liilliliilitiil mMmMm^,v ^^^m^'^myM- '..'!■(' '':< '/-"■'•.' f 1 '\'' •'■■ '■' ; '!; y , ''O'- '■\' \\ •■ ^:v;i'; ■■iM'iA'' fi. i ates !-^i '^^^^. A^'^ (^, Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/courseinalgebrabOOvanvrich ootjk.se IN ALGEBRA Bking Course One in Mathematics IN THE UniveRvSity of 'Wisconsin. BY' C. A. VAN VELZER and CHAS. S. SLIGHTER. MADISON, WIS.: Capital City Pud. Co., Printers, 1888. Copyrighted 1888 BY C. A. VAN VELZER and CHAS. S. SLtCHTER. PREFACE The present volume originated in a desire on the part of the authors to furnish a text of Course I. as was previously mapped out by the department of mathematics at the University of Wisconsin. The orginal intent was to produce a syllabus for the use of students in this institution, but it was subsequently thought that a work which w^ould be useful here might also be found useful elsewhere, and hence it was decided to give the w^ork more the character of a treatise than a syllabus. To insure the best results it was thought desirable to print the present preliminary edition and put it to the test of class room work, and at the same time to invite criticism and suggestions from teachers and others inter- ested in mathematics, and then from the results of the authors' tests, and from the experience of others, to rewrite the work, changing it freely. For these reasons the treatment of many subjects in the following pages should be understood as merely tentative. The final form will depend entirely upon the results of experience. An examination of the text will reveal many deviations from the beaten path, but the idea was not to deviate simply for the sake of being different from others ; on the contrary the authors have freely drawn from other works. The sources from which material has been most largely drawn are the following: For problems, Christie's Test Questions and Wolstenholm's Collec- tion ; for various matters in the text, Kempt' s Lehrbuch in die Moderne Algebra, and the algebras of Chrystal; Aldis; Hall and Knight; Oliver, Wait and Jones; and Todhunter ; for historical notes, Marie's Histoire des Sciences Mathematiques et Physiques, and Matthiesen'sGrundzuge der Antiken und Modernen Algebra. mSOBOSS IV Part II. of the present work, containing chapters on Iniagin- aries, the Rational Integral Function of .r, Solution of Numerical Equations of Higher Degree, Graphic Representation of Equa- tions, and Determinants, has already appeared and for this part, as well as for the present volume, suggestions are invited. Several modifications have already suggested themselves to the authors, but it is hoped that any into whose hands either volume may fall will communicate with the authors with reference to any changes before the work is put in permanent form. University of Wisconsin, Madison, Wis, 1888. TABLE OF CONTENTS. Pa OK. CHAPTER I. Introduction, . . _ . j CHAPTER II. Theory of Indices, - - - - i$ CHAPTER III. Radicai. Quantities and Irrational Expressions, 35 CHAPTER IV. Quadratic Equations Containing One Unknown Quantity, - - - - - 51 CHAPTER V. Theory of Quadratic Equations and Quadratic Functions, - - - - 64 CHAPTER VI. Single Equations, - - - - 74 CHAPTER VII. Systems of Equations, - - - 89 CHAPTER VIII. Progressions, - - - - - ic6 CHAPTER IX. Arrangements and Groups, - - - 114 CHAPTER X. Binomial Theorem, - - - - 130 VI CHAPTER XI. Theory of Limits, - - - - 139 CHAPTER Xn. Undetermined Coefficients, - - - 153 CHAPTER XIII. Derivatives, - - - - . 164 CHAPTER XIV. Series, - - - - - - 183 CHAPTER XV. Logarithms, - - - - 196 ALGEBRA CHAPTER I. INTRODUCTION. 1, Definitions. When we wish to use a general term which shall inclnde in its meaning any intelligible combination of alge- braic symbols and quantities, the word Expression will be adopted. Thus ,2 j\ r 2 I i I \ c^ -{■ d^ -\- abed (x^—d) Cax^-\-bx-\-e); — - — — ^ \/h^,-l. / — may be called expressions. It includes the words polynomial, fraetio7i, and radical and more besides. When we wish to call attention to the fact that certain specified quantities appear in an expression it may be called a Function of those quantities. Thus if we desire to point out that x appears in the first expression above, it would be called a ftinction of x. If we wish to state that a, b, c, and d occur in the second expression, we would call it 2, fimction of a, b, c, and d. If we wish to say that y occurs in the last expression, it may be called o. function of y, or if we wish to say that a, b, and y occur in it, we would speak of it as afmction of a, b, and y. A formal definition of the word function would be : A Function of a quantity is a name applied to any mathemati- cal expression in which the quantity appears. 2. Definition. An expression is Integral with respect to any quantity or quantities, that is, is an integral function of those quantities, when the quantities named do not appear in any man- ner as divisors. Thus z^x^-\-\x—y/2 is integral with respect to X ; that is, is an integral function of x. a—b-\-a-b ab x^-\-xy X is integral with respect to a and b, but fractional with respect to 2 Algebra. jc and J/ ; that is, is an integral function of a and d, but a./radwfial function of jr and y, the word fractional meaning just the opposite to integral. 3. Definition. An Expression is Rational with respect to any quantity or quantities, or is a rational function of those quan- tities, when the quantities referred to are not involved in any manner by the extraction of a root. Thus i^TlT^ is rational with respect to x, but irrational with respect to c and d\ that is, it is a rational function of x, but an irrational function of c and d, the term irrational being used in just the opposite sense from rational. 4. An expression may be both rational and integral with re- spect to certain quantities, in which case it may be spoken of as a Rational Integral Expression with respect to those quantities, or as a rational integral function of the quantities. In the sam6 way we may speak of an expression which is rational and frac- tional with respect to certain quantities as a Rational Fractional Expression with reference to those quantities, or as a rational fractional function of the quantities. In like manner we may use the terms Irrational Integral Expression and Irrational Fractional Expression, or Irrational Integral Function or Irrational Frac- tional Function. In the following examples the student is expected to answer the question. What kind of an expression? with reference to the quantities specified opposite each. /. ax^-\-a^x''-\-a^x. With respect to ;t- ? to a ? to Jt: and a~i c 2. 2« ( \-\-—y I ~ ^) With respect to <2 ? to r? a J. bx" T+^y- With respect to jt ? to j^' ? to ;t: and j'? >/a x^-+^b+\/c With respect to ;»;? to^y? ay^-\-by-\-c ' to x andjj'? to a, b, and r? 5. Definition. If by any operation we render an expres- sion integral with reference to certain quantities, in respect to Introduction. 5 which it was previously fractional, we are said to Integralize the expression with respect to those quantities. Thus the expression a'x' 2ab b'x' b x^ a is integralized with respect to a and b if it is multiplied by ab. Similarly, if, by any operation, we . render an expression rational with reference to certain quantities, in respect to which it was previously irrational, we are said to Rationalize the expres- sion with respect to the quantities named. Thus if we square the irrational expression '^^ x'-\r^~ab xy-^-y'' it is rationalized with respect to x and y. 6. Definitions. The Degree of a term with respect to any quantity or quantities is the sum of the exponents of the quan- tities named. Thus ab'^x^y is of the third degree with reference to X, of the first degree with reference to y, of the fourth degree with reference to x and y, of the third degree with reference to a and b, etc. But the degree with reference to any quantities is not spoken of unless the term is rational and integral with respect to those quantities. Thus we do not speak of the degree of such a term ^^ — , with respect to either a or x. The Degree of a polynomial with respect to any specified quan- tities is the degree of that one of its teniis whose degree (with re- spect to the same quantities) is highest. Thus, x^—alxy^-^-cxy is of the third degree with respect to x, of the second degree with respect to y, and of the fourth degree with respect to x and y. But the degree of a polynomial is not spoken of unless the poly> nomial is rational and integral with respect to the quantities specified. It can easily be seen that the degree of the product of several polynomials is the sum of their separate degrees. Thus {^x'-^xy-\-f) (xy-j-bx^y) is of the fifth degree with respect to x and y ; of what degree is it with respect to jf ? with respect to^? The Degree of an Equation is the degree of the term of highest degree with respect to the tinkyiown quantities. But both mem- 4 Algebra. bers of the equation must be rational and integral with respect to the unknown quantities and the indicated operations must be per- formed ; otherwise the degree is not spoken of. What is the degree of (ji" +y ) (xy -}- 1 ) = 2o8jf y ? Instead of speaking of expressions as being of the first or of the second or of the third degree, they are commonly designated by adjectives borrowed from geometry as linear or quadratic or cubic expressions respectively. An expression of the fourth degree is sometimes called bi-quadratic, meaning twice squared. In place of the expression, "of the second degree in respect to .r," it is common to say, "of the second degree in jr." 7. DefinitioNvS. a polynomial is Homogeyieous with respect to certain quantities when all its terms are of the same degree with respect to those quantities. Thus a^-\-a^b-\-ab'-\-b^ is homo- geneous with respect to a and b. An equation is Homogeneous when all the terms are of the same degree with reference to the unknown quantities. Thus the equa- tion xy-\-j>^-\-x^=o is homogeneous, but xy-^y^-\-x^=2o is not homogeneous. It is to be noted ^ here that we use the term homogeneous equa- tion in the strict sense, following the established use of the term. But in some American text books homogeneous equation includes equations like x-y-\-y^-{-x^=20, that is, no account is taken of terms involving nothing but known quantities. 8. Definition. An expression is Symmetrical with respect to two quantities if the expression is unaltered when the two quantities are interchanged. Thus x^-{-y^ is symmetrical with re- spect to X and J' ; for putting j/ for x and x for j we obtain j^+j*:^, which is the same as the original. Also x" -\- ax + a'' is symmet- rical with respect to a and x. Is x'^ + 2xv—y symmetrical with reference to x and y ? An equation of two unknown quantities is symmetrical when the interchange of the unknown quantities throughout does not modify the equation. Such is X + xy -f xy' +y= 1 024. Introduction. 5 9. Incommensurable Numbers. Algebraic numbers* may be divided into two kinds, depending upon the relation which they bear to the unit or unity. If a number has a common measure with unity, it is called a commensurable number. Thus 7 is a commensurable number ; also J is a commensurable number, since one quarter of the unit is a common measure between f and unity. Commensurable numbers thus include both integers and fractions. If a number has no common measure with unity, it is called an incommensurable number. Thus >/~2 is incommen- surable. A little consideration will show that v^ 2 cannot be an integer nor a fraction. It is not an integer because (0)^=0, (1)^=1, and (2)^=4, and there are no integers intermediate be- tween these. It cannot be a fraction, for if possible suppose that some irreducible fraction, represented by—, equals v^ 2 . Then "^ 2 = -7-, or squaring, 2——^, which is absurd, for an integer cannot equal an irreducible fraction. Therefore -v^ 2 is not a fraction. But it is an exact quantity, for we can draw a geomet- rical representation of it. Take each of the two sides, CA and CB, of a right angled triangle equal to i. Then AB, the hypoth- enuse,willequalN^(i)^ + (i)^=N^ 2 Thus A^ v^ 2 is the exact distance from A to B, which is a perfectly definite quantity. Thus the idea that incommensurables are indefinite or inexact must be avoided, (l) This notion has arisen because the frac- tions we often use in place of incommen- surables, such as 1. 4 142-}- for v/ 2, are 7nerely approximations to the true value. (^ ("fj ^ We now give a property of incommensurable numbers which will serve to make their separation from the class of commensur- able numbers (integers and fractions) more apparent. It is that an incommensurable number when expressed i?i the decimal scale never repeats, while a comtnensurable number so expressed always repeats. * As here used the term Algebraic number does not include the so-called imaginaries, which, strictly speaking, are not numbers at all. Imaginaries ai-e treated in Chapter 1. Part II. 6 Algkbra^ ^^ I ( ^A« ; Thus, 75=75.0000000000+ repeating the o. ^= .5000000000+ repeating ike o. i= -3333333333+ repeating the 3. yyy= .279279279279+ repeating the 279. >/ 3 = 2.7320508+ never repeating , "^20= 2. 7 1 441 77+ 7iever repeating. -•=. 3.1415926+ Wd'Z'^r repeating. The student should endeavor to get a fair notion of what is meant by an incommensurable number. It is a difficult idea to grasp at once, but it is one which the student should continue to consider until the conception takes a definite and rational shape. POSITIVE AND NEGATIVE QUANTITIES. 10. In Algebra we are often called upon to distinguish between quantities which are directly opposite each other ; as, for instance, degrees above zero from degrees below zero on a thermometer scale, distance north of the Equator from distance south of the Equator, distance east of a given point from distance west of the same given point, etc. The distinction is made by means of the signs + and — , e. g.^ if +10° means a temperature of 10° above zero, then —10° would mean a temperature of 10° below zero, and if + 10 miles means 10 miles w^r/// of the Equator, then — 10 miles would mean 10 miles south of the Equator, and if +10 rods means 10 rods east of a given point, then — 10 rods would mean 10 rods west of the same given point, and if +10 be ten units of any kind in a^iy sense, then —10 would be ten units of the same kind in just the opposite sense. These two kinds of quantities are called />^«VzV^ and negative. 11. The distinction between positive and negative quantities is made by means of the same signs as are used to denote the opera- tions of addition and subtraction, and it might seem that it is un- fortunate and unnatural that the same signs are used in these two ways. It may be unfortunate, but it is not unnatural, as we pro- ceed to show. 12. Suppose that, by one transaction, a man gained $500, and by another he lost $700 ; then he lost all he gained and $200 more, Positive and Negative Quantities. 7 or his capital suffered a diminution of $200. If his original capital was $1,000, then the first transaction increased it to $1,500, and the second transaction diminished it to $800. Thus an addition of $500 followed by a diminution of $700 is equivalent to a single diminution of $200, or $i,ooo-}-$50o— $700= $1,000— $200. Hence %^qo—%'j 00 when joined to $1,000 may be replaced by — $200 joined to $1,000. Now, as any other original capital would have answered as well as $1,000, we may neglect that original capital and write $500— $700= —$200. Thus we see, by this illustration, that it is natural to prefix the minus sign to the $200 to indicate a resultant loss of $200. 13. We might have used an illustration involving some other kind of quantity than money, as titne, distance, etc., and have ob- tained an equation similar to the one just written. We may then make an abstraction of the $ sign and write simply 500— 700= — 200. 14. In Ari theme tic we are concerned only with the quantities o, I, 2, 3, 4, etc., and intermediate quantities, but in Algebra we consider besides these the quantities o, —I, —2, —3, —4, etc., and intermediate quantities. 15. We may represent these two classes of quantities on the following scale, —5.-4. —3. —2, —I, o, I, 2, 3, 4, 5, which extends indefinitely in both directions from zero. The sign -f perhaps ought to precede each of the quantities at the right of o in this scale, but when no sign is written before a quantity the + sign is always understood. 16. Quantities to the right of o in the above scale are positive and those to the left of o are negative, or we might say Arabic nuynerals preceded by a + sign or by no sign at all are positive quantities, and Arabic numerals preceded by a — sign are yiegative quantities. 8 Algebra. 17. In Algebra quantities are represented by letters, but a letter is just as apt to represent a quantity to the left of o in the above scale as it is to represent one to the right of o ; so that, while in the case of a numerical quantity, /. e. one represented by figures, we can tell whether the quantity represented is positive or nega- tive by the sign preceding it, yet, it the case of a literal quantity, /. e. one represented by letters, we cannot tell by the sign before it whether the quantity represented is positive or negative. If we speak of the quantity 5 we know that it is positive, but if we speak of the quantity a we do 7iot know by the sign before it whether it is positive or negative. We know that —5 is negative, but we do 7iot know that —a is negative. A mifuis sign before a letter always represents a quajitity of the opposite kind from that represented by the same quantity with a plus sign or no sig7i at all before it. Thus, if «=3, then — <2=— 3, and if ^= — 3, then — a = 3. 18. Looking at the above scale it is evident that of any two positive quantities the one at the right is greater than the other or the one at the left is less than the other, e. g. io>6 or 6' definition of multiplication is repeated addi- tion, yet, even in arithmetic, the word outgrows its original meaning, for, by no stretch of language, can the operation of mul- tiplying I by ^ be brought under the original definition. According to the original definition, multiplication, in arithme- tic, is intelligible so long as the multiplier is a whole number. 3 can be repeated 4 times, and so can \ be repeated 4 times but 4 cayinot be repeated \ a time. A— 2- lo Algebra. ^ repeated 4 times is ^ multiplied by 4, yet, in arithmetic, 4 multiplied by ^ is a familiar operation. Let us inquire how this comes to have a meaning, and how it happens that 4 multiplied by \ turns out to be ^ <^4. 21. As long as a and b are positive whole numbers it is easy to see that a b=ba. Suppose, to fix the ideas, that <3!=3 and b=^, then we may write down 5 rows of dots with three dots in each row, thus — and we have in all 5 times 3 dots. But we may look at vertical rows instead of horizontal ones and we see three rows with 5 dots in each row, and of course the number of dots is the same ; so we may say 5x3=3x5. Any other positive whole numbers would do as well as 3 and 5, and so if a and b are any positive whole numbers, ab=ba, i. e. , in the product of two numbers, if is indiffereyit which is the multiplier and which the multiplicand, so long as both numbers are integers. ^ 2.2.. Now, in arithmetic, the operation of multiplication is so extended that eve^i when one of the quayitities is a fraction it shall still be indifferent which of the two quantities is the multiplier and which the multiplicand. This gives a 7nea7iing to multiplication when the multiplier is a fraction, and thus it happens that 4 multiplied by \ is take?i to mean the same as \ multiplied by 4. 23. In exactly the same way in algebra, the operation of mul- tiplication is extended so that whatever numbers, positive or neg- ative, integral or fractional, are represented by a and b we shall always have ab'^ba. Introduction. i i and since we know what is meant by —3 multiplied by 5, the equation ab=^ba gives a meaning to 5 multiplied by —3. .•.5 multiplied by —3= — 15. From this we are led to say that when the multiplier is neg- ative, the product is just the opposite from what it would be if the multiplier were positive. Therefore, if a and b are any two positive quantities, we may WTite the following four equations : a.b=ab (i) {—a).b=—ab (2) a.{—b)=-—ab (3) {-a).{-b) = ab. (4) From the ist and 4th we conclude that the product of two posi- tive quantities or tzvo 7iegative qua?itities is positive, and from the 2d and 3d, the product of 07ie positive and o?ie negative quantity is neg- ative. 24. The four equations just written are true whether a and b are positive nor not. Consider, for example, the second equation under the supposi- tion that a is negative and b positive ; then {—a).b becomes the product of two positive quantities and is therefore positive, but —ab is also positive in this case, as it should be, rendering the equation still true. And so of the other equations, whether a and b are positive or not. Therefore, directing our attention to the signs, we may say that the product of two quantities preceded b}- like signs is a quantity preceded by the + sign, and the product of two quantities preceded by unlike signs is a quantity preceded by a — sign. This statement is usually shortened into the following — In multiplication like sig?is give plus and imlike sig-fis give minus. This is often confused with the statement in italics in the pre- ceding article. They are not identical, but both are true. 25. As division is the inverse of multiplication, it easily fol- lows that the quotient of tivo positive or two negative quantities is positive, and that the quotient of a positive by a negative quantity, 12 AI.GEBRA. or a negative by a positive quantity, is yiegative. It also follows that in division like signs give plus and unlike signs give mi7ius. The proof of these two statements is left as an exercise for the student. 26. Theorem. The difference between like powers of two quanti- ties is exactly divisible by the difference of the quantities themselves. It is easily seen on trial that ia' — jr')-i-(a — x) = a-^x. {a^ — x^)-7-(a—x)==a^-\-ax-hx'. {a'—x')^{a—x)=^d'-\-a'x-^ax^ + x\ (a^—x^)-^(a—x)=a*-\-a^x-\-a^x^-\-ax^-\-x*. Observing the uniform law in these results it would be at once suggested that the theorem is universally true ; that is, that what- ever be the value of n, ^~J^.^a"-'-\-a"-^x-{-a"-'x'-\- . . -^-a^x^'-'+ax^-'-^-x"-'. (i) a—x This can easily be shown to be true, for multiplying the right hand side of this equation by a—x it becomes a"—x", as follows : a"-'-}-a"-'x-^a"-'x'-{- . . . -j-a'^x"-^-^ ax"-'+x"-' a—x a'' '" -\-a" —a" -'x-j-a"-'x'-h . ~'x—a"~''x^— . , . . -\-a'x''-'-\-a'x"-'-\-ax"- . . —a'x"-^—a'x"-'—ax"- I ' — X' :« -ho +o + . . . . . -f-o -f-o +0 — X'' But multiplying the left hand side of equation (i) by a — x we obtain a"—x" also. Hence equation (i) reduces to a" — x"=a" — x'\ and hence must be correct. 27. Theorem. The difference of like even powers is exactly divisible by the sum of the quantities themselves. It will be found on actual division that {a" — x^^ — {a-\- x)=:=a— X . (a*—x^)-7-(a-\-x')=a^—a-x-j-ax^—x^. (a'—x^)-i-(a-\-x)=a'—a*x-\-a'x'—a'x'-\-ax*—x\ Introduction. 13 The obvious uniformity in these results forces the suggestion that the law of fonnation of the quotient will hold in any similar case. That is, that ^ ~'^ =a"-'—a"-'x-\-a"-^x'— . . . —a'x"-'-^ax"-''—x"-\ (i) a-\-x where we have given the — sign to the odd powers of x, n being any even number. Multiplying the right hand side of the equa- tion by a-h^ we obtain a"—x", thus: a"-'—a"-'x-j-a"-'x'— . . . —a'x"-'-\-ax"-'—x"-' a-\-x a" —a"~'x-\-a"~''x^^ . . . —a^x"~^-\-a''x"'~^—ax"~' -{■a"-'x—a"-'x'+ . . . -\-d'x"-^—a' x"-'-\-ax"-'—x" a" +0 +0 -f . . . +0 +0 -fo —x" But multiplying the left hand side of equation (i) by a-f .r we obtain a"— x" also. Hence equation (i) must be true, since it reduces to a"—x"=a"—x". 28. Theorem. The sum 0/ like odd powers of two quantities is exactly divisible by the sian of the quayitities themselves. By trial we find this theorem holds in the first few cases as follows : {a-\-x)-^(a-^x)=\. {a^-{- x^) -r-(a-^ x) = a'—ax -\- x'' . (a'-j-x')^(a-^x)=a*—a'x-\-a'x'—ax^+.t'. (a'+x')--r(a+x)=a^—a^x-^a'x'—a'x'-\-a'x-'—ax^-\-x^. The simple law in the formation of these results would naturally suggest the general truth of the theorem. That is, that ^^"=a"-'-a"-'x-\-a"-'x-'- . . . -\-a'x"-'-ax'-'-^x'-\ (i) a-f-x where the terms containing the odd powers of x have the minus sign, n being any odd number. Multiplying this equation through by a-{-x it becomes a"-hx"=a"-\-x", and hence must be true. 14 Algkbra. 29. The last three theorems have such a variety of applications that it is important that they should be committed to memory. We suggest the following scheme for keeping them in mind : x—a divides the difference of like powers. ,. .' ^, ( difF(frr, and formula (d) if n<,r. Suppose «r, and, if we like, we might retain formula (c) and entirely dispense with formula (d). Again, it may be seen, in a similar manner, that if <^~''=— form- ula (d) could be used when /z>r as well as when n<.r, so that we might, if we like, retain formula (d) and entirely dispense with formula (c). If we find that we may use negative exponents upon the above interpretation, then we will for the most part dispense with form- ula (d), using it only now and then, if at all, when it comes a little handier than formula (c). 18. Again, by the above interpretation formula (c) can be used when one or both of the exponents are negative. First, suppose r negative and equal to —q, then a"^a-''=a" -. — = a"a''= a"^". But substituting in (c), the same result as before, so that formula (c) may be used when Second, suppose ?i negative and equal to —s, then a" ' a' a'' a' a'' a'^''' But by substituting in (c), the same result as before if our interpretation of negative ex- ponents be correct, so that formula (c) may be used when n is negative. Third, suppose both 71 and ^negative and let /z=— ^and r=— ^, then \ 1 a a '-^a ^= — ; — =— =d;* \ a a'' a' But by substituting in the formula, the same result as before, hence formula (c) may be used when both 71 and r are negative. 24 Algebra. 19. Formula (a) niaj^ be used when either or both of the ex- ponents are negative if the above interpretation be correct. First, suppose r=—q, then But by substituting in the formula, Second, suppose n=—q, then «^ But by substituting in the formula, so that formula (a) may be used when n is negative. Third, suppose ?i=—s and r=—q, then -V -^_ ^ ^ _ ^ _ ^ But by substituting in the formula, ^ +-?' r^"/ /.„w/ .«.- so that formula (a) may be used where both n and r are negative. 20. Formula (b) may be used when either ?i or r or both are negative. First, suppose r negative and equal to —q, then ' I _ I (a")'~a^ But by substitution in the formula, so that formula (b) may be used when r is negative. Second, suppose n negative and equal to —q, then But by substituting in the formula, so that fonnula (b) may be used when 7i is negative. Theory of Indices. 25 Third, suppose both exponents are negative and let 71=— s and r=—q, then a' But by substituting in the formula, so that formula (b) may be used when both exponents are neg- ative. 21. Thus we see that if we interpret a~'^ as being - -, a being any number whatever (not zero), and q being any whole number, the exponents in all our formulas may be any whole numbers, positive or negative, and this makes our formulas considerably more general than they were before. Now, because the supposition a~^^=~^ leads to no inconsistency it is permissible, and because it gives greater generality to our formulas it is advantageous. Therefore we adopt the equation «~^= — as defining the mean- ing of a"'. 22. Since <2~^=- , and therefore -— ,=<3!^ it follows that in a'^ a ^ any fraction d^ny factor may be transferred from the numerator to the denominator, or vice versa, by simply changing the sign of the exponent. Hence, if in formula (c) we transfer «'' from the denominator to the numerator, — becomes a"a~'', which by formula (a) equals a'' ^"-''•, so that fonnulas (a) and (c) are really identical, but, for the sake of convenience, both are retained. A— 3 26 Algebra. examples. „, . ab' . ,. 7. Write -rr. ni one line. TiJC^V" 2. Write . „ in one line. x'v^(a — by . ?. Wnte "^ m one line. -^ I ^. Write *^~ all iii the loivcr line. 5". Write- , r- so that all exponents are preceded by the + sign. 6. Write ^—-^ — with all positive exponents. 23. Having now dispensed with formula (d) and extended formulas (a), (b), (c) so that the exponents may be a/iy ivholc members, positive or negative, the question arises, can we give still greater generality to our formulas by using exponents which are fractions ? 24. If quantities with fractional exponents have an}- meaning and if we can use them in our formulas, we must have by form- ula (b) n being here any positive zvhole niunber; i. e. a>' is a quantity which raised to the ?ith power equals a. Raise both sides of this equation to the rth power, r being a positive whole number, and we get so that, if we are permitted to use fractional exponents, a>' denotes the ;'th power of the ;^th root of a. Theory of Indices. 27 25. Again, by the definition of a quantity with an ex- ponent — I, and by formula (b) U" '')-«" I or, taking the reciprocal of both sides, -'^ I a~': .26 Thus we have suggestions of meanings for both positive and negative fractional exponents, and if we introduce fractional exponents into our formulas with the meanings suggested, these formulas will be found to give consistent results, as we shall see. 27. Before substituting in our formulas it is necessar>' to stop and show that, with the meanings suggested, a quantity with a fractional exponent has the same value whether the exponent is in its lowest terms or not. 1 Let «^"=JL- then ^=jt-^"=^ji-?;" i_ In a similar manner it may be shown that _ '' _'/'" a "—a f" 28. Examination of formula (a). r . p . Let — and - be any two positive n q two negative fractions. Then there are four cases to consider. Let- and - be any two positive fractions and and n q ^ ^ n q 28 Algebra. FiJ'st case. L. L First, a"a'^= what? Second, a" a '^ = what? Third, a~^a^= what? Fourth, a "a '^= what? a"a'^ =a"fa'"^ by art. 27. and by direct substitution in the formula we get a" a'^ =a"^ 'f Second case. a" a '/ = a"''a '"!= j <2"'/ a'"^ ] and b}^ direct substitution in th? formula wc also get a" a '^=a" '' Third case. This is the same as the second case, only the order of the fac- tors is changed, and therefore as in the second case the result will be the same as given by direct substitution in the formula. Fourth case. --^-^11 I a "a ''= -v -/= , 7 a'' a'^ a"a'f I -(- + ^) ----^ • a"^ 'i , V ^"^ ^^y direct substitution in the formula we also get Thus we see that by using fractional exJponents according to the suggestions before obtained, the result of multiplying two fractional powers of a is, in ever}^ case, in perfect accord with for- mula (a). Theory of Indices. 29 29. Examination of formula (b). As before, let - and - be any two positive fractions, and n q — - and — - any two negative fractions. Then there are four n q cases to consider. ( First, k" "'= what? First case. Second, [«" ) ^= what? Third, \a " \' = what? Fourth, {a "\ '= what? Let or {a"\ '^x'-'; From ('i j «"'.==jt-''^; (2) (3) (4) (5) ((>) (1) .-. from (6) and (-]) \a" J '=fl«^ and by direct substitution in the formula we also get Second case. rp a"\ " /— rp—^ "^' a" Mi' a"f and bv direct substitution in the formula we also get Third case. a"\ '=a 1 1 I- 9 a"f 30 AI.GEBRA. and by direct substitution in the formula we also get Fourth case. . I I I I r "^ a« a" and by direct substitution in the formula we also get 'z ■=a'"J. Thus we see that by using fractional exponents according to the suggestion before obtained, the result of raising any frac- tional power of a to any other fractional power is, in every case, in perfect accord with formula (b). 30. Examination of formula (c). As before, let - and - be any two positive fractions and — n Q '^ P and — any two negative fractions. Then we have four cases to consider. First, a" -^a'l = what ? Second, a" -^a what ? Third, <2 " -r-^! '-^ = what ? First case. Fourth, a '• —a " = a" -—a'^ =a" a '' =a" what ? by Art. 29, second case, and by direct substitution in the formula we also get 1 / /- _ / a" -7-a'^ =a" '' . Second case. ^ _ A !L t. ^ • Z a'' -—a 9—a''af=a"^'f, and by direct substitution in the formula we also get a" -r-a ''^=<2" ^ ^. Theory of Indices. 31 Third case. _ ^- / _ '; _ / _ L _ L a "-^af=a "a ^=^ " '^ , and by direct substitution in the formula we also get _ Ji t _ 1 _ z*^ a "—a"— a " ^. Fourth case. a ''- -^a '''=« "af=a " f ^ and by direct substitution in the formula we also get _ -!L _ A _ il 4. j^ a "--a '^ ==a " f. Thus we see that by using fractional exponents according to the suggestion before obtained, the result of dividing one frac- tional power of a by another fractional power of a is, in every case, in perfect accord with formula (c). 31. Now, because the suppositions a" ={ Va Y and a " = --^ :_ - lead to no inconsistency they are admissible, and because they give greater generality to our formulas they are advantageous. Therefore we adopt these equations to define the meaning of quan- tities affected with fractional exponents. 32, The formulas (a), ( b), (c) are now so generalized by the above definitions that they can be used when the exponents are any positive or negative whole numbers or fractions, and it might naturally be asked, is this the greatest generality of which they are capable ? • Excluding the so-called imaginaries, there is no kind of alge- braic numbers not yet discussed except incommensurable num- bers, and the consideration of quantities affected with incommen- surable indices is reserv^ed for chapter XI. In the meantime, however, it should be remembered that the formulas are to be used only when the indices are commensurable. 32 Algebra. 33. By means of the meanings now given to negative and fractional exponents it is easy to see that the formula a"b"c" . . . =(a b c . . . )" holds whether ;/ is positive or negative, integral or fractional. I — ^ First, let ?/= , a positive fraction, and let a''=-x and b'' =^y\ r .-. a=^x" and b=^y'' 1 _i_ then a'' b'' =^xy, and ab=x''y''=^(xy)'\ 1 .-. (ab)'-=xy y \ 1 ^ .-. a'' b'' = (ab)'' . 1 Multiply both sides by c and we get L 1 A 11 A a '■ b'' C =(ab)''c'' = (abc) '' , and so on, evidently, for any number of factors. This is quite an important formula, stated in words it is, The product of the r th roots of several quantities equals the rth foot of their product. Second, let 7i=—r, a negative quantity, either integral or frac- tional, then a-'b-'=~ \^= \^ = (abr\ a"^ b (ab) Similarly a~''b~''c~''= (abc)~'\ and so on, evidently, for any number of factors. 34. The formula 7„==' t[ also holds good whether 7z is posi- tive or negative, integral or fractional. 1 1 1 First, let n=j^, a positive fraction, then --=^'-^-1 = -]- . .. -^ , — yb ) ib ) 0'' Stated in words, this is. The quotierit of the rth roots of two qiiayitities equals the rth root of their quotieyit. b-" a" \a\ \b Theory of Indices. 33 Second, let ?i=—r, a negative quantity, either integral or frac- tional, then a-'' b'- \bV wy \a \b EXAMPLES. 1. Write the following expressions, using fractional ex- ponents in place of the radical signs : ^^ a\x \jlry >J"> ' 2. Write the following expressions, u.sing radical signs in place of fractional exponents : . . 5 jt^ J. Multiply ^ ^ by Jtr'-^. 1 _ .1 4. Multiply x'^ by Jt: '\ 5. Multiply -^ by -^. x^ X ^ 6. Multiply [ip'^by-^. (xj n \T"(J 7. Divide (x+yjUhxCx-^j')''^- 8. Divide M^xy'+x^y~y^ by x^—y p. Divide jf^-fy^ by x'- -\-y" . 1- L ?L L \. 10. Multiply X- —x'^ -\-x'' -x"- -^x- -x-^x"- -\ byy^'+i. 11. Simplify Xx'^-^x'] '^. 12. Simplify (.r^jz-'^-^J . x^y~'2^. i/f.. Find the continued product of 34 Algebra. 75. Multiply Jf ^ —X - by [jt"^— Jt- ^J 16. Simplify [{J^^}"']^' _ j_ ( x''" ) " ly; Simplify ^. .3. _i i 18. Simplify —^ ^ { , /p. Sii ^. ,.^ 2>('2x)^(%bx)^ 20. Simplify ^/ ^ ' (ax)'^\^ 6a CHAPTER III. RADICAI. QUANTITIES AND IRRATIONAI, EXPRESSIONS. I. From the last chapter the student has learned that there are two methods in use for indicating the root of a quantity, one by the ordinary radical sign and the other by a fractional exponent. Of course it is entirely unnecessary to have two modes of writing the same thing, and in this sense either one of the two ways may be considered superfluous. But practically each method of nota- tion has an advantage in special cases, and the student will feel this as he proceeds. This fact that both methods are better than either one, accounts for the retention of both in mathematics. 2. HiSTORiCAii Note— The introduction of the present symbols into alge- bra was very gradual, and the use of a particular symbol did not generally become common until some time after its suggestion. The signs -f- ^^^^ — were first used at the beginning of the 16th century in the works of Gramma- teus, Kudolf and Stifel. Recarde (born about 1500) is said to have invented the sign of equality about this time. Scheubet's work (1552) is the first one containing the sign ^Z . and Vieta (born 1540) first used the vinculum in con- nection with it. Before this, root- extraction was indicated by a symbol some- thing like I]^. /XStfixLn (born 1548) first used numbers to indicate powers of a quantity, and lic-even suggested the use of fractional exponents, but not until Descartes (born 1596) did exponents take the modern form of a superior figure. The development of the general notion of an exponent (negative, fractional, incommensurable) first appears in a work of John Wallis (pubUshed in 1665) in connection with the quadrature of plane curves. To show the appearance of mathematical works before the introduction c;f the common symbols, we give the following expression taken from Cardan's works (1545) : 1^ V. cu. 1^ 108 p7 10 I mj^ cu. J^ 108 m 10, which is an abbreviation for "Radix universalis oubica radicis ex 108 plus 10, minus radice universali cubica radicis ex 108 minus 10." Or, in modern sym- bols, ■ ^>/i'o8-fio-^^^io8~io Here is a sentence from Vieta's work (1615). Et omnibus perE cubum ductis et ex arte concinnatus, E cubi quad, -f Z solido 2 in E cubum, acquabitur B plani cubo. This translated reads : Multiplying both members (" all ") by K-t ami imit- ing like terms, 36 AI.GEBRA. 3. Definitions. In the following pages, by the word Radical may be understood the indicated root of an expression, whether that root is indicated by the ordinary radical sign or by a frac- tional exponent. By the Index of a radical may be understood either the number written in the angle of the radical sign or the denominator of the fractional exponent. A multiplier written before a radical will sometimes be called the co-efficient of the radical. A Simple radical is the indicated root of a rational expression. A Complex radical is the indicated root of an irrational expression. A monomial Surd is the name applied to the indicated root of a commensurable number, when that root cannot be exactly taken; as x/|, or V3- If all the irrational terms in a binomial or polynomial are surds, it is called a binomial or polynomial surd, as the case may be. It should bo noticed here that we make a distinction between the terms irrational expression and surd, a distinction which is not commonly made, the two terms being generally defined as identical. According to the above definition, ^4 ^'^-\-\/t ^^l3, '^ ~ are not surds. But they are irra- tional by the definition of I, Art. 3, This limited meaning of the word surd is convenient and is growing in use. It is found in both Aldis' and Chrystal's algebras. Radicals are said to be Similar when they have the same index and the expressions under the radical signs are the same ; that is, two radicals are similar when they differ only in their coefficients. Such are ^\^ ab and ms/ ab; also f^y and f^y. 4, Definition. For a radical to be in its siinplest form it is necessary (i) that no factor of the expression under the radical sign is a perfect power of the required root ; (2) that the expres- sion under the radical sign is integral ; (3) that the index of the radical is the smallest possible. It will be seen from the following pages that every simple radi- cal can be placed in this form without changing its value. The transpositions necessary to effect the reductions depend upon cer- tain principles, or theorems, established in the last chapter, which we collect here for reference. Radicals. 37 5. The 71 th root of the product of several quantities is equal to the product of the 71 th roots of the several quantities. That is, VabF^'.=Va Vl V~c . . 1 A I. Jl or (a b c . . )''=a" b"c" . . 6. The 71 th root of the quotient of tivo 7iumbers is equal to the quotient of their 71 th 7vots. That is, __ „ i_ 1 f<2 ) " a or Sb^^Tb l_ 1 \aY _a" ib) - -L b> 7. The 7irth root of a qua7itity equals the 11 th I'oot of the rth root of the qua7itity. That is, V^=Vv^, or (ay-^\(a)^\ 8. To REMOVE A Factor from beneath the Radical Sign. When any factor of the quantity beneath the radical sign is an exact power of the indicated root, the root of that factor may be taken and written as a coefficient while the other factors are left beneath the radical sign. Thus v/128 may be written v/64 X 2, which, by iVrt. 5, equals v^64X v^2, which equals 8v/2. As an- other case take "^ \6ax\ which equals ^8^^ X 2ax= ^8-r^ X -^ 2ax — 2 x\^^2ax. It is readily seen that this same process may be ap- plied to any similar case. 9. Examples. Remove as many factors as possible from be- neath the radical signs in the following : W50. x/8io. ^87o. \\^']2x^y. v^'^2808. 38 AlvGEBRA. p. - >/ aj(f -\- 2X^ . 10. \^iSa^b^ II. V 192^^-rj}/" 12. V(a—b)"(b-\-a)''^\ 10. ^o Intkgralize the Expression under the Radicae Sign. Suppose we wish to transform the radical 3 lab"" sjxf so that there shall be no fraction under the radical sign. Multi- ply both numerator and denominator of the fraction by a quantity that will render the denominator a perfect cube, thus : 3 M'_3 \ab' x^y__z \ab^x^y x^y ^ x^y^ But, by Art, 6, siab'xy ^'ab'xy i , - = ,^^ab\vy. In general, to integralize a radical of the form j-, multipl numerator and denominator by b"~^ and we obtain n \ab"-' ^ w> ' which, by Art. 6, equals Vaiy- V^« and this is equal to iv..»-, which is in the required form. Radicals. 39 II. Examples. Integralize the expressions under the" radicla signs in the following, simplifying the result in each case by Art. 8, if necessary: I. Process V.47 N49X3~W49X9~^'''^^- 4 5 6 7 8 9 10. (a-b)"- 12. To LOWER THE Index OF A RADICAL. It is plain that V'25 by Art. 7 = n/~^ 4/ 25- >/ 5 ; also that V ^a^=\^ ^'7~^^=^\/2a; similarly ^y ^b^=^^ s/ Ab^'="^^ 2b ] and in general, "v^<2"='^^V'^== ''^a . From this we see that the index of a radical can be lowered if the expression under the radical is a perfect power correspond- ing to some factor of the original index. 13. Examples. Reduce the following to their simplest fonns. See Art. 4. I. >/ '^^x'y^. 6/ „ 3' 4- 1000 9 1^ 40 AI.GEBRA. 6. gx^—iSxj-i-gj''' 7. 2^a—d—V64d'-\-64a'—i28ad. 14^ To INTRODUCE A COEFFICIENT UNDER THE RadICAI^ Sign. It is sometimes convenient to have a radical in a form without a coefficient. The coefficient can always be introduced under the radical sign by the inverse of the method of Art. 8. Thus, 2X'^ 2ax=^^ S x^"^ 2ax—y^^ i6ax* ; similarly, a'\^ c=V a"c. 15. Examples. Place the coefficients in the following under the radical sign without changing the value of the expression : I. TyttX^s/ 2)CIX. x^ a—x. 50^50- a-b^\^ X — y- 16. Addition and Subtraction of Radicals. Similar rad- icals (Art. 4) can be combined by addition or subtraction ; and if they are dissimilar no combination can take place. Take for ex- ample the expression, V^^'+2v/yV-2j^— 4-^^IO. Reducing each expression to its simplest form, it becomes ax'''^ T^a-\-\^ \o—^ax^\^ 2f^-\-\^ 10. It is now noticed that the first and third and the second and fourth radicals are similar to each other ; whence, grouping sim- ilar terms, the expression becomes (ax'-^ax^-)^2>^ + (i+\)^/io, or \ax''V 2)a-\-i\'>/ 10. We observe here the necessity of reducing each of the radicals in any given expression to its simplest form, for then it can be told whether or not any number of the radicals are similar to each other and consequently whether or not they can be combined together. Radicals. 41 17. Examples. Give the value of each of the following ex- pressions in as simple a form as possible : /. lov^f-f V 1000. 3. 2 n/48 + 3^ 147-5^75- 4.. v^98 + 'v^72 + 'v/ 242. 5- ^7^*+ ^^b^-\- \^4Sa'd\ 6. |v4ox=— 3V625Jt:^+io^500oJr". ^ • 5 ^ 5^8^147 2 ^ ^• P- JO. Prove h+^^-^+ Igfl +^^+^^^ ^Jgl+j-U X 18. Multiplication and Division op Radicals. The pro- duct of several radicals of the same index may be expressed as a single radical by means of Art. 5. Thus ^^2X \/3X V ^ — V 2 X3X5=v^30 ; ^r^r^X ^ mx X ^V^i=^^ x^ r^frf=rx^rnf ; VaxVbxVc. . .—Vabc. . . The result should always be reduced to its simplest form. If there are coefficients they should be multiplied together for a new coefficient, for ay^^x b'y/y c^/ z=-abc>/ x Vj/ 'V z=^abc"y xyz. The quotient of one radical by another of the same index may be expressed as a single radical by means of Art. 6. Thus Vb sjb b The result should always be expressed in its simplest form. If we wish 'to multiply or divide radicals of different indices we must first reduce them to a common index. This can be done by A— 5 42 Algebra. expressing the radical by means of fractional exponents and then reducing them to a common denominator, Thus = 4"^^ 3^'^- 2TV (by ^^j-^ 8)='^7'X3'X2'=2'i/io8. ji_ Also ^ = 5;^_i.f5^_,i^-^,o. \J^3 ^T-w ^3" " 19. Examples. Find the value of each of the following ex- pressions : /. \^ T,acX ^y 2a7n X "^bax. 2. N/|xx/fXv/i. 3. N/iX^'^f 4. a'^y a — xy.x^s/ a-\-x. 5- ^ TO X ^ TT- /5. 2x'^ X 3X^. 7- '"Nf^^""Ji S. ^24X6^3. 7^0. 2^/ 6-f-6'v^ 2. Z^. -V^ 2 -7-^/2. ij. ^ a'—x^-^^ a—x. J5- 5-^^3- 7<5. v^fl5 — X \^a-\-x A polynomial involving radicals is generally more easily multi- plied or divided by another such polynomial by first expressing the radicals by fractional exponents. As shown in Chapter II, the work will then be no different in principle from the case when the exponents are integral. But in a few of the simpler instances it is unnecessary^ to pass to fractional exponents, e. g. RaDICAIvS. 43 I J. Multiply 3 — \/6 by v/ 2—^/3. Process: 3— v/6 v/2— v^3 3V/2 — 2V/3 -3^ 3 + 3^2 6v/2~5v/3 Find the value of 18. (v/3-v/2)(>/3 + n/2). i-^^b^ since (^3 — ^5)(^34-^^5) = 3-' 3 — ^- ^'^~ For larger polynomials it may be that the ratfonalizing factor is itself composed of several factors. Take the quadratic tri- nomiai surd ^^2 — ■v^3./ The rationalizino; factor is /^ (v/2 — n/3 + v/7)(2 — 2v/6)for (v/2 — v/3 + \/7)(\/2 — v/3 — v/7)(2 — 2'V^6) = |(v/2-v/3r-(N/7T](2-2N/6) = — (2H-2V^6)(2 — 2V^6)=20. Radicals. 45 23. Problem. To Rationalize any Binomial Quadratic Surd. Any binomial quadratic surd may be represented by a^p-^b^q, where a and b may be either positive or negative. The rationalizing factor is plainly a\^p—b\^q, for {a\^p + bs/q){a\^p—bs^~q)=^a-p—b'q^ which is rational. 24. Problem. To Rationalize any Trinomial Quad- ratic Surd. Any trinomial quadratic surd may be represented by a^p-\-b^q-\-c'^r, where a, b, and c are supposed to be any rational quantities whatever, positive or negative or integral or fractional. Multiply first by ay^p-\-b\^q—c\^r, and we obtain (a^p-^b\^q-\-cs/'r){a-/p^b\^q—c\^'r) = {a^p + b^~qy—{c\^rY =a'p-\-b^q—c^r-\-2ab^pq, (i) which is rational as far as r is concerned. Now multiply this by {a-p-\-b~q—c'^r)^2abs/pq (2) and we obtain {{a'p-\-lfq—c^r)-\-2ab>'^pq\{{a'p-\-b''q—c^r) — 2abyjq\ = (^arp-Vb^q-c^rXf-_^a^b^pq, (3) which is rational with respect to all the quantities. The ration- alizing factor for the original trinomial quadratic surd is thus seen to be {a yp + byq—c^l^){a'p-\-b'q—c''r—2ab^p}^ (4) 25. The second parenthesis in (4) above will be found to be composed of the two factors (aVp—b'yq-^c^r)(a\^p—b\^q—c\^r) Hence the rationalizing factor of aVp -{-b\/jif^C\/^ may be written (aV]) J^bV'q —cs/ r )(as^p—bV q -{-%^/r_)(a>/p—b^q —c>^r ) Observe that the terms of each of the component trinomial fac- tors of this expression are those of the given irrational quantity and the signs are those exhibited in the scheme — + -h - + - + + ^<--'^ 46 Algebra. Now it is evident that, keeping the first sign unchanged, there is no other arrangement of signs than those written in this scheme, except the arrangement + + + , which is the arrangement of the given trinomial. Therefore The ratio7ializing factor for any trinomi7ial quadratic surd is the product of all the different trinomials which caii be made from the original by keeping the first term unchanged a?id giving the sig?is + and — to all the remai7iing terms in every possible order, except the order occurri?ig in the given tri?iomial. As an example, find the rationalizing factor for ^S~^7~^'^3- The above method shows it to be and multiplying the original trinomial by this the rationalized re- sult is found to be —40. The above problem is cabable of generalization, but its proof cannot be practically given here. The generalized statement is as follows: The ratiofializing factor for any polynomial quadratic surd is the product of all the differe7it polyjiomials which ca7i be made from the origi7ial by keepi7ig the first teri7i imchanged a7id givi7ig the sig7is -j- a7id — to all the re77iai?ii7ig terms in every possible 07'der except the order occurring i7i the give7i polyno77iial. 26. Problem. To Rationalize any Binomial Surd. A bi- nomonial surd will either take the form a^r-^c~q ox a r— cl. Now since these fractional exponents may be reduced to a common de- nominator so that the expressions become a^-fx"-y+ . . ,y-^)=x"-y\ by substitution, I, Art. 26. But which is rational. Therefore (i) is "the rationalizing factor for a" — c" . jL J (b) To rationalize the form a" -\- c" . — -L. 1 L As before, let a"^=x and c"=^y\ whence a" -\- c" =^x-\-y. Multiply x-\-y by ♦ x"-'—x"-'y-\-x"-^y— . . . zhy"-\ (2) The product is f x''—y" if 71 is even, ( .r"+y if wisodd, (x-\-y)(x"-^-x"-y-\-x"-y- . ■ -^y-'^^x. by I, Arts. 27, 28. But ■^' —y = \a" { t u Both of these results are rational ; therefore (2) is the rational- izing factor. 27. Examples. I. Rationalize d'^- With a common denomininator for the exponents this becomes 4 .3 11 d^ — f^'; whence ;/ = 6, 5=4, /=3 ; then x=d^ , y=r^ . . ^^-yj(^.r-^ -^x4j/+.r3_j'2 -f x2j/-^ +.ry'* -f^^; ('4 11 I 2-0 _1 ri 3 1_:2 fi. 8 9 4 12 \V\ f -2 X I f J 0_ 8. 1, 0. '2_ 4 ;', 2 4 = \d-'-r'\ [d'^~ +d^r' ^d""?^ + d^r^ + d^r^ ^7^ =id^ — r'^, which is rational. 2. Rationalize 6 + 3 v^5. With a common denominator for the fractional exponents this .4 1^ } becomes 6"^ + (3* X5)^; whence w=4, ^=4, ^=1 : then .v=6'* and 1 1/= (34 X 5)"^. Therefore (x^y)(x^-x''y^xy''--y^ ) 48 ' Algebra. = [6*+(3'*X5)*J f6-'^-6^X3X5^+6X3'^X5^-3-^X5*J 3- = 6*-3*X5=89i Rationalize \^ 2 + 2^^9. ( rta irr . 28. Rationalization of the Denominators of Frac- TiONS. The most common application of rationalizing factors is in the rationalization of the denominators of irrational fractions. Considerable labor is saved in computing the value of a numeri- cal irrational fraction if we first rationalize the denominator. Thus, to compute the value of — _ - -_i correct to five decimal places, three square roots must be taken and one of them must be divided by the difference of the other two. Now, it will be obvious on reflection that these square roots must be taken to nearly ten places of decimals if we are to be absolutely certain that five deci- mal places of the quotient are correct. It will be easily seen how much more readily the value can be found after the denominator has been rationalized. Multiplying both numerator and denomi- nator by the rationalizing factor for the denominator, wt have >^5 V z^{-y 'j-\-\^ 2) v^35-fv/io v^7-x/^~"(>/7-v/2) {'s/~^-\-V~2)~ 5 Now but two square roots need be taken, and these to no more than five decimal places, since the exact value of the denominator is known. 29. Examples. 1. Rationalize the denominator of — ::= -. v/3-f-\/2 2. Rationalize the denominator of ' — - — ;=. 1 3. Prove I :_ =(2 — V^2)"2. [2 + V2 4.. Given v^ 3= 1.7320508, find the value of —. Rationalize the denominator of Rationalize the denominator of Radicals. 49 v/,8 cS'. Rationalize the denominator of — 3 + 2n/2 p. What relation must hold between a and x in order that a -{- \- X s/ 2 — '^'^~}-'^S 10. Rationalize the denominator of - _— - _ -^. V 2-HN/3-f>/5 //. Compute the value of the following to three places of decimals, having first reduced it to its simplest form : ^34-v^S+^5-x/ \^x'-\-i-\-\^x''—i '^sc'^i — s/x'—i 12. Prove ^ -u_-^_ ^_:"_-;_-{- --^r =2Ji:^. V x=+ I — ^x"— I s/ x'-\- 1 -j- Vx"— I 30. Theorem. If a, b,p, q are commensurable a7id >/ b and^q incommensurable, and (f a-\-\^ b =p-{-\^q , then a=pands^ b =v^^. \i a does not equal/, suppose a=p + d. Substitute this value for a in the given equation, and we have p+d+\^b=p-]-\^q or d-{- \^b — >/q squaring both sides whence d- + 2ds/b-\-b=q 2d That is, an incommensurable quantity equals a commensurable, which is absurd. Therefore a cannot differ from p. And if-fl'=/>, \^b must equal ^ q . A— 6 50 Al^GKBRA. 31. ExAMPLKS. We append a few miscellaneous examples on the last two chapters. JL _L i /. Does (a-\-x)''=a''-j-x'' f 2. Multiply together ^a , y^\ ^r, ^ yi^ and a~'^. 2 2 J. Simplify -3— -^r X 1 - I y-\-x" 4. Multiply together ^x'^^+x"y^y'\ s^J(f—y\ s/x^'^—x^y +V" and \^x''-\-y\ 5. Multiply together 1 1^ 1 i_ (a'-^ab+tf-')", (a—b)", (a—b)'- and (d'-^ab + b')- (f — x:^ a — X 6. Simplify TT^ s/a- "+i 7, Simphfy y^:^.:.. 2 ~+i X— ^ x'—y y ^. Cube the expression a"v/x—%/<^av^jj', 9. Prove 2 + ^3 is the reciprocal of 2— "v/3 ; and find what must be the relation between the two terms .r and x^y so that jtr-fx/ y shall be the reciprocal oi x—^ y. 10. Simplyfy ! — . . (i-\-xy^(\-xf 11. Simphfy the expression ^ 1 (iJrxf-(\-xf first by rationalizing the numerator, and then by rationalizing the denominator. 12. Prove that if p=i and ^=5, Pq-^e/'-'^-2-\-qp-^e'^-' 3 + 5^'' CHAPTER IV. QUADRATIC EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 1. Definition. An Equation of the Second Degree, or a Quadratic Equation, is one where the highest degree of any term with reference to the unknown quantities is two. It must be remembered that the degree of an equation with reference to any quantity is not spoken of unlCvSS the equation is rational and integral with reference to that quantity. See I, Art. 6. 2. We will consider in this chapter quadratic equations con- taining but one unknown quantity, such as — ^ — 3^+5^=24, (i) 2x^— |jr=.346, (2) ^3 ^^-^- (f- ^^l\x=^- v^ 5, (3) m-\-^-\ X' + {d— t)x=p-^ \^k. (4) These equations are all obviously quadratics. But some equa- tions, which are irrational or fractional with reference to x in their present form, drop into the quadratic type as soon as the proper transformations are performed. Thus the equation, y/ 1,— ^ a I . V.a—>/b ^x+ = -f - .. (5, a) \^ b ^x \/ a may be made integral with reference to x by multiplying through by s/ X, the resulting form being (y/ fj-y/aWx , Wa-s/ bWjt "+- ^1 = ' ^ —^--~ ' Transposing and uniting terms, \^ab Transposing the rational parts to the right hand side of the equa- tion, we obtain the form {b —a)^x _ _ y^al ~' 52 Algebra. Now rationalizing with respect to x, by squaring both sides of the equation, it becomes at? Finally transposing and collecting terms, we have , a' + ^' which is a quadratic equation. While the equation (5, a) has been reduced to the quadratic form (5, d) by apparently legitimate processes, yet we will find that the integralization and rationaliza- tion" of an equation with reference to the unknowai quan- tity has in general an effect on the solution of the equation which it is necessary to take into account, and which renders it pos- sible that the values of -r which satisfy (5, b) may not be identical with those that satisfj^ (5, a). For this reason the treatment of those equations which require the operation of integralization or of rationalization before they are in the quadratic form, is reserved for Chapter VI. 3. Typical Forms ok the Quadratic. It is evident that equations (i), (2), (t,), (4.) and (^5, d), or any other quadratic equations which can be imagined, may all be said to be of the typical form, ax''-\-dx=c, (6) where a, d and c are supposed to stand for any numbers whatever, either integral or fractional, positive or negative, or commensur- able or incommensurable. Hence ax'-[-dx=^c is said to be a typical form of the quadratic equation. If we suppose the quadratic equation to be divided through by the coefficient of x:^ the result will be of tlie form x'-\-px=q, (7) where / and q are supposed to be any algebraic quantities what- ever, fractional or integral, positive or negative, commensurable or incommensurable. This is the second typical form of the quadratic equation, and one which is much used. 4. Definition. A /vV(?/of an equation is any value of the un- known quantity which satisfies the equation. Thus \ isa root of the equation 3;^— 6=0, for when substituted Quadratic Equations. 5,-^ for X it satisfies the equation. Also, both 2 and 3 are roots of the equation .r^— 5.r+ 10=4, for either of these values when substitu- ted for X will satisfy the equation. The student must carefully note that this is an entirely different use of the word 7vot from that occuring in the expres.sions square root, cube root, etc. 5, Equations of the second degree are often divided into the two classes of complete and mcoynplete quadratics. A complete quadratic is one which contains both the first and second powers of the unknown, as x~-\-px^q. An incomplete quadratic has the first power of the unknown quantity lacking, and hence can al- ways be placed in the fonn x"—q, where q is any algebraic quan- tity conceivable. By some the adjectives affected and pure are used in place of the words complete and ijicomplete respectively. 6. Problem. To Solve anv Incomplete Quadratic. First, reduce to the form x'^=q by putting all the known quantities on the right hand side of the equation and all the terms containing x'^ on the left hand side, then dividing through by the coefficient of x\ Then take the square root of both sides of the equation, remem- bering that every quantity has two square roots, and we obtain x=±^q and the equation is solved. It might be thought that in taking the square root of both sides of x'^q we should write But, by taking the signs in all possible ways, this givesi j^x= + s/q —x=^ — ^q Jrx^-s/q —x=-\->/q. 54 Algebra. Each of the first two of these is equivalent to x= ^ q , and each of the last two is the same as x= — \^q, and, on the w^hole, we merely have Whence it is seen to be sufficient to write the sign ± on but one side of the equation. 7 . Examples of Incomplete Quadratics. Solve the following equations : 2. (ax~.b)(ax-\-b)=c. J. (x+2)^-h(x-2r=24. /. (x-\-d/-j-(x-dr=r. 5. (ax-\-d)^-{-(ax — bf=^c. 6. (x+T)(x-g) + (x-'j)(x-^^)=^'j6. 7. (x-{-a)(x-b) + (x-a)(x-^b)=c. 8. (x-^a)^=q. Show that examples i , 3 and 6 may be solved by proper substi- tution in the results to examples 2, 5 and 7 respectively. 8. We have solved the equation x"=^q, and also the equation (x-{-a)^—q (Ex. 8) in a similar manner. Now it is evident that the equation x'-\-px=q can be solved if it can be put in either of the above forms. It can be placed in the form (x-\-a)^=q if the first member can be made the square of a binomial. On inspection it is seen that x"-\-px are the first two terms of the square of a binomial, the third term of which must be ^/>^ Hence, if we add \p'^ to both sides of the equation x^J^px=q. it takes the form x'+px+\p^=q+\p\ or ^ r-^+i/>/=^+i/^ which is of the form (x-\-a)-=q. The process of putting a quadratic equation in this form is called completi7ig the square. Quadratic Equations. 55 9. Probi^km. To Solve thk Typical Quadratic x'-{-px—q. Add ^/>^ to both members and we obtain The left hand member is seen on inspection to be the square of the binomial (x-\-yP) ; whence taking the square root of both members, Solving this simple equation for x we have x=-\P^^q+\p% w4iich gives the two values of .1 , -\P+^(1-^\P' and -^p-Vq-^\p\ Hence, to solve an equation in the form j^-\-px=-q, add the square of 07ie-half the coef[icie7it of x to each side of the equation. Take the square root of both me7nbers, and an equatio?i of the first decree is obtained, from jvhich x can be found in the usual way. 10. Problp:m. To vSolve thk Typical Quadratic ax^-\-bx Multiply through by 4<3! and obtain ^cC\x^ 4- \abx^ ^^ac. Adding b^ to both members it becomes \a^x^ + d^a bx -f- b- = ^ac-\- b^ . The left hand member is seen on inspection to be the square of a binomial ; whence, taking the square root of both members, we obtain 2ax -{- b—±^ /i,ac-\- b\ whence, solving this simple equation, — b^sf'^c'^lf 2a which gives the two values of .v, — ^-f v^4«r+A-' , —b—y/Atac-\-b' and 2a 2a Hence, to solve an equation in the form ax'-\-bx==c, multiply through by four times the coefficient of .v- and add the square of the coefficient of x to each side of the equation. Then take the square root of both members, aiid an equation of the first degree will be ob- taiyied, from ivhich x can be found. 56 Algebra. i Ife »b^ 11. HiSTOiiiCAii Note. The origin of the solution of the quadratic equa- tion cannot be definitely traced to any one man or any one race. Algebra, as we now have it, has been a slow growth, and, as we pass ba(jk in time, it grad- ually shades off into the arithmetic of antiquity. Diophantus, an Alexandrian Greek of the fourth century, A, D., who wn te a treatise cm arithmetic, could undoubtedly solve - 4- 27^=529. Taking the square root of both members, 20-X"— 27=^1=23, and solving this simple equation The first task is to place the eqiiation in one of the typical forms. Now "complete the square." ■9-'i+5 = Since it hi been proved that thi meth- od will give a complete square, it is not nec- essary to work out the value of the co- efficient of X, butmere- ly to indicate it by (). 2o.x-=5o or 4 x= 2^- or i. Quadratic Equations. 57 Solve the following : /. .v-^ 4-7.^4-15 = 5. 2. X- 4- 6x— I = 5 — 20A-. 3' 3^^"+5-v==ioo. 4.. 1 5^-"— 28.^4-10=5. In completing the square in a case like this where the coeffi- cient of jf is divisible by 2, fractions can be avoided without multipljang the equation through by 4. Thus: 15^^— 28x= — 5, multiplying through by 15 (instead of 4X15), and adding the square of one-half of 28, we obtain (i5r-i:=- 15x28-1-4- (14)'= -5x15 + (14)' w^hich is a complete square. 5. A-+i2yV^'=38|. 7. x^4-6.5i = 5.2x. 8. x^--\-\—ax—-=o. 9' (■'^—?>)(^^—S) = o. 10. (x—a)(x—d)=o. 11. (zx—^)(^x—2,)=o. 12. (ax—b)(bx—a)=o. ij. (x-\-a — b)(x—a-\-b) = o. 14.. a'—x^=^( a—x )( b-\-c—x ) . /5- r33+io.r/+r56+io-t-/=r65+i4^C. 16. r7-4v/3;.r=4-r2-\^3A=?- 18. x^-\-qax=a'' — b'. ip. (x—a/=(x—b)(a + b). 20. dez'—(d'-^e')2^-de=o, 21. x''—2a(x-\-b)=^2bx—a'—b'. 22. .r'4-ioji--f 30=5. 2j. a-^b-\-x— a'b'x'. x^ x"" 24.. -\-ax=' +bx. ^ a b 25. ax'-\-bx-\rC=x'-^px-\-q. K—l \ \ 58 Algebra. 26 . x~ — I = k(kx' — ^x—k) . 2y. (x—a)(x—b)-\-(x—b)(x—c)-\-(x—c)(x—a)=o. Result, x=\(a + b+c)±iya' + b'' + r-—ab—bc—ca. 28. ji-'^ + 6x4-2i = io. This becomes, in the typical form, x^ + 6;i"= — II. Completing the square, we obtain x^-^6x-\-g— — 2. Now we cannot obtain the square root of the right-hand mem- ber of this equation ; for it is a negative quantity, and the square of no algebraic number can be negative. But, if we were to go through the operation of finding x as has been done in the other cases above, and indicate the root of —2 as if we coicld take it, we would have x= — 3±v/— 2. Thus we have had forced upon us in the solution of the quad- ratic equation, something which, whatever interpretation it may have, is evidently 7iot a?i algebraic quantity in the sense in which the term is commonly used. Such an expresion is called an im- aginary, and its treatment is reserved for Part II of the pres- ent work. In the next chapter will be found a discussion of the circumstances under which such expressions occur. 2g. 4Jt:^ + 4-r+4=jr^. 13. Problems Requiring the Solution of Quadratic Equations. The student in his previous study has probably already noticed that the first task in the algebraic solution of a problem is always an attempt to express the language of the prob- lem in algebraic symbols ; that is, to cast the relations and condi- tions expressed by the words of the problem into an equivalent statement in the form of one or more algebraic equations. This w^ork is called the statement of the problem, and is generally a difficult one for the beginner to perform. When the statement of a problem is complete, all that remains to be done is the solution of the equation or equations obtained thereby by processes already familiar. We wish to strongly emphasize the fact that the equation obtained by the translation of the words of most of the algebraic problems in the books is often 7iot an exact equivale?it to the condi- Quadratic Equations. 59 tions and relatio7is told in the language of the problem. In fact, the equation often eynbraces more than the problem itself. We will illustrate this by the following problem : A certain number consists of two digits whose sum is lo. If we reverse the digits and multiply this new number by the original number, the product will be 2944. Required the number. I^et -r=the digit in unit's place ; then 10— -v=the digit in ten's place, and 10(^10— Aj = the value of the digit in ten's place ; whence 10(^10— ;rj-{-;r= the value of the orig- inal number ; also lox+i^io— -rj=the value of the number with the digits reversed. I But, by the problem, I [lorio— -i-;-f.r] [loi-f ("lo— -^vj]=2944. (\) \ That is. Statement, or trans- lation of the language into an algebraic equa- tion. Solution of the cqim tion. (^ 100— 9JiJ(^ lo-j- 9.rj= 2944. Expanding left member, 8 IX''— 8 lox— 1000= — 2944. Transposing and uniting, 8iA^— 8iox= — 1944. Dividing through by 81, x^— iox= — 24. (2) Completing square and solving, -t==4 or 6. The number is therefore either 46 or 64. Now consider equa- tion (i) as a translation of the problem into algebra. As far as is stated by the eqiiation ( i ) the unknown quantity x may be an}' algebraic quantity conceivable, — positive or negative, integral or fractional, rational or irrational, or, in fact, it may possibly be what we have called an imaginary. As far as the equation ex- presses the nature oi x, it may as likely turn out in the solution one kind as another of those enumerated. But, as expressed in the language of the problem, x must be a digit ; that is, a positive integral 7iumber less thaji ten. The equation does not express this fact and cannot be made to do it. The reason why the prob- 6o Algebra. lem really works out all right is that it was made to order ; that is, the number 2944 was especially selected so that the problem would * ' work out " . If we wish this problem stated in words so that it is more nearly identical with its expression in the form of an equation, we must throw out the word "digits" as follows: There are two numbers whose sum is ten. If ten times the first plus the second is multiplied by ten times the second plus the first, the product will be 2944. Find the numbers. This is nearly as general as the algebraic equation. It permits of either positive or negative, integral or fractional, commensur- able or incommensurable, results, and indeed as the word num- ber is often used it would permit of imaginary results. This prob- lem can be made identical with the original by adding at its close some such caution as this : Do not obtain a fractional, a negative, nor an incommensur- able result, nor any result greater than 9. It is such conditions as these that we fail to incorporate into an algebraic equation. The algebraic statement, as far as the un- known is concerned, is always the most general possible and con- tains in it no restriction of the unknown to any particular class of numbers, and for this reason the algebraic statement of a problein is often more general thaii the problem itself. This fact should be re- membered, as it will help to explain many apparent difficulties which arise in some problems. These non-algebraic conditions in a problem must be ignored until after the solution is had, and then if a result is obtained like a fractional number of live sheep or a negative price per head, it must be cast out, not because the mathematics is unreliable, but because the problem is cramped and does not fill up the full measure of generality which algebraic methods provide for. The greatest breadth and elegance of algebraic analysis would be observed in the treatment of problems in geometry, mechanics and physics, but since we cannot presume any considerable famil- iarity with these, only problems involving the simplest geomet- rical principles have been inserted. While the elegance of alge- braic methods is best seen in the solution and discussion of problems of equal generality with their algebraic statement, yet those we give are not entirely of this class. Quadratic Equations. 6i PROBLEMS. /. The hypothenuse of a right angled triangle is io,and the excess of the perpendicular over the base is 2. Find the sides of the triangle. 2. The hypothenuse of a right angled triangle is h, and the excess of the perpendicular over the base is c. Find the sides of the triangle. Can e in this problem be assigned a^iy value whatevei ? J. The perimeter of a rectangle is 16 feet, and its area is 15 square feet. Find the dimensions of the rectangle. 4. The perimeter of a rectangle is p feet, and its area is a square feet. Find the dimensions of the rectangle. Show, from the result, that the square is the greatest possible rectangle which can be made with a given perimeter. 5. The sum of the squares of three consecutive odd numbers is 83. Find the numbers. What would you say in case the number 56 was given in place of 83? Make the problem read so that 56 will be allowable. 6. The sum of the squares of four consecutive even num- bers is 120. Find the numbers. 7. If 962 men were drawn up in two squares, and it were found that one square had 18 more ranks than the other, what would be the size of each square ? 8. A boat's crew row 3^ miles down a river and back again in I hour and 40 minutes. Supposing the river to have a cur- rent of 2 miles per hour, find the rate at which the crew would row in still water. What do you say about tlie negative result ? p. A boat's crew row d miles down a river and back again in / hours. Supposing the river to have a current of r miles per hour, find the rate of rowing in still water. Show from the result that the problem will always give one positive and one negative value of x for all values of d, t or r. 62 Algebra. lo. The total area of two squares is a square feet. A side of one square is found to differ from a side of the other by d feet. Find the side of each square. Is this problem possible for all values of (T^? //. Two trains are dispatched from a station, one starting an hour before the other. The rate of motion of the later train is 5 miles per hour more than that of the other, and it overtakes the first train at a distance of 150 miles from the station. Find the rate of motion of each train. 12. Generalize the foregoing problem and solve it. DiscUvSS the results. 13. A rectangular metal plate is 20 inches longer than wide. It is expanded by heat until each dimension increases by 2V ^^ ^^s former length, thereby increasing the area of the plate 246 square inches. Find the original dimensions of the plate. 14.. A man, bom in 1806, died at the age of x in the year x^ When did he die ? 75. Two trains pass at a junction. One is traveling south at the rate of 30 miles an hour and the other is traveling west at the rate of 40 miles per hour. How long before the two trains are 100 miles apart? Interpret the two results. 16. Two trains, A and B, are traveling on roads at right angles to each other, each approaching the crossing. A is 10 miles from the crossing and traveling uniformly 30 miles an hour, while at the same instant B is 20 miles from the crossing and traveling uniformly 40 miles an hour. When will they be 5 miles apart ? Explain the two results. ly. Two trains, A and B,['Sir^ traveling on roads at right angles to each other. A is J40 miles from the crossing and is moving towards it at the ||uniform rate of 30 miles an hour. B is 20 miles from the crossing and is moving frofji it at the uni- form rate of 25 miles an hour. At what times are the trains 90 miles apart ? Interpret the results. Quadratic Equations. 63 18. Along the sides of a right angle two bodies, A and /y, move with unifonii velocity. A \^ a miles from the vertex and moving p miles per hour, while at the same instant B '\^ b miles from the vertex and moving q miles per hour. At what times are the two bodies d miles apart ? Show that the result obtained can be used as a formula to solve Prob. 16. Show that by giving the proper interpretation to q, as to its positive or negative character, that the formula can be made to solve either Prob. 16 or 17 at will. Under what conditions will the bodies 7iev€r be d miles apart ? 7^. Two circles, A and B, move wdth their centers always on the sides of a right angle. A, whose radius is -R feet, is a feet from the vertex and moving uniformly p feet per second. B, whose radius is rfeet, is b feet from the vertex and moving uni- formly q feet per second. At w^hat times are the circles tangent to each other? Result : Tangent externally in ap^bq^s/(RJrrnp^±^)^-(apJjqr ^^^^.^ p^-^q^ ~ seconas. Tangent internally in ap^ i;q±,V(/^-rr(p^ + q^) + (ap-^bqr- ^^^^^^^^ P" + f Show that it is possible for them to be tangent externally and not tangent internally. Show that it is impossible for the circles to be tangent inter- nally without first being tangent externally. Show that the known quantities, may have such values that the two circles will never be tangent at all. 20. Find the side of an equilateral triangle, knowing that a side exceeds the altitude by d feet. CHAPTER V. THEORY OF QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS. I. It follows immediately from the definition (I Art. 4 ) that every rational integral quadradic function of x is of the form /x^-\-72X-\-r where f, n and r stand for any algebraic numbers whatever, posi- tive or negative, integral or fractional, commensurable or incommensurable. If we take the typical quadratic equation ax''-\-bx=^c and transpose the c to the left-hand side of the equation it becomes ax'^ -\- bx — r= o . This can obviously be said to be of the form lx^-\-nx-\-r=^o and consequently a quadratic equation may be defined as an equa- tion which can be placed in the form of a rational integral quad- ratic function equal to zero. Since a root of an equation has been defined as any expression which substituted for the unknown will satisfy the equation, there- fore it is evident from the form ax^-\-bx — <:=o that a root of a quadratic equation may also be stated to be an expression which substituted for x causes ax--\-bx—c to equal zero ; that is, causes the function* oi x to vanish. Hence we may say: A quadratic equatioyi is a7iy equation ivJiich _^ — ^ can be put in the form of a rational ititegral quadratic fu7ictio7i'7quaT \ffi^ to ze7'-o, a7id a root of it is a7iy expression ivhich, substituted for x, causes the fu7ictio7i of x to vanish. Thus the equation ..r^— 3Jt:=io, whose roots are 5 and —2, when placed in the form of a function of x equal to zero, becomes x'^ — 3X — 10=0. It is now seen that the roots are such quantities that, when sub- *Because of the array of adjectives in the expression ' rational integral qnadrati( function of x " we shall often, for the remainder of this chapter, use the expression "func- tion of X " in its place. Theory of Quadratics. 65 stituted for x, cause the function of x to vanish. For the func- tion of X is X^—'TfX—lO and putting 5 for x it becomes 25—15—10 which is zero. Putting —2 ior x the function of x becomes 4+6—10 which is also zero. If anything else than a root is put for x the function will not vanish ; thus when -r=— 4, function of x becomes 16+12—10= 18 x=- -3, ' " 9+ 9—10= 8 [x=- -2, ' " 4+ 6—10= 0] x=- -I, 1+ 3—10=— 6 .r= 0, ' *' 0+ 0— io= — 10 x= I, " * ' " I— 3—10= — 12 x= 2, ' *' 4— 6— io= — 12 x= 3, ' " 9— 9—10= — 10 x= 4- ' " 1,6—12—10=— 6 lx= 5, 25—15—10= 0] x= 6, 36—18—10= 8 2. If we suppose the quadratic function divided through by the coefficient of x'' it may be represented by x^-]-ex-\-f. If we take the quadratic x^-\-px=q, and transpose the q to the other side of the equation, we obtain x'^-\-px—q=o where the left member is seen to be of the form x^-^ex-{-f. Then, since every quadratic may be put in the form x--\-px=q, it may also be placed in the form x^-]-px—q=o or better x^-\-ex-{-f=o. In either of the quadratic functions lx--\-7ix-\-r or xr-\-cx-\-/ the term which does not contain x, that is r or /, is called the ab- solute term. A— 8 66 AI.GEBRA. 3. By solving the equation it will be found that its roots are -i^+ V-K-/ and -}^e-\/\e^-/. 4. ThEORKm. Every quadratic fun ctio7i of x can be i-esolvcd into the product of tivo linear functioiis of x. Take the function of Jt: in the form x^-\-ex-^f. add and subtract \e:' from the function, thus not altering its value. We obtain then _ ^ x^-^ex+\e^-\e^+f This may be written (x+\er-(\e^-f), or, if we please, as the difference of two squares, {x+w-WV^T- Writing this as the product of the sum and difference, it takes the form or (x+\e-\^\r-f){x+\e+ Vi^-/), which is the product of two linear functions of x. 5. Examples. Resolve the following quadratic functions into the product of two linear functions of x : 1. x^— jr— 2IO. 2. 3.r"+2Ji:— 85. J. x''—6bx-\-gb-. 4.. /\.a^x^—4ax-\-i. 5. .r=-i4.r+33. 6. Theorem, /f the roots of a quadratic equation are a and b, then the equation may always be put i?i the form (x—a)(x—b)^o. By Art. 4, the equation x''-\-ex-\-f=o (i) may always be placed in the form U-+if + \/i^-/)Gr+i^- Vi?=7)=o. (2) Theory of Quadratics. 67 If we represent the two roots of (i) by « and b for the sake of brevity, we see from Art. 3, that Substituting these in equation (2), it becomes (x—a)(x—b) = o. 7. C0ROI.LARY. If all the terms of a quadratic be transposed to one side, that member is exactly divisible by x minus a root. 8. Corollary. The form (x—a)(x—b')=^o may be used iyiter- changeably with x'' -\- ex -\-f= o to represent any quadratic equation. 9. Theorem. Every quadi^atic equation with one unknown q2ia7itity has two roots and only two. It has been shown that every quadratic equation can be placed in the form (x—a)(x—b)=o. This equation is satisfied when the left member is zero. But the left member becomes zero when either one of its two factors is zero ; that is, when x=^a or x=b. Because each of these two values of x satisfies the equation it has two roots. But the equa- tion can have no other root ; for if any other value than a or bh^ assigned to x, neither of the factors will be zero, and consequently their product will not be zero. Hence there can be no more than two roots. It is not claimed that there must be two different roots. In fact, there is nothing in any of the reasoning thus far which shows that a and b must always have different values. In general, they are different from each other, but a special case would be where they are alike. In this case the quadratic takes the form (x—a)(x—a)^o, and we still speak of two roots because there are two factors and because it is merely a special case of the general truth. To say that an equation has two roots equal to each other is merely an- other way of saying that there is but one value which satisfies the equation. 68 Algebra. 10. Theorem. Whe7i a quadratic eqiiatio7i is in the form x^ -\- ex -\-/= o , the coefficie?ii of x with its sign changed equals the sum of the two roots,. and the absolute term equals the product of the tzvo roots. The two roots of the equation x^-\-ex-\-f=o are —\e-\->f\e^—f and —\e—s^\(f —f — e-\- o Adding them, their sum is seen to be — ) or, dividing through by a—b, a-\-b+e=o, or e= — (a^b). (4) That is, the coefficient of x is the sum of the roots with opposite signs. Now substitute this value of ' and z. Then we know by Art. lo, y-\-z=^—e (i) and y^=f^ (^) squaring (i) we obtain _>'" + 2y2 -\-z^=e\ r 3 j Subtracting four times (2) from this y^-—2yz-^z'=e'—4f or, extracting the root, r— ^=±V^ — 4/i * and since y-^z=—e, —e±\/e^—4.f J'=~ — T ■ 2 Solve in this manner the equation 3:1" — 5.^+2=0. Theory of Quadratics. 71 13. Discrimination of the Roots of the Quadratic Equa- tion. The roots of the equation :x^-hex-\-f=o are x=—^e-h^/{e'—/and x^—\e—sj\e''—f. (a). \i\e'—f is positive there are two real and unequal roots. (b), \i\e—f\s negative there are two imaginary roots. (c). If \e'—f is zero the two values of x each reduce to —\e and the two values oi x are real and equal. (d). \i\r—f is a perfect square the two roots are rational, if e is rational. (e). If \e-—f\s> not a perfect square the roots are irrational. The expression \e''—f is called the Discriminant. The case where \e^—f is zero deserves further attention. If J(?^— y=o then \e'=^f and the equation ji--f^-^'+y==o becomes x^-\-ex-i-\e'=o or (x-\-^e)(x-hie)=o. Whence we see that when a quadratic equation has two equal roots the function of x is a complete square. 14. To Find the Conditions that a Quadratic Equation MAY HAVE TWO POSITIVE RooTS. Represent the roots by a and d. Then since —(a-\-b)^=e if the roots are both positive the coefficient of x must be negative. Also since CLb=f if the roots are both positive the absolute term must be positive. Hence the full co7idition that both the roots of a quadratic be posi- tive is that the coefficient of x be negative and the absolute term positive. 15. To Find the Condition that a Quadratic Equation MAY HAVE Two Negative Roots. Represent the roots as before. Then since — (a-{-b)^=e if both roots are negative the coefficient of x must be positive. And since (ib=f if both roots are negative, the absolute term must be positive. Hence the full condition that both the rootsofaqimdroHc be neg- ative is that the coefficieiit of x^be positive^id the absolute ternj/ we^ 72 Algebra. 16. 'To Find the Condition that a Quadratic Equation MAY have: onk Positive and one Negative Root. Since ^^=/ if the roots are of opposite signs the absolute term must be negative. Since —(a-{-d) = e if the positive root is numerically the greater, e is negative and in case the negative root is numerically the greater, e will be pcsitive. The conditio7i that a quadratic have roots of opposite signs is merely that the absolue term be negative, but if the coefficient of x is nega- tive the positive root is numerically the greater and if the coeffi- cient of X is positive the negative root is numerically the greater. 17. BxAMPivES. Discriminate the roots of the following equa- tions; that is, tell by inspection whether the roots are real or im- aginary, and if real, tell whether they are positive or negative. I. x^'-f-Sjc— 9=o. -T^ -f 7 O-T -f- 1 2 GO = O. :r^— 4Jt-+4=o. x^-{-iox-\-/^^=o. X'^ — ^X-\-20 = 0. x''=iojt"— 25. X^—\2X= — 2^. 18. In a manner similar to that of Arts. 14 — 16 the student may determine the following : 1. Find the condition that a quadratic equation may have two roots numerically equal but of opposite signs. 2. Find the condition that a quadratic equation may have two roots which are reciprocals of each other. J. Find the condition that a quadratic equation may have one root equal to zero. 19. M1SCE1.1.ANEOUS Exercises in the Theory of Quad- ratics. I. If a and d are the roots of x^-\-ex-^f=o, find the value of ^^-f-<^^ in terms of =-^Y-^"— 5^* + -^"— 25 (I) are — i and 5. Either of these when substituted for x will satisfy the equation. But divide the equation through by x— 5. The resulting equation is 3=.v + .r-f5. (2) Now this equation is not satisfied for x=z^. The sole root is — I. Hence, although equation (2) mUvSt be t^-ue if (i) is, yet the equations are not equivalent, since their solutions are not iden- tical. One root has disappeared in the transfonnation. Just how this occurs will be best seen after we place (i) in the form (x—a)(x—b) = o. Since the roots of (i) are — i and 5, by the principle of V, Art. 6 it is equivalent to (x-s)(x+i)=^o. (3) Now, if we divide this through by -^"—5, we remove that factor in the left member which is zero for -r=5. Consequently the equation will be no longer satisfied for .1 = 5. If we should divide through by x-\-i the equation will be no longer satisfied for x= — I . Also consider the equation .1-"— 6.r+8=o. (4) It is satisfied for .1 = 2 or -f=4. Now multiplying both members t>y -^" + 3 we obtain (x+:,)(x--6x-\-^)=o. (5) But this equation is satisfied for either .v=— 3, or .1 = 2, or .1 = 4. Hence, although multiplying both members of (4) by .i"-{-3 has not altered the equality, yet a value of x extraneous to the orig- inal equation has been introduced. Again the equation 2-r— i=.i--j-5 (6) 76 Al^GKBRA. is satisfied only by the value -r=6. Now square both sides of the equation, obtaining 4Jt-^— 4.r-j- 1 =A-=-|- io.r+ 25, (7) which is satisfied for either x=6 or x=—\. Here, obviously, an extraneous solution has been introduced by the operation of squaring both members. In a like manner notice the effect of taking a root of both members of an equation. Thus suppose ^■^=(x—6)\ (8) This is satisfied for either x=2 or —6. Take the square root of each member and we obtain 2x=--x—6, (9) which is satisfied only by x=—6. We have lost one of the solu- tions of the equation during this transformation, Equation (?>) is really not equivalent to (()), but to the two equations ^2X= + (x-6)\ . {2X=-(x-6) \ ^^^^ We have given examples enough to show that certain opera- tions upon an equation may modify the solution. Thus we see that during a series of transformations which sometimes an equa- tion must undergo before we can reach the values of the unknown it is possible that the solutions that satisfy the original equation may all be lost and that any number of new ones may be intro- duced, so that the final results may have no relation at all to the problem in hand. It is now proposed to formulate certain propo- sitions which will enable us to tell the exact place in the process of any solution where roots may be lost or new ones may enter. We will then be able to perfonn the different operations on the members of an equation if we will note at the time their effect on the solution and finally make allowance for it in the result. This fact must be emphasized : ^/le test fo?- a?ty solution of aii equation is that it satisfy the original equation. '' No matter how elaborate or ingenious the process by which the solution has been obtained, if it do not stand this test it is no solution ; and, on the other hand, no matter how simply obtained, provided it do stand this test, it is a solution." — Chrystal. When one equation is derived from another by an operation which has no effect one way or another on the solution, it may be SiNGivE Equations. 77 spoken of as a legitimate transformation or derivation ; when the operation does have an effect upon the final result, it may be called a questionable derivation, meaning thereby that the operation requires examination. If there are two equations such that any solution of the first is a solution of the second, and also that any solution of the second is a sokition of the first, the two equations are said to l)e equivalent. 3. Theorem. The trayisformatioii of an equation by the addition or subtraction from both members of either a kyiown quantity or a functio7i of the unknown is a legitimate derivation. An equation containing one unknown quantity, as it commonly appears with quantities on each side of the equation, may be generalized in thought by the expression A function of x= Another functioti of x. Or, using L to represent the left-hand side of the equation, what- ever it may be, and R to represent the expression on the right- hand side, we can represent any equation very conveniently by L = R. (I) Now suppose that T, which ma}^ be either a known quantity or a function or the unknown, be added to both members of the equa- tion, making L+7 = R-\-T. (2) Now it is plain that (2) cannot be satisfied unless L=R and that it is satisfied if L — R. Hence (2) means no more nor less than (i). Therefore the derivation is legitimate. 4. CoROLivARY. Transposition of terms from one member to the other, changing the si^ns at the same time, is legitimate. Thus \i L — R, to pass to L—R=o is merely subtracting A' from both members. 5. Theorem. Multiplying both members of an equation by the same exp^rssion is legitimate if the expression is a known quantity, but questionable if the expression is a function of the unknown. 78 Algebra. Represent the equation by L = R. (I) Multiply both members by T, obtaining LT=RT. (2) Now this may be written as (L-R)T^o. (3) If T is a known quantity this can only be satisfied by the sup- position that L = R, that is, the equation is equivalent to (ij. But if 7^ is a function of the unknown (for example, 2x or A-I-5, or -r^+8) then (3) may be satisfied by any value of the unknown that Will make 7^=o (such as x=o, or ^ = 5, or ,f= — 2, respect- ively, in the three examples given), whence (3) would not be equivalent to (i) but to the two equations. \ L = R} \ T=o. ) 6. Corollary. //* any equation involves fraetions 7vith only know)i qnantities in the denominators, it is legititnate to elear of fractions. The multiplier in this case is a known quantity. 7. ThEORKM. It a^i eqiiation involves irreducible fractions zvith unknown quantities in the denominators, and the denominators are all prime to each other, it is legitimate to integralize by multiplying through by the least common multiple of the denominators. To illustrate the reasoning take the equation I 2 3 where the fractions are supposed to be in their lowest tenns and X ^, X ^, X^ represent diffei^ent functions of the unknown and where A, B and C are either known quantities or functions of the unknown. Multiplying by the least common multiple of the denominators we obtain AXX^-^BXX^-^CX^X^^o, (2) Now, since X^ , X^ and X^ are prime to each other no common factor has been introduced by multiplying by X X^X^, and con- sequently no additional solutions can appear. Single Equations. 79 8. As an example under the above theorem take the equation II — 2Jl- 3-V— I These fractions are in their lowest terms and their denominators are prime to each other. The least common multiple of the de- nominators is f II — 2.rX3-v— I j. Multiplying through by this we obtain (^X—l)('J — x) + (ll — 2X)(4X~s)=2(ll—2X)(T,X—l). (2) Now we can see that although ( i ) has been multiplied through both by (i\ — 2x) and (2>^—i), yet neither of these has been in- troduced as a factor through the equation. Hence there is no ad- ditional solution introduced. The roots of (^2J will in fact be found to be 4 or —10, which values also satisfy (\). But an extraneous solution may be introduced if the denomin- ators are not prime to each other, or if some of the fractions are not in their lowest terms. Thus has two denominators alike, and consequently not prime to each other. Multiplying through by the common denominator x^—g, we obtain ^x(x + 2.) = 6(x-^)^^(x+2>) (4) or, reducing, .1'— 2.1=4 (s) whose roots are 3 and — i. Now if we put the original equation (t,) in the form X— 3 x-\-3 that is 3= , (6) it is seen that it is satisfied only for .1 = — i. Hence a solution was introduced in clearing (t,) of fractions. It is easy to see that (t,) is really equivalent to (6) and hence that in clearing (t,) of. fractions by multiplying by .r-— 9 we multiplied by .v— 3 when it was not necessar\^ ; this is where the solution .r=3 was introduced. 9. Theorem. /szrrr equation can be inte^ralized kiritimately. For if the several fractions in the equation are not in their low- est terms thev can be so reduced. Then these fractions can all 8o AlyGKBRA. be transposed to one side of the equation, their common de- nominator found and then added together. This will now give but one fraction in the equation, and, when this is reduced to its lowest terms, we will have an equation of the form N which, since - is in its loivest terms by supposition, will take on no additional solutions when multiplied through by D^ according to Art. 7. 10. Theorem. The raishig of both members of an equation to the same poiver is equivalent to 7nultiplying throni^h by a funetiofi of the unk7ioum ayid hence is a questionable derivation. Take the equation 7. = ^ (1) and raise both members to the «th power, obtaining IJ'^R". (2 Now r O is equivalent to L-R^o and (2) is equivalent to But (^) can be derived from (^3 j by multiplying both members by /."-'+Z«-=7?-f-/."-^y?'--f . . .-^L'R"-'^LR'-'+R"-' wdience f 2 j is equivalent to the t7co equations ( L = R ) {L"-' + L"-'R-\-L"-'R'- + . . .^L'R"-^ + LR"-' + R"-' = o. \ 11. Theorem. Dividing both members of an equation by the same expression is legiti7nate if the expression is a knoum quantity, but questionable if it is afunctio7i of the imknoum.\ Suppose both members of the equation to be divisible by T and write the equation LT^RT. (i) Now if T'ls a known quantity, then by Art. 5 this equation is equivalent to L=R (2) whence division by 7^ w^ould be legitimate. But if 7" is a func- tion of the unknown quantity, then (i) \s equivalent to the two equations ^ L=R T=o. SINGI.E Equations. 8i Division by T would give us but one of these, and consequently solutions would be lost. Hence the division by a function of the unknown is a questionable derivation. 12. Theorem. The extraction of the same root of both members of an equatio7i is equivalent to dividing by afnnction of the unknown and hence is a questionable derivation. For we can pass from to L=^R (2) by dividing both members of ("i^ by L"-'-^L'-^R-JrL"-'R'-\- . . . ^L^R"-^-JrLR"-^^R"-\ Hence, by Art. 11, root extraction is a questionable derivation. 13. Examples of the Integrauzing of Equations. In the following equations the student should note the precise effect of all questionable operations at the time they are performed. I. Solve \(x-^)(x-^)(x-2)==(x~^)(x-^2>)(^+2). X -h 2 X— 2_5 X— 2 X 4- 2 6' 2. Solve J. Solve --i - + ^l- = i. 12 x-^i 4. Solve -H — = . X X X c^ - X'^ XI 5. Solve-— ,= i 6. SolvL i = ^. x—i X 6 7. Solve fjf2£-- i_5 ^ -r + 5 x'—x — 6 x-\- 2' 14. Examples of the Rationalization of Equations. The most expeditious method for rationalizing any given equation de- pends upon the peculiar make up of the equation, and can only be determined by the student after a little experience with this class of equations. A— 10 82 Algebra. /. Giveji \/g^- x+x==ii. (i) Transpose ever>'thing but the radical to tbe right-hand side of the equation and we obtain \/g + x=\\~x. (2) Squaring both sides gives 9 + -i'=i2i — 22.A-I-X' (t^) and solving this quadratic we find x=7 or 16. From (2) to (t^) is a questionable derivation ; for squaring both members of an equation, L=^R, we have found ("Art. 10) to be equivalent to multiplying through by L-\-R, and that the result- ing equation is equivalent to the two equations Therefore (^3 j is equivalent to the two equation f \ i or to s/ (^-\-x—\\—x 1 \/9 + -^H-ii--^=o I ^^^ j — V9+-^^+-^=ii) ^^^ Hence, if we understand equation (\) X.o read The positive square root of (<:)-\-x)-\-x=^\\ then a new solution has been introduced between (2) and ('3J. But if we understand equation (\) \.o read A square root of (g-{-x)-\-x= 1 1 then it is equivalent to both the equations in (5), and no solution has been introduced. This is because the introduced equation, ±/.-f /v'=o is identical with the original equation :^L—R. In these cases the student will always find that ratioyialization may or may not be considered as a questionable derivation according us we consider the radicals to call for a particuIvAR root or AiSiY root of the expressions involved. It is more in accordance with the generalizing spirit of algebra to consider the radical sign, wherever it occurs, as calling for any of the possible roots. This will be better appreciated by the stu- dent when he learns in Part II that every expression has three different cube roots, four fourth roots, five fifth roots, etc. SiN(;i^E KquatioNvS. S3 2. Solve \/-*-+V-r + 6=3. (i) Squaring each side of the equation, obtain .r-f 2V^'-i"6i+xH-6=9. .(2) Transposing all but the radical to the right-hand side this becomes Squaring, we obtain 4-1"+ 24.r=9— 1 2Jf +4.r (/^) or -^=i- What are the questionable steps ? What is their effect ? The above solution is really equivalent to the following : \/ x-\- sj X -f- 6=3. Transpose the 3 to left member, obtaining V -^ ■+ V -^ + 6— 3=0 ; Multiplying through by the rationalizing factor of left member, (III, Art. 26) we obtain Wx-V \/^rT6-3)( V-^+ V.r + 6+3) x(V-i^— V.v"-f 6— 3)(\/^— V-^t;-{-6-f 3)=o, which reduces to .4.1-— 1=0. (5) or ^ = 4- The introduced equations arc V.r-f- \/x 4-6 + 3—0 ; \/-^"— V-^' -I- 6— 3=0 V -^'— V -^ -f 6 + 3 = o. Here, then, is an apparent paradox : three solutions seem to have been introduced, yet there is only ^;?^ in all ! This can be explained in the following manner. If we regard the radical signs as calling for any one of the two roots of the ex- pression underneath, then the introduced equations are all iden- tical with the original equation ana hence could not give rise to a differe7it solution. If we restrict ourselves to using that square root in each case which has the sign given before the radical, then none of the introduced equations have any solution Tvhatever^ and hence no solution is introduced in this case. ^ . s/ x-\-s/ a-\-x 3. Solve —^ ^^ _^.=.=.v. sJ X— sf a -\- X ^4 Algebra. Sometimes such equations are best simplified by first rationaliz- ing either the numerator or denominator of the fraction. Ration- alizing the denominator of this fraction, the equation becomes 2 -r -j- <2 + 2 V ^-^' 4- ■^^_ X — a — X ' or 2sJ ax-\-x^^= — (ax-\r2x-\-a), whence \ax ■\- ^x' =(ax-\-2x-\-a)\ etc. Solve .r-f- s/ X -\- 3=4-^' — i . Solve \/'l4 — X-\- ^/ II — Jf= ^-rr^=r- V 1 1 — -^ Solve s/ ^x -^ g— s/ X — i= \f x-\- 6. y. Solve V-^ — 9+ V-^ + 12= V-^ — 4+ V-^ 4- 13. .r 3 Extracting the square root, \/^+V^-3= ,^ . (\) V-^"— 3 Single Equations. 85 Clearing of fractions, V^"— 3-^"+-^-3=3 ; (5) whence x^^ (6) 2 2 Tfy. Solve .1== 1 -- ^. X -^ Sf 2-\-X' X—\/2-hX' 16. Solve \/ X -}- \/ 2 -\- X V2 + X //. Solve V 5-^ + 10= V 5-^+ 2. iS. Solve \/i+-^-+ ViH-;ir + >/i+.r= Vi— -^■. /p. Solve -^—-~=(x-\-2)\ X — V -^ — 9 Rationalize the denominator of the fraction. 15. Equations which can be Solved as Quadratics. The method employed in the solution of quadratic equations will some- times enable us to solve equations of other degrees, or even irra- ttonal equations. Thus consider the equation 3-^— 5-^' + 4=2. (i) Multiply both members by four times coefficient of .x"* and add the square of the coefficient of x' to each side, as in the solution of a quadratic equation. It then becomes 36^1:^— 6ojt:''-f25=r. (2) The left-hand member is now a perfect square. Whence, extract- ing the square root of both members, equation (2) becomes 6jr"— 5==ti whence x^== i or |. Therefore x= -f i or — i or -f Vl or — V| • As another example consider the equation 4\/x +3A-=4. (i) Put 1'= \/x, whence it is seen that (1) becomes 4.^+3/'— 4- (^) Solving this quadratic we find j'=| or — 2. Whence, since y= \^ x \/-^ = | or —2. Therefore x==^ or 4. These examples suggest the following theorems : 86 Algebra. 16. Theorkm. ."/;/>' equatio7i ivhich can be placed in the form x'^"-\-ex"-\-f=o can be solved as a quadratic. x'^''-\-ex"-\-f=o (\) may be written which, if we regard (x") as the unknown qantity, is seen to be in the quadratic form. Completing the square of (2) it becomes (x'^-'^e(xn -i- \e^Y-f r 3) Whence x"-\-\e= =b sj \e—f or .r" = — .}r ± V \e —f. 1 Therefore x=( —\edti,s/ \e'-f)" (a) which is the solution of the equation of the proposed form. 17. Theorem. Ajiy equation ivhicli can be placed in the form X~ -^eX"-\-/=o, tvhere X starids for arty linear or quadratic function of the unhioivn, can be solved as a quadratic. For, by the last article, it will be found that 1 Now, if .Vis a linear function of x, this equation is of the form 1 which can be easily solved for x. If A' is a quadratic function of r equation {\) must be of the form 1 ax'^bx-\-c={^-U±is/\r-fY (^) Now this is a quadratic equation in terms of x, since all other quantities in the equation are known, and hence the equa- tion can be solved. In treating examples which come under these two theorems it may be possible that we will not find all the values that will sat- isfy the given equation. This happens because we are not always able to find n different n th roots of a quantity, while that num- ber really do exist. Thus from the equation we will find by considering x'^ the unknown quantity that jt-3=27 or —8 whence -^=3 or — 2. Single Equations. 87 But really 27 and —8 have each ///r^d' different cube roots instead of merely the ones we have written above. The full considera- tion of this matter involves subjects vSomewhat more advanced, and more than the mere statement above given will not be at- tempted until Part TI of the present work is reached. 18. Examples. The following five are examples under the theorem of Art. 16 : / . Solve x'' -f 1 6.V ■ =225. 2. Solve x""— ^x' +8=0. J . Solve 6x* —35=11 X-. 4. Solve x'-f| = J:^Ji:^ ^ . >^.t-3 — 2 V X -f X=0. The following five are examples under the theorem of Art. 17. 6. Solve -^H-5V37— -^ — 43- Process : Subtract 37 from each side of the equation, obtaining -V— 37-+-5V37— ■''^— 6 M^hich may be written — r37--v; + 5V37--^-6 or r37--^>'-5V37--^"=-6. Putting )' for \/t,7—x this becomes r=_y/=_6. Solving, j'==3 or 2. That is V37— •^ — 3 or 2 whence 37— A-=9 or 4 and .r=2S or 33. The same example may be treated by the method of Art. 1 1 . 7. Solve v^—V-^— 9=21. S. Solve 2\/^'— 5-V+2— -^^"-f-8A-=3.v— 78. ^. Solve (2X'—T,X-{-l)^=22X'''—T,;^X-\- I I. /o. Solve 4x'—^x-\-20^/2x^—5x-i-6=6x-\-66. The following. are examples of either the theorem of Art. 16 or of Art. 1 7 : 88 AlvGEBRA. TT. Solve axV x + 7^7^^ = c. y/ X 12. Solve x~^—2X~^=%. 2 i I J. Solve Ji:"— 5^-"H-4=o. /^. Solve 3-r— 20=7 V-^- 75. Solve iox'^"-\-xn-\-24.=o. 16. Solve T T ox~'' +1 = 21 x"^. ij. Solve sj x-\-^x ^ = 5. 18. Solve 3.r"— 4x-f\/3-^'— 4^— 6=18. 19. Prove that the equation x^—97Ji'*+ 1500=204 is equiva- lent to the equation (x^—\b)(x^~%\) = o. 1 -? A ^o. Solve 2x'^ — '}^x'^-\-x'^=^o. Result : -r=o, or i, or 8. 21. Solve fjL— «)" + >— r='^. ^^. Solve 8-v2"— 8.I.- 2«=63, i I- ± ^j. Solve -r^ -f 8.r ^ + g^r ^ =0. CHAPTER VII. SYSTEMS OF EQUATIONS. 1. Definition. If a number of equations containing several unknown quantities are supposed to be so related that they are all satisfied simultaneously by the same set of values of the un- known quantities, the equations are said to constitute a 5y.y/d'/w, or a System of Simultaneous Equations. Thus the equations 2-r-f y-\- 5^=19"^ 7^x-\-2y+ 45'= 19 - are satisfied simultaneously by the set of values, jr=i,jj/=2, ^=3, and are said to constitute a system. This set of values, or the process of finding them, may be called the Solution of the system. The reader is supposed to be already familiar with methods of solution of a system of simple equations containing as many equa- tions as different unknown quantities, such as the system given above. The systems we propose to consider in this chapter are those involving quadratics or equations of higher degrees. 2 . The student should not suppose that every system of equa- tions which may be proposed is capable of solution. It is one requirement that the number of unknown quantities be just equal to the number of equations in the system. But even this is not all. Some of the equations in the system may contradict some of the others, in which case a solution is impossible. For example, take the system 34-2J/=2X (l)\ X- r=i (2)\ From equation (2) x= I -hj'. Substitute this value of .r in equation (\) and we obtain 3+2V=2 + 2J/, or 1=0, 90 Algebra. r=-2 J and by no other method of elimination can we get anything but an absurdity from the given system. Equations of this kind are said to be incompatible because one equation affirms what another denies. We will see this to be so in the above system if, by proper transformations in equation (\), the system be written x—v= I (4) \ These equations are necessarily contradictory and can have no solution. Another example of an incompatible system is x—y=4. - X-^y — 2: =2 ) From the second of these equations it is seen that Substituting this value of x in the first and third of the equa- tions in order to eliminate x, we obtain the system 2J/— -2-= — 2_y—2 and, since these are incompatible, we can go no further. There is still another case in which a system may have no solution. Consider the equations 4X=2>(2 — 3y) j From the first equation we find x — ^—^y. Substituting this value of x in the second equation we obtain which reduces to 0=0, and we get no solution. Equations of this kind are said to be dependent because the equations really make the same statement about the unknown quantities. This will be seen when, by proper transformations in the equations, the above system is written. 4-^4-97=6) 4j»;-f 9j^/=6 j It is now seen that the equtions of the system do not state independent truths, and consequently the system has no more meaning than a single equation containing two unknown quantities. Systems of Equations. 91 It will also be found that the system 4-^+ 3)'+ 22-= I X-h7,(2-\-2)=2(l-y)) is a dependent one, the dependence being between the first and third equations. We may then enumerate three conditions which must be ful- filled by a system of equations in order that a solution may exist : There must be just as many equations as there are unknown quantities. The equations must be compatible. The equations must be independent. 3. Of course if any equation of a system be operated upon in any manner during the solution, care rnust be taken that the transformation be with a due regard to the theorems in VI, Arts, 3 — 12. Obviously, no operation which it is questionable to per- form on an equation standing alone can be legitimately performed upon one belonging to a system. But in addition to the reduc- tions which single equations may undergo, equations of a system permit of certain transformations peculiar to themselves, and it remains to investigate the possible effect of these on the solution of the system. The following theorems are designed to point out the effect on the result of the ordinary steps in the process of elimination. 4-. Theorem. // from the system of equatiotis L R, 2i'e derive the system, L=R, L.S + L,T=R.S + R,T (a) (b) U=R„ where all but the second equation rejnain unchanged, the derivation is legitimate if T is a known quayitity, 7iot zero, but questionable if T is a function of the unknown quantities, it bei?ig indifferent whether S is a hiown qua7itity or a function of the unknown OJies. 92 Algebra. Write system (a) so that it will read L -R =o (I) L,-R,=o (2) L,-R„=o and system (^(^ j so that it will appear L,-R =o (3) S(L.-R,)-T(L.-RJ=o (4) (c) l„-r/ First, suppose T a known quantity. Then any set of values that will satisfy (c) must make L^— R,, L^ — R^, . • • and Iy„— R„ each zero. But any set that makes these zero must satisfy (d) also. Hence any solution of {c) is a solution of (d). It is seen from ('3/ that any set of values that satisfies (d) must makely_ — R^ zero. Equation (^) will then become TrL-RJ=o. (s) Now since T is a known quantity, not zero, this cannot be sat- isfied unless ly^— R^ is zero. Hence any set of values, in order to satisfy (d), must make L^— R^ and L^— R, and also . . . L„ — R« each zero. But any set of values that makes these zero will satisfy (c). Therefore any solution of (d) is a solution of (c). Now we have shown, Jirst, that any set of values that will satisfy (c) will satisfy (d), and second, that any set of values that will satisfy (d) will satisfy (c). Hence the two systems are equivalent. Second, suppose T a function of some of the unknown quanti- ties. In this case equation (^) may be satisfied by any set of values that will satisfy the equation T=o without assuming that L^—R^ is zero. Consequently (^^j can be satisfied without equation (2) being satisfied ; that is, without (c) Systems of Equations. 93 being satisfied. Therefore (d) is not equivalent to (c) but to the two systems L-R=o L-R.=o L,-R„= I.-R=o] T=o I U-R„=o] 5. Examples. The derivation discussed in the above theorem is the one so frequently used in elimination. Thus take the system 2.r-hj'=i7 (y)\ ^x—\oy= 5 (2) S Multiply (\) through by 5 and (2) through by 2 and obtain a new equation b}^ subtracting the former from the latter and the system becomes 2.r+r=i7 (7i)} We have eliminated x from the second equation and consequently y is readily found to equal 3. From (■})), X is then found to equal 7. The theorem shows it is also legitimate to transfonn into ^ ' o . , 6— 3.r=o j by multiplying the first equation through by x and subtracting the resulting equation from the second. An example of the use of the following theorem will be found in V, Art. 12 (c). 6, Theorem. It is legitimate to derive from the system the system SL/+TL =SR-"-f TR. \ ' ^^ //" T is a known quantity, not zero. 94 Algebra. L -R =o (I) I Iv -R =0 (2) \ ^ ^ Rewrite (a) and (d) so that they shall read L,-R=o Iv-R=o (: and L-R, = o (:,) / s(i.r-R,o+T(L.-Rj=o r4; ^' ^^ It is evident that any set of values which will satisfy (^) will satisfy (d), for whatever makes ly, — R^ and L^— R„ each zero will satisfy (d). It is seen from (t,) that any set of values that satisfies (d) must make ly,— R, zero. Equation (4.) will then become Ta.-R,)=o. (5) Now, since T is a known quantity not zero, this cannot be sat- isfied unless L^— R^ is zero. Hence any set of values that satisfies (d) must make L,— R, and L,— R, each zero ; that is, must be a solution of (c). Now, since any solution of (r) is a solution of (d) and any solution of (d) is a solution of (r), the two systems are equivalent. 7. Theorem. //' from a system containing tivo unknowji quantities zve derive the systetn L=R, (2) \ ("^ L=R. (z)l the derivation is questionable if ly^ and R^ both involve unknoivn quantities, but legitimate if either is a kyiozvn quantity 7iot zero. First, suppose that L, and R^ each involve unknown quantities. Any value of the unknown quantities which will satisfy the equation L=o must satisfy equation (^), since the relation L,=Ri must hold if system (b) is to be satisfied. Also any value of the unknown quantities which will satisfy the equation R=o must satisfy (\), since the relation L,=R^ must hold if the system is to be satisfied. Systems of Equations. 95 Moreover, any value of the unknown quantities which will satisfv L,= R, mitst satisfy (4.) since the relation L, = R, must hold. Therefore, from these considerations, it is evident that the system (d) is not equivalent to system (a), but to the three systems Second, suppose that either ly^ or R^ is a known quantity not zero. One of them, say R^, is the known quantity. Therefore L^ cannot be zero, since the relation Lj=Rj must hold. Hence the introduced system (bj and (bj are absurdities, since they require that Iv^ and R^ be zero. Consequently the derivation is legitimate since the introduced systems are incompatible. 8. Examples. As an illustration of the theorem, consider the sj^stem . • jf— 4=6— r / 2.r+i'=i3 \ This is satisfied by .1 = 3 and j'=7. Now form the system .V— 4=6— J^' ) which is satisfied by either of the sets of values, .1 = 3, )'=7 or .r=4, j/=6. The additional solution may be obtained from either of the systems .1 — 4=6—1' ) .1—4=0 ) .V— 4=6— r ) 6— r=o \ As another example consider the system ,r-f 2j'=7 j 96 Algebra. which is satisfied by -v=5, y=i. From this we may obtain the system X—2J'=2> } From the first equation of the system Substituting this value for x in the second equation, it becomes 9 + 1 2v + 4.y — 4 J'" = 2 1 ; whence j'= i . Therefore, from the first equation of the system, x=s. In this case we see that no solution has been introduced. In fact, the introduced systems become X— 2r=3| A'— 2JI'=0 j X—2J=2>\ 7 = oj which are incompatible. 9, Theorem. If /mm the system we derive the system lCr,;} ^^^ the derivatio7i is questionaale if both L^ ayid R^ involve tuiknoivn quantities, but legitimate if either is a known quantity 7iot zero. First, suppose that L, and R^ both involve unknown quantities. Then, by Art. 7, if we pass from (b) to (a) we gain solutions. Hence to pass from (a) to (b) is to lose those solutions. Second, suppose that either L, or R^ is a known quantity. Then, by Art. 7, if we pass from (b) to (a) no solutions are gained. Hence none are lost if we pass from (a) to (b). 10. Examples. According to the above theorem it is legiti- mate to divide one equation by another, member by member, if one member is a known quantity not zero. Systems of Equations. 97 Thus take the system and derive the system The only set of values which will satisfy (a) is ji:=4, jk=i. This set satisfies (b) and no solution is lost. The system (a) is equivalent to the system (b) and to two other systems (see Art. 7), but the other two systems are incom- patible. As an example of the case in which solutions may be lost, consider the system which is satisfied by either of the two sets Jt:=o, j/=3 and ^==3, jj/=o. If we divide the second equation by the first, member by member, we pass to the system x=2,-y which is satisfied only by the values ^=3, y=o. II. Solution of a Linrar-Quadratic System. We now propose to take up the solution of those systems involving two unknowns which consist of one linear and one quadratic equation. It is convenient to call this a linear-quadratic system. We will proceed by first working the following particular example : jr=— 2y=i (2)) '^ From equation ( i ) the value of x in terms of j is easily seen to be Substituting this value of .r in equation (2) we obtain or 25— loj'+y— 2^=1 (^) Unitijig and transposing terms y+ioj=24, . (6) whence, solving this quadratic, _>'=2 or —12, and from equation (\) ' .r=3ori7. A— 12 98 Algebra. Consequently there are hvo sets of values which will satisfy system (a), namely, and x=i7, )/= — 12. Now the method here used may be applied to the solution of any linear-quadratic system containing two unknowns. In fact, take the general case* x-\-ay=b \ .^. x''-{-c}r-\-dxy-^ex-\-fy=g \ ^ where a, b, c, d, e, f, and g are supposed to stand ior any real quantities whatever. The value of x in terms of y from the first equation of the system is x=b—ay (-J) Substituting this for x in the second equation of the system, that equation becomes b"—2aby-\-a'y^'-\-CT^bdy—ady~-\-eb—aey^fy^=g. Combining together those terms which contain jk^ and those which contain y and transposing all the known terms to the right hand side of the equation, this becomes (a" -f c—ad)y''-^ (bd— 2ab—ae-\-f)y=g— b'—eb, which is a quadratic in which y is the only unknown quantity, whence it can be solved. The values which may be found from this can be substituted in equation (']) above, and the values of .V will be determined. 12. Examples. Solve the following systems : J f x4-j'=7 (a-^ -I- 21/^=74. \2x-i-xy-\-2y=^i6. I -r-f r=4 ^* I xy=g6. *It might be thought that this is not a general case, since x in the first equation aufl x^in the second do not appear with coefficients. But if either of them had a coefficient the equation could be reduced to the given form by dividing through by that very coefficient. Systems of Equations. 99 r -v— r ■>- ,-=4 \^x .v—v 13. Solution of Systems of two Quadratics. If we have a system of two quadratic equations containing two unknown quantities and attempt to eliminate one of the unknown qiyintities it will be found in general that the resulting equation is of the fourth degree. Thus take the system y'-^xy—\o. 1 We find from the first equation that SubvStitute this value for y in the vSecond equation, and it be- comes or, expanding and collecting terms, .^^-f.r^ — 5JI* — 5J' — 25= 10. Now, since we are not yet familiar with the solution of equa- tions of a degree higher than the second, the treatment of sys- tems of two quadratics in general cannot be taken up at this place. But there are two important special cases of svstems of two quadratics whose treatment will involve no knowledge be- yond the solution of quadratic equations, and these we will now consider. The cases referred to are \ I. Where the terms in each equation containing the un- known quantities constitute a homogeneous expression with respect to the unknown quantities. II. Where the equations are symmetrical.* 14. Case I. We will illustrate the first case, and also the method of elimination which may be applied to any example of it, by the following solution : Solve the svstem ^- .. -^ ^^ > \ 3-V'— ioy=35. (2) ♦For the deflnitione of boniogeueous and symmetrical see I, Arts. 7 and 8. lOO A1.GEBRA. Suppose x=z>j', where z' is a new unknown quantity. Then, substituting this in the equations, the system becomes From equation (2,) we find that and from equation (4:) (a) Whence, from (5J and (b) 5 35 r=,-75^ re; V^ — 2V 327'— 10. Clearing of fractions, 15-^^— 50=352'=— 7oz\ Transposing and uniting terms, and dividing through by lo 2z;=-7Z;=-5. Solving this quadratic z'=|- or T. Substituting the first of these values in (^3J we obtoin Whence- jj/ = d= 2 and since x^=vy Now, substituting the second value of v in (2,) we obtain y_2r==5. Whence j'= db V — 5 and, since :t-=z_;;/, x=±\/— 5- Therefore we have, as the solution of the original syvStem, the four sets of values. x=5, y=2\ x= — 5, jj'= — 2; x=\/ — 5, j'=V — 5; A-= — V— 5, j/= — V— 5- The general system of two|equations of the above class may be represented by x" + axy -\r byr^c \ , , > x'^-dxy^cy-'^^f.] ' -^ (F. &^h-^ Systems of Equations. lOI The student may show that the method set forth above will solve the general system and hence any possible example under it. 15. Examples. Solve the following systems : J ^ f ^+^=3* 2. t Jf X - I xy '■- 3- {x^+xy=is \xy-f=2. 4-' \ xy=ii. ^x-\-y ^x-y lo 5- ^x-y x-\-y 3 ( x'+y=45. 16- Case II. To show that any system of two sj^mmetrical quadratics can be solved, we will start with the general case, which is evidently. x'-j-ax-\-dxy-\-ay-4-y^=c \ , . x'-\-dx+exy-\-dy+y'=/\ ^^^ If x^ and jj/^ appeared in either of these equations with coeffic- ients the system could be reduced to the given form by dividing the equation through by that coefficient. Through the giv^en system substitute 21 -{-w for x and 21— w for y, where ti and w are two new unknown quantities. Then (a) becomes 22i^-\-2Zi^-\-2au-{-b2i''—bz(f=^c\ ,,v 22r-\-2u<^-\-2dii-\-eic' — ew'^f ) ^ ^ Subtracting the second of these equations from the first we obtain 2(a—d)2i-\-(b—e)ie—(b—e)ur=c—f, or 2(a—d)u-\-(b—e)(u'—'iif)=c—f. Whence c—f—(b—e)(u'—iif) 2(a—d) Now if the right-hand side of this equation be substituted for 21 in the terms 2a2c and 2d2i of system (b), that system will con- tain no powers of the unknown quantities but the second and will I02 Algkbra. therefore come under Case I. When // and ?»' are thus found, x and J' can be detennined from the equations. The above work shows that Case II can ahva^-s be solved, but we do not pretend that the method used is always the most eco- nomical one to employ. The insight and ingenuity of the student will often suggest special expedients for particular examples which are prefera])le to a general method. 17. KxAMPLKS. Solve the following systems: ^ ( x-\-xy-\-y=6s \ xy =50. Ur= -f- X -\-y 4-J'^ = 2 6 . 1^+^ = 14 S- I X y 45 ^' I .i:r=6. A common expedient for readily j^oh-ing such a system is to first transform it into the svstem 1 x^- — 2xy-\-y'=^6\ i A"+2.i:r-fj'-'=3oJ- from which the values of x—y and x-\-y can be found and conse quently the values of x and y. :^±fj.^ _ x-\-y 9 ^ • -i-+j'_i8 '"^y ~ii 18. Miscellaneous Systems. We have enumerated all the classes of systems involving equations of a degree higher than the first which can invariably be solved without a knowledge of the solution of cubic and higher equations. The solvable cases embrace but a small fraction of the S3'Stems which ma}^ arise. Of the numbers remaining a still smaller proportion can be solved by special expedients. The great mass of systems involving quad- ratic or higher equations are thus irreducible by straightforward methods of solution, /. e., as a rule, such systems are insolvable, not- Systems of Equations. 103 withstanding a chance exception. In those systems which may be solved, special expedients are more to be sought for than gen- eral methods. In fact, sharp inspection of the equations and a knowledge of algebraic forms will often be the means of discover- ing an ingenious solution for an apparently insolvable system. The theorems of this chapter will be found useful either in just- ifying or in throwing doubt upon many of the common transfor- mations during the ordinary solution of a system. As far as possible the student should endeavor to take account of all ques- tionable derivations at the time they are made and make allow- ance for them in the result. In this connection the remarks at the beginning of the last chapter should not be forgotten. No matter how skillfully or ingeniou.sly a set of values may have been obtained, they must satisfy the original system, ot it is no solu- tion. Whenever am- derivation not allowed by the theorems is used, however plausible it may seem, this ultimate test must be applied. The following systems include examples of the cases already considered, besides others requiring special treatment. The method of Case II will be found to solve many symmetrical sys- tems of high degree. . ! -r + r=5 rO • i.v5-fjp=275. (2) Put .r =//-frt' and r=// — zf, whence the system becomes \ 2H^-\-20U\i^-\- 10^71^=275. From the first of these equations ?^=f . Whence, substituting this in the second, we obtain ^125 , 62s . , 12s ^ 16 2 2 which is a quadratic in terms of w\ When iv is found x and r can be found from the equations x=u-\-iu and )'=7/-— rr. ( X- 1 >^ == I ox\ ' -f 9000 I .r--hj'=200. -> I ■v-J'=2 ^' |.v^— y=8. , I -^-fj'=5 + ■ ].r3 + r^=65. I04 Al^GKBRA. 5-. 6. 7- 8. 9- lo. II. 12. 14. 15' 16. I?' 18. 19. 20. 21. 22. t xy=b. ( x^-\-xy-\-2y^=2 {2x^-\-xy-\- y=2. \x^— xy-\-y=2i. Cxy+yx=2o V, X y 4. f (x-\-y)xy=24.o \ x^-\-y^=28o. j x^-hr-\-x+y=iS j x-—y^-\-x—y=^6. \ (^-4J(y—7)=o. j x^-\-y^—i2=x-\-y I xy-i-8=2(x+y) i jr*— Jt'^+y— j/^=84 l9(y-x)=is I x^—y''=(f, ( .r^ — xy=2 ( 2Jt:"+y=9 jf - 4-jF^ — -^ — j^' = 7 8 jrK+x4-jF=30. f .r=4-y=34 (2J«;^— 3:1:^=23— 2^. ^ V-^+Vj^=3 f sj X— sf y=-2 \ (x-\-y)^^=^jo^ Result, X- Result, jr=4 or |. y=2 or ' ;^=± ^ V3 ^>/3^ Result, ^-=9 or — ^/+JV^39- J^/=3 or — '/-• i-V-39. Result, j»;==b5, or d=3. jj/==b3, or ±5. Result, x=^. y=i. Result, x=^, or— 3. y=S, or-5. Systems of Equations. 105 r xy=i22^ Result, ^=49 or 25. lV-^+Vj=i2. j=25or49. ^x-\-y^a—s/ x-\-y I x—y= b. Result, ^=^r^±A^+i^yV±l 4 24^ y A— 13 CHAPTER VIII. PROGRESSIONS. 1. Definitions. An Arithmetical Progression is a series of terms such that each differs from the preceding by a fixed quan- tity, called the common differeiice. The following are examples : 74-9+11 + 13+15+ • • . 31 + 26 + 214-16+11+ . . . a-^r(a-\-d)^(a^2d)^(a^-^d)^ ... (x—y)-\-x-^(x-\-y)-\- . . . (jc-2,y)-^(x-y)^(x^y)-^(x^iy)-V . . . The first ana last terms of any given progression are called the Extremes, and the other terms the Means. 2. To Find the «th Term of an Arithmetical Pro- ORESSION. Represent the first term of the progression by a and the common differeiice by d. Then we have Number of ter 711. i. 2. 3. 4. 5. Progressio7i. a -\- (a-\-d) -\- (a-\-2d) + (a-\-2fd) + (a-\-^d), etc. We notice that by the nature of the progression every time the number of terms is increased by i the coefficient of d is increased \>y I also ; hence to get the 71 th term from the 5th term, the com- mon difference must be added to it n — ^ times. Whence, representing the ;^th term by /, l=a-\-/^d-\-(7i — ^)d, or l=a-\-(7i-i)d. (i) 3. To FIND THE Sum of n Terms of an Arithmetical. Progression. Representing the sum of the arithmetical pro- gression by s, we have s=a-i-(a + d)-\-(a + 2d)-\-(a-h2,d)+ ...+/, (i) or, writing this progression in revense order, we have s=l-{-(l-d)-^(l-2d) + (/-sd)-h . . . -\-a (2) Now adding (i) and (2) together term for term, noticing that the common difference vanishes, we have 25=r«+/;+r^+/;+ra+/;+r«+/;+ . . . +(a+i). Progressions. 107 If the number of temis in the original progression be called n^ this becomes 2S=^n(a-\-l)^ whence s=-\n(a-]rl). 4. To Insert any Number of Arithmetical Means be- tween TWO GIVEN Quantities. Suppose we are to insert p arithmetical means between the two terms a and /. The whole number of terms in the progression consists of the r means and the two extremes. Hence the number of tenns in the progression is/> + 2. Therefore, substituting in (\), Art. 2, we obtain /=^ + (^^-f 2— iX - / — a and now, since the common difference is known, any number of means can be found by repeated additions. 5. The two equations l^a-\-(n—\)d ■ ' (1) s^\n(a-^l) (2) contain five different quantities. If any two of them are unknown and the values of the re5t are given the values of the two un- known can be determined by a solution of the system. As an example, suppose that a and d are unknown and the rest known. Putting X for a and y for d so that the unknown quantities will appear in their usual form, the system becomes l=x+(n-i)y (z) s=\n(x-^l) (^) Finding the value of x in each equation the sj^stem becomes ^x=l-(n-Y)y (s) |.= --/, (6) Whence, equating right-hand members of ($) and (b), we obtain /-r«-ii>'="-/, 2?ll—2S . . whence y— -> . (']) n{n—i) io8 AlvGEBRA. Therefore, restoring a and dva (6) and (-j), 2S «= /, 71 - 2nl—2S d=— r. n{n — i) where a and d are expressed in temis of the three known quan- tities, n, /, snd s. In Hke manner we may suppose a7iy two of the five quantities unknown and find their values in terms of the known ones. In all, ten different cases may arise, which are given in the following table, each of which is to be worked by the student. No Given. I. a, d, n, 2. /, d, 71, 3- a, /, 71, 4- a, n, s, 5- 71, d, s, 6. /, 71, s, 7- a, d, /, 8. a, /, s, 9- a, d, s, lO. /, d, s, quired /, s, a, s, d, s, d, /, a, /, a,d, l=a-i-(?i—i)d ; a=/—(?i—i)d; . I— a «= : d= 71— \ 2S—2a7l a= 71(71-1)' S {7l—l)d 2S J 71 I— a , 71, s, '71=—^ — hi ; Result. s=\n[2a-\- {71— i)d\ s^=\n[2l— {72 — i)d~\. d= I, 71, / = —^d^s/2ds-\-ia—y.r:n-- a, 71, a = \d±is/{l^\dY—ld^ 7i{n—i)' {l-a){l-a-{-d) 2d I'— a" 2S — a—V d—2adt:^(2a—d)^Sd8 2d 2/+ dzt x/W-fclf-^^Sdi^ 2d 6. Examples. 1. Find the sum of g tenns of the progsession 3 + 7 + 1 1 , etc. 2. The first term is 96, the common difference —5 ; what is the 1 3 th term ? Progressions. i 09 J. The first term is 8^, the common difference — f, and the number of terms 29 ; what is the sum ? 4. The first term is f, the common difference f, and the number of terms 1 2 ; what is the sum ? 5. Insert 10 arithmetical maeans between —J and -f J. 6. Find the sum of the first n odd numbers 1+3 + 5 + 7, ^^c. 7. Find the sum of 7i terms of the progression of natural numbers 1 + 2 + 3 + 4, ^tc. S. Find the sum of fi terms of the progression of even numbers + 2 + 4 + 6 + 8, etc. p. The first term is 11, the common difference —2, ^nd the sum 27. Find the number of terms. 10. The first term is 4, the common difference is 2, and the sum 18. Find the number of terms. 11. The first term is 11, the common difference is —3, and the sum 24. Find the number of terms. 12. The sum of ?t consecutive odd numbers is s. Find the first of the numbers. ij. Select 10 consecutive numbers from the natural scale whose sum shall be 1000. 14. Sum Vi+ V2 + 3Vi+ etc., to twenty terms. zy. Sum 5 — 2 — 9—16— etc., to eight terms. 16. Find the tenth term of the arithmetical progression whose first and sixteenth terms are 2 and 48 ; and also detenn- ine the sum of those eight terms the last of which is 60. 77. Insert five arithmetical means between 10 and 8. iS. Insert four arithmetical means between —2 and —16. ip. How many terms must be taken from the commence- ment of the series 1 + 5 + 9+13+17 etc. , so that the sum of the 1 3 succeeding terms may be 741 ? 20. Wnat is the expression for the sum of ;/ terms of an arithmetical progression whose first term is f and the difference of whose third and seventh terms is 3 ? 21. The sum of the first three terms of an arithmetical pro- no Algebra. gression is 15 and the sum of their squares is 83 ; find the com- mon difference. 22. There are two arithmetrical series which have the same common difference ; the first terms are 3 and 5 respectively and the sum of seven terms of the one is to the sum of seven terms of the other as 2 to 3. Determine the series. 7. Definitions. A Geometrical Progression is 3. s&ri^s oi t^rms such that each is the product of the preceding by a fixed factor called the Ratio. The following are examples : 3 + 6+12 + 24 + 48, etc. 100+50 + 25+ i2|-+6^, etc. i + i+i+xV+sV, etc. i+i + TVH-2V+4V etc.^ The first and last terms of any progression are often called the Extremes and the remaining terms the Means. 8. To Find the n th Term. I^et a represent the first term of the geometrical progression and r the ratio. Then the pro- gression may be written : Nnmber of term. i. 2. 3. 4. 5. Progressioji. a -\-ar-\- ar^-\- ar^ + ar'^. We notice that, by the nature of the progression, every time the number of terms is increased i the exponent of r is increased by I also ; hence to get the ?i th term from the 5th term it must be multiplied by the ratio ^—5 times. Whence, reoresenting the nth term by /, and /=(ar^)(n—2,), or l=ar"-\ (i) 9. To Find the Sum of ?^ Terms. Represnting the sum of the geometrical progression by ^ we have s=^a-{-ar-\-ar^-^ar^-\- . . . -\-ar"~''-\-ar"~\ (i) Multiplying this equation through by r— 1 , we obtain (r — i)s=a?'" — a. Whence ar"— a r—i (2) Now ar" — r(ar" '). Therefore, since ar" '=/, ar"=al. Progressions. i i i Whence, substituting this value of ar" in (2) we obtain as an- other expression for ^ rl—a s= (2,) 10. To Insert any Number of Geometrical Means Be- tween TWO Given Quantities. Suppose we are to insert p geometrical means between the two terms a and /. The whole number of terms in the progression is therefore p-\-2. Hence, substituting /-f^ for 7i in (1), Art. 8, l=aa"-\ Consequently and now, since the ratio is known, any number of means can be found by repeated multiplications. 11. The two equations ' • I ar" — a s= V, r — I contain five different quantities. If any two of them are unknown, and the values of the rest are given, the values of the two un- known can be determined by a solution of the system. But if r is an unknown quantit}^ the equations of the system are of a high degree, since n is usually a large number and always greater than 2 at least. In this case we will be unable to solve the system, as it is one beyond the range cf Chapter VII. Also if n is an un- known quantity, we will have an equation with the unknown quantity appearing as an exponent, which is a kind of equation we have not yet discussed. Hence there are a limited number of cases in which we can solve the above system. The following ta- ble contains the ten possible cases, with the solutioni as far as possible. The values of n in the last four are printed merely to make the table complete, for the manner of obtaining them is not explained until Chapter XV is reached. 112 AlvGEBRA. No I. Given. 'J ' 1 ""I 2. /, r, ;^, n, r, s, quire, d Result, /, s, l=ar"-'] (r-i>_ a, s, «= a, L a= r, 5, /, 71, S, a, r, / a. L s. 1- 8. 9- lo. /, r, 5, <2. r, ^, 5, ;?, r, 71, r"— I ' s= ar" — a r — I la"— I {r—i)sr"-' r"~i l''^^—a^i s= _L 1 ar^' — rs—a—s ; l(s—iy' '=a(s—ay' \ a(s—ay-'=l(s—iy-'; (s—l)r"—sr"-'=—l. Ir—a _log ^— log a r— I log r ^— a _ log /—log « '5—/' log (^—«;— log (^—/) + 1. / .. ,_^ + (r- i>^ log [^ + (r-i)^]-log g r log r log /— log [/r— (r— 1)>?] a, 71, a=lr—{r—\)s\ 71=—^^ ^-^ ^ ^ +1. log r 12. Examples and Problems. 1. Find the sum of 10 terms of the progression 3 + 9 H-27H-etc. 2. Find the sum of 10 terms of the series fV+xf o^ + tAtt* etc., or the series .333 + j. Find the sum of 100 terms of the progression .3333+ etc. 4. Sum 5 terms of the progression 27 + 270+2700, etc. 5. Sum 10 terms of the progression 4—2+1— etc. 6. Sum the series V3 + >/6 + vi2+ etc., to eight terms. 7. Sum the series 3 — 2 + |— f + etc., to nine terms. 8. Sum the series —4 + 8—16 + 32, etc., to 6 terms. p. The fourth term of a geometrical progression is 192 and the seventh term is 12288 ; find the sum of the first three terms. Progressions. 113 10. Prove that if quantities be in geometric progression their differences are also in geometrical progression, having the same common ratio as before. 11. The first and sixth terms of a geometric progression are I and 243 ; find the sum of six terms, commencing at the third. 12. The first term of a geometric progression is 5 and the ratio 2. How many terms of this series must be taken that their sum may be equal to 33 times the sum of half as many terms ? A— 14 CHAPTER IX. ARRANGEMENTS AND GROUPS. 1. Definitions. Every different order in which given things can be placed is called an Arrangement or Pertmitation, and every different selection that can be made is called a Group or Combination. Thus if we take the letters a^ b, c two at a time there are six arrangements, viz : ab, ac, ba, be, ea, eb, but there are only three groups, viz : ab, ae, be. If we take the letters a, b, c all at a time, there are six arrangements, viz : abe, acb, bae, bca, cab, cba, but there is only one group, viz : abc. 2.. probi.em. to find the number of arrangements of 71 Different Things taken All at a time. First. If we take one thing, say the letter a, there can be but one arrangement, viz : the thing itself. Second. If we take two things, say the letters a and b, there are two arrangements, viz : ab, ba. Third. If we take three things, say a, b, r, there are six arrangements, viz : abc, acb, bac, bca, cab, cba. Notice that there are two arrangements in which a stands first, two more in which b stands first, and two more in which c stands first. Fourth. If w^e take four things, say «. b, c, d, then we may arrange the three letters b,'C, d in every possible way and place a before each arrangement, then arrange the three letters a, c, d in every possible way and place b before each arrangement, then arrange the three letters a, b, d in every possible way and place Arrangements and Groups. 115 the letter c before each arrangement, and finally arrange the three letters a, b, r in ev^ery possible way and place the letter d before each arrangement. It is evident that all four letters a, b, c, d appear in each arrangement thus formed, and it is also evident that the number of arrangements in which a stands first is exactly the same as the number in which b stands first, and so on. Hence there are in all four times as many arrangements of four things taking all at a time as there are of three things taking all at a time, or there are four times six or twenty-four arrangements of four things taking all at a time. hi geyieral, if we have n things, say the letters a, b, r, d, e,f, . . then we may suppose all the letters but a arranged in every pos- sible order and then a placed before each of these arrangements ; then we may suppose all the letters but b arranged in every pos- sible order and then b placed before each of thCvSe arrangements, and so on. It is evident that all 71 letters appear in each arrangement thus formed, and it is also evident that the number of arrangements in which a stands first is exactly the same as the number in which any other letter stands first. Now the number of arrangements in which a stands first is evi- dently the number of arrangements of (n—i) things taken all at a time, and hence the total number of arrangements of n things taking all at a time is w times the number of arrangements of ;^— i things taking all at a time. Let us represent the number of arrangements of ?i things taking all at a time b}^ A„ and the number of arrangements of ?i—i things taken all a time by A;,_i, etc. Then by what has just been shown we have A„_2 = (?i-2)A,,_^, A3 = 3A2, A2^2Ai , Il6 Al^GEBRA. Now multiply these equations together, member b}^ member, and we get Aj Ao A3 . . . A„=2Ai 3A2 . . . ??A„_i = 1x2x3 . . . ;/ Ai A.2 . . . A„_i. By cancelling common factors we get A;,= i X 2x3 . . . 71. The product of the integer numbers from 71 down to i or from I up to 71 is often represented by j^^ or ;?! , and is read factorial w, or 71 admiration. With this notation we may write 3. ProbIvEm. To find the Number of Arrangements of ?^ Things taken r at a Time. ivCt us first take a particular case, say the number of arrange- ments of five things, say the five letters a, b, c, d, e, taken three at a time. . Suppose the arrangements all made and we select those which begin with a and put them by themselves in one class, then those which begin with b and put them by themselves in another class, and so on. We then divide the whole number of arrangements into five classes, and it is evident that the num- ber in any one class is just the same as in any other class. Consider those which begin with a. Then every arrangement in this class contains besides a two of the four letters b, c, d, e, and since a is fixed and the other letters arranged in every possible way, therefore the number of these arrangements must equal the number of arrangements of the four letters b, c, d, e taken two at a time. Iti g€7ieral, if we have 7z things, say the letters a, b, c, d, e,f, . . to be taken r at a time, we may select all those arrangements which begin with a and put them by themselves in one class, then those which begin with b and put them by themselves in another class, and so on. We thus divide the whole number of arrange- ments into 71 classes, and it is evident that the number of arrangements in any one class is just the same as the number of arrangements in any other class. Consider those which begin with a. Arrangements and Groups. 117 Then every arrangement in this class contains besides a, (r—i) of the letters b, c, d, . . , and since a is fixed while the remain- ing letters are arranged in every possible order, therefore the number of arrangements in the class considered must equal the number of arrangements of 71 — \ letters b, c, d, . . , taken r— i at a time. As there are 71 such classes and as the number of arrangements in each class equals the number of arrangements of n—\ things taking r— i at a time, therefore the total number of arrangements of n things taken r at a time equals ;2 times the number of arrange- ments of ?^— I things taken r— i at a time. Let us represent the number of arrangements of ?i things taken r at a time by A(") and similarily any number of things taken any number at a time, say s things taken / at a time (s being greater than by A(;), then by what has just been proved A(';) = nAOz\) A(':rl) = (n-i)A(':r?:) A('{-'"^') = (n-r+i). Multipl}^ these equations together, member by member, and cancel common factors and we get AC'.) = 7l(?l-j)(7l-2) . . . (7i-r+i). Multiply and then divide the right-hand member by (n — r)(7i — r-{-i) . . . i and we get (7f~r)(7i — r—i ... I It is easily .seen that the numerator is 1 71 and the denominator is \n—r, hence A(';)=,'- 71 — 7' 4. PROBI.KM. To FIND THE NUMBER OF GROUPS OF ;/ DIF- FERENT Things taken r at a Time. . Take the letters a, b, c, d, c and suppose the groups all written down ; then, fixing our attention upon any one group, it is evident that there could be several different arrangements made from that group by changing the order of the letters. ii8 Algebra. It is further evident that if we form all possible arrangements in each group we thereby obtain the total number of arrangements of the 71 letters taken r at a time. The total number of arrangements then equals the number of arrangements in each group multiplied by the number of groups. Hence, representing the number of groups of ;/ things taken r at a time by GC-) and remembering that the number of arrange- ments in each group equals the number of arrangements of r things taken all at a time, that is ' r, and further remembering that the total number of arrangements equals n(n—\)(n—2) . . (7i — r^\), we have 71 r G» = , '- hence GC") = n 5. The form of this result shows that the number of groups of u things taken r at a time is the same as the number taken w — r at a time. This is also evident in another way, for every time we select 7 things from ii things we leave out ;/ — r things ; hence there must be as many ways oi leaving- out w — r things as of selectiTtg r things, but of course there are as many ways of selecting 7i — r things as there are of leaving out ii — r things. 6. In all that precedes, it was supposed that the given things were all different and that in forming the arrangements or groups none of the given things were repeated. Now we will consider arrangements and groups in which the things may be repeated and those in which the given things are not all alike. 7. Problem. To find the Number of Arrangements op 71 Things taken r at a Time, Repetitions being Allowed. Suppose first we wi.sh the number of arrangements, including repetitions, of .the four letters a, b, r, d taken one at a time. Evidently there are four arrangements, viz : a, b, c, d. Next suppose we wish the arrangements, including repetitions, of the four letters a, b, c, d taken two at a time. Arrangements and Groups, 119 The arrangements are the following : aa ab ac ad ba bb be bd ca cb cc cd da db dc dd Thus we see that there are sixteen arrangements, that is, 4^ arrangements. In exactly the same way if we have n letters a, b, c, d, e,f, . . . , the a may be followed by each of the n let- ters, giving 71 arrangements beginning with a ; the b may be fol- lowed by each of the ?i letters, giving n arrangements beginning with b, etc. So there are evidently 71 arrangements beginning with eac/i letter ; hence in all there are w"" arrangements of ti things taken two at a time, allowing repetitions. Let us now find the number of arrangements, allowing repeti- tions, of n things taken three at a time; and first to give definite- ness to the ideas, consider the number of arrangements, allowing repetitions, of four letters a, b, c, d taken three at a time. We have written out the sixteen arrangements of four letters taken two at a time, and now we may suppose each of these sixteen arrangements to be preceded by the letter a, then each of these sixteen arrangements to be preceded by b, etc. We then have sixteen arrangements of three letters each, beginning with each letter, and as there are four letters there are in all four times six- teen, or sixty-four, arrangements of the letters a, b, c, d taken three at a time, repetitions being allowed. Now, in the same way, if we have n letters a, b, c, d, e,f, . . , we may suppose each of the n" arrangements two at a time to be preceded by a, then each of these same 7f arrangements to be pre- ceded b}^ b, etc. Thus we get ir arrangements beginning with a, li" arrange- ments beginning with b, if arrangements beginning with c, etc. Hence in all we obtain 71 times ?/^ or w\ arrangements of 71 letters taken three at a time, repetitions being allowed. hi ge7ieral, if we know the number of arrangements of 71 letters taken 5 at a time, repetitions being allowed, we may find the num- ber of arrangements of the 71 letters taken ^+1 at a time. Representing the number of arrangements, with repetitions, of I20 Algebra. ;/ letters a, b, i\ d, . • ri4; From (\^) and (^gj we get \n—\ \n — I r «— /--f I r— I U^-^r (n—r)\n—\ r\n^\ nln^^i \ii n^r \r \n — r r \ n—r v^y which by Art. (^) equals G("), hence Gc;)=G(r^)+GC;zj). (1^) We have obtained a few relations connecting arrangements with arrangements in equations (\), (2), (6), (y), also a few relations connecting groups with groups in equations fii), (14), (^i6j. We now obtain a few relations involving both arrange- ments and groups in the same equation. We have already found in Art. 4 AC)=|rG(;), ^17; and as I r=A(r) we may write (ij) in the form Ao-AOG&o. ris; From (y), A(;:)=A(;:_i) and writing this value in ('iSj we get Aao=A(;_i)G(';). ri9; In (^i8j substitute the valufe of G(") given in (16) and we get A(';)=A(;) [G(rM+G(;'z})]. (20J But it readily follows from (^i8j that A(r^)=A(r)G(rM, Substitute this value of G("~M ift (^20^ and we get A(;')=A(rM+A(;) G(;'zj). r2i; 128 Algebra. Since by Art. 8, groups where repetitions are allowed can be expressed in terms of groups when repetitions are not allowed, it would be an easy matter to obtain equations involving groups with repetitions. 13. Examples and Problems. 1. How many different groups of two each can be made from the letters a, d, I, n, sf See VIII, Art. 5. 2. How many arrangements of five each can be made from the letters of the word Groups f J. How many different signals can be made with five flags of different colors hoisted one above another all at a time ? 4. How many different signals can be made from seven flags of different colors hoisted one above another, five at a time ? 5. How many different groups of 1 3 each can be made out of 52 cards, no two alike? 6. How many different signals can be made from five flags of different colors, which can be hoisted any number at a time above one another ? 7. How many different signals can be made from seven flags of which 2 are red, i white, 3 blue, i yellow when all are dis- played together, one above another, for each signal. 8. A certain lock opens for some arrangement of the num- bers o, I, 2, 3, 4, 5, 6, 7, 8, 9, taken 6 at a time, repetitions allowed. How many trials must be made before we would be sure of opening the lock ? p. In how many ways can a committee of 3 appointed from 5 Germans, 3 Frenchmen and 7 Americans, so that each nation- ality is represented ? 10. How many different arrangements can be made of nine ball players, supposing only two of them can catch and one pitch ? 11. How many different products of three each can be made from the four letters a, b, c, df 12. In how many different ways can the letters of the word algebra be written, using all the letters ? Arrangements and Groups. 129 /J. In how many ways can a child be named, supposing that there are 400 different Christian names, without giving it more than three Christian names ? /^. In how many ways can seven people sit at a round table ? 75. There are 5 straight lines in a plane, no two parallel ; how many intersections are there ? 16. On a railway there are 20 stations of a certain class. Find the number of different kinds of tickets required, in order that tickets may be sold at each station for each of the others. 77. Find the number of signals that can be made with four lights of different colors, which can be displayed any number at a time, arranged either above one another, side by side, or diagonally. 18. From a company of 90 men, 20 are detached for mount- ing guard each day ; how long will it be before the same 20 men are on guard together, supposing the men to be changed as much as possible ? How often will each man have been on guard during this time ? ig. A lock contains 5 levers, each capable of being placed in 10 distinct positions. At a certain arrangement of the levers the lock is open. How many locks of this kind can be made so that no tw^o shall have the same key ? 20. There dre n points in a plane no three of which are in the same straight line. Find the number of straight lines which result from joining them. 21. There are n points in a plane, no three of which are in the same straight line except r, which are all in the same straight line ; find the number of straight lines which result from joining them. 22. There are n points in space, no four of which are in the same plane with the exception of r which are all in the same plane. How many planes are there, each containing three of the points ? A— 16 CHAPTER X. BINOMIAI. THEOREM. 1. The Binomial Theorem enables us to find any power of a binomial without the labor of obtaining the previous powers. In order to observe the law of formation of a power of a binomial we first observe the law of formation of the product of several binomial factors of the form x+a, x-\-d, x-\-c, etc., and we will afterwards be able to arrive at the power of a binomial by the supposition that a=b=c, etc. 2. IvAw OF THE Product of Factors of the form x-\-a, x+d, x-\-c, Etc. By actual multiplication it is seen that (x-\-a)(x-\-b)=x'-\-(a-\-b)x-\-ab, (x-\-a)(x-\-b)(x-\-c)=x^-\-(a-\-b-\-c)x^-\-(ab-\-ac-\-bc)x-{-abc, (x+a)(x+b)(x+c)fx-\-d)=x'-{-(a-\-b-j-c-j-d)x' + (ab + ac-\-ad-\- bc-\- bd-\- cd)x- + (abc-\-abd-\- acd-\- bcd)x -\-abcd. By a careful inspection of these products we will discover the presence of two uniform laws — a law for the exponents and a law for the coefficients. The law of the exponents is readily seen to be as follows : The expoyient of x in the first term of the product is equal to the number of binomial factors, and in the remining terms it continually decreases by 07ie until it is zero. The law of the coefficients may be stated thus : 7^he coefficient of the first term is unity; the coefiicicnt of the second term is the sum of the second terms of the binomial factors; the coef- ficient of the third term is the su?n of all their different products take7i two at a time; the coefficient of the fourth term is the sum of all their different products taken three at a time, and so on. The last term is the product of all the second terms of the binomial factors. 3. Proof that the Laws are General. We will now show that if the laws observed above hold in the product of a given number of binomial factors, they will hold in the product of any number of binomial factors whatever. BiNOMiAi^ Theorem. 131 For, assume that we have tested the above laws in the case of the product of a certain number of factors, suppOvSe n, and have found them to hold true. To facilitate the discussion we will reprCvSent the n second terms of the binomial factors by a^, a,^, a^, a^, . . . «„* instead of a, d, c, d, etc., and accordingly the product of the /i binomials (x~\-aJ(x-\-a.J(x-\-a^)(x-\-a4) . . . (x-\-a„_-^)(x-\-aJ =x"-\-(a^-\-ao-{-a..-\-a^ . . -}-a„)x"~'^ -\-(a-^a.^-^a^a^-\-a^a^-\- . . -\-a„_^a„)x''~'^ In order to abreviate this expression it is convenient to let Pi equal the Jirs^ parenthesis, or the sum of all the different second terms of the binomial factors. P2 equal the second parenthesis, or the sum of all the different products of the second terms of the binomial factors taken two at a time. P3 equal the third parenthesis, or the sum of all the different products of the second terms of the binomial factors taken three at a time ; and so on. P„ equal the ?? th parenthesis, or the product of all the second terms of the binomial factors. • With these abbreviations the second member of the above equation reads jr" + PiX"-i+P2-^-"~' + P3^"-^-f . . . +P„. Multiplying this expression, which represents the product of n binomial factors, by a new binomial, x-{-a„^^, we derive the fol- lowing result for the product of ;^-f i binomial factors : x''-^^-^(F,-ha„,Jx"-^(F,-\-a„^,FJx"-'^ + (F^-{-a„,,FJx''-''-\- . . . H-a„,-iP.,. It is seen from this result that the law of exponents still holds. For there are n-\-i binomials and the exponent of x begins, in the first term, with n-\-i and decreases continually by one through the remaining terms until the value zero is reached. ♦This notation presents many mechanical advantages. It must not bo supposed, however, that there is any relation subsisting between n^ and a.2 or any other two of the symbols ; they are as independent as distinct letteis. 132 Algebra. The law of coefficients holds good also. For : The coefficient of the first term is unity. The coefficient of the second term is F^-\-a„.^. Now P^ stands for the sum of the n terms a^-^a^-i-a^-l-. . . a„. Hence Pj-f <2„^ 1, or the coefficient of the second term, is the sum of all the different second terms of the binomial factors. The coefficient of the third term is P2+«;h i^^i- Now P2 represents the sum of all the different products of the ?t letters a^, a2, a^, a^^, . . . a„ taken two at a time. (i). That is, P2 represents the sum of all the different products of the n-\-i letters a^, a^, a.,^, . . . a,,^^ taken two at a time which do not contain «„+i . Again, (2). That is, ^„+iP2 equals the sum of all the different prod- ucts of the 7^-f-I letters a^, a.^_^, a^^,, a^, . . . a,^^^, taken two at a time, which contain a,,;^. Therefore, putting (i) and (2) together, P2H-^„. iPi equals the sum of all the different products of the ?i -f i letters «i, <2 2, ^3, «4, . . . «„4-i, taken two at a time, both those which do and those which do not contain ^„+i. Th^ coefficient of the fourth term is Pg+a,,^ 1P2. Now P3 equals the sum of all the different products of the ?i letters a^, a^, a^, a^, . . . a,„ taken three at a time. (^3J. That is, P3 equals the sum of all the different products of the ?i-\-i letters «i, a^, a^^, a^, . . . a„.^.^, taken three at a time, which do not contain a,,.^^. Again, «„+lP2=«i«2^«-M+«1^3^«+l+^1^4^«fl+ • • +'^;/-l^"^//l 1- (\). That is, «„4 1P2 equals the sum of all the different products of the n-\-\ letters a^, a^, a^, a^, . . . a,,_,_^ , taken three at a time, which contai7i a,,^.^. Therefore, putting (t,) and (4.) together, Pi^+a„_^^F2 equals the sum of all the different products of the ?i+i letters a-^^, a^, a^, <24, . . . «„-; I, taken three at a time. In like manner we may treat the coefficient of the fifth term, and so on. The last term is the product of all the n-\-i letters «!, ^2) ^3» «4. • • • <^«+l- Binomial Theorem. 133 Therefore, we have proved that if the laws of exponents and coefficients hold in the product of n factors, they will hold also in the product of ^2+ i factors. But they have been proved by actual multiplication to hold when four factors are multiplied together, therefore they hold when five factors are multiplied together, and if they hold when five factors are multiplied together they must hold when six are multiplied together, ana so on indefinitely. Hence the laws hold universally, 4. Deduction of the Binomial Formula. We have now proved that the equation (x-\-a^)(x-\-a^J(x-\-ar,) . , . (x-^a„._^)(x-\-aJ = x" + (a,+a.,-\-a.,+ . . . +ajx"-^ '\-(a^a2-{-ci-^a.;^-\-a-^a^-\- . . . -\-a„_-^a,Jx"~'^ is true for all positive values of n. Since a^, a2, a.^, a^, . . . a„ are any numbers whatever, we may assume that they are all alike and we may suppose each equal to the quantity a. Then each of the factors in the left- hand side of the above equation will become equal to x-\-a, and consequently the left-hand member will become On the right-hand side of the equation the term x" remains un- changed. I^n the second term the parenthesis becomes the sum of n a's ; that is, it is equal to ??a, so that the second term itself becomes nax"~^ . In the third term the parenthesis reduces to the sum of as manj^ <2-'s as there are groups of 71 things taken two at a time : that is, 7i( n — I ) the parenthesis becomes ^=r5-f ioy-f4oy 4- 8oj'^-f 80^+32. 6. Binomial Theorem. The binomial formula may be stated in the form of a theorem as follow^s : hi a7iy poiver of a bi7i077iial x-\-a, the exp07ient of x begi7is 171 the first term with the expohcTit of the poiver, a7id in the followi7ig ter77is co7iti7iually decreases by one. The exp07ie7it of a com77iences u'ith 07ie in the seco7id ter77i of the power, a7id co7iti7iiially i7icreases by one. The coefficie7it of the first te7'7n is 07ie, that of the seco7id is the ex- p07ie7it of the power ; and if the coefficie7it of any ter7n be 77iultiplied by the expo7ie7it of x i7i that ter77t and divided by the exp07ient of a i7icreased by 07ic, it will give the coefficie7it of the siicceedi7ig term. 7. Historic aIj Note. The first rule for obtaining the powers of a bi- nomial seems to have b(?en given by Vieta (1540-l{i03). He observed as a necessary result of the process of multiplication that the successive coefficients of any power of a binomial are : first, unity ; second, the sum of the first and BiNOMiAi. Theorem. 135 secon(i coefficients in the preceding power , third, the sum of the second and third coefficients in the preceding power, and so on. Vieta noticed also tlie uniformity in the product of binomial factors of the form x-\-(i, x-\-h, x-\-c, etc. But Harriot (1560-1621) independently and more fully treated of these prod- ucts iu showing the nature of the composition of a rational integral equation. See VI, Art. 1. In this connection it is interesting to note that Harriot was the first mathematician to transpose all the terms of an equation to the left member. The binomial formula as now used ; that is, the expansion of the n th power of a binomial, expressed with factorial coefficients, was the discovery of Sir Isaac Newton (1642-1727) and for that reason it is commonly called Sir Isaac Newton s Binomial Theorem. 8. Number of Terms in the Expansion. The exponents of « through the binomial formula constitute the following scale : o, I, 2, 3, 4, . . . ?i. The number of terms in this scale is 7i-\-i. Therefore the number of terms in the expansion of (x-\-a)" is ?^+ i. .9. Value of the mi Term. The value of the rth term in the expansion of (x-\-a)" can be easil}^ found By the law of exponents, the exponent of j; in the /irs^ term is ;^ ; in the second, n — i ; in the third, n — 2, and so on; conse- quently in the rth term it is n — (r—\), or ?? — r-f i. Also by the law of the exponents, the exponent of a in the second term is i ; in the third term, 2, and so on ; conseqtiently in the rth term it is r— I. So, without the coefficient, the rth term must be By inspection of the coefficients in the expansion in Art. 4, it is seen that the numerator of the coefficient of any term is the product of the natural numbers from ;/ to a number one greater than the exponent of a. Since the exponent of .r in the rth term has been found to be ;?— r-fi, this numerator of the coefficient vciVL^\,h^ 7i(n—\)(n—2) . . . (7i — r-\-2). An inspection of the binomial fonnula will also show that the denominator of any coefficient is the product of the natural num- bers from unity to a number equal to the exponent of a. Whence the denominator of the coefficient of the rth term must be I, 2, 3, . . . (r—\). Therefore the complete ?'th tenn is 71(11— \ ) (11 — 2) . . . (n — r-^2) ^,._^ ^ „._^., , 1.2.3.4 . . • i^'—"^') 136 Algebra. Multiplying numerator and denominator of the coefficient by I n — r-f I , this becomes r— I \n — r+ i 10. Theorem, hi the expansion of (x^ a)" the coeffide^it of the rth term from the beginyiing equals the coefficient of the rth teim from the end. Since there are ?z+i terms all together (Art. 8), the /th term from the end has 7i-\-\—t, or n—t-\-\, terms before it. Hence the /th term from the end is the same as the ;^ — /4-2th term from the beginning. From the preceding article the ;z— /-|-2th term equals \n—t-\-\ \t—\ But from the preceding article the /th term from the beginning equals |/— I \^n — t-^\ It is plainly seen that the coefficients are identical. 11. Expansion of (x—a.)" If we substitute —a for a in the binomial formula^we will obtain the following result for the expansion oi x—a : (x—a)"=x"~nax"-'-\-- ^a\x"'' * ^' \a^x"-^4- . . 1.2 1.2.3 12. Theorem. /?i the binomial formula the s?im of the coeffi- cients of the even terms equals the sum of the coefficients of the oda terms. In the expansion of (x—a)"^\x\. x=i and a=\. We then obtain , n(n—i) 7i(n—i)(n—2) , 0= I — ?z -f -^ ^ <^ ^ 4- etc. , 1.2 1.2.3 which shows, since the negative on the right side of this equation must equal the positive, that the sum of the coefficients of the first, third, fifth, . . . terms equals the sum of the coefficients of the second, fourth, sixth, . . . terms. Binomial Theorem. 137 13. Theorem. The sum of all the eoejjicients in the expansion 0/ (x-\-a)" equals 2". In the expansion of (x-\-a)" ^nt x—i and a=\. We then have n(n—i) n(n—i)(?i—2) 2=1+;/+ - +- +etc, 2 1-2.3 14. Examples. /. Expand (x-\-a)^. 2. Expand (b—e)^. J. Expand 6'+3/- ^. Expand (b'—e")^. 5. Expand (x-\-af. 6. Expand (x-\-2c)^. 7. Expand (2>b+i)\ 8. Expand (x'-\-a')\ g. Expand (2ax—x')\ 10. Expand Wab—^abf- 11. Expand y-i--\ • 12. Expand (5—ix)'\ 13. Find the 5th term of (xy-\-x"-y\ 14. Find the 9th term of [^x'^+^x I I ]" 75. Find the ?iih. term of \n"-\ \ . {. 'M 16. Expand (x''-\-2ax-\-a'')^. ly. Expand (V<^^— 2 J^)'^ 18. Find the 1000 term in (x-\-ay"^\ 15. Expansion OF A Polynomial. The power of a polynomial can be obtained in the following manner. Suppose it is required to expand (a-\-b-^cy. We can proceed thus : (a^b^ey^{a^(b-^c)J = a' + 2>a^(b + e) + 2>a(b+e)^-\-(b-\-e)\ which, when the powers of ^ + ab'-\-6abe^ 3^^' + <^'+ 3^V+ 3^r" +r^ Notice that the result is a homogeneous symmetrical fu7iction of a, b, and c. —17 138 Algebra. 16. ExAMPIvKv^i. Expand (a-^d—c)\ Expand (ad-{-dc+ac)^. Expand X — a Expand (a-\-d-^t^-\-d)\ Expand (i-^x+x")'. CHAPTER XI. THEORY OF UMITS. 1. Definition. When a quantity preserves its value un- changed in the same discussion it is called a Constant, but when under the conditions of the problem a quantity may assume an indefinite number of values it is called a Variable. Constants are usually represented by the first or intermediate letters of the alphabet and variables by the last letters. The notation by which we distinguish between constants and variables is the same as that by which we distinguish between known and unknown quantities, but it must not be thought that any analogy is intended to be pointed out by this fact. When we are discussing a problem in which both constants and variables appear we usually do not care whether the constants are known or unknown. 2. Definition. When a variable in passing from one value to another passes through all intermediate values it is called a contmuous variable; when it doe's not pass through all intermed- iate values it is called a discontinuous variable. 3. Definition. When a variable so changes in value as to approach nearer and nearer some constant quantity which it can never equal, 3^et from which it may be made to differ by an amount as small as we please, this constant is called the Limit of the variable. 4. Illustrations. If a point move along a line AB, starting A B ■I • II at A and moving in such a wav that the first second the point moves one-half the distance from A to B, the second second one- half the remaining distance, the third second one-half the distance which still remains, and so on ; then the distance from A to the moving point is a variable whose limit is the distance AB. For, no matter how long the point has been moving, there is still some 140 Algebra. distance remaining between it and the point B, so that the dis- tance from A to the moving point can never equal AB, but as the moving point can be brought as near as we please to B, its dis- tance from A can be made to differ from the distance AB by an amount as small as we please. Thus we see that the distance from A to the moving point ful- fills all the requirements of the definition of a variable, and the distance AB all the requirements of the definition of a limit. The student must note that it is not the point B that is the limit of the moving point, although the moving point approaches the point B, but it is the distance AB that is the limit of the distance from A to the moving point. If we call the distance the point moves the first second i (then of course the whole distance AB would be 2), the distance trav- ersed the second second would be \, that traversed the third .second would be \, and so on, and the entire distance from A to the moving point at the end of n seconds would be the sum of n terms of the series T 1 1 1 1 ^» 2» 4' 8"' 16"' • • • Now it is sure that the more terms of this series that are taken the less does the sum differ from 2 ; but the sum can never equal 2. Hence we say that the limit oi the sum of the series I+-2"+4+"8+T(5" • • • as the number of terms is indefinitely increased is 2. Again consider any regular polygon inscribed in a circle, and then join the vertices with the middle points of the arcs subtend- ing the sides, thus forming another regular inscribed polygon of double the number of sides. From this polj^gon form another of double its number of sides and so on. Now the polygon is always ivithi7i the circle, and hence the area of the polygon can never equal the area of the circle, but as the process of doubling the number of sides is continued, the less does the area of the poly- gon differ from the area of the circle. Hence, we say that the limit of the area of the polygon is the area of the circle. Again as a straight line is the shortest distance between two points, any side of the inscribed polygon is less than the sub- tended arc, hence the sum of all the sides or the perimeter of the polygon is less than the sum of all the subtended arcs or the cir- Theory of Limits. 141 cumference of the circle, or in other words the perimeter of the polygon can never equal the circumference of the circle, but as the process of doubling the number of sides is continued, the perimeter of the polygon differs less and less from the circumfer- ence of the circle, hence the circumference of the circle is the limit of the perimeter of the inscribed polygon. 5. The student should not infer from what has been said that all variables have limits! In fact, the truth is quite the contrary, for most variables do not have limits. Thus, in the illustration of the moving point given above, the variable does not have a limit if we suppose the point to move at a uniform rate. For, if the velocity is uniform, it is a mere question of time until the moving point passes B, or, in fact, any other point to the right of B, how- ever remote. Much more would this be true if the point moved with increasing instead of uniform velocity. Again, consider the fraction X If X be supposed to change in value, the value of the fraction changes and is itself a variable. Now suppose x to decrease in value. It is plain \)ci.2i\. the value of the fraction increases without limit as x decreases. In other words, the value of the fraction can be made as large as we please by taking x small enough. Hence, as X decreases, the value of the fractioyi has 710 limit. 6. It follows immediately from the definition of a variable, that the difference betiveen a variable and its limit is a variable ivhose limit is zero. For if X be a variable whose limit is a, then x may be made to differ from a by as small a quantity as we please, hence a—x may be made as small as we please; yet as x can never equal a, a—x can never equal zero, hence a—x is a variable, whose limit is zero. 7. Theorem. //* two variables are always equal a?id each approaches a limit, the limits must be equal. Let X and v be the variables, and let limit x=a and limit r=^. 42 Algebra. We are to prove that a^b. If a and b are not equal, suppose a greater than b and let a-b=d I^et a—x=u and b—y=^i\ then a=x-{-u and b=^y-\-i\ and a — b=d becomes by substitution, (x + u)-(y+v)==d or (^.r —y)-\-( u —v) = d Since lim x^=a, lim u=o and as lim y=b, lim t'=^, or ?/; and -v are each variables which can be made as small as we please, and hence the difference u — v can be made as small as we please, and so can be made so small as not to cancel d, hence x—y would equal something, or x andjF would differ, which is contrary to the hypothesis; hence a cannot be greater than b, and in the same way it may be shown that b cannot be greater than a. Therefore a^=b. 8. Theorem. The limit of the algebraic sum of several variables equals the algebraic sum of their separate limits. Let the variables be x,y, z, etc., and let lim .v=<7, lim r=<^, lim .5-=^, etc., we are to prove lim ^r+j/+2-t- . . . ; = C^ + ^+r+ . . . j I^et a — x=^u .'. x=a — u, etc. , etc. ; thenjt-+r + -+ . . . =(a-\-b-\-c+ . . )-(u + v-h7C'+ . . ;. Suppose i/ to be numerically the greatest of the quantities u, V, zv, . . . and suppose that there are n of these quantities. Now, since x may be taken so near a as to differ from it by an amount as small as we please, we may take x so that k n or nu<^k, however small k may be. Then u-{-v-\-w-\- . . . <;^^/ (since u is the largest of the quantities u, v, u\ . . . ). Hence u -\- v -\- zu -\- . . .jf, y—x is positive, and therefore «'"' is greater than unity, for a positive power or root of a quantity greater than unity is itself greater than unity. Hence, a-' -r-a' > i hence a^^a' . Second Case. Where a is less than unity. As before a^' -r-a" =<2^~' and «^~''«'''> «^'', so whether a is greater or less than unity, rt^is intermediate between a* and a^ . Now consider x and j' variables, but always commensurable, and let x increase and y decrease, and suppose them lo approach the same incommensurable limit 71, As x and j/ are commensur- able, a'' and a^have definite meanings, and as x ahdjj/ approach equality, (one increasing and the other decreasing), «^ and «-^also approach equality, or in other words there is some quantity be- tween ^'^and «^ from which each of these quantities may be made to differ by an amount as small as we please. But ^* and a'' can never become equal, since x and y cannot become equal, hence each of these quantities approaches the same limit. Since we have now proved that both ^^''and a^ approach a limit, as X and y themselves approach a limit, we may if we choose neglect J and «^and fix our attention upon x and a" remembering, however, that x varies just as it varied before, and hence just as before a"- approaches a limit and indeed the same limit. This limit we will represent by ^" . Thus we have a meaning for «" where n is incommensurable, viz: it is the limit approached by a' {x being commensurable) as x approaches a limit n. 18. Extension of Formula {a) of Chapter II to Incom- mensurable Indices. So long as x and y are commensurable we know that a-a'^a""^' (i) Let X approach an incommensurable limit n, and y approach an incommensurable limit r. Then x-^y approaches a limit n-\-r which is usually incom- mensurable, but may possibly be commensurable. Also a"- ap- proaches a limit a" , a' approaches a limit a'' and a' ^ approaches a limit a"+^ Now in equation (i) the left-hand member is one variable, and the right-hand member is another which is equal to the first. Hence, Art. 7, lim «*^«-''=lim a'''^'' . But the left-hand member is the product of two variables, hence by Art. 10, lim (a"" a" )=^\\r\\ a' lim a^ or \im(a' a^ )=a" a'^ , hence a"a'^=^a"+''. 148 Algebra. 19. ExTP^NSiON OP Formula (b) of Chapter II to In- commensurable Indices. With the same notation as in the previous article we have {a^)y^a^y (i) hence Xxva ( a"" y =-X\vci a^^' (2) and b}^ Art. 7 lim '' = T^" «" / (4) and because y is commensurable (a^^ aJ' y = (a" y (a '' y = (a^' y a''\ (5) Substitute in (4) and we get (a^y=(a'^ya"\ (6) hence lim ("«">"= lim [fa" >"«"-^], Art. 7, (7) and lim l(a"ya"y']=.\im (a"y lim a"-\ Art. 10 (8) therefore lim ("«" y =lim (a" y lim a"-\ (g ) But since J/ approaches r, therefore lim (a")^'=(a"y, (10) and because // approaches o and y approaches r, and hence uy approaches o, therefore lim a"-'=a''= i . (11) Substitute for the right-hand member of (9) the values found in (10) and (11) and we get lim (a" y=(a" y. (12) Substitute for the two sides of equation (2) the values found in (12) and (3) respectively and we obtain (a''y=a"''. 20. Extension of Formula (c) of Chapter II, to In- commensurable Indices. With the same notation as in the two previous articles we have a- -^^•" =«—-'■ (i) Since lim x=n and lim j'=r, lim (x—j)^=n—r and lim a'''=a" , lim a\=a'' and lim <2 '"-'==«""'. Theory of Limits. 149 From (i) lim (a' -7-a' ) = \\m ^'"•''by Art. 7. But lim (a'' —a' )=\\m. a' -r-lim a^ =a" -r-a ami lim «'~''=a""'', therefore a" -k-a'' =a"~'' . 21. It is also eas}' to see that where 71 is incommensurable a"b''=(ab)" and Hence But and hence Again hence and hence lim a' lim a' a F lim b"" b' a ' a " X [b a" a\" b"~ b\ by Art. 7. For let t7 and b be tvvo bases and x a variable which remains commensurable, but approaches an incommensurable limit 7t, then a''b'=(ab)' . lim d;"<^=lim (ab)' lim a'' br' =lim a' lim b'- =a" b" . lim (aby =-(al)" , a"b^=(ab)" . b^ Ui 22. Examples of Limits. In the following examples an expression like , \ r MS to be read: the limit of\—~^\ h 'l a {a-\-h) -^ iw^h) as h approaches a as a liinit. The symbol ^ stands for the word approaches. limit ( (x-\-Ji)-'—x^ \ Find h : o T Process. limit {(x-k- /!/— .v"| limit f >^ :: ol S~h : o I x^-\-2hx—h^—x'' I _ limit J ~/^ : ol 2X-\-h }=.,. 150 Algebra. " ^. , limit ( mx ") 2. Find ^ \^ ^ > ?. Find "'"*|?l--:q x^ a ( a—x J ^.. , limit (^r^-/^;'--^^-M ^ h ^ o { h \ ^. . limit ( -r^+i ) 5". Find ^ \ - „- .- ^ ,, limit f.r"— a") ^. Prove ^ - =na" '. .V ^ ^ ( .r— ^ I 23. lyiMiT OF THE Sum of a Decreasing Geometrical Progression as n Increases. It was noticed in Chapter VIII that if the ratio of a geometrical progression is less than unity, each term of the series is necessarily less than the one preceding it. In this case the series is called a decreasing progression. In the case of a decreasing geometrical progression, it is a little better to write the expression for the sum of the series in the form: I — r Now if we like we may consider n a \'ariable, and then the two sides of this equation are two variables that are always equal. Therefore, their limits are equal. Whence we may write lim ^ as n increases=lim-i * - — :as n increases. I I — r J Now since r is less than i, the term ;-" continually approaches the limit o as n increases. Whence taking the limit of the right hand member of the equation, we may write : ,. . a hm .V as ;/ increases= I — r. 24. Examples. I. Find the limit of the progression T-SSSS + J ^"^ ^' increases. Here ^«=to and r=-f^. Whence, lim .y= ' ^ ^ =|. ^~T0 Therefore, lim .3333+ =15. Theory of Limits. 151 2. Find the limit of the progression ir-|-| + | + TV+ ^^c, as n increases. J. Find the limit of .272727+ as n increases. ^. Find the limit of .2 792 792 79 -f as n increases. 5. Find the limit of the sum of i~>y + TV~A+ ^t<^-' ^^ ^ increases. 6. Find the limit of the sum ofV8+\/4+V2-fV 1 + as ;/ increases. 25. Theorem. The limit of the sum of the series i-\-r-\-r'-\- ^-3_j_^.4_j_ ^^^^ ^^ y decreases and n increases is i. In the equation ,. a lim .T= — I— /' the expression will of course have different values for different I — r values of r. Hence we may, if we choose, look upon this expres- sion as a variable. But as ;- approaches o as a limit the fraction approaches a as a limit. Therefore we ma}* say that in a decreas- ing geometrical progression as the number of terms increases with- out limit, and as the ratio approaches zero as a limit, the sum approaches ' term after the third contains x^, and so on ; hence every term after the n th will contain the factor x'\ and so it is natural to assume the remainder after 71 terms are written to be of the form R^jt". Instead of writing limit of A^-t-A^.r-f- ... as the number of terms increases without limit we write A„+A A-+A x=+ . . . A„_i-t"-' + Rx" and in the same way write B,,+Bx+BX+ . . . +B„_iX"--hR,-^t-" instead of writing limit of B^ -f B^.v + B.x^ . . .as the number of terms increases without limit. Using this notation we may write A,,+A A-+ . . . -f A„_iX"-'-|-Rx" = B„+Bx+ . , . +B„_i.v"-'-hRx". (i) If now we consider .v as a variable approaching zero we have here two variables which are always equal, and therefore by Chapter IX, Art. 7, their limits are equal. By Chapter IX, Art. 26, the limit of left-hand member equals A^, and the limit of the right-hand member equals B^* hence A^^=B^. 156 Algebra. Subtract A,^ from the left-hand member and B_^ from the right- hand member of (^ i J and we get A,-r+A,j»:^+ . . . +Rx"==Bx+BX-+ . • • 4-RX' (2) Divide both members of (2) by x and we get A^-\-A'x+ . . +A,_iJt-"-=+Rjtr""' = B^ + B^jt--f . . . +RX'-'. (3) As before, we have two variables always equal, hence theii limits are equal. But as X approaches zero the limit of the right-hand member equals A^ and the limit of the left-hand member equals B^. Hence, by Chapter XI, Art. 7, A=B,. Repeating the reasoning, we may show successively that A^=B^, a=b;, etc. 6. The theorem of the last article will enable us to change the form of a function. The method of doing this consists in assuming a function of the required form with unknown coefficients and then determin- ing the coefficients so that the function assumed shall be identical with the function proposed. The unknown coefficients are deter- mined b}^ placing the proposed function equal to the assumed function, reducing to the rational integral form, and equating the coefficients of like powers of the variables on the two sides of the equation. If the proposed function can be placed in the assumed form it will be found that there are as many independent compatible equations as there are unknown quantities to determine. 7 . Definition. A function is said to be Developed or Ex- panded when it is expressed in the form of a series, the sum of whose terms when the number of terms of the series is limited, and the limit of the sum when the number of terms is unlimited, equals the given function. 8. The development of functions is one of the most common applications of the method described in Article 6. The process Undetermined Coefficients. 157 is usually referred to as the method of undetermined coefficients. We will illustrate the method by working an example. I^et us develope the fraction -. Assume _^^_=A„-f AA-+A^.r=-f . . . -f A,_i.r"-i4-Rjt-". (\) Multiply both sides of (^ij by \—x and we get i=A +rA,-AJ.r+rA^-A.X+ . . . + rA_i-A„_2>-"-i-t-rR-A„_Jjt-"-R;i-"-i. We see that the left hand member contains no power of x ex- cept the zero power, or, in other words, in the left hand m^iber, the coefficients of the various powers of x except the zero power are each zero. Hence equating coefficients we get A=i A-A=o.-. A=A, A-A=o.-. A^=A^, A3— A^=o .-. A3=A^, etc., etc. From these equations the law of the series is so evident that we can wTite as many more equations as we please without further calculation. We thus see from the second column of equations that each coefficient equals the preceding one, and as the coefficient of x", or the absolute term, equals i; therefore each of the other coeffi- cients equals i . Hence we obtain — -- = i+-r+-r^+-v^ + -r^+ . . . I — X As we usually determine only a few of the coefficients, and then discover if we can the law' of the series, so it is usual in the assumed series with undetermined coefficients to write only a few terms and indicate the others including the remainder by dots thus: ~- =A, + AA-+A.r^-fA-r^+ . . . I— .r . ' - ^ Instead of using the method of undetermined coefficients we might have proceeded by ordinary long division as follows: 158 Algebra. A- I ;i+-v + a---f A- + .r^ 1— X X X—X' Jtr* X^—X^ x^ x^—x* x\ Here as before we obtain I— .r This series on the right side of the sign of equality is conver- gent if x<,i, but not otherwise, and therefore this series cannot be called the development of unless .v is less than unity. See Art. 7. When .r is equal to or greater than unity the fraction ^ cannot be developed. 9. Examples. Develope the following fractions both by the method of undetermined coefficients and by actual division, and in each case discover the law of the series. Also in each case state for what values of x the series is a true development. I i-^x \-\-x \—x-\-x- Ijf-V I + 2.V-}- 3JI:' I — X' l—2X-\-T,X^' ^ « ^, 1-2X±SX^^ . 4. '"^3-^'. o. _5£+7^. 5- 7 ^- ^o. SX"" — T,X l — 2X-\- TfX" 3 — 4.x'' 4- 2X^ Compare the laws of the series in the developments of the frac- tions in examples i and 4 ; also compare the laws of the series in examples 5 and 7 ; also in examples 9 and 10. Undetermined Coefficients. 159 Query : What controls the law of the series in the develop- ment of a fraction ? Query : How does the numerator affect the development of a fraction in the fonn of a series ? Query : What would the results to examples 7 and 8 suggest about the development of fractions which are reciprocals ? 10. It sometimes happens when we try to develope a fraction by the method explained that some of the equations are absurd or contradict one another. The reason of this is because the fraction cannot be developed into a series of the form assumed. Thus if we try to develope I we assume x—x- X — X- 0.2 3 4 Multiply by x—x" and we get i=AXA AJx^-|-fA^AjA-^+ ... hence 1=0, A=o, A=o, A^===o, etc. But the first of these equations is false, so we consider that the function cannot be developed into a series of the assumed form. But we note the denominator of the given fraction contains a factor A-, and that hence the fraction proposed equals • , the second factor of which has already been developed. From this we would infer that the development of ^ could be obtained from the development of by dividing every term in that development b}- .v. i6o A1.GEBRA. Hence — ^=x~^ -j-i-\-x-{-x^-{-x^^-h. . . x—x^ and we would obtain this very result if should assume ^:^,^Ax-^-hA^-\-A^x-\-Ax-+. . . or in other words if we de^in the assumed series with a term con- taining x~^ instead of beginning with an absolute term. If the fraction we wish to develope is in its lowest terms, and if the lowest power of x that appears in the denominator is the rth power then we must begin our assumed series with a term con- taining x~''. This is a safe rule whether the fraction is in its lowest terms or not, but it i's not always necessary when the fraction is not in its lowest terms. In any case when we form an equation by putting a given frac- tion on the left and an assumed series on the right side of the sign of equality, the assumed series must begin with such a power of X that when the equation is integralized the lowest power of x on the right side of the equation will be as low as the lowest power on the left side. 11. Examples. \-\-x-\-x^--\-x^ 1. Develope , , — . X^-\-2X^ ^ , X+'IX'^ 2. Develope — —^,. ^ X^-\- 2X^ 12. Not only fractions but some irrational expressions may be developed by the method of undetermined coefficients. Let us develope Vi — ■^• Assume ^/Y^x=A^-]^Ax-\-Ax' + Ax^ + Ax'-{-A_x^-\- . . . Square each side and we get i-x=A-f2AA^jt-4-(2AA+A;K+(2A>.3+2AA>^ + (2A A^+2A A3-f A;>-^4-(2A„A^ + 2A A^+2A^A3>5+ . . Equating coefficients of like powers of x we get Undetermined Coefficients. i6f A=i, 2A^A,= — I, 2A^A,+ 2A A^=o, 2A^a^+2AA,+a;=o, 2AA^+2A A^4-2A^A^=o, 2AA+2A,a^4-2AA^+a;=o, etc. From these we get A=i, _ 2 A, A, 2A„ • A _ ^A,A,+2A,A.+A/ etc. From these the law of the series can be seen. Taking these equations in order, we find the numerical v^ahie of the undetermined coefficients to be as follows : Ao=i» A^=— J, A^=— -|-, A^=— jV, A^= — i|-g-, A — 7 A 21 ^5— TTe' ^6— r(5"2T- Making these substitutions in the assumed development, we obtain . _ X jr x^ ^x^ yx^ 2i-V^ 2 8 16 128 256 1024 13. Examples. I ^ I. Develope 1+2 x— • 2. Develope \/x-j-x'\ J. Develope (i-\-x)^. A-20 1 62 Algebra. 14. It is interesting to note that the development of an irra- tional expression 7?iay turn out to be a series of a limited number of terms. Suppose, for example, we wish to develope s/ i — 2x-\-x^ and do not recognize that i — 2x-\-x^ is a perfect vSquare, then assume as before s/ i — 2x-]-x^=A^^-\-Kx-\-Kx'+ . . . Square both sides and we get I — 2.:»;— jL'^=A^,+ 2A^Aa'-|-(A,^+2A^AJx^ ■ +(2A^A^-}-2A A^>-^-f .... Bquating coefficients of like powers of x and we get A„=i, 2A^Aj= — 2 .'. A^= — I, A;-f2AA=i . . A =o, 2A^A^-f 2A A^=o .-. A^=o, A;H-2A A^+2A A =o .-. A^=o, etc., etc., and each of the subsequent coefficients will turn out to be zero, hence we ^et S/ l — 2X-\-X''^==l—X. 15. In developing irrational expressions it sometimes happens that we should begi7i our assumed development with some negative power of X. An inspection of the p:op33ed example will show with w^hat power of jr the development should begin ; for the assumed series must be such that, when the equation obtained by putting the given function equal to the assumed series is reduced to the rational integral form, then the lowest power of x on the side which contains the undetermined coefficients must be as low as the lowest power on the other side of the equation. Thus, to develope ^M+ 2 we would begin the assumed series with a term containing x~\ for when this is squared the lowest power of X is x~^ and when both sides are multiplied by x"" to reduce to the integral form then the series on the right side of the equation will begin with an absolute term as it should. Undetermined Coefficients. 163 16. If^we wish to develope the algebraic sum of two or more radicals it is best to develope each one by itself and then find the algebraic sum of the results. 17. Examples. /. Develope 1 1 — \. 2. Develope Ax-\- --+ sj i -f-i". ' . Develope %/ i -f- 4A- + 6-r" -|- 6-r ^ + 5^1- * -}- 2-r ^ + x"". 4. Develope |.r^4- 2. V -1-3 + ^ +— . X x^ CHAPTER XIII. DERIVATIVES. I. Notation. A definition of a function of a quantity was given in I, Art. i. To designate a function of x we use the notation /fxj. A function of a quantity is denoted by writing the quantity in a parenthesis and. writing the letter f or F or some other func- tional symbol before the parenthesis, e. g-. f(x), F(x), F/x) denote functions of ;i', f(y), F(y),fjy) denote functions of r, f(x-\-h), F(x-\-h),f'(x-\-h) denote functions oix-{-k, f(a), F(a),f„(a) denote functions of «. The student must be careful not to look upon the expression f(x) as meaning /times x. The symbol /as used here is not a quantity at all, but simply an abbreviation for the words ftindion of. It frequentl}' happens that in the same discussion we wish to refer to different functions of x, in which case we use different functional symbols, as F( x ) , f( x ) , fj x ) , /„( x ) , F„(x), etc. It also frequently happens that in the same discussion we wish to refer to the same function of different quantities, in which case we use the same functional symbol before the parenthesis but dif- ferent quantities within the parenthesis, e. g. \i f(x) denotes jtr^+i then f(a) denotes a^-\-\, f(.z) denotes .J^-f i, etc., and if F(x) denotes /v/-^4-3 then F(y) denotes Vj'4-3, F(x+/i) do:- notes V -^' 4- // + 3 , -etc . A function of two quantities is any expression in which both of the quantities appear. If we have to deal with a function of two quantities, say x and y, we use the notation /(^jf, y) or F(x, y), and if, in the same dis- cussion, we wish to speak of two or more functions of x and y, different functional symbols are used, a.s//x', y), f,(x, y), etc. Derivatives. 165 2. In such an expression as /(^.r, j/^ the two quantities x and y are entirely unrestricted in value and independent of each other ; but if we have an equation like f(x, y)=-o, then x and y are to some extent restricted ; any value may indeed be given to 07i€ of the quantities but then the equation fixes the value of the othei\ or in other words, either one of the quantities x or y de- pends upon the other one; e. g., \i f(x,y) stands for x—y-\-2 then when this is not put eaual to anything there is no relation between .v and y. We may let ^ = 3 and j'=5 or 7 or 10 or any other number. But if we put this same function equal to zero, then there is some relation between x and y and they are to some extent restricted in value. We may let ^'=3, but then ^r=5 and nothing but 5. 3. If the equation F(x, y)=^o can be solved for j', we can ex- press y in terms of x, or y can be determined as a function of x. If we thus determirue J we havey=/(x). In this equation, y=/(x), we may look upon x as a variable, and of course if x varies r will also var\'. We may consider x to vary in any way we please, but then the equation determines the way in which r varies. For the reason just stated x is called the independent variable, and r, which is a function oi x, is called the dependen t i 'a Ha ble. 4. In the equation y=f(x) if a value be given to .v then r will have some corresponding value, and if .r be given another value different from the first one then y will have some value different from the one // had at first. Moreover, the amount by which y thus changes in value will depend in some way upon the amount by which .r changes, or in other words, there is some relation connecting the change in value of y with the change in value of X. This relation we will examine, and it will be found to be a very important relation in all that follows. 5. Suppose f(x) to stand for 2x-\-\, then putting this equal to r we have 1 66 Algebra. Let us now give to .r a series of values, sav the successive in- tegers from I to lo, and in each case compute the corresponding vahie of r. The results may be expressed in the form r I 6 8 lo 12 14 16 18 20 22 24 A- I I 2 3 4 5 6 7 8 9 10 where any number in the lower line is one of the values of .v and the number immediately above it is the corresponding value of j'. If ji:= 2 the corresponding value of r is 8, and if .v— 10 the corresponding value of j' is 24, and if X be considered to increase from 2 to 10 then at the same time y will increase from 8 to 24, or, starting at -v=2, if x in- crease by 8, J' will increase by 16, or if the increase of .r is 8, the corresponding increase of r is 16. Still starting at .v=2, let us increase x b}^ various amounts and determine the corresponding increase of r. The results may be arranged in the form increase of J' | 16 14 12 10 8 6 4 2 increase ofx|8 7 6 54321 We might have started with some other value of x than 2 and have obtained similar results. In every obsen^ed case we .see that the increase of j is just twice the increase of .v, or in ever>'' observed case increase of v increase of x It is easy to see that this is necessarily the case whatever the value of .V with which w^e start and whatever the amount by which X is increased, for if x increases by any amount, 2X will increase by just twice that amount and the change in the value of X does not affect the 4, therefore 2.1-1-4, or y, will increase twice as much as .r increases, or increase of v increase 01 x 6. Notation. In what follows we deal largely with equa- tions formed by putting r equal to a function oi x, and as we will make extensive use of the increase in the value of .v and the cor- DerivativEvS. 167 responding increase in the value of v it is well to have a conven- ient notation by which these amounts of increase are denoted. So in future we will use Jx to denote the increase in the value of x and Jv to denote the corresponding increase in the value of r. In this notation the fraction at the end of Art. 5 would be written The student is cautioned not to think of J.v as being J times X, for the symbol J as here used does not stand for a quantity at all, but simply for the words increase of. 7, Let us now consider the equation y=^x'-\-\. In this equation giv^e x the successive integer values from —3 to 7 and compute the corresponding values of y. We may ar- range the results as in Art. 5, y_ I iQ 5 2 I 2 5 10 17 26 37 50 A- I —3 —2 —I o I 2 3 4 5 6 7 If ;t:=i the corrCvSponding value of r is 2, and if -i-=7 the cor- responding value of J is 50, and if x be supposed to increase from I to 7, at the same time y will increase from 2 to 50, or starting at .r=i, if x increase by 6 then y will increase by 48, or when J-v=6, ^y=\%. Still starting at x= i , let us give to ^x various values and de- termine the corresponding values of ^y. The results may be arranged in the form jj_| 48 35 24 15 8 3 ^x\ 'b 5 4 321- ji/ Here we have a case where the ratio ~ is not always the same as it was in Art. 5, but at one time it is Y"* o^" ^. ^^ another time it is -V-, or 7, etc. As can be seen by the above scheme, the fraction -j- takes suc- cessively the values 8, 7, 6, 5, 4, 3 as J.v takes ;the successive values 6, 5, 4, 3, 2, i. We now give to x values intermediate between i and 2 and 1 68 Algebra. compute the corresponding values of r. The results may be arranged in the form y I 2 2.000020000I 2.0002000I 2.00200I 2.02CI 2.21 X I I I.OOOOI I.GGOI • I.OOI I.OI I.T As before, let us start at x=^ i and give to Jx various fractional values and determine the corresponding values of Jr. The results may be arranged as before in the form Jj/ I .21 .020 1 .002001 .00020001 .0000200001 J-T I .1 .01 .001 .0001 .00001 An examination of this scheme shows that J r .21 when J.r=.i, then -~- = — = 2.1 Jjt- . I 1. < ^x, -^y -^^^oi when J.r=.oi, then -7-= =2.01 Jjt .01 . Jr .002001 when Jjf=.ooi, then -^= =2.001 Jjf .001 , , Jy .00020001 when Ja"=.oooi, then -^= = 2.0001 Jx .0001 , , Ay .0000200001 when Jx=.ooooi, then -^^=: =2.00001 Jx .00001 From the first part of the Article it appears that ~f-_ is a var- iable, and from what we have just obtained it further appears that as J.r is taken smaller and smaller the fraction -{- approaches J.r nearer and nearer the value 2, or in other words, it appears that the fraction -^ approaches 2, /. c. 2 times i, as -x approaches zero. In obtaining the result it is to be noticed that we consider x to in- crease/r^w the value i, but if we let x increase b}^ various amounts, beginning to count the increase in x from the value 2, reasoning exactly as we have just done would lead to the conclusion that Ay -y^ approaches \, i. e. 2 times 2, as Ax approaches zero. Again, if we begin to count the increase in x from the value 3 we would be led to the conclusion that -~ approaches 6, /. e. 2 times 3, as Ax approaches zero. Derivatives. 169 8. In general, if a be taken as the value of x from which we be- gin to count the increase of Jt: we would judge from analogy that the Ji/ fraction— approaches 2a as J-r approaches zero, or, using the notation of Chapter XI, Art, 22, limit (Jy) This we will now prove. Since y=x^-{-i, (i) whatever value be assigned to x the equation will enable us to compute the corresponding value of y. First, let x=^a and represent the corresponding value of jk by b, then b^^a'-^-i. (2) Now let x=a-\-Jx and represent the corrCvSponding value of^by b-\-^y, then from equation (i) we get ^4-4y=r^ + J-r/+i, (3) or simplifying, b-\-Sy=a'^-2aJx-\-(Jxf^\. T^) Subtract (2) from (\) and we get Ay^2a^x-V(^x)\ (5) Divide (^5 J by J.r and we obtain ^=2^ + J;r. (6) As Jji" varies, of course the the two sides of equation (d) are variables, and, indeed, they are two variables that are always equal, and as J.r approaches zero these two variables each ap- proach a limit. Hence by Chapter XI, Art. 7, their limits must be equal. Therefore }^^^ ^ \ f^ = 2a. Ay 9. Definition. The value of the fraction -f- when that frac- . Jr tion is constant, or the limit of the fraction -~ as Jx approaches zero when that fraction is a variable, is called the Derivative of y tvith respect to x, and is represented by the notation D,j', where j' is a function of x. A— 21 170 Algebra. 10. The general method of finding the derivative oiy with re- spect to X is that used in Art. 8, viz : give to x some value, say a, and find the corresponding value of y, then give to x a new value, a + Jx, and again find the corresponding value of j^ Sub- tract the first of the equations thus obtained from the second and we have the value of Jj. Divide both sides by Jx and we have the value of -^. Jx Then finall}^ find the limit of this fraction as Jx approaches zero. 11. We will now exemplify the method in a few examples. Firsf. Find D,-j^' when j'=4.x^-\-^. (i) I^et x==a and represent the corresponding value of r by d and we get ^=4«'-f-5. (2) Now let x=^a + Jx and the corresponding value of j' will be the value d plus the amount by which j/ has been increased, or b-{-Jy, hence d-{-Jj'=4.(a + JxJ'+s. (3) Expanding, b-\-Jy=/\^a^-\-SaJx-\-/^(Jx)^-\-^. (4) Subtract (2) from (^) and we get Jy^^aJx^^(Jx)\ . (5) Divide ( s) ^Y ■^^' ^^^ we obtain -^=8a + 4rJx;. (6) Taking the limit of each side as Jx approaches zero we get limit (Jy\ or D,j=8«. (8) Seco7id. Find Y)^y when y=^cx--\-e. (i) Let x—a and represent the corresponding value ofj'by b, then b=ca^-\-e. (2) Now let x=-a-^Jx and we get b^Jy=.c(a^Axy-^e, (3) or expanding, b-^Jy^ca'-\r2acJx-\rc(Jxy-\-e. (4) Subtract {2) from (\) and we get J>/= 2acJx + c( Jx)\ (5) Derivatives. 171 Divide both sides of ($) by Jx and we get Jr/ -~- = 2ac-i'cJx. (6) Taking the limit of each side as Jx approaches zero and we get limit (Jjy] or D,r=2«r. (8) Third. Find D,.^r when y—cx'^-\-eX'\-f, (i) IvCt x=^a and represent the corresponding value oi y by b and we get l^cd'-\-ea-\-f. (2) Now let x=a-\-^x and we get bJ^^y^c(a-VAxr^-e(a-^^x)-irf. (3) or expanding and arranging, b-^Ay=.cd'^ea^f-^(2ac^e)^X-^c(^x)\ (4) Subtract (2) from (\) and we get Jy^(2ac-\-e)^x^-c(^xy. (5) Divide both sides by ^x and we get ■^- = 2ac-\-e-\-cJx. (6) Taking the limit of each side as J.r approaches zero we get limit \^y\ , ' . . ^'•O oijl^| = 2^^+^' ^7) or D..i'=2«r+^'^-\-2X. 6. c(x-\-\y. 3. (x+l)(x-{-2). ■ 7. ~. 4. (x+c)\ 8. s/x. 14. Extension of Meaning of T).^y. In Art. 10 we found that when j=4^^ + 5, D,.y=8« when y=^cx'^-\-e, Dj,y=2ac and when j=cx^-\-ex-\-/, T>^y=2ac-\-e In each case D^ r is of course a constant as it should be by the definition in Art. 9, where T)_^y is defined to be a limit, and the limit of a variable is by definition a constant. * In each case here noticed D^^ is a constant whose value de- pends upon the value a from which we begin to count the increase of X, or, as we may say, Ti^y is a function of a, while in Art. 5 T>^y was a constant which does not depend upon a. In any case D^ji' is either a function of a or it is independent of a, and when it is a function of a the a is the value from which we begin to count the increase of x. Now, as we nia}^ begin to count the increase from a7iy value of jf, a is of course a?ty value of x\ and so we may represent it by X instead of a and relieve D^y from being a constant, or in- other words, wherever T>^y was a function of a by the original definition we regard it now and hereafter as the same function of X that, by the original definition, it was of a. 15. I^et us now work out a case that was worked in Art. 11, using X now where a was used before. Take ^.= 4^-4-5.- (0 If we write x-\-Jx in place of x and therefore y-j- M' in place of y we have ■ j/+Jj/=4r-r+Jx/ + 5. (2) Derivatives. 173 Expanding, y-\-Jy=4X^+SxJx-^4(Jxr-\-5. (3) Subtract (1) from (t,) and we obtain Jy=SxJx+^(JxJ'. (4) Divide both sides of (4.) by Jx and we get ^=8-r + 4'J-^^ (5) Taking the limit of each side as Jjr approaches zero we get limit fJy) or B,y=8x. (7) We notice that the result is exactly the same as equation (S) in the first example under Art. 10, except that x appears here where a appeared before. We will hereafter proceed as we have just done and will usually find D.r y as a function of x, but occasionally, as in Art. 5, D^y will turn out to be a constant. 16. Derivative of a Constant. Let j'=a constant, then as x does not appear in the expression for r, X maybe changed at pleasure and the change does not affect J/, or J-v may have any value whatever, but Jy is always zero. Jy Hence -f-=o, ^x therefore D.v^r=o. 17. To FIND THE Derivative with Resect to x of the Algebraic Sum of two Functions of x. Let one function of x be represented by u and the other by v, and let their sum be represented by ]'; then y=u-\-r. (i) When X is increased by Jx suppose that u is increased by J«, V is increased by Jv and y is increased by J)', then after x is increased by J.r we have y-\-Jy=^U-[-Au-\-V-\-Av. (2) Subtract (i) from (2) and we get Jr=J//-hJr. (2^) 174 Algebra. Divide both sides of (^3 j by Jx and we get Jx Jx- Jx * ^^ Taking the Uniit of each side of (4) as Jx approaches zero we get limit j-^y\_ limit f-^^^) , limit \ Jz> \ -^x - olJ^rl'-^x ^ o\-j^j-^Jx - oJXrj ' ^^ or D^y=D,rU-\-T>A'. In the same way if y=?^ — z' we would get The result may be expressed thus : T/ie derivative of the algebraic sum of two functions of x equals the algebraic sufu of their separate derivatives. 18. I'o FIND THE Derivative with Respect to x of the Algebraic Sum of any Number of Functions of x. Let there be any number of functions of x represented by ?/, v, 7v, etc., and let their sum be represented by r; then we have y=u^-v-\-iv-^ ... rO Increase x by the amount Jx and suppose that u, v, it\ etc., are increased by the amounts J?/, Jv, Jw, etc., respectively andj' is increased by Jy, then we have after x is thus increased yJ^Jy=U-^Ju-}-V-{--iv-{-W-j-J7t'-\- ... (2) Subtract (i) from (2) and we get Jy==J^,^J^,^J,c'-\- . . . (:,) Divide both sides o{ (t,) by Jx and we have Jy Ju Jv Jw Jx Jx Jx Jx Taking the limit of both sides of (4) as Jx approaches zero we have D,i'=D.,?^H-D,2' + D,7r4- ... (^) If some of the signs \n (\) had been negative the same process could have been applied and the result would have had negative signs in the same positions as they appeared in the original functions. The result may be expressed thus : The derivative of the algebraic sum of several functions of x equals the algebraic sum of their separate derivatives. Derivatives. 175 19. Examples. Find the derivative with respect to .v of the following ex- pressions : 1. 2X^-\-\X^-\-X. ^. Jt^-f 3A--I-2. 2. X^ -\- X^- -\- X -\- 1 . 5. X-^ — X-^\. 3. -r''— I. 6. x^-\-i. 20. 'To FIND THE Derivative with Respect to x of the Product of two Functions of x. Let u and v be the two functions of x, and let y be their prod- uct, then we have y—uv. (i) Now increase x by J.v and suppose the corresponding amounts by which y, u, and 7' respectively increase are Jr, -^u, and Jz^, then we have y^Jy=(u-^Ju)(v+Jv). (2) Expanding (2) we get y-{-Ay=^iiv-\-uM'-\-vJu-\-JuM\ (^) or y-\-Jy=uv-\-uJv-^(v-\-Jv)J/(. ( ^j vSubtract ( i ) from (^) and we get Jr= uJv+ (v^ JvJJzi. C^J Divide both sides of (^5 j by Jx and we get Taking the limit of each side of (6), remembering that the last term of the right-hand member is the product of two quantities, hence its limit equals the product of their separate limits, and that J?' approaches zero as J-v approaches zero, hence the limit of v-\-J2'=7\ we get limit f JjM ,. . Jz' , ,. . J// J-v ^ o1j^| = '' l^"^^t 3^ + z^ hmit --, (y) or D,y=2iD,z'i-vDji. (S) 21. To FIND THE Derivative of the Product of any Number of Functions of x. First, take three functions of -v, say «, z\ and zc, and let y be their product, then we have y=uz'Zji'. (i) 176 Algebra. Let vw=v', ih.^n y=uv', and hence D,.jj/= v'B^u + iijy^v'. (2) But Tij)'=zvT>,v-\-vT>.,zv, (^) hence by substitution in (^2 j we have Now if we had any number of functions of x, say u, r, w, . . . and if we let j' be their product we have y=iwivz ... (\) Let the product of all the functions after the first be represented by a single letter, that is, let v'=^vwz . . . then y=uv'. Find D.,.y as the product of two functions. Then T>,y=v'T)ji + uT>,,v'. (2) Find D^t'' by letting v'=vw\ where w' represents the product wz . . . Substitute the value thus found in (2). The result will con- tain one temi involving T).,w'. Find the derivative by considering iv' to be the product of hvo factors. Continue this process until finally we reach the product of the last two factors of the expression with which we started. The result may be stated thus : The derivative with respect to x of the product of a7iy yiiunber of functions is equal to the sum of all the products obtained by multi- plying the derivative of each factor by the product of all the other factors. If the equation here described be divided through by the prod- uct of all the given functions, the result may be represented in quite a convenient form, viz : ■" - -h— ^4-- — -h— ^+ ... y 21 V w 22. ExAMPLKs. Find the derivative with respect to x of the following expres- sions without performing the multiplications indicated : 7. (x-\-i)(x-\-2) compare with example 4, Art. 19. Derivatives. 177 2. (Af—x-j-i)(x^i) compare with example 6, Art 19. J. (x^-\-x-{-i)(x—iJ compare with example 3, Art, 19. 4. (x'-j-j)(x^-j-ax-\-d), 23. ^o FIND THE Derivative with Respect to x of the Quotient of two Functions of x.. Let u and 2' be the given functions of x, and let y be their quo- tient, then we have V From (i), by multiplying by v, we get u=zy, (2) hence T),.u=i>D_,.j'+y'D^v, (;^) u or Tiji=vY)^y-\-~'V^^v. (^) V Multiply both sides by v and we get vJ^^u^v'^jy^y+ul^.v. (s) Transposing and dividing by z'-" we get Expressed in words this is T/ze derivative of a fraction equals the denominator into the deriv- ative of the numerator minus the numerator into the derivative of the deyiominator all divided by the square of the denominator. 24. Examples. Find the derivative with respect to x of the following ex- pressions : x^-\- 1 1. ' compare with example 5, Art. 19. X-Y I ' x^—Sx^'+iix—S ., , * 2. — - — compare with example 4, Art. ig. ■^ 3 x+j • ^' X'+l' A— 22 178 Alge^bra. 25. To FIND THE Derivative with Respect to x of a Function of Another Function of x. Suppose y is some function of 3, and .z is some function of .r, then ultimately y is a function of x, hence it has a derivative with respect to x. But as y is directly a function of z it has a derivative with re- spect to z. Moreover, as s is a function of x it has a derivative with respect to -v. We hav^e identically Jx Jz Jx ' ^ Taking the limit of each side as Jx approaches zero, remem- bering that the limit of the product of two variables equals the product of their limits, we have limit ( dy \ _ limit \ Jy ] limit \ -Jf / ^-r ; o ] jjc \ ~'^-^^ ^ o) j^ \ ' Jx^ o i jjt^ ( ' ^^ Now z being a function of x we may write 2=f(x). and if x be increased by Jx we have z-^Az^f(x-{-Jx), and from this it is evident that, as Jx approaches zero, Jz must also approach zero. Hence t |Jj.) limit |Jj.| Substitute from (t^) in (2) and we have limit \ Jy\ _ limit f Jy ] limit \ Jz } The left-hand member of (/\) is D,j'; the first factor of the right-hand member is T>~y, for it is just the same as the left-hand member except that z everywhere takes the place of x; and the second factor of the right-hand member is D, 2-. Hence T>^y=T>,y .Ti^z. (^) If y=pf and .5'=jr'-f-2, then l)^y=2z and D,z=2x. Hence by equation (s) D , y=2z . 2X= ^zx= ^x(x^ -\-2) = /\.x ' + 8x. Derivativks. 179 It is easy to see that this result is correct, for in the equation 1'=^-, substitute the value of r and we have Hence D, r=4.r^+8-r. 26, Examples. Find the derivative with respect to x of the following ex- pressions : /. (x^ + ax-hdj\ 5. ^x= + 3A--f2/-5rA-+3Ji-H-2;4-5. 6. 2(x'-Y)^-\-^(x'-ir-^(x^-l). 27. To KIND THE Derivative with Respect to x of any Positive Integral Power of x. Let y=-x". ' (ij Give to X the value x-\-Jx and we get r-hJj/^/jr-f J.rj". (2) Expanding the right-hand member of (^2 j we gel r+jv=.v'--f;/A-"-'jA-+ -''"''" 'V-Y-^t-/+ . . . +r-'-r;-. (?>) Subtract (i) from ^3^ and we get Divide both sides of ( /^) by - .r and we get Taking the limit of each side as Jx approaches zero we have D^y=nx"-\ (6) Reasoning exactly as above we could show that when ]'=,y=nax"~\ This fonnula may be expressed in words thus — The deriratiTC with respect to x of ax" is found by multiplying the exponent by the eoefficierJ and reduci^ig the exponeiit \ . i8o Algebra. It is to be noticed that this formula applies to the derivative with respect to x of a power of x. Of course any other letter be- sides X could be used to denote a variable. Thus, whenj/=«r", T>^y=na2"~\ But we must be careful not to use this formula to find the de- rivative with respect to some quantity, of a power of some other quantit3% Or, in oth,er words, in order to be able to use this form- ula the quantity which is raised to a power must be same as that with respect to which the derivative is taken. 28. "To FIND THE Derivative with Respect to x of any Negative Integral Power of x. Let j,=a-»=l (i) T).y=^^^^^^^^ by Art. 23. (2) Simplifying, remembering that the derivative of a constant is zero, we get ^^y=--^.n=-^^' ' (3) It may be objected to this method that we have used the form- ula for the derivative of a fraction whose numerator is i when that formula supposed that numerator and denominator were each functions of x. We may then take j= "„^^ and uo7t' use the formula of Art. 23 and we get as before D, r= — nx~"~\ It easily follows that if y=:ax'~", then D,^y=—nax~"~\ Here, as in Art. 27, in order to use the formula the quantity raised to a power must be the same as the one with respect to which the derivative is taken. We may express this formula in words thus — T/ie derivative ivith respect to x 0/ ax~" is found by ?nultiplyin^ the expojient by the coejficie^it and reducing the exp07ient i. Derivatives. i8i 29. I'o FIND THE Derivative with Respect to a- of a Fractional Power of x. ut z=xf (I) and let j/==^^=A-^ (2) Then T>,y-=T>,y.D,z, Art. 25. (3) But J}^y^qz'^-\ (A) hence I)^y=gz''-\ B^z. (5) But from (2) D,j'=/>a:^-', (6) hence from (^) ' and (^) qz"-'. J}_,z=px^-\ (7) Divide by qz', which is the same as qx^, and we get z-'D,z=^x-\ (^) p Multiply the left membei by z and the right member by a"'', which is the equal of z, and we obtain p -t-, T>,z—~x'f The same reasoning would show that if r=«A:^, then _ ap ^-i T>^y=^ x'' 9 Hence, as in the two preceding articles, the result is obtained by multiplying- the exponent by the coefficient a?id reducing the ex- po7ient I . 30. Examples. Find the derivative with respect to x of the following expres- sions : I. ( x-—x-\- 1 ) -\- 2( x^—x-\- 1). ix^—i] , (x^—i Vx^—\ 5. {x-\-sJ \-x^)\ l82 A1.GEBKA. (). 7. 9- JO. \/ i+.r— \/ i—x 1 1 + V I —'^" I i Jill 1 \/ a-\-A s/ a-\- sf X J;iy \ .r X' 12. (a-j-xj". (d+x)'-. CHAPTER XIV. SERIES. 1. A definition of a series was given in XII, Art. 4, and it was there noticed that infinite series are divided into the two classes of convergent and divergent. Convergent series have definite limits as the number of terms is increased without limit, but from their nature divergent series are wholly indefinite, and hence // is not safe to use divergent series or to base any reasoning upon them. In all that follows, and indeed in all that precedes, it is to be understood wherever infinite series are used that the results hold as long as all the series used or obtained are convergent. In many cases a series is convergent or divergent according to the value of some letter in the series, and it is always understood in such cases that the letter concerned is limited to those values which make the series convergent, and no inference is to be drawm for any other value. It would be fortunate if some simple and universal criterion were known whereb}- we might determine w^hether any given series is convergent or divergent, but unfortunately no such cri- terion has been found. There are, however, some cases in which we can determine whether a series is convergent or divergent and we gi\'e a few of these. 2. L-t the terms of a series be represented by u^, u^, u,, etc., in each case the subscript being the same as the number of the term; and let R^ be the remainder after the first term, R„ the re- mainder after the second term, R, the remainder after the third term, etc. ; in each case the remainder after any number of terms are taken is represented by R with a subscript equal to the num- ber of terms already taken; and further let the sum of any number of terms be represented by S with a subscript equal to the num- ber of terms taken, /. e. the sum of two terms will be represented bv S., the sum of three terms bv S,, and so on. 184 Algebra. 3. With the notation just explained, the sum of a series which has a limited number of terms will be represented by S,^ + R^, whether ^ is i or 2 or 3 or any other number not exceeding the whole number of terms of the series. In an infinite convergent series S„ approaches a limit as n in- creases without limit, and the value of this limit is S^ + R,^, where q is any positive whole number whatever. It is easy to see in this case that ^..^ o as n increases without limit. In an infinite divergent series S„ does not approach any limit neither does R,, approach any limit, and S^ + R;^ has no definite value at all. 4. It is evident that a series cannot be convergent unless after a certain number of terms are taken the successive terms grow smaller and smaller, or, in other words, unless ?/„;: o as n in- creases without limit. But while this is necessary it is not suf- cient, for a series viay be divergent and still u,,'^ o as n increases without limit. Take for example the series i + R-Hi+ . . . ■ where the nth term is — , which evidently approaches zero as n increases without limit. If this series be grouped thus : + a + TV + TV + TV + A + T'4+TV+TV)+ • ■ • then in no group is the sum less than ^ and as there are an un- limited number of groups, the series evidently does not approach any limit, but increases without limit as the number of terms in- creases without limit, therefore the series is divergent. 5. ThkorKm. a series, all of whose ter?ns are positive, is diver- gent if nu,^ does not approach zero as n increase's without limit. Since all the terms are positive, 7iii,^ is positive, and since 7111,^ does not approach zero, we may take r a quantity so near zero r that n2i,^r, then ?/„>—. n Series. 185 bimiiarilv, /^,.i> , , ?^;,^9> , , etc. «-f I "■ n-\-2 Hence ;2 ?/ 4- 1 n-\-2 But the quantities in the parenthesis form the terms of the series i-{-\ + \-\- . . . after the ?ii\i term, and this latter series has been shown to be divergent, or in other words, the quantity in the parenthesis increases without limit, ^ and therefore increases without limit ; therefore the series is divergent. 6. Theorem . If the terms of a series are alternately positive and negative and after a certain number of terms each term is numeri- cally less than the preceding one^ and the n th term approaches zero, as n increases ivithout limit, the series is convergent. Let the series be u^-—u^^-{-u^ — u^-]r . . . and let the series be represented by S ; then we may write either S = S,-f r«,,+ i— ?^^+.J+r2^./+3— 2Vf4^+ ... (t) or s=Sv,i— r^Vf'i— ?^^+:j— r^^^+4— ?wJ— . • . (2) After a certain number of terms, say k, each term is less than the preceding one, so if q be larger than k, each parenthesis in (i) and also in (2) is positive, and therefore from {1) S>Sy, and from (2) SS and SS,/ and. ^4't 2''+ 1 "^^^H 1 • Therefore S is intermediate in value between two quantities, the larger of which grows smaller and the smaller of which grows larger. 186 Algebra. Moreover, S,,, 2-i i and S^, 2 r differ by /^„4.2hi> which approaches zero as r increases ; therefore the two quantities between which S is always found approach equality as r increases. Therefore S has a definite value, or, in other words, the series is convergent. 7. Theorem. If all the terms of a series are positive and after a certain 7inmber of terms each term is less than the one before it and the limit of the nth tert?i is zero ; the?i if the limit of the ratio of the ^-fi^i^f/i tern I to the nth term is less than i the series is convergent. If all the terms are positive and after a certain number of terms each term is less than the preceding one, then, anywhere after this certain number of terms, the ratio of any term to the preced- ing one is positive and less than i. Now, since each of these ratios is less than i and the limit of the ratio is less than i, we may take some quantity, /', less than i but so near i that each ratio will be less than k. Hence ' 2^/.:i^ _ ^ ^^ See XII, Art. 8. I— A- and this is a definite quantity. Therefore the right-hand side of the last inequality is a definite multiple of u,„ and u„ ^ o ; therefore the right-hand side of the last inequality approaches zero ; and as the left side is less than the right side and -neither side can be negative, therefore the left- Series. 187 hand side of the last inequality approaches zero ; therefore the remainder after ri— i terms approaches zero : therefore the series is convergent. 8. In the theorems of the two preceding articles the student should note the force of the words ''after a certain number of terms. ' ' The first few terms of a series may not give any indica- tion as to whether the series is convergent or divergent. Take, for example, the series where the rth term is rx''~\ and suppose x=-^-^ ; then the succes- sive terms grow larger up to the ninth term, which =^3. The tenth term has the same value as the ninth, but every term after the tenth is less than the preceding one. Moreover, as n increases without limit, the // th term approaches zero and the ratio of the (7i-\- 1 jth term to the 7^th term equals i + x, and this evidently L ^^ J approaches x as a limit. Hence the series is convergent. 9. Theorem. A series is convergeyit if the series obtained by making all its terms positive is convergent. Let the limit of the sum of the positive terms be represented by Uj, and the limit of the sum of the negative terms be represented by U,; then the limit of the sum of the series will be U,— U^. Now consider a new series formed from the given series by making all its terms positive ; then the limit of the sum of this new series will be U, + U^, and as this new .series is convergent by hypothesis, U,-f U,, has a definite value. Again, as U, and U^ are both positive and as their sum has a definite value, therefore each of these quantities U^ and U^ has a definite value, therefore their difference, U^— U^, has a definite value ; therefore the .series is convergent. 10. Theorem. The series a^-\-a^x-\-a^x^-\-a^x'^-\- . . . is convergent when a'- 2^; + 3>'^r-f 3, etc., therefore Ur.^ ^ -\- u^j^ 2 + ^'-+ 3 + ^'-+ 4 + • • • approaches a definite limit as the number of terms increases with- out limit. Now as the sum of the first r terms of (i) is a definite quantity and the sum of the terms after the rth approaches a def- inite limit, it follows that the whole series U -{-11 -\-2l ^u -\- . . . approaches a definite limit as the number of terms increases with- out limit, or in other words, the series is convergent. SERIEvS. 189 12, Examples. Determine whether the following infinite series are convergent or divergent : /. 1 4- '+'..+ ',+ . . . 2 3' 4^ I" 2^ ^^ i^ !3 1 4 X X' x^ 3' 1 4-"- 4-- -4-" ,4- . . . when .i-^.ax" =T>^nax"~'=n( n— I )ax"~^ D3<7.r"= D^n(n — i )ax"~''= n(n —\)(n — 2)ax"~^ etc. Series. 191 15. Examples. /. Find 5 successive derivatives of x\ 2. Find 4 successive derivatives of x^-\-x*'-\-x^-\-y-\-x-\- 1. J. Find 3 successive derivatives of ^. Find 3 successive derivatives of \^ i-j-x. 5. Find 3 successive derivatives of V i + 2x. 6. Find 3 successive derivatives of sf \-\-x". Find 3 successive derivatives of — x 16, Theorem. In a fiindiou of a binomial x-\-/i, sayf(x-\-h), ihe derivative with respect to x, wheri h is regarded consta7it, is equal to the derivative with respect to h ivhen x is regarded consta?it. l^etv=f(x-{-hJ, and let x-\-h=2, then D, r=D, r. D,5. See XIII, Art. 25, Kq.fs) Eut D,,2'=i. Hence Dj'=D..j'. (i) Again, D/,j=D,j'. I)i,z. SeeXIII, Art. 25, Eq.(5) But T>,,z=i. Hence Df,v='D,v. (2) From (i) and (2) it follows that 17. Taylor's Formula. We are now prepared to take up Taylor's formula. If f(x + h J can be developed into a series arranged according to positive increasing powers of //, let us assume /('.v-f/^;=A,-f A// + Ay/--fA>^+ . . . (i) A\iiere A , A,, A.., etc., do not contain /;, but are in general func- tions of .r. Take the derivative with respect to x of each side of ( i ) and we have D,J(x-\-h)==D,,Aj-hB,,A^-^lrD,A^ + h'D,A^+ . . . (2) Also take the derivative with respect to // of each side of (i) and we have D„/rr+//; = A, + 2A//-f 3A/r + 4A/^+ . . . (3) 192 Algebra. By Art. 16 these two expressions must be equal, therefore equating coefficients of like powers in (2) and (3) we get A=DA (4) 2A^=D,A, . ■. A =,VD,A =-D?A„ (5) 3A=D,A, . •. A=iD,A=_'_D.?A. ■ (6) 4A=D.,A, . •. A =iD.A.= ^^D;^A ^ ' ■ ^ 2.3.4 (1) 5A=DA • •. A.=iD,A= ^ D^A ^ ' 2.3,4.5 ■ ' etc. (8) NowMf we make h = o in (i) it is easy to see that X^=f(x), w\\.&r^f(x) means the same function of ,r that the given function is of .r+/?, or in other words, f(x) is what the given function be- comes when h is put equal to zero. If, in (i), we substitute /(^xj for A., and for the coefficients of the various powers of// the values found in equations C4) to (8), we get f(x^-h)==f(x) + hJ}jX^x) + ~^''^I)lf(x)-^ ^'lj)-^f(x)-^ This result is Taylor's formula and is often spoken of as Ta}'- lor's theorem. 18. AppIvICATion of Taylor's Formula. Let us develope (x-\-hf by Taylor's formula. Here f(x-\-h) = (x-\-h)\ Therefore f(x)=x\ Finding the successive derivatives of x^ w^e get TiJ(x) = 6x\ T}lf(x) = 6 . 5Jr^ mf(x)==6.SA^\ Dyr-^; = 6.5.4.3-x-=, Dj/f.^;=6.5.4.3.2Jt^ D«/r-^; = 6.5.4.3.2.i, and every derivative after the seventh will equal zero. Therefore by substitution in Taylor's formula we get (x-\-h)'=x'+6xyi + ^ ' ^x'Jr + ^'~^'^x%^ + 6,5.4.3^.^,^6,^4,3:2^^,^6.5^,2^,^ 2.3.4 2.3.4.5 2.3.4.5.6 or (x^Iif=x^-\-6x^h-^iSx'h'-\-20xyp-{-iSxVi' + 6xh''-^x\ Series. 193 This result is seen to be the same as that obtained by a direct application of the Binomial formula, which of course is as it ought to be. As a second example, let us develope ^ x-\-/i by Taylor's formula. Here f(x+h)=:(x-^h)^ . 1 Therefore f(x)=x'^. Finding the successive derivatives of x^ we get etc. Making the substitutions in Taylor's formula we get j_ _x _ii _A —L s/x-^h=x--\-\x '^h—\x Vz^+yV-^ V/^— yfg.r V/'-h . . . If in this equation we make jr= i we get If in this equation we change the sign of h we get Compare this with the result obtained -in XII, Art. 12. It was stated in Art. 13 that Taylor's formula could be used to develope a7iy function of a binomial which is capable of being de- veloped into a series arranged according to positive increasing powers of one of the quantities. It is indeed a matter of substi- tution, but care must be taken that the substitution be such that the development obtained is arranged according to positive in- creasing powers of the proper quantity. If, for example, we wish to develope V^'+i into a series ar- ranged according to positive increasing powers of .r, it might at first appear that, in the development of sf x-\-h, we could simph' make h=i ; but this would give us a series arranged according to positive increasing powers of i, not x. The proper course is as follows : First, develope V x-\-h according to positive increasing powers of h ; then in this result make .i = i and we have the de- velopment oi \/i-\-h arranged according to positiv^e increasing powers of//; then, finally, in this result, change // into .i* and we obtain the result sought. V 194 Algebra. 19. Binomial Theorem for any Exponent. Let us apply Taylor's formula to the development of (x-\-h)" according to positive increasing powers of //, where n is either positive or negative, iyitegral or fractional. Here, then, f(x^h) = (x^h)\ Therefore f(x)=^x". Finding the successive derivatives of x" we obtain jyJ(x) = nx"-\ T)lf(x) = 7i(?i-i)x"-% J)lf(x) = n(?i-i)(n-2)x"-\ B^/(x) = n(n-i)(n-2)(n-3)x"-' etc. Therefore by substituting in Taylor's formrfla we get _^^^,Xn-22(n-3)^„_^^^,^ ... (I) li Thus we arrive at the Binomial formula, where, however, the exponent is not restricted, as in Chapter X^, to being a positive whole number. From this we see that the Binomial formula in its greatest gen- eralit}^ is a special case of Taylor's formula. The series (i) will be finite if ?i is a positive whole number, but not othenvise. When the series (i) is infinite it should be examined to see whether it is convergent or divergent, for values mar be given to X, //, n, which will render the equation (i) untrue. For example, let x=i, /^ = — 3, and ;/= — 2 ; then the left-hand member of (i) becomes (1 — 3)"^ which equals (— 2)"^ which equals -, which equals \, a definite quantity. But the right-hand member of (i) becomes 1+6 + 27+108 + . . a sum of positive whole numbers each greater than the one be- fore it, and evidently the sum does not approach |. 20. Examples. 1. Develope (^i— AJ~^ by Taylor's formula. 2. Develope (i—x)~'^ by the Binomial formula. Compare f ij and (2') with XII, Art. 8. SERIEvS. 195 J. Develope (i-\-xj^ by Taylor's formula. 4. Develope (i-\-x)'^ by the Binomial formula. Compare (t^) and (^4^ with XII, Art. 13, Ex. 3. 5. Develope (^1 — 2^-/'^ by the Binomial formula. <5. Develope (a~'2x) '' by the Binomial formula, -i 7. Develope (c^-^.x'' ) "^ according to positiv^e increasing powers of x by Taylor's formula. 8. Develope (c^-\-x'') '^ according to positive increasing powers of c by Taylor's formula. -}- g. Develope (c^-\-x^) '^ according to positive increasing powers of X by the Binomial formula. _i 10. Develope (c^-\-x^) ^ according to positive increasing powers of c by the Binomial formula. 11. Find the first negative term in the development of (i-\-x) '^ by the Binomial formula. CHAPTER XV. LOGARITHMS. I. After the extension of the theory of indices in Chapter XI so as to embrace incommensurable exponents, we are enabled to give an interpretation to the expression a"" for all possible values of x, integral or fractional, commensurable or incommensurable. Since x appears in this expression in such an unrestricted form it is common to speak of the expression as an exponential function of x, intending to call attention thereby to the fact that X may be considered a continuous variable as in any ordinary algebraic function. If in the equation we assume x to pass from one extreme of the algebraic scale to the other, taking in every possible value, then we are able to give a meaning to this equation in two variable ; because for every pos- sible value of X, «', that is, r, has a definite meaning and value. In this connection it must be remembered that we are using a and a' under the restrictions mentioned in XI, Art, 15. So that when 7ve speak of a'' we mean that a is a positive number, and by the value of a"" we mean that one of its values which is positive. Hence in the equation «"'=_>' we are to think of but one value of y re- sulting when any particular value is assigned to x. Thus in io*^5=y ^rg are to understand i'= + ^^ 10 and not r= — >^io or any other possible value of y. Of course the very restrictions just mentioned prevent y from having a 7iegative value. Moreover, it is net evident that y can have every positive value we please. For example, is it not plain that a value of x exists which satisfies the equation 10* = -. In general, while it is easy to see that in the equation there always exists a value of y for any value assigned to x, it is far from evident that there exists a value of x corresponding to every value which may be assigned to r. Whence the necessity for the following theorems. / Logarithms. 197 2. Theorp:m. 77/6' expression w can be made to differ from i by less than any assigned quayitity if x be sufficiently increased. Suppose it be required to increase x so that a^ — y- . d 1 Hence to make a' less than i+a^take «— I 1 Example: Find .v such that 10' , or Qoooo. .0001 3. Theorem, The expression a' is a contijinous foinction of x. Suppose a^'-^y and let x take on any increase, s, and suppose the corresponding value oi y bej' + /, so that We are to prove that as x passes continuously from x to x-\*s that y passes continuously from y to y-\-t; that is, as x changes from X to x-\-s by passing over every intermediate value that y changes from r \.o y-\-t by passing over every intermediate value. The equation a'''=y-\-t may be written a"a'=y-\-t, ^ (2) and .since «'^=j', this may be written a^a'=a^-\-t. (7,) or a'-'(a'—i) = t. (4.) 198 Algebra. Now, by the last theorem, by taking .s^ small enough a' may be made to differ from i by an amount as small as we please. Hence in equation (^) t may be made as small as we please by taking .9 small enough. That is, the difference between two successive values of a"-' can be made as small as we please. Therefore it is a continuous function of x. 4. It follows directly from the above theorem that for every positive value ivhich may be assigned to y iji the equation a'==y, a correspo7iding value of x exists zvhieh will satisfy the equation. For the last article shows that as x is increased continuoush' from the value o without limit that r increases continuously from the value i w^ithout limit. That is, y may have every value greater than i . It is also seen that as x is decreavsed continuously from the value o without limit that r decreases continuously from the value i . That is, y may have every fractional value. The above shows that if an}^ value be assigned to j' in the equa- tion a''=y that a value of .v exists which will satisfy it, but it does not explain how to find that value. Thus it does not show how to find X in the equation 10' = 5. The method of finding this will be explained later. 5. DefinitioNvS. In the equation a'—y, where a is some chosen positive number not i ; The constant quantity a is called the Base. The quantity^ is called the Exporientia! of .v to the base a. The quantity x is called the Logarithm of y to the base a, and is written x—Xogay- The use of the w^ord logarithm may be kept in mind by remem- bering this sentence : In the equation a'=^y, x is called the Ex- ponent of the power of a or the Logarithm of y. Of course the two equations a'^y (i) .r=logaj' (2) express the same truth respecting the relation between x and y. The second equation uses the logarithmic notation and is always to be interpreted by means of the first equation. If in the equation a'=y, where a is some positive number not I, different values be assigned to v and the corresponding values Logarithms. 199 of .V be computed and tabulated, the results constitute a System of Logarithms. Since any positive value except i may be chosen for the base a, the number of different possible systems of logarithms is un- limited. In fact, however, only two systems have ever been tabulated ; the Natural or Napcrian or Hyperbolic Syste?n, whose base is approximately 2.7182818 + , and the Common or Briggs' System, whose base is 10. The Naperian logarithms of all numbers from i to 20,000 have been computed to 27 places of decimals. The common logar- ithms of all numbers from i to over 200,000 have been found. They are usually printed to seven decimal places, but they have been computed to many more. The great value of a table of logarithms is the immense amount of labor which can be saved by its use in multiplication, division, evolution, or involution of numbers, as will be explained here- after. 5. Exa:\iplEvS. Write the following equations, using the log- arithmic notation : I. 10* = -. ^• io--^= 1.77828-1- 2. t''==1'. 9' a""'=!=a"a''. J. 11''= 121. JO. a' = a. 4. 1 0'= 1000. II. ^^^^-''=r. 5. 1 6 --^=2. 12. J^'^-'^y. 6. io'=i. 13- eJ'=a. 7. io~^=.ooi. ^4- I Q .30:030^ 2. i^xpress the following, using the exponential notation : ^5- log^;(i)=-.3333 + 19- logjo24= 10. 16. log, 4=. 602060 20. loge r= I . ^7- log,,ioooo=4. 21. log/,^'^ = ^. 18. logj^.ooooi== — 5. 22. \ogoa = B. 6. PkoitErties of Logarithms. Inasmuch as logarithms are merely the exponents of a fixed base, the properties of logar- 200 Algebra. ithms are entirely dependent upon the properties of exponents in general, which have already been established. Among the fundamental properties of logarithms are these : The logarithm in any system of the base itself is i . For a^—a, that is, loga«=i. The logaritlun of U7iity in all systems is o. For <2"=i, that is, logai=o. Negative numbers have 7io logarithfns. For in the equation a' =j', a is positive by supposition and by the value of «' we mean that one of its values which is positive. Hence j^ cannot be negative. See Art. i. If we understand the same S3^stem of logarithms to be used throughout, then the following four theorems hold. 7. Theorem. The logarithm of the product of several minibos equals the sujn of the logay ithms of the separate Jaetors. Let n and r be any two positive numbers and let log;,7?=A- and loga;^=.2'. fi) Then, by the definition of a logarithm, 71= a' and r=a~. Multiplying these equations together, member by member, . nr=a'''^~. That is, \ogan7'=x-^s, or, from ("i;, log;,;^r=log,.,;^^-log,,r. (a) In the same way, if log;, 5=?/, then 7irs=a''"^'''^'". That is, logn /zr^= log,, n + log,^ r+ log« .?. 8. Theorem. The logarithn of the quotie7it of two 7iii7nbe7's equals the loga7'ith77i of the divid-end minus the logarithm of the divisor. Let 71 and r be any two positive numbers and let log,, 11= X and logar=z. (i) Then, by definition, ^^— <2'and 7'=a\ Logarithms. 201 Consequently -= — =^' r a^ Therefore, by definition, or, by equation (\), log, n r =logfl«— logar. (b) 9. Theorem. The logarithm of a powei^ of a number is equal to the logarithm of the number multiplied by the exp07ient of the power. Let 71 be any number, and let \o-r. That is, loga;z^=/ loga;2. (c) _ 10. Theorem. The logarithm of any root of a number is equal to the logarithm of the riumber divided by the index of the root. Let n be any number, and let \Q%an---x. Then, by definition, n = « ". Consequently 's/n—a^. Therefore, by definition, log„(V»)=:^. That is, log„(V„)='°?«". (d) 1 1 . Theorem. If several numbers are in geometrical progression!^ their logarithms are in anithmetical progressioii. Let the numbers which are in geometrical progression be rep- resented by 71, 7ir, 7ir^, nr^, . . . Then their logarithms to the base a form the series loga n , \ogn n -f log,, r, log,, n + 2 log,, r, log,, w -}- 3 log;^ r, . . . which is an arithmetical progression with the common difference equal to log«r. A— 25 202 Algebra. 12. Examples. In these examples and in all the following pages the Common LogaiitJun is designated by the symbol log instead of log,^. Hence when no subscript appears we are to un- derstand that the base is lo. 1. log (1888 X476-M492)=log 1888 -flog 476— log 1492. 2. log [V789X_(|ftf)^=ilog 789 + 5 log 239-5 log 930. 3. log i--^^^ =what? L ^ ^^ J 4. log/,(<:V^-^//i';/2)=what? 5. logzJ(/^^^^^^^) = what? /• log^>J":w,^=wbat? 8. log/, ^^= what? 9. Prove loga(loghi5'') = loga^. 10. Prove loga^= logert' 13. Characteristic and Mantissa. For reasons which will appear later the common logarithm of a number is always writ- ten so that it shall consist of a positive decimal part less than i and an integral part which may be either positive or negative. Thus the common logarithm of .0256 is really —i. 591 76, since ■°""""-ro-W«='o256. 59176 But instead of writing log.o256= — 1.59176 we write the equivalent equation log .0256= — 2 -f. 40824, or, as is the universal custom, with the minus sign over the 2, log .0256=2.40824 The minus sign over the 2 shows that 2 is alone affected ; that is, the decimal fraction following it is positive. The student must always take especial car e to coi'rectly iyiterpret this method ofyiotation. When the logarithm of a number is arranged so that it consists of a positive decimal part less than i and an integral part either Logarithms. 203 positive or negative, special names are given to each part. The positive or negative integral part is called the Chai^aderistic of the logarithm. The positive decimal part is called the Maiitissa. 14. The following table is self-explanatory: ic* =10000, whence log 10000=4 10'' =1000, " log 1000=3 10- =100, " log 100=2 10' =10, " log 10=1 10" =1, " log 1=0 io~'=.i, " log .1 = — I io~-=.oi, " log .01 = — 2 io~^=.ooT, " log .001 = — 3 io~'*=.oooi, " log .0001 = — 4 Here we observe that as the numbers pass through the series loooo, 1000, 100, 10, etc., the logarithms pass through the series 4, 3, 2, I, etc.; that is, continuous division of the number by 10 corresponds to a continuous subtraction of i from its logarithm. This can easily be shown to hold in any case. 15. Theorem. Multiplyirig any ymmbcr by 10 increases the com- mon logarithm by- i, and dividing any number by 10 decreases its common logai-ithm by i. Let r be any number and x its common logarithm. Then logj'=jf, or io^=j'. We are to prove log iov=.r-fi, and log T J' = -V — I . By formula (a). Art. 7, log ioj'=log j'.+ log 10. But log 10=1 (Art. 6) and log i'=a-. Hence, substituting, log ioj'=-t-j-i. Also by formula (b). Art. 8, That is, log iVj'=-^ — T- 204 Algkbra. 16. Corollary. Moving the decimal point iyi a numbei' one place to the right increases its common logarithm by i, and moving it one place to the left decreases its logarithm by i. 17. Corollary. The common logarithms of all 7iumbers con: sisting of the same significant fgnres have the same. ma?itissa. For moving the decimal point merely adds or subtracts i from the logarithm ; that is, merely affects the characteristic. Thus log 256=2.40824 log 25.6=1.40824 log 2.56=0.40824 log .256=1.40824 log .0256=2.40824 log .00256=3.40824 18. Theorem. If a member has its frst significant figure in units' place, the characteristic of its common logarithtn is o. If the number has its first significant figure in units' place, the value of the number must lie somewhere between i and 10. But the logarithm of i is o and the logarithm of 10 is i. Hence the logarithm of the proposed number must lie somewhere between o and I. (Art. 3.) That is, its characteristic must be o. Thus log 2.56=0.4082400 and log 9-99=o-9995655 19. Theorem. The characteristic of the conwiori logarithm of a number eguals the mimber of places the first significant figiire of the number is removed from units' place, and is positive if the first sig- nificayit figure sta?ids to the left of units' place and is negative if it stands to the right of units' place. By the previous article, if the first significant figure stands in units place the characteristic, is o. If the first significant figure stands in the ni\i place to the left of units place, then the char- acteristic of its logarithm must be a number such that it can be made from o by adding i to it n times. (Art. 15.) In other words, the characteristic must be n. If the first significant figure of the given number stands in the ;^th place to the right of units place, then the characteristic of its Logarithms. 205 logarithm must be a number such that it can be made by sub- tracting I from o n times ; that is, it must be —n. 20. Examples. The above enables us to tell by inspection the characteristic of the common logarithm of any number. Thus the characteristic of the logarithm of 237945.834 is +5, because 2, the first significant figure, stands in the fifth place to the left of units place. In the same way, the characteristic of the log- arithm of .0007423 is —4, because 7 stands in the fourth place to the right of units place. In determining the characteristic, care must be taken that we count from the units place and not from, the decimal point ; for the decimal point stands to the right side of units place. Find the characteristic of the logarithms of the following num- bers : T. 1888. 1 19 5. 3000.0303 ^. .3724 6. .00000000849 J. 783294.009 7. .00010000849 5 6G (>7 68 69 25 70 71 72 78 74 75 76 77 78 79 25 80 81 82 88 84 85 86 87 88 89 25 90 91 92 93 94 95 96 97 98 99 26 00 01 02 03 04 05 06 07 08 09 Num. 3 408 2400:2569 40964265 5791 5961 7486 7656 t 9180 9350| 409 0874 1043 2567 2736' 42594428 5950:6119 7641:7810 2739 4435 6130 7825 9519 1212 2905 4 597 6288 7979 t 9331 950ul9669 410 1021 1190 1359 2710 28788047 4808 45(57 4785 6085 6254 1 6 423 7772i7941i8nO t 9159 9627 979() 411 1144 1313 1481 2829 :^998 8166 4518 4682 4850 619716865 6534 7880 8048 8217 t 9562 97819899 412 124411412 I58u 29i;5 8098|3261 46O5I4773U94I 6285 '6458 6621 7961 8182 88U0 + 9648i9811 9978 413 1321 '1488 1656 2909 3078 3248 3417 3587 4604 4774 4944 51 13 '5283 6300|6469 6639 6808 6978 7994 8 164 18338 8503 8672 9688 985cS* 027 1*196 * 366 3757 5452 7147 8841 *535 J382 155111720^1889 2059:2228 8074 3243 8413 3582 3751 8920 4766 4985 5105j5274 5448 5612 6458 6627 6796 6965 7184 i7808 8148 8317 8486 8655 8824 8998 9838 *007 176*845*5141*683 1527 1696 18(35:2034 2203 3216 3885 8554 3723 8891 4904 5078 5242l54i0 5579 6592 6760 6929 7098 7266 8278 8447 8616 9964 * 188 *3()1 1»)50 1818 1987 8384 85Uh!8671 5019 15187 5355 6702|6870 8885' 8558 *()67 *285 1748^1917 8429 8597 5109:5277 5445 6789 «)957 7125 7039 8721 * 403 2085 8765 8468 8636 *14U *;U4 1824 1991 299813165 3333 3501 3668 4674!4842i5(M)9 5177 5845 6850 ti5 18 668516858 7U20 8025 8193!8360!8528 «695 9700:9867 *085 *202 *369 414 1374 3047 4719 6891 806ci f 9733 415 1404 3073 4742 6410 8077 t 9744 416 1410 3076 4741 1541 1708 821 41338 1 4887 5054 655916726 8230; 8897 1876 8549 5221 6898 8564 9901 * 068 *235 1570 1737|1904 3240 34<»7:8574 4909 5075 '5242 6577 6743 '69 10 8244 9911 1577 3242 49u7 841118577 *077|*244; 1743 '19 10 8409 3575 5074 5240 2043 871 6 5388 /060 8731 *402 2071 3741 54U9 7077 8744 *4ll 2077 8742 5407 8804 *482 2159 8784 *4ro 2155 3840 5524 7207 8890 *571 2253 8938 5613 7298 8971 *649 2327 8958 *639 2;:l24 4008 5692 7375 9058 *740 2421 4101 5781 7461 9139 *817 2495 3836 4004 4171 5512 5680 5847 7188 8868 *537 2210 8883 5556 7227 889S 7855 7528 2872 4060 5748 7485 9121 *807 2492 4177 5860 7544 9226 *9()8 2589 1269 5949 7629 9807 *985 2662 3926 5622 7817 9011 *704 2397 4089 5781 7472 9102 *852 2541 4229 5917 7604 9290 *976 2661 4 34 5 6029 7712 9894 1076 2757 4487 6117 7796 9475 1153 2830 Diff. j Dilf, A: Mul. 4339 4507 6015 6182 7690|7858 9030 9197 9865 9582 *704*872'l039'l206 2378 2545 4051 42l« 5723 5890 7894i7561 9065 9282 * 569 * 786 *903 2 238 2405 2572 3907 4074 4 241 557615743 5909 7244i7410 7577 I I 1 8911 *577 2248 8908 9077 9244 *744*911 2410 2576 40754241 2712'2S80 4385 4552 6057 6224 77>9 7896 9899 9566 1070 1237 2739 2906 4408 4575 607(5 (3248 7744 7911 9411 9577 1077 1244 2743 2909 440.S 4574 6072 6239 167 1 169 2 338 3 507 4 676 5 845 () 1014 7 1183 8 1352 9 1521 1 168 2 33(> 3 504 4 672 840 6 1008 7 1176 8 1844 9 1512 1 167 2 384 3 501 4 668 5 835 (> 1(J02 7 1169 8 1836 9 1 ^^^y.^ 3 9 Diff. Diff. & Mul. Logarithms. 207 page on the same horizontal line with 2596 until we come to the column headed 4, at which place will be found the figures 3716, which are the last four figures of the required mantissa. The first three figures are found in the column headed o and are seen to be 414. Whence the mantissa of the logarithm of 25964 is .4143716, and therefore log 25964=4.4143716. Of course a decimal point belongs before the mantissa of each logarithm, and since this fact is understood, it is unnecessary to print the decimal points in a table. Inasmuch as the first three figures of the mantissa are only printed in the column headed o, it is necessary to mark the point at which these first three figures change. This is done by an asterisk ( *) standing in the place of a cipher in the last four fig- ures. Thus to find the logarithm of 25646 we must note that the three figures change from 408 to 409 at the point 25645 (which is indicated by printing ^027 in place of 0027), and consequently log 25646=4.4090196. The dagger (t) which appears in column o is intended to cau- tion us that the first three figures change at some place in the same horizontal line with it. If we wish to find the logarithm of a number consisting of more than five figures, say 25705.84, then we must take the nearest number whose logarithm is given in the table, that is to say, 25706.00. Thus log 25705.84=4.4100345, nearly. Greater accuracy may be secured by means of tables of dif- ferences and multiples, as is explained in connection with any good table of logarithms. A table of logarithms of numbers from i to 1 00000 can be used to find the logarithm of any number consisting of five significant figures. Thus to find the logarithm of 25.964 we entirely neglect the decimal point- in finding the mantissa, as the decimal point affects the characteristic alone. Thus log 25.964=1.4143716. A table of logarithms also enables us to find the number corre- sponding to any given logarithm by a mere reversal of the process already explained. Thus all numbers the mantissas of whose 2o8 Algebra. logarithms lie between .4082400 and .4166239 are on the speci- men page we give. Suppose, as an example, that w^e wish to find the number corresponding to the logarithm 2.4127469. The characteristic merely affects the decimal point, and consequently the problem is merely to find the significant figures which cor- respond to the given mantissa. The nearest mantissa printed in the table is .4127461 and this corresponds to the figures 25867. Hence, pointing oif the number by means of the characteristic, we find that the number whose logarithm is 2.4127469 is 258.67. In connection with tables of logarithms methods are explained by means of which more figures of this number could be found by means of tables of multiples and differences, or of proportional parts. 22. KXAMPLKS. 1. Find the logarithm of 25734. 2. Find the logarithm of 26000000. J. Find the logarithm of 25.999 /. Find the logarithm of .02578411 5. Find the logarithm of .260099 6. Find the number whose logarithm is 3.4147561 7. Find the number whose logarithm is 0.4104400 8. Find the number whose logarithm is 2.415999 g. Find the number Avhose logarithm is 1.4094094 10. Find the number whose logarithm is 7.4100000 23. Multiplication by Logarithms. Formula {a) (Art. 7) enables us to find the product of several numbers by means of a table of logarithms. Thus, suppose we wish the product of 98 by 265. From a table of logarithms we find log 98= 1.9912261 log 265= 2.4232459 log 98 X 265=4.4144720 From the table of logarithms (see sample page) it is found that 4.4144720 is the logarithm of 25970. Therefore 98 X 265 = 25970. Logarithms. 209 24. ExAMPi^ES IN Multiplication. /. Log 327.45=2.5151450 and log 79.493=1.9003839; find the product of 327.45 x 79.493. 2. Log .53927=1.7318063 and log 4.7655=0.6781085: find the product of .53927 X 4.7655. J. Log 6.3274=0.8012253 and log 1645.6=3.2163243 ; find the product of 6.3274 x 1645.6. 25. Examples in Division. See Art. 8. 1. Find the quotient of 327.45 by 1645.6. From a table of logarithms we find iQg 327.45= 2.5151450 log 1645.6= 3.216 3243 log 327.45^1645.6=1.2988207 It is seen from a table of logarithms that the number corre- sponding to the logarithm 1.2988207 is .19901+ Therefore 327:45 -T- 1645.6= . 19901 + 2. Log 53.927=1.7318063 and log 2.0724=0.3164736; find the quotient of 53.927-^2.0724. 3. Log 33333=4.5228744 and log 13001 = 4.1139768; find the value of fetf ^. Log 54321 = 4.7349678 and log 20.877 = 1.3196681 ; find the value of 54321^20.877. 26. Examples in Involution. See Art. 9. 1. Find the third power of 1373.3. From a table of logarithms we find that log i373-3=3-t377654 3 whence log (1373.3)^=9.4132962 From the sample page it is seen that the number whose logar- ithm is 9.4132962 equals 2590000000, nearly. Therefore ■^ (1373-3)^=2590000000, nearly. 2. Find the fifth power of 1.9201, whose logarithm is 0.2833238. A-2(J' 2IO Algebra. J. Find the tenth power of .69353, whose logarithm is 1. 8410653. /. Find the seventh power of 15.926, whose logarithm is 1. 2021067. 27. ExAMPLKS IN Evolution. See Art. 10. /. Find the cube root of 26. From a table of logarithms we find log 2^=1.4141374 Therefore log x^^26=o.47i379i The number whose logarithm is 0.471 3791 is found to be 2.9606 + Hence ^26=2.9606-!- 2. Find the square root of 668.63, whose logarithm is 2.8251859. J. Find the fifth root of 11 09600000000, whose logarithm is 12.0451664. ^. Find the tenth root of 1.384, whose logarithm is 0.1411675. 28. Exponential Series. The Exponential Series, or the Exponential Theorem, as it is often called, is an expansion of ^' in terms of the ascending powders of x. The following demonstra- tion* of this important theorem is due to Mr. J. M. Schaeberle, of the Lick Observatory, and is inserted here with his pennission. We are required to expand a"" in a series of ascending powers of X. Assume «^ = A + B.r+C-r^ + D-x-3 + E-:i-'+ ... (i) where A, B, C, etc., are undetermined coefiicients. The limit of the left-hand side of this equation as x approaches o is plainly i. The limit of the right-hand side of this equation as X approaches o is A. (See XI, Art. 26.) Therefore A=i. Substituting this value of A in (\) and then squaring both members, ij ^^^-*•=I-f2B.^--f(2C-|-B0-^"+(2D-f 2CB).r3 4-(2E + 2DB + 0'»+ . . . (2) *See Annuls of Mathematics, Vol. Ill, p. 15J. < Logarithms. 211 But if we substitute 2X in place of x in equation (i) we obtain Therefore, equating like powers of jr in equations (2) and (2,)y we obtain B= B^ B^ B=B; C= ; D= - ; E= ; etc. Whence, on substituting these values of B, C, D, etc., equation (\) becomes |_2 1 3 1 4 Now, there must exist some quantity, c, at present unknown in value, such that c^'=^^ (5) or, m other words, such that loge«=B. (6) Substituting loge^a- for B throughout equation (/[) we obtain a^=i4-^log.. + ":^^^^^^%-:^^^^+^^^^^ . . (7) \3 [3 |4 which is called the Expo7icntial Theorem or Series. 29. To FIND THE Value of the Base e. The base a in the last article is any chosen positive quantity not i, and its value is therefore at our disposal. Hence in the exponential series (equa- tion 7) we may put a==e, $0 that loge^ becomes loge^ ; that is, i. Equation (-] ) then becomes X^ X^ X* ^='+-+L.-+i:3 + |4 + --- ^'^ This important result is convergent for all values of .v, (see XIV, Art. 12, Ex. 13,) and consequently the equation is true when x= i . Therefore we have .= : + :+ |>|i + ,;+... (.) By taking a sufficient number of terms of this series we may approximate the value of e to any desired degree of accuracy. Thirteen terms of the series give ten places of decimals correctly and we h^ve ^=2.7182818284 ... (2,) 212 AI.GEBRA. This number is one of the most important constants in mathe- matics. It is called the Naperia7i Base and is always represented by the letter e. Its value is known to more than 260 decimal places. 30. Logarithmic Series. The Logarithmic Series is the ex- pansion of logeri+-^^ in terms of the ascending powers oi x. From the exponential series .-'==i + rlog..+-^^^^7^^^-V^^^^^ . .(.) . Whence, transposing the i and dividing through byj', -^-^-=log..-fr-i^^^V-^-^^^^^^^^ ... I r^' y ( h 1 3 S Therefore, since these variables are always equal, , limit (a-'-i, _limit \ , r(log£fl:+-y(l°S?fI'+ '\\ / .,) Whence it is easy to see ' limit f^' — I) , , . Now put I -f-^t: for <2, then we have . ,, , limit {(i-^xy-i-) logeri+x;-,, : oj — -y — j- Expanding (\-\-xy' by binomial formula, .J ' o( 1 . 2 1.2.3 } rs; The limit of the right member as r ]: o can be plainly seen ; whence we obtain the equation iog.rn-x;=x--V---^+ ... r6; 234 This is the Logarithmic Se7'ies. 31. CoNVERGENCY OF THE SERIES. The above seri,f;s is not convergent for values of x greater than i , and hence cannot be used for computing the logarithm of any integral number but 2. The following scheme will give a series which is available for computing the logarithms of all integers. 1.2.3.4 IvOGARITHMS. 213 32, A Logarithmic Series Convergent for Integral. \'ai.uks of .v. In the logarithmic series 234 Substitute —x for x and we shall have X^ X^ JT* \oge(i—^')=—JC — —.. . (2) Subtracting (2) from (i ), observing that \oge( ^ -\-x)—\oge(i—^) = loo:e , we obtain I —X loge.'-"^-^"=2U- + V+'-^^H-'A'^-f- ... I (Z) i--^' 13 5 7 J I 2Z-\-2 2Z Now put A = — -— , whence i4--i'= — ; — , i— -r= , and 2,3 -f I 23- -f- I 22-\-\ " = ". Therefore we obtain I— .V Z- Whence, since loge /^=loge(^i-f^y)— loge^-, by substituting and transposing logeS" we have This series converges rapidly for integral values of z. Its use in computing the logarithms of numbers will now be explained. 33. To Coinipute the Naperian Logarithms of Numbers. The logarithm of i is o in all S3'stems. To compute loge2, put 3=1 in equation (^) above. We then obtain Now put 2 = 2 in equation (^). Then we have loge3=. 6931472 + 2 " + --S+ '+-'--+ '-^+. =1.0986123 5 3-5' 5-5^ 7-5' 9-5' J To find log(>4 we know loge4=loge2^=2 loge2 ; whence loge4 = i-3862944 "H ,4- ,+-'"-+ . . . ! = 1.6094379. 9 3-9' 5-9-^ 7-9' J 214 Algebra. To find logt>5, put ^=4 in equation (5). We then have loge5=i-3862944+2 In like manner the logarithms of all numbers may be found. The logarithms of composite numbers need not be computed by the series, since the logarithm of any composite number can be found by adding the logarithms of its component factors. 34. REiyATioN Between the IvOGarithms of the Same Number in Different Systems. Consider the systems whose bases are a and e. Then if 71 is any number, we wish to find the relation between logon and \og,.,n. Let x=loge?? and r=log;,;z. Then 71= e"" and 71 = a''; whence ^' = «-'. (i) Therefore a^e-''. (2) If we write this in logarithmic notation we have loge«=-, (2,} or, substituting the values of x and y, we obtain Therefore \ogan=- ■ -\ogen, (^) loge« which is the relation between log..,w and \ogen. 35. Modulus of Common Logarithms. If in equation (^)' above we understand e to represent the Naperian base and a the common base, then equation ( ^) becomes ^^^^^=iog;io^^^^^^- ^'^ But Iogeio=loge2 + loge5 = (by Art. 33) 2.3025851 and - log e I O = .43429448 Therefore representing .43429448 by M we have log 71=-1A \oge7i. (2) The decimal represented by M is known to 282 decimal places. and is called the Modulus of the system of common logarithms. Logarithms. 215 Equation (^2J is seen to^ express the important truth that the common logarithm of any niunber can be obtained by multiplyini^- the Naperian logarithin of that number by the modulus of the com- mo7i system. 36. Computation of Common Logarithms. We can now compute the common logarithms of numbers. We merely need to multipl}^ each of the Naperian logarithms already found by the modulus ,43429448 . . In this manner we find log 2=0.3010300 log 3 = 0.4771213 log 4=0.6020600 log 5 = 0.6989700 etc. etc. How can you find- log 6 ? 37, Historic aIj Note. The invention of logaritliins is regarded as one of the greatest discoveries in mathematical science. The honor of the inven- tion as well as of the construction of the .^rst logarithmic table belongs to a Scotchman, John Napier (1550-1617), baron of Merchiston. His first work, Mirifici logarifhmorum canonis descriptio, appeared in 1(514 and contained an account of the nature of logarithms (from his standpoint) and a table of natural sines and their logarithms to seven or eight figures.\But Napier's logarithms were not the same as those now called Naperian logarithms. The base of his system was not e, although closely related to it. Henry Briggs, professor of geometry at Gresham College, London, was much interested in Napier's invention and in lO 15 visited Napier and suggested to him the advantages of a system of logarithms in which the logarithm of 1 should be and the logarithm of 10 should be 1. Napier, having already thought of the change, gave Briggs every encouragement to compute a system of the new logarithms and made many important suggestions, among which was that of keeping the mantissas of all logarithms positive by using negative char- acteristics. In 1()17 Briggs published the common logaiithms of the first 1000 numbers, the book being QaWtid. Logariihmornm chilias priim. Briggs con- tinued to labor at the calculation of logarithms, and in 1624 published his ArHhmetica Logarithmiea, which conlaimMl the logarithms of the numbers from 1 to 20000 and from 90000 to 100000 to 14 places of decimals. This gap between 20000 and 90000 was filled up by Adrian Vlacq, who published in 162h the logarithms of the numbers from 1 to 100000 to ten places. Vlacq's table is the source from which nearly all the tables have been derived which havo ^subsequently been published. 2l6 Al^GEBRA. The moaning of logarithms to Napier and Briggswas enth-ely differoiit from that we now have. They never thouglit of connecting logarithms with the idea of an exponent, and consequently had no conception of what we call the base of the system. Their idea of logarithm is contained in the meaning of the term itself, which comes from two Greek words meaning the number of the ratios. This idea of a logarithm is thus explained: Suppose the ratio of 1 to 10 be divided into a large number of equal ratios (or factors), say 1000000. Then it is true that the ratio of 1 to 2 is composed of 301030 of these equal ratios (or factors), and 301030, the number of the ratios, is the logarithm of 2. In the same way the ratio of 1 to 3 is composed of 477121 of these equal ratios (or factors), and the logarithm of 3 is hence said to be 477121. The first methods used for computing logarithms were very tedious . The great work of computing was finished long before the discovery of the log- arithmic series. The above note is derived from J. W. L. Glaisher's article on Logarithms in the Encyclopedia Britannica. THE END. / ^ .-aX m ,b c\\ "* i :. .-■»..--:'. .:-^-^-.^iit^ :^-.^>b?-^-: './:^.,^,jj^ M306055 I'^ji. THE UNIVERSITY OF CALIFORNIA LIBRARY mA \m^M IM-^ mm i ,!'>R