806 6 .8623 .8375 .8135 .7903 .7679 .7462 .7253 .7050 .6853 .6663 .6302 7 .8413 .8131 .7860 .7599 .7348 .7107 .6874 .6651 .6435 .6227 .5835 8 .8207 .7894 .7594 .7307 .7032 .6768 .6516 .6274 .6042 .5820 .5403 9 .8007 .7664 .7337 .7026 .6729 .6446 .6176 .5919 .5674 .5439 .5002 10 .7812 .7441 .7089 .6756 .6439 .6139 .5854 .5684 .5327 .5083 .4632 11 .7621 .7224 .6849 .6496 .6162 .5847 .5549 .5268 .5002 .4751 .4289 12 .7436 .7014 .6618 .6246 .5897 .5568 .5260 .4970 .4697 .4440 .3971 13 .7254 .6810 .6394 .6006 .5643 .5303 .4986 .4688 .4410 .4150 .3677 14 .7077 .6611 .6178 .5775 .5400 .5051 .4726 .4423 .4141 .3878 .3405 16 .6905 .6419 .5969 .5553 .5167 .4810 .4479 .4173 .3888 .3624 .3152 16 .6736 .6232 .5767 .5339 .4945 .4581 .4246 .3936 .3651 .3387 .2919 17 .6572 .6050 .5572 .5134 .4732 .4363 .4025 .3714 .3428 .3166 .2703 18 .6412 .5874 .5384 .4936 .4528 .4155 .3815 .3503 .3219 .2959 .2502 19 .6255 .5703 .5202 .4746 .4333 .3957 .3616 .3305 .3022 .2765 .2317 20 .6103 .5537 .5026 .4564 ' .4146 .3769 .3427 .3118 .2838 .2584 .2145 21 .5954 .5375 .4856 .4388 .3968 .3589 .3249 .2942 .2665 .2415 .1987 22 .5809 .5219 .4692 .4220 .3797 .3418 .3079 .2775 .2502 .2257 .1839 23 .5667 .5067 .4533 .4057 .3634 .3256 .2919 .2618 .2349 .2109 .1703 24 .5529 .4919 .4380 .3901 .3477 .3101 .2767 .2470 .2206 .1971 .1577 25 .5394 .4776 .4231 .3751 .3327 .2953 .2622 .2330 .2071 .1843 .1460 26 .5262 .4637 .4088 .3607 .3184 .2812 .2486 .2198 .1945 .1722 .1352 27 .5134 .4502 .3950 .3468 ,3047 .2678 .2356 .2074 .1826 .1609 .1252 28 .5009 .4371 .3817 .3335 .2916 .2551 .2233 .1956 .1715 .1504 .1159 29 .4887 .4243 .3687 .3207 .2790 .2429 .2117 .1846 .1610 .1406 .1073 30 .4767 .4120 .3563 .3083 .2670 .2314 .2006 .1741 .1512 .1314 .0994 31 .4651 .4000 .3442 .2965 .2555 .2204 .1902 .1643 .1420 .1228 .0920 32 .4538 .3883 .3326 .2851 .2445 .2099 .1803 .1550 .1333 .1147 .0852 33 .4427 .3770 .3213 .2741 .2340 .1999 .1709 .1462 .1252 .1072 .0789 34 .4319 .3660 .3105 .2636 .2239 .1904 .1620 .1379 .1175 .1002 .0730 35 .4214 .3554 .3000 .2534 .2143 .1813 .1535 .1301 .1103 .0937 .0676 36 .4111 .3450 .2898 .2437 ■ .2050 .1727 .1455 .1227 .1036 .0875 .0626 37 .4011 .3350 .2800 .2343 .1962 .1644 .1379 .1158 .0973 .0818 .0580 38 .3913 .3252 .2706 .2253 .1878 .1556 .1307 .1092 .0914 .0765 .0537 39 .3817 .3158 .2614 .2166 .1797 .1491 .1239 .1031 .0858 .0715 .0497 40 .3724 .3066 .2526 .2083 .1719 .1420 .1175 .0972 .0805 .0668 .0460 41 .3633 .2976 .2440 .2003 .1645 .1353 .1113 .0917 .0756 .0624 .0426 42 .3545 .2890 .2358 .1926 .157* .1288 .1055 .0865 .0710 .0583 .0395 43 .3458 .2805 .2278 .1852 .1507 .1227 .1000 .0816 .0667 .0545 .0365 44 .3374 .2724 .2201 .1781 .1442 .1169 .0948 .0770 .0626 .0509 .0338 45 .3292 .2644 .2127 .1712 .1380 .1113 .0899 .0727 .0588 .0476 .0313 46 .3211 .2567 .2055 .1646 .1320 .1060 .0852 .0685 .0552 .0445 .0290 47 .3133 .2493 .1985 .1583 .1263 .1010 .0808 .0647 .0518 .0416 .0269 43 .3057 .2420 .1918 .1.522 .1209 .0961 .0765 .0610 .0487 .0389 .0249 49 .2982 .2350 .1853 .1463 .1157 .0916 .0726 .0576 .0457 .0363 .0230 50 .2909 .2281 .1791 .1407 .1107 .0872 .0688 .0543 .0429 .0339 .0213 258 Table F— American Experience Table of Mortality Ags Nttmbeb Living Number Dying Age Number Living Number Dying Age Number Living Number Dying X k dj ae Ix dx X Ix dx 10 11 12 13 14 100 000 99 251 98 505 97 762 97 022 749 746 743 740 737 40 41 42 43 44 78106 77 341 76 567 75 782 74 985 765 774 785 797 812 70 71 72 73 74 38 569 36178 33 730 31243 28 738 2391 '2448 2487 2505 2501 15 16 17 18 19 96 285 95 550 94 818 94 089 93 362 735 732 729 727 725 46 46 47 48 49 74173 73 345 72 497 71627 70 731 828 848 870 896 927 75 76 77 78 79 26 237 23 761 21330 18 961 16 670 2476 2431 2369 2291 2196 20 21 22 23 24 92 637 91914 91192 90 471 89 761 723 722 721 720 719 50 51 52 63 64 69 804 68 842 67 841 66 797 65 706 962 1001 ,1044 1091 1143 80 81 82 83 84 14 474 12 383 10 419 8 603 6 955 2091 1964 1816 1648 1470 26 26 27 28 29 89.032 88 314 87 596 86 878 86 160 718 718 718 718 719 66 56 57 58 69 64 563 63 364 62 104 60 779 59 385 1199 1269 1325 1394 1468 86 86 87 88 89 5 485 4193 3 079 2 146 1402 1292 1114 933 744 . 555 30 31 32 33 34 85 441 '84 721 84 000 83 277 82 551 720 721 723 726 729 60 61 62 63 64 57 917 56 371 54 743 53 030 51230 1546 1628 1713 1800 1889 90 91 92 93 94 847 462 216 79 21 385 246 137 58 18 36 36 37 38 39 81822 81090 80 353 79 611 78 862 732 737 742 749 756 65 66 67 68 69 49 341 47 361 45 291 43 133 40 890 1980 2070 2158 2243 2321 95 3 3 INDEX Abel, 143. Abscissa, 52. Absolute term of quadratic equa- tion, 79. Absolute value, 127. Acceleration, 6, 69. Addition, 2, 9 ; geometric or vector, 128 ; of complex numbers, 127 ; of radicals, 29. Algebraic function, 225 ; limit of, 225. Algorithm, Euclidean, 155. AUoys, 102. American Experience Table, 217, 258. Amortization, 200. Amount, compound, 196, 197 ; table for compound, 256. Amplitude of complex number, 127. Annuity, 198 ; amount of, 198 ; present value of, 199. Antilogarithm, 44. Approximate value, of radicals, 31 ; of real roots, 136. Arrangements, 159. Associative Law, 7. Average, simple, 193 ; weighted, 194. Axis, of a function, 52 ; of imaginary numbers, 126 ; of real numbers, 126 ; of variables, 52. Bacteria, growth of, 204. Baikstow, form of Bryan's aero- plane equations, 148. Base, 22, 26. Beam, built in, 147 ; reenforced concrete, 148. Binomial theorem, 179, 210. Biquadratic equation, graphic solu- tion of, 118; transformation of, 143. BocHEB, Higher Algebra, 176. Boiling point of water, 103. Boyle's law, 70. Beyan, aeroplane equations, 148. Burnside and Panton, Theory of Equations, 131, 136. Catjchy, General Principle of Con- vergence, 222. Center of gravity, 194. Certainty, 213. Characteristic of logarithm, 40; de- termination of, 41. Chemical compounds, 102. Circle, equation of, 102, 106. Coefficient, of radical, 26, 27 ; of simple term, 14. Coefficients, binomial, 181 ; in terms of roots, 88, 149 ; of polynomial, 131; of quadratic equation, 79; undetermined, 138. Co-factor, 164. Cologarithm, 44. Combination, 206. Combinations, formulas for, 209. Common denominator, reduction to, 4. Common difference, 185. Commutative law, 6, Comparison series, 240. Conaparison test, 238. Complex number, 35, 125 ; amplitude of, 127 ; geometric representation, 126 ; modulus of, 127. Complex numbers, 125-130 ; con- jugate, 127 ; fundamental oper- ations on, 127. Compound interest, 185, 196-205; instantaneous, 203 ; table for, 256. Compound interest law, 202. Computation, by series, 248 ; of determinants, 166 ; with loga- rithms, 45. Conditional equality, 12. Conic section, 106. Conjugate complex numbers, 127. 259 260 INDEX Conjugate radicals, 30. Consistency of linear equations, 1.74. Constants, table of important, 255. Convergence, absolute, 244 ; Catjchy's criterion for, 222, 238 ; comparison test for, 238 ; General Principle of, 222, 237 ; of sequence, 222 ; of series, criterion for, 237 ; test-ratio test for, 242. Cooling, law of, 204. Coordinates, 52. Coordinate paper, 54. Cosine, series for, 249. Cramer's Rule, 93, 100, 171. Cube roots, tables for, 32, 254. Cubic equation, graphic solution of, 117 ; transformation of, 142. D'Alembert, test-ratio test, 242. Degree, of a polynomial, 14 ; of a rational fraction, 14 ; of a simple term, 14. Denominator, common, 4 ; radicals in, 30. Depreciation, 200. Depreciation charge, 200. Depreciation fund, 200. Derivative, 156 ; geometric inter- pretation, 231 ; of polynomial, 230 ; of power function, 229. Descartes, rule of signs, 152. Determinant, 91, 158; application to systems of linear equations, 170 ; computation of, .166 ; expan- sion of, 159 ; general properties of, 160 ; of order 2, 91 ; of order 3, 99 ; of system of equations, 92, 170 ; rule of signs for, 1.59. Determinants and linear equations, 158-178. Dickson, Elementary Theory of Equa- tions, 131, 136, 177. 'Discount, 204 ; table for compound, 257. Discriminant of quadratic equation, 85. Distributive law, 7 ; generalized, 8. Division, 2, 3, 9 ; by zero, 11; har- monic, 195; of complex numbers, 127 ; of logarithms, 43 ; of radicals, 29 ; synthetic, 138. Du Bois Retmond, 222. Elements of determinant, 92. Eliminant, 172 ; use of in geometry, 172. Ellipse, 106 ; equation of, 108. Equation, conditional, 12 ; graphic solution of biquadratic, 118 graphic solution of cubic, 117; graphic solution of quadratic, 116 of fifth degree, 143 ; of uniform motion, 57, 59 ; quadratic, 79 quadratic in two variables, 106. Equations, containing radicals, 89 having common root, 177 ; systems linear in x^ and y^. 111 ; systems of linear, 91-104, 170-176; sys- tems of linear homogeneous, 172 ; systems of non-linear, 105-119; systems of symmetric, 113 ; sys- tems solved by various devices, 115; systems with terms of degree 2 or 0, 114; transformation of, 141. Euclidean algorithm, 155. Existence of a limit, 222. Expansion, adiabatic, 65 ; of deter- minant, 159. Expectation, mathematical, 218. Exponential function, 202 ; series for, 247, 249. Exponential series, 247. Exponents, 22 ; fractional, 24 ; laws for, 22 ; negative, 23 ; zero, 23. Factorial, 181. Factors, rational, 17 ; prime, 19. Factor theorem, 132. Fahrenheit thermometer, 58. Fine, College Algebra, 1, 177. Form, indeterminate, 227. Fractions, 3 ; geometric represents^ tion, 3 ; fundamental operations upon, 9. Function, 49-78; algebraic, 225; discontinuous, 70 ; exponential, 203, 247, 249; graph of, 53; implicit, 78; limiting value of, 223 ; linear integral, 54 ; power, 64, 229 ; quadratic integral, 60 ; rational integral, 131 ; with un- known coefficients, 101 ; zeros and infinities of, 77. INDEX 261 Fundamental laws, for exponents, 22 ; of algebra, 6-8. Fundamental operations, 2, 9 ; upon complex numbers, 127 ; upon radi- cals, 29. Fundamental theorem of algebra, 131. General Principle of Convergence, 222, 237. Geometric addition, 128. Geometric representation, 1, 2, 3 ; also see graph. Grade, 56. Graph, 49-78 ; of linear equation, 56 ; of linear quadratic system, lllf of polynomial, 133, 151 ; of quad- ratic equation in two variables, 106 ; of quadratic expression, 60 ; of system of quadratic equations, 112; statistical, 70. Graphic solution, 94 ; of biquadratic equation, 118; of cubic equation, 117; of inequality, 122; of linear quadratic system, 111 ; of quad- ratio equation, 116; of system of linear equations, 94 ; of system of quadratic equations, 112. Gravitation, 6, 69. Gravity, center of, 194: Gunter's scale, 47. Highest common factor, 19, 154 ; Euclidean algorithm for, 155. Homogeneous linear equations, 172. Hookb's Law, 58. Hokner's method, 143-146. Hyperbola, equation, of, 109 ; equi- lateral, 67. Identities, 12; important, 16. Imaginary number, 35, 125. Imaginary roots, 84, 116, 119, 150; interpretation of, 84. ' Impossibility, 213. Independent events, 215. Indeterminate form, 227. Index of radical, 26, 27. Index law, 22. Induction, mathematical, 179, 186, 189, 207, 208. Inequality, 3, 120-124 ; graphic solution, 122 ; solution of, 121 ; with two unknowns, 123. Infinite series, 235-250. Infinity, 66, 225 ; of a function, 77. Integer, positive, 1 ; negative, 2. Intercept, of curve, 110; of line, 57. Interest, compound, 6, 196; exact, 46 ; instantaneous compound, 203 ; ordinary, 46 ; simple, 5, 58. Interpolation, 74-76 ; of logarithms, 43. Inversion of subscripts, 158. Irrational numbers, 26 ; approxi- mate value for, 31. Kelvin, Lord, 203. Least common multiple, 4, 20. Lever, principle of, 94. Light, transmission of, 205. Limit, 192, 221, 224; criterion for existence of, 222 ; improper, 226 ; of algebraic function, 225; of function, 223. Linear equation, graph of, 56. Linear equations, 91-103, 158, 170- 177 ; application of determinants to, 170 ; homogeneous, 172 ; non- homogeneous, 174 ; with two un- knowns, 91 ; with three unknowns, 99. Linear-quadratic system, 105 ; graph- ical solution of. 111. Line through two points, 173. Literal symbols, 5. Loci through given points, 101. Logarithm, 37; base of, 38; charac- teristic of, 40 ; common or Briggs- ian, 39 ; mantissa of, 40 ; Na- pierian, 203, '248; of power, 39; of product, 38 ; of quotient, 38 ; of unity, 37. Logarithms, 37^8 ; four-place table, 252; multiplication and division of, 43 ; system of, 39 ; theorems relative to, 37-41. Logarithmic scale, 47. Logarithmic series, 248. Mantissa of logarithm, 40. Mathematical expectation, 218. 262 INDEX Mathematical induction, 179, 186, 189, 207, 208. Maueek, see Tubnbaure and Maubeb. Maximum, 62, 232. Mean, arithmetic, 193 ; geometric, 194; harmonic, 194. Means, arithmetic, 185 ; geometric, 190. Mebbiam, Textbook on Mechanics of Materials, 147. Minimum, 62, 232. Minor, 165. Minors and cofactors, 163-166. Modulus, of complex number, 127 ; of system of common logarithms, 248. Mortality table, 217; American Ex- perience, 258. Motion, accelerated, 64 ; uniform, 57. Multiple, common, 20 ; least com- mon, 20. Multiplication, 2, 9 ; of complex numbers, 127 ; of logarithms, 43 ; of radicals, 29. Napierian logarithms, 203, 248. Newton's law of cooling, 204. Number, complex, 35, 125; imagi- nary, 35, 125 ; irrational, 26 ; natural, 1 ; negative, 2 ; real, 35. Numbers, system of rational, 4 ; system of real, 27. Numerical value, 12. Operations, fundamental, 2 ; ex- tension of meaning, 9 ; upon fractions, 9 ; upon complex num- bers, 127 ; upon negative numbers, 10 ; upon radicals, 29. Ordinate, 52. Parabola, 106 ; equation of, 108. Parentheses, 11. Pascal's triangle, 182. Pendulum, 33. Permutations, 159, 206 ; formulas for, 207; and combinations, 206- 211. Plane, equation of, 103 ; through three points, 174. Platinum wire, resistance. of, 103. Polynomials, 131-157; derivative of, 230 ; graph of, 133, 151 ; number of roots, 133 ; root of, 131 ; zero of, 131. Potentials of electrical condenser, 204. Power, 22, 24, 25. Powers and roots, table for, 254. Power function, 64 ; derivative of, 229 ; graph of, 65 ; with negative exponents, 66. Power series, 244. Present value, 196 ; of annuity, 199. Probability, 212-219 ; compound, 214; partial, 214; simple, 212, 215 ; total, 214. Progressions, 185-205 ; arithmetic, 185 ; geometric, 188 ; harmonic, 194; infinite geometric, 191, 240. Pulley, 103. Quadratic equation, 79-90 ; charac- ter of roots, 84 ; formation of, 83 ; geometric interpretation, 86 ; graphic solution of, 116; in two variables, 106 ; product of roots, 87 ; solution of, 80, 81 ; sum of roots, 87. Quadratic expression, 18 ; definite, 87 ; indefinite, 87 ; reduction to sum or difference of squares, 18. Quadratic function, 60; definite, 87; graph of, 60, 61; indefinite, 87 ; maximum and minimum values of, 62. Radian, 249. Radicals, 26-36 ; approximate value for, 31; conjugate, 30; equa- tions containing, 89 ; fundamental theorem for, 34 ; similar, 29 ; simplest form for, 28. Radicand, 26, 27. Radium, dissipation of, 204. Rational numbers, system of, 4. Ratio of geometric progression, 188. Reciprocal, 23. Reckoning, rules of, 7. Remainder theorem, 137. Repeating decimals, 193. Resultant, 177. Root of equation, 22, 79, 131. INDEX 263 Roots, approximate determination of, 143; character of, 84, 150; common to two equations, 156, 177 ; extraneous, 89 ; imaginary, 150; irrational, 85; multiple, 156, 234 ; number of, 133 ; number of real, 152 ; rational, 85, 140. Salmon, Higher Algebra, 168. Scale, complete, 2 ; Gunter's or logarithmic, 47 ; natural. 1. Sequence, 32, 220, 222, 235; con- vergent, 222. Sequences and limits, 220-234. Series, absolutely convergent, 244 ; as a means of computation, 248 ; binomial, 247 ; comparison, 240 ; convergent, 237 ; divergent, 237 ; exponential, 247, 249 ; for cosine, 249; for sine, 249; geometric, 191, 240; infinite, 235-250; logarith- mic, 248 ; oscillating, 237 ; power, 244 ; semi-convergent, 244 ; with negative or with imaginary terms, 244. Signs, Descartes's rule of, 152 ; rule of, for determinants, 159. Similar radicals, 29. Simplest form of radical, 28. Simple term, 14. Sine, of one radian, 249 ; series for, 249. Sinking fund, 200. Slide rule, 47, 48. Slope, of curve, 232 ; of a line, 56. Solution, trivial, 172'. Speed, 57. Square root, table for, 32. Steepness of a stairway, 56. Straight line, equation of, 55-57. Sturm : Theorem on real roots, 136, 152. . Substitution, 141. Subtraction, 2, 9 ; and negative numbers, 2 ; of complex numbers, 127 ; of radicals, 29. Synthetic division, 138. Systems of linear equations, 91-104, 170-174; applications of, 101, 102 ; consistent, 97 ; Cramer's rule for, 93, 100, 171 ; dependent, 97 ; determinant of, 92, 170 ; ehm- inant for, 172; solution of, 91, 92, 94, 97, 170; homogeneous, 172 ; inconsistent, 96. Systems of non-linear equations, 105-119. Table, American Experience, 258; for compound discount, 257 ; for compound interest, 256 ; for cube roots, 254 ; of important con- stants, 255 ; for square roots, 254. Tangent, 50, 56. Test-ratio test, 242 ; for, binomial series, 247 ; for power series, 245. Tetrahedron, volume of, 170. Thermometer, Centigrade, 58 ; Fah- renheit, 58. Transformation, of biquadratic equa- tion, 142 ; of cubic equation, 142 ; of equations, 141-143 ; real rational, 143. Trial divisor, 146. Triangle, area of, 169 ; Pascal's, 182. Trivial splution, 172. TttHNBAURE AND Matjber, Prin- ciples of Reinforced Concrete Con- struction, 148. Type term of series, 235. Undetermined coefficients, 138. Uniform motion, 57, 59. Value, numerical, 12 ; of mathemat- ical expectation, 218 ; present, 196. Variable, 49, 221 ; dependent, 49 ; independent, 49, 237 ; limit of, 221-223. Variation, 55. Variation of sign, 152. Vector, 128. Vector addition, 128. Velocity, 57 ; of water, 74. Water, boiling point of, 103 ; velocity of, 74. Weber, Lehrbuch der Algebra, 131. Weir, 33. Zero, 1 ; division by , 1 1 ; of a func- tion, 77 ; of a polynomial, 131. Printed in the United States of America. THE following pages contain advertisements of a few of the Macmillan books on kindred subjects. ELEMENTARY MATHEMATICAL ANALYSIS BY JOHN WESLEY YOUNG Professor of Mathematics in Dartmouth College And frank MILLET MORGAN Assistant Professor of Mathematics in Dartmouth College Edited by Earle Raymond Hedrick, Professor of Mathematics in the University of Missouri Cloth, i2mo, S42 pp. A textbook for the freshman year in colleges, universities, and technical schools, giving a unified treatment of the essentials of ■trigonometry, college algebra, and analytic geometry, and intro- ducing the student to the fundamental conceptions of calculus. 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