THE [JMBER-SYSTEM OF ALGEBRA TREATED THEORETICALLY AND HISTORICALLY BY HENRY B. FINE, Ph.D., Professor of Mathematics in Princeton College. LEACH, SHEWELL, & SANBORN, BOSTON AND NEW YORK. Copyright, 1890, By HENRY B. FINE. Typography by J. S. Gushing & Co., Boston. Presswork by Berwick & Smith, Boston. PREFACE. The theoretical part of this little book is an elementary exposition of the nature of the number concept, of the posi- tive integer, and of the four artificial forms of number which, with the positive integer, constitute the "number- system " of algebra, viz. the negative, the fraction, the irra- tional, and the imaginary. The discussion of the artificial numbers follows, in general, the same lines as my pam- phlet : On the Forms of Number arising in Common 'Algebra, but it is much moro exhaustive and thorough- going. The point of view is the one first suggested by Peacock and Gregory, and accepted by mathematicians gen- erally since the discovery of quaternions and the Ausdeh- nungslehre of Grassmann, that algebra is completely defined formally by the laws of combination to which its funda- mental operations are subject ; that, speaking generally, these laws alone define the operations, and the operations the various artificial numbers, as their formal or symbolic results. This doctrine was fully 'developed for the negative, the fraction, and the imaginary by Hankel, in his Complexe Zahlensystemen, in 1867, and made complete by Cantor's beautiful theory of the irrational in 1871, but it has not as yet received adequate treatment in English. Any large degree of originality in work of this kind is naturally out of the question. I have borrowed from a iii i rvooo. IV PREFACE. great many sources, especially from Peacock, G-rassmann Hankel, Weierstrass, Cantor, and Thomae (Theorie dei andlytischen Functional einer complexen Verdnderliclien) . 1 may mention, however, as more or less distinctive features of my discussion, the treatment of number and counting (§§ 1-5) and the equation (§§4, 12), the prominence given the laws of the determinateness of subtraction and division, and the demonstration of the one-to-one correspondence be- tween numbers defined by regular sequences and the points of- a line (§40). Much care and labor have been expended on the his- torical chapters of the book. These were meant at the out- set to contain only a brief account of the origin and history of the artificial numbers. But I could not bring myself tc ignore primitive counting and the development of numeral notation, and I soon found that a clear and connected account of the origin of the negative and imaginary i possible only when embodied in a sketch of the early his tory of the equation. I have thus been led to write i resume of the history of the most important parts of ele- mentary arithmetic and algebra. Moritz Cantor's Vorlesungen uber die Oeschichte der Mathe matik, Vol. I, has been my principal authority for tin entire period which it covers, i.e. to 1200 a.d. For tin little I have to say on the period 1200 to 1600, I hav< depended chiefly, though by no means absolutely, or Hankel: Zur QescMchte der Mathematik in Altertum unci Mittelalter. The remainder of my sketch is for the mosi part based on the original sources. HENRY B. FINE. Princeton, April, 1891. CONTENTS. I. THEORETICAL. I. The Positive Integer. PAGE The number concept 3 Numerical equality 3 Numeral symbols 4 The numerical equation 6 Counting • 5 Addition and its laws 6 Multiplication and its laws * 7 II. Subtraction and the Negative Integer. Numerical subtraction . 8 Determinateness of numerical subtraction 9 Formal rules of subtraction 9 Limitations of numerical subtraction 11 Symbolic equations' 11 Principle of permanence. Symbolic subtraction 12 Zero 13 The negative 14 Recapitulation of the argument of the chapter 16 III. Division and the Fraction. Numerical division .18 Determinateness of numerical division 18 Formal rules of division 19 Limitations of numerical division 20 Symbolic division. The fraction '. 21 Negative fractions 22 General test of the equality or inequality of fractions 22 Indeterminateness of division by zero 23 v Vi CONTENTS. PAGE Determinateness of symbolic division 23 The vanishing of a product 24 The system of rational numbers . . . 25 IV. The Irrational. Inadequateness of the system of rational numbers 26 Numbers denned by " regular sequences." The irrational 27 Generalized definitions of zero, positive, negative ; 2{T Of the four fundamental operations 29 Of equality and greater and lesser inequality 31 The number denned by a regular sequence its limiting value 31 Division by zero 33 The number-system defined by regular sequences of rationals a closed system , .. 34 V. The Imaginary. Complex Numbers. The pure imaginary fc 35 Complex numbers , 36 The fundamental operations on complex numbers . . _ . . . 37 Numerical comparison of complex numbers 38 Adequateness of the system of complex numbers. ., 39 Fundamental characteristics of the algebra of number 39 VI. Graphical Eepresentatton of Numbers. The Variable. Correspondence between the real number-system and the points of a line . • \ 41 The real number-system " continuous "" 43 The variable 43, 45 Correspondence between the complex number-system and the .points of a plane 44 Definitions of modulus and argument of a complex number and of sine, cosine, and circular measure of an angle 45, 48 Demonstration that a + ib — p (cos + i sin 6) = pe^ 45, 48 Construction of the points which represent the sum, difference, product, and quotient of two complex numbers 46, 47 VII. The Fundamental Theorem of Algebra. Definitions of the algebraic equation and its roots 50 Demonstration that an algebraic equation of the nth degree has n roots 51, 53 CONTENTS. Vll VIII. Numbers Defined by Infinite Series. page I. Heal Series. Definitions of sum, convergence, and divergence 54 General test of convergence 55 Absolute and conditional convergence 55, 57 Special tests of convergence 57 Limits of convergence 59 The fundamental operations on infinite series 61 II. Complex Series. General test of convergence 62 Absolute and conditional convergence 62 The region of convergence 63 A theorem respecting complex series 64 The fundamental operations on complex series 65 IX. The Exponential and Logarithmic Functions. UNDETERMINED COEFFICIENTS. INVOLUTION AND EVOLUTION. THE BINOMIAL THEOREM. Definition of function 66 Functional equation of the exponential function 66 Undetermined coefficients 67 The exponential function 68 The functions sine and cosine 71 Periodicity of these functions 72 The logarithmic function 72 Indeterminateness of logarithms 75 Permanence of the laws of exponents 76 Permanence of the laws of logarithms 77 Involution and evolution 77 The binomial theorem for complex exponents 77 II. HISTORICAL. I. Primitive Numerals. Gesture symbols 81 Spoken symbols 82 Written symbols '. . . .♦ 84 Vlll - CONTENTS. II. Historic Systems of Notation. page Egyptian and Phoenician 84 Greek 84 Roman 85 Indo- Arabic 86 III. The Fraction. Primitive fractions 90 Roman fractions 91 Egyptian (the Book of Ahmes) 91 Babylonian or sexagesimal 92 Greek 93 IV. Origin of the Irrational. Discovery of irrational lines. Pythagoras 94 Consequences of this discovery in Greek mathematics 96 Greek approximate values of irrationals 98 V. Origin of the Negative and the Imaginary. THE EQUATION. The equation in Egyptian mathematics 99 In the earlier Greek mathematics 99 Hero of Alexandria 100 Diophantus of Alexandria 10-1 The Indian mathematics. Aryabhatta, Brahmagupta, Bhaskara 103 Its algebraic symbolism 104 Its invention of the negative 105 Its use of zero 105 Its use of irrational numbers 106 Its treatment of determinate and indeterminate equations 106 The Arabian mathematics. Alkhwarizmi, Alkarchi, Alchayyami 107 Arabian algebra Greek rather than Indian 110 Mathematics in Europe before the twelfth century Ill Gerbe- J . Ill Entra.^e of the Arabian mathematics. Leonardo 112 Mathematics during the age of Scholasticism 113 The Renaissance. Solution of the cubic and biquadratic equations 114 The negative in the algebra of this period. Eirst appearance of the imaginary 115 Algebraic symbolism. Vieta and Harriot 116 The fundamental theorem of algebra. Harriot and Girard 117 CONTENTS. - ix VI. Acceptance of the Negative, the General Irrational, and the Imaginary as Numbers. PAGE Descartes' Geometrie and the negative 118 Descartes' geometric algebra 119 The continuous variable. Newton. Euler 121 The general irrational 121 The imaginary, a recognized analytical instrument 122 Argand's geometric representation of the imaginary 122 Gauss. The complex number 123 VII. Recognition of the Purely Symbolic Character of Algebra. quaternions. the ausdehnungslehre. The principle of permanence. Peacock 124 The fundamental laws of algebra. " Symbolical algebras." Gregory 126 Hamilton's quaternions 128 Grassmann's Ausdehnungslehre 129 The fully developed doctrine of the artificial forms of number. Hankel. Weierstrass. G. Cantor 130 Recent literature 131 PRINCIPAL FOOTNOTES. Instances of quinary and vigesimal systems of notation 82 Instances of digit numerals 83 Summary of the history of Greek mathematics 95 Old Greek demonstration that the side and diagonal of a square are incommensurable 96 Greek methods of approximation . . 98 Diophantine equations ... 102 Alchayy ami's method of solving cubics by the intersections . conies 109 Jordanus Nemorarius 113 The Summa of Luca Pacioli ,......,.. 113 Regiomontanus . . C ............ . 114 Algebraic symbolism , 113, 116 The irrationality of e and 7r. Lindemann ... .......... 70 I. THEORETICAL. I. THE POSITIVE INTEGEK, AND THE LAWS WHICH REGULATE THE ADDITION AND MULTIPLICATION OF POSITIVE INTEGERS. 1. Number. Separateness or distinctness is a primary cognition, being necessary even to the cognition of things as individuals, as distinct from other things. The notion of number is based immediately on this pri- mary cognition. Number is that property of a group of distinct things which remains unchanged during .any change to which the group may be subjected which does not destroy the distmct- ness of the individual things. Such changes are changes of the characteristics of the individual things or of their arrangement ; for these do not cause one thing to split up into more than one, nor more than one to merge in one. This characteristic of number may be stated in a theorem which is the fundamental postulate of arithmetic : The number of things in any group of distinct things is, independent of the characters of these things, of the order in ^ which they may be arranged in the group, and of the manner in which they may be associated with one another in smaller groups. 2. Numerical Equality. The number of things in any two groups of distinct things is the same, when for each thing in the first group there is one in the second, and reciprocally, for each thing in the second group, one in the first. Thus, the : aber of letters in the two groups, A, B, (7; D, E, F, is cxxc sarre. In Jie second group there is a letter I 4 NUMBER-SYSTEM OF ALGEBRA. which may be assigned to each of the letters in the first : as D to A, E to B, F to C\ and reciprocally, a letter in the first which may be assigned to each in the second : as A to D, B to E, C to F. Two groups thus related are said to be in one-to-one (1-1) correspondence. Underlying the statement just made is the assumption that if the two groups correspond in the manner described for one order of the things in each/ they will correspond if the things be taken in any other order also ; thus, in the example given, that if E instead of D be assigned to. A, there will again be a letter in the group D, E, F, viz. D or F, for each of the remaining letters B and C, and recipro- cally. This is an immediate consequence of § 1. The number of things in the first group is greater than that in the second, or the number of things in the second less than that in the first, when there is one thing in the first group for each thing in the second, but not reciprocally one in the second for each in the first. 3. Numeral Symbols. As regards the number of things which it contains, therefore, a group may be represented by any other group, e.g. of the fingers or of simple marks, I's, which stands to it in the relation of correspondence described in § 2. This is the primitive method of repre- senting the number of things in a group and, like the modern method, makes it possible to compare numerically groups which are separated in time or space. The modern method of representing the number of things in a group differs from the primitive only in the substitu- tion of symbols, as 1, 2, 3, etc., or numeral words, as one, two, three, etc., for the various groups of marks I, II, III, etc. These symbols are the positive integers of arith- metic. A positive integer is a symbol for the number of things in a group of distinct things. THE POSITIVE INTEGER. 5 For convenience we shall call the positive integer which represents the nnmber of things in any group its numeral symbol, or when not likely to cause confusion, its number simply,— this being, in fact, the primary use of the word " number " in arithmetic. In thj3 following discussion, for the sake of giving our statements a general form, we shall represent these numeral symbols by letters, a, b, c, etc. 4. The Equation. The numeral symbols of two groups being a and b ; when the number of things in the groups is the same, this relation is expressed by the equation a = b; when the first group is greater than the second, by the inequality a>b; when the first group is less than the second, by the ine- quality a V. a(b + c) = ab-\-ac. Or, III. The product of a by b is the same as the product of b by a. TV. The product of a by be is the same as the product of ab by c. V. The product of a by the sum of b and c is the same as the sum of the product of a by b and of a by c. These laws are consequences of the commutative and associative laws for addition. Thus, III. The Commutative Law. The units of the group which corresponds to the sum of b numbers each equal to a may be arranged in b rows containing a units each. But in such an arrangement there are a columns containing ft units each ; so that if this same set of units be grouped by columns instead of rows, the sum becomes that of a numbers each 8 NUMBETt-SYSTEM OF ALGEBRA. equal to b, or ba. Therefore ab = ba, by the commutative and associative laws for addition. IV. The Associative Law. ^b]c = c sums such as (a-fa-i to b terms) = a + a + aH to be terms (by the associative law for addition) = a(bc). V. The Distributive Law. a(b + c) = a + a -f- a -f • • • to (b -f- c) terms = (a + cH to & terms) -|-(a + aH tp c terms) (by the associative law for addition), = ab + a c. The commutative, associative, and distributive laws for sums of any number of terms and products of any number of factors follow immediately from I-V. Thus the product of the factors a, b, c, d, taken in any two orders, is the same, since the one order can be transformed into the other by successive interchanges of consecutive letters. II. SUBTRACTION AND THE NEGATIVE INTEGER. 8. Numerical Subtraction. Corresponding Tjq^very,math- ematical operation there is another, commonly called its in- verse, which exactly undoes what the operation itseff does. Subtraction stands in this relation to addition, and division to multiplication. To subtract b from a is to find a number to which if b be added, the sum will be a. The result is written a — b ; by definition, it identically satisfies the equation VI. ( a -b) + b = a-, SUBTRACTION. 9 that is to say, a — b is the number belonging to the group which with the 6-group makes up the a-group. Obviously subtraction is always possible when b is less than a, but then only. Unlike addition, in each application of this operation regard must be had to the relative size of the two numbers concerned. 9. Determinateness of Numerical Subtraction. Subtrac- tion, when possible, is a determinate operation. There is but one number which will satisfy the equation x 4- b = a, but one number the sum of which and b is a. ui other words, a — b is one-valued. For if c and d both satisfy the equation x + b = a, since then c-\-b = a and d + b = a, c + b = d-\-b\ that is, a one-to- one correspondence may be set up between the individuals of the (c + b) ajid {d -f- b) groups (§4). The same sort of correspondence, however, exists between any b individuals of the first group and any b individuals of fche secon|Lk it must, therefore, exist between the remaining c of the fltrst and the remaining: d of the second, or c = d. This characteristic^of subtraction is of the same order of importance as the commutative and associative laws, and we shall add to the group of laws I-V and definition VI — as being, like them, a fundamental principle in the follow- ing discussion — the theorem ( a = b, which may also be stated in the form : If one term of a sum changes while the other remains constant, the sum changes. The same reasoning proves, also, that ( As a + c > or < b + c, 1 a > or < b. 10. Formal Rules of Subtraction. All the rules of sub- traction are purely formal consequences of the fundamental 10 NUMBER-SYSTEM OF ALGEBRA. laws I-V, VII, and definition VI. They must follow, what- ever the meaning of the symbols a, b, c, +, — > = ; a fact which has an important bearing on the following discussion. It will be sufficient to consider the equations which fol- low. For, properly combined, they determine the result of any series of subtractions or of any complex operation made up of additions, subtractions, and multiplications. 1. a — (b + c) = a — b — c = a — c — b. 2. a — (b — c) = a — b-\-c. 3. a-\-b— b = a. 4. a + (& — c) = a + b — c = a — c + 5. 5. a (b — c) = ab — ac. For 1. a — b — c is the form to which if first c and then b be added; or, what is the same thing (by I), first b and then c ; or, what is again the same thing (by II)/5-}-c at once, — the sum pro- duced is a (by VI)r~a — b — c is therefore the same as a — c — b, which is as it stands the form to which if b, then c, be added the sum is a; also the same as a— (6 + c), which is the form to which if b -+- c be added the sum is a. 2. a — (b — c) = a — (b — c) — c -f c, Def. VI. = a — (b — c -f- c) + c, Eq. 1. = a ■— b + e. Def. VI. 3. a + b — b + b = a -f- fc. But a -f 6 = a + 6. Def. VI. .-. a -j- 6 — & = a. Law VII. 4. a + & — c = a + (& — c -f c) — c, Def. VI. .^ =a + (6 — c). Law II, Eq. 3. 5. a& — ac = a (6 — c + c) — ac, Def. VI. = a (b — c) + ac — ac, Law V. = a(6—c). Eq. 3. SUBTRACTION. 11 Equation 3 is particularly interesting in that it defines addition as the inverse of subtraction. Equation 1 declares that two consecutive subtractions may change places, are commutative. Equations 1, 2, 4 together supplement law II, constituting with it a complete associative law of addi- tion and subtraction ; and equation 5 in like manner supple- ments law V. 11. Limitations of Numerical Subtraction. Judged by the equations 1-5, subtraction is the exact counterpart of addition. It conforms to the same general laws as that operation, and the two could with fairness be made to interchange their rdles of direct and inverse operation. But this apparent equality vanishes when the attempt is made to interpret these equations. The requirement that subtrahend be less than minuend then asserts itself as a fatal limitation. It makes the range of subtraction much narrower than that of addition. It renders the equations 1-5 available, for special classes of values of a, b, c only. If it must be insisted on, even so simple an inference as that a — (a -f- 5) -f-2 b is equal to b cannot be drawn, and the use of subtraction in any reckoning with symbols whose relative values are not at all times known must be pro- nounced unwarranted. One is thus led perforce to ask whether int.erpretability •is after all necessary to the validity of reckonings and, if not, to seek to free subtraction and the rules of reckoning with the results of subtraction from this crippling limi- tation. 12. Symbolic Equations. Principle of Permanence. Sym- bolic Subtraction. In pursuance of this inquiry one turns first to the equation (a — b) + b — a, which serves as a definition of subtraction when, b is less than a. This is an equation in the primary sense (§4) only when a — b is a number. But in the broader sense, that 12 NUMBER-SYSTEM OF ALGEBRA. An equation is any declaration of the equivalence of definite combinations of symbols — equivalence in the sense that one may be substituted for the other, — (a — b)-\-b = a may be an equation, whatever the values of a and b. And if no different meaning has been attached to a — b, and it is declared that a — b is the symbol which associated with b in the combination (a — b) + b is equivalent to a, this declaration, or the equation (a — b)-\-b = a, is a definition * of this symbol. By the assumption of the permanence of form of the numerical equation in which the definition of subtraction resulted, one is thus put immediately in possession of a symbolic definition of subtraction which is general. The numerical definition is subordinate to the symbolic definition, being the interpretation oi which it admits when b is less than a. But from the standpoint of the symbolic definition, inter- pretability — the question whether a — b is a number or not — is irrelevant; only such properties may be attached to a — b, by itself considered, as flow immediately from the generalized equation {a — b)-\-b = a. In like manner each of the fundamental laws I-V, VII, on the assumption of the permanence of its form after it has ceased to be interpretable numerically, becomes a declaration of the equivalence of certain definite combi- nations of symbols, and the formal consequences of these laws — the equations -1-5 of § 10 — become definitions of addition, subtraction, multiplication, and their mutual * A definition in terms of symbolic, not numerical addition. The sign + can; of course, indicate numerical addition only when "both the symbols which it connects are numbers. SUBTRACTION. 13 relations — definitions which are purely symbolic, it may be, but unrestricted in their application. Now with reference to the legitimacy of such definitions as these there can be no question. They are consistent with each other, and of course consistent with the numerical defi- nitions, which are indeed but special interpretations of them. If used consistently, there is no more possibility of their lead- ing to false results than there is of the more tangible numeri- cal definitions leading to false results. The laws of correct thinking are as applicable to mere symbols as to numbers. What the value of these symbolic definitions is, to what extent they add to the power to draw inferences concerning numbers, the elementary algebra abundantly illustrates. One of their immediate consequences is the introduction into algebra of two new symbols, zero and the negative, which contribute greatly to increase the simplicity, compre- hensiveness, and power of its operations. 13. Zero. When b is set equal to a in the general equation (a — b) + b = a, it takes one of the forms (a — a) + a = a, (b-b) + b = b. It may be proved that a — a = b — b. For (a — a) + (a + b) = (a — a) -f a + b, Law II. = a + by since (a — a) -{- a = a. And (b - b) + (a + b) = (b - b) +6+a, Laws I, II. = b + a, since (b — b) -j- b = b. Therefore a — a = b — b. Law VII. 14 NTJMBEB-SYSTEM OF ALGEBRA. a — a is therefore altogether independent of a and may properly be represented by a symbol unrelated to a. The symbol which has been chosen for it is 0, called zero. Addition is defined for this symbol by the equations 1. -f- a = a 7 definition of 0. a + = a. Law I. Subtraction (partially) , by the equation 2. a — = a. For (a - 0) + = a.' Def . VI. Multiplication (partially), by the equations 3. ax0 = 0xa = 0. For a X = a(b — 6), definition of 0. = ab — ab, § 10, 5. = 0. . definition of 0. 14. The Negative. When b is greater than a, equal say to a + d, so that b — a = d, then a — b = a — (a + d), = a — a — d> § 10, 1. = — d. definition of 0. For — d the briefer symbol — d has been substituted ; with propriety, certainly, in view of the lack of significance of in relation to addition and subtraction. The equation — d = — d } moreover, supplies the missing rule of sub- traction for 0. (Compare § 13, 2.) The symbol — d is called the negative, and in opposition to it, the number d is called positive. Though in its origin a sign of operation (subtraction from 0), the sign — is here to be regarded merely as part of the symbol — d. — d is as serviceable a substitute for a — b when a < b, as is a single numeral symbol when a > b. SUBTRACTION. 15 The rules for reckoning with the new symbol — definitions of its addition, subtraction, multiplication — are readily deduced from the laws I-V, VII, definition VI, and the equations 1-5 of § 10, as follows : 1. b + (-b) = -b + b = 0: Tor — b -f- b = (0 — b) + b, definition of negative. = 0. Def . VI. — b may therefore be defined as the symbol the sum of which and b is 0. 2. a + (—b) = — b + a = a — b. For a + ( — b) = a + (0 — b), definition of negative. = an- 0-b, §10,4. = a-6.' §13,1. 3. _ a + (-&) = - (a + &), For — a + (— b) = — a — 6, by the reasoning in § 14, 2. = 0-0 + 6), §10,1. = — (a -f- 6). definition of negative. 4. a — (— b) = a + b. For a — ( — b) = a — : (0 — &), definition of negative. = a - + b, § 10, 2. = a + 6. § 13, 2. 5. (-a)-(- &) = &-«. For — a — (— 6) = — a + b, by the reasoning in § 14, 4. = b - a. § 14, 2. Cor. (-a)-(-a) = 0. 6. a( — b) = ( — b)a= — ab. For = a(6-6), §13,3. = ab -f a( — 5). Law V. .-. a(- &) = - a&. ' § 14, 1 ; Law VII. 16 NUMBEli-SYSTEM OF ALGEBRA. 7. (-a)xO = Ox(-a)=s=0. For (- a) x = ( - a) (b - 6), definition of 0. = (-a)6-(-a)6, § 10,5. = 0. § 14, 6, and 5, Cor. 8. (-a)(-6) = a&. For = (— a)(6 — 6), §14,7. = (- a) 6 -i-(- a) (- 5), Law V. == _ a& + (_ a )(_ & ). §14,8. ... (__ a ) (_ &) = 5. § 14, 1 ; Law VII. By this method one is led, also, to definitions of equality and greater or lesser inequality of negatives. Thus 9. — a >, = or < — by . according as b >, = or < a.* For as &>,=,< a, _ a + a + b >, =, < -b + & + a, §14,1; § 13, 1. or — a >, =, < — by Law VII or ^Il\ In like manner — a < < b. 15. Recapitulation. The nature of the argument which has been developed in the present chapter should be care- fully observed. From the definitions of the positive integer, addition, and subtraction, the associative and commutative laws and the determinateness of subtraction followed. The assump- tion of the permanence of the result a — b, as defined by (a — &) + &== a, for all values of a and b, led to definitions * On the other hand, — a is said to be numerically greater than, equal to, or less than .— 6, according as a is itself greater than, equal to, or less than b. SUBTRACTION. 17 of the two symbols 0, — d, zero and the negative ; and from the assumption of the permanence of the laws I-V, VII- were derived definitions of the addition, subtraction, and multiplication of these symbols, — the assumptions being just sufficient to determine the meanings of these operations unambiguously . In the case of numbers, the laws I-V, VII, and definition VI were deduced from the characteristics of numbers and the definitions of their operations ; in the case of the sym- bols 0, — d, on the other hand, the characteristics of these symbols and the definitions of their operations were deduced from the laws. With the acceptance of the negative the character of arithmetic undergoes a radical change.* It was already in a sense symbolic, expressed itself in equations and inequali- ties, and investigated the results of certain operations. But ' its symbols, equations, and operations were all interpretable' in terms of the reality which, gave rise to it, the number of. things in actually existing groups of things. Its connec- tion with this reality was as immediate as that of the ele- mentary geometry with actually existing space relations* But the negative severs this connection. The negative is a symbol for the result of an operation which cannot be effected with actually existing groups' of things, which is, therefore, purely symbolic. And not only do the fundamen- tal operations and the symbols on which they are performed lose reality ; the equation, the fundamental judgment in all mathematical reasoning, suffers the same loss. From being a declaration that two groups of things are in one-to-one correspondence, it becomes a mere declaration regarding two combinations of symbols, that in any reckoning one may be substituted for the other. * In this connection see § 25. 18 NUMBER-SYSTEM OF ALGEBRA. III. DIVISION AND THE PKACTION. 16. Numerical Division. The inverse operation to multi- plication is division. To divide a by & is to find a number which multiplied by b produces a. The result is called the quotient of a by b, and is written — • By definition P VIII. f^)b = Like subtraction, division cannot be always effected. Only in exceptional cases can the a-group be subdivided into groups each containing b individuals. 17. Determinateness of Numerical Division. When divis- ion can be effected at all, it can lead to but a single result ; it is. determinate. For there can be but one number the product of which by b is a ; in other words, IX ( If cb = db, X c=d* . For b groups each containing c individuals cannot be equal to b groups each containing d individuals unless c = d (§4). This is a theorem of fundamental importance. It may be called the law of determinateness of division. It declares that if a product and one of its factors be determined, the remaining factor is definitely determined also; or that if one of the factors of a product changes while the other remains unchanged, the product changes. It alone makes division in the arithmetical sense possible. The fact that * The case b = is excluded, not being a number in the sense in which that word is here used. DIVISION AND THE FBACTION. 19 it does not hold for the symbol 0, but that rather a product remains unchanged (being always 0) when one of its factors is 0, however the other factor be changed, makes division by impossible, rendering unjustifiable the conclusions which can be drawn in the case of other divisors. The reasoning which proved law IX proves also that ( As cb > or < db, IX . -N (. c > or < d. 18. Formal Rules of Division. The fundamental laws of the multiplication of numbers are III. ab = ba, IV. a(bc)=r a $c, V. a(b + c) =ab + ac. Of these, the definition VIII. (r)b = a, the theorem ( If ac = be, IX ' \ a = b, unless c = 0, and the corresponding laws of addition and subtraction, the rules of division are purely formal consequences, dedu- cible precisely as the rules of subtraction 1-5 of § 10 in the preceding chapter. They follow without regard to the meaning of the symbols a, b, c, =, +, — , ab, -• Thus : b 1. For and bd a c _ac b d bd ? .t .bd = -b - b d b d Laws IV, III. = ac, Def. VIII. — • bd = ac. Def. VIII. 20 NUMBER-SYSTEM OF ALGEBRA. The theorem follows by law IX. 2. W = ad 'c\ be d) For ^s. c * Def. VIII. c\ d b cl) and — • - = - • - , § 18, 1 ; Law IV. be d . b cd _ a since — cd = dc = lx cd. Def. VIII, Law IX. cd The theorem follows by law IX. o a , c _ adjkbc^ b cl bd Tor f2 ± 5.^ M = -b • cZ± -cZ . b, Laws III-V; § 10, 5. \b d) b d = ad ± 6c, Def. VIII. and f ad ± bc \d = ad± be. Def. VIII. V bd J The theorem follows by law IX. By the same method it may be inferred that b > '~' < d' as ad >, =, < be. Def. VIII, Laws III, IV, IX, IX'. 19. Limitations of Numerical Division. Symbolic Division. The Fraction. General as is the form of the preceding equations, they are capable of numerical interpretation only when -', - are numbers, a case of comparatively rare occur- rence. The narrow limits set the quotient in the numer- ical definition render division an unimportant operation as DIVISION AND THE FB ACTION. 21 compared with addition, multiplication, or the generalized subtraction discussed in the preceding chapter. But the way which led to an unrestricted subtraction lies open also to the removal of this restriction ; and the reasons for following it there are even more cogent here. We accept as the quotient of a divided by any number 6, which is not 0, the symbol - defined by the equation b regarding this equation merely as a declaration of the equivalence of the symbols (-)b and a, of the right to sub- stitute one for the other in any reckoning. Whether - be a number or not is to this definition irrele- b a vant* When a mere symbol, t is called a fraction, and in opposition to this a number is called an integer. We then put ourselves in immediate possession of defi- nitions of the addition, subtraction, multiplication, and division of this symbol, as well as of the relations of equal- ity and greater and lesser inequality — definitions which are consistent with the corresponding numerical definitions and with one another — by assuming the permanence of form of the equations 1, 2, 3 and of the test 4 of § 18 as symbolic statements, when they cease to be interpretable as numerical statements. The purely symbolic character of - and its operations detracts nothing from their legitimacy, and they establish division on a footing of at least formal equality with the other three fundamental operations of arithmetic* * The doctrine of symbolic division admits of being presented in the very same form as that of symbolic subtraction. The equations of Chapter II immediately pass over into theorems 22 NUMBER-SYSTEM OF ALGEBRA. 20. Negative Fractions. Inasmuch as negatives conform to the laws and definitions I-IX, the equations 1, 2, 3 and the test 4 of § 18 are valid when any of the numbers a, b, c, d are replaced by negatives. In particular, it follows from the definition of quotient and its determinateness, that a __a m — a _ _a . —a_a ^b~ P ~V~ b l —b~b' It ought, perhaps, to be said that the determinateness of division of negatives has not been formally demonstrated. The theorem, however, that if ( ± a) ( ± c) = ( ± b) ( ± c), ± a = ± b, follows for every selection of the signs ± from the one selection +, +, +, + by § 14, 6, 8. 21. General Test of the Equality or Inequality of Fractions. Given any two fractions ± -, ± — b d ±f>,=or<±^, b d according as ± ad >, = or < ± be. Laws IX, IX'. Compare § 4, § 14, 9. respecting division when the signs of multiplication and division are substituted for those of addition and subtraction ; so, for instance, a — (6 + c) = a — 6 — c = a — c — b , (i) (?) gives — = -l_i- = -l_u be c b In particular, to (a — a) + a = a corresponds - a = a. Thus a purely symbolic definition may be given 1. It plays the same role in multipli- cation as in addition. Again, it has the same exceptional character in involution — an operation related to multiplication quite as multipli- cation to addition — as in multiplication; for 1™ = 1", whatever the values of m and n. Similarly, to the equation (- a) + a = 0, or (0 - a) + a = 0, corre- sponds f-\ a = 1, which answers as a definition of the unit fraction - ; and in terms of these unit fractions and integers all other fractions a may be expressed. DIVISION AND THE FRACTION. 23 22. Indeterminateness of Division by Zero. Division by does not conform to the law of determinateness ; the equa- tions 1, 2, 3 and the test 4 of § 18 are, therefore, not valid when is one of the divisors. * The symbols -, -, of which some use is made in mathe- matics, are indeterminate.* 1. - is indeterminate. For - is completely denned by /0\ the equation ( - j =' ; but since x x = 0, whatever the value of x, any number whatsoever will satisfy this equation. 2. - is indeterminate. For, by definition, j-JO = a. Were - determinate, therefore, — since then ( - ) would, a x W by § 18, 1, be equal to , or to -, — the number a . . would be equal to -, or indeterminate. Division by is not an admissible operation. 23. Determinateness of Symbolic Division. This excep- tion to the determinateness of division may seem to raise an objection to the legitimacy of assuming — as is done when the demonstrations 1-4. of § 18 are made to apply to symbolic quotients — that symbolic division is determinate. It must be observed, however, that -, - are indetermi- nate in the numerical sense, whereas by the determinateness of symbolic division is, of course, not meant actual numerical determinateness, but " symbolic determinateness," conform- ity to law IX, taken merely as a symbolic statement. For, as has been already frequently said, from the present stand- point the fraction - is a mere symbol, altogether without numerical meaning apart from the equation I - \b = a, with * In this connection see § 32. 24 NUMBER-SYSTEM OF ALGEBRA. which, therefore, the property of numerical determinateness has no possible connection. The same is true of the prod- uct, sum or difference of two fractions, and of the quotient of one fraction by another. As for symbolic determinateness, it needs no justification when assumed, as in the case of the fraction and the demonstrations 1-4, of symbols whose definitions do not preclude it. The inference, for instance, that because b dj \bd a c _ac b d bd which depends on this principle of symbolic determinate- ness, is of precisely the same character as the inference that b dj b d which depends on the associative and commutative laws. Both are pure assumptions made of the undefined symbol a c b d with that of the product of two numerical quotients.* for the sake of securing it a definition identical in form b d 5 24. The Vanishing of a Product. It has already been shown (§ 13, 3, § 14, 7, § 18, 1) that the sufficient condition for the vanishing of a product is the vanishing of one of its factors. From the determinateness of division it follows that this is also the necessary condition, that is to say : If a product vanish, one of its factors must vanish. Let xy — 0, where x, y may represent numbers or any of the symbols we have been considering. * These remarks, mutatis mutandis, apply with equal force to sub- traction. DIVISION AND THE FRACTION. 25 Since xy = 0, xy + xz = xz, § 13, 1. or x (y -\-z) = xz, Law V. whence, if x be not 0, y + z = z> Law IX. or y = 0. Law VII. 25. The System of Rational Numbers. Three symbols, __ d - have thus been found which can be reckoned with by the same rules as numbers, and in terms of which it is possible to express the result of every addition, sub- traction, multiplication or division, whether performed on numbers or on these symbols themselves ; therefore, also, the result of any complex operation which can be resolved into a finite combination of these four operations. Inasmuch as these symbols play the same r61e as numbers in relation to the fundamental operations of arithmetic, it is natural to class them with numbers. The word " number," originally applicable to the positive integer only, has come to apply to zero, the negative integer, the positive and nega- tive fraction also, this entire group of symbols being called the system of rational numbers* This involves, of course, a radical change of the number concept, in consequence of which numbers become merely part of the symbolic equip- ment of certain operations, admitting, for the most part, of only such definitions as these operations lend them. * It hardly need be said that the fraction, zero, and the negative actually made their way into the number- system for quite a different reason from this ; — because they admitted of certain " real " interpre- tations, the fraction in measurements of lines, the negative in debit where the corresponding positive meant credit or in a length measured to the left where the corresponding positive meant a length measured to the right. Such interpretations, or correspondences to existing things which lie entirely outside of pure arithmetic, are ignored in the present discussion as being irrelevant to a pure arithmetical doctrine of the artificial forms of number. 26 NUMBER-SYSTEM OF ALGEBRA. In accepting these symbols as its numbers, arithmetic ceases to be occupied exclusively or even principally with the properties of numbers in the strict sense. It becomes an algebra whose immediate concern is with certain opera- tions defined, as addition by the equations a + b — b + a, g ' -f- (p + c) = a + b + c, formally only, without reference to the meaning of the symbols operated on. # IY. THE IEKATIOtfAL. 26. The System of Rational Numbers Inadequate. The system of rational numbers, while it sufhces for the four fun- damental operations of arithmetic and finite combinations of these operations, does not fully meet the needs of algebra. The great central problem of algebra is the equation, and that only is an adequate number-system for algebra which supplies the means of expressing the roots of all possible equations. The system of rational numbers, however, is equal to the requirements of equations of the first degree only ; it contains symbols not even for the roots of such elementary equations of higher degrees as a? == 2, x 2 = — 1. But how is the system of rational numbers to be enlarged into an algebraic system which shall be adequate and at the same time sufficiently simple ? The roots of the equation X n + p l X n ~ 1 +J?2#*~*H hPn-lX +Pn == * The word " algebra" is here used in the general sense, the sense in which quaternions and the Ausdehungslehre (see §§ 127, 128) are alge- bras. Inasmuch as elementary arithmetic, as actually constituted, accepts the fraction, there is no essential difference between it and elementary algebra with respect to the kinds of number with which it deals ; algebra merely goes further in the use of artificial numbers. .The elementary algebra differs from arithmetic in employing literal symbols for numbers, but chiefly in making the equation an object of investigation. THE IRBATIONAL. 27 are not the results of single elementary operations, as are the negative of subtraction and the fraction of division 5 for though the roots of the quadratic are results of " evolution/' and the same operation often enough repeated yields the roots of the cubic and biquadratic also, it fails to yield the roots of higher equations. A system built up as the rational system was built, by accepting indiscriminately every new symbol which could show cause for recognition, would, therefore, fall in pieces of its own weight. The most general characteristics of the roots must be discovered and defined and embodied in symbols — by a method which does not depend on processes for solving equations. These symbols, of course, however character- ized otherwise, must stand in consistent relations with the system of rational numbers and their operations. An investigation shows that the forms of number neces- sary to complete the algebraic system may be reduced to two : the symbol V— 1, called the i magina ry (an indicated root of the equation x 2 -f- 1 = 0), ancl the class of symbols called irrational, to which the roots of the equation 0^—2 = belong. 27. Numbers Defined by Regular Sequences. X -The Irra- tional. On applying to 2 the ordinary method for extracting the square root of a number, there is obtained the follow- ing sequence of numbers, the results of carrying the reck- oning out to 0, *1, 2, 3, 4, ... places of decimals, viz. : 1, 1.4, 1.41, 1.414, 1.4142, ... These numbers are rational ; the first of them differs from each that follows it by less than 1, the second by less than' — , the third by less than , ••• the nth by less than 10' J 100' J lO'* 1 And ■■ , is a fraction which may be made less than any assignable number whatsoever by taking n great enough. ( 28 • NUMBER-SYSTEM OF ALGEBBA. This sequence may be regarded as a definition of the square root of 2. It is such in the sense that a term may be found in it the square of which, as well as of each fol- lowing term, differs from 2 by less than any assignable number. Any sequence of rational numbers a 1? a 2 , a 3 , ••• a M , CL^+ly "' a M+^ "** in ivhich, as in the above sequence, the term a^ may, by tak- ing fx great enough, be made to differ numerically from each term that folloivs it by less than any assignable number, so that, for all values of v, the difference, a M+v — a M , is numerically less than 8, however small S be taken, is called a regular sequence. The entire class of operations which lead to regular sequences may be called regular sequence-building. Evolution is only one of many operations belonging to this class. Any regular sequence is said to "define a number" — this " number " being merely the symbolic, ideal, result of the operation which led to the sequence. It will sometimes be convenient to represent numbers thus defined by the single letters a, b, c, etc., which have heretofore represented posi- tive integers only. After some particular term all terms of the sequence a 1? a 2 , ••• may be the same, say a. The number defined by the sequence is then a itself. A place is thus provided for rational numbers in the general scheme of numbers which the definition contemplates. When not a rational, the number defined by a regular sequence is called irrational. The regular sequence .3, .33, •••, has a limiting value, viz., - ; which is to say that a term can be found in this sequence which itself, as well as each term which follows it, differs from - by less than any assignable number. In other words, o THE IBBATIONAL. the difference between r and the /xth term of the sequence o may be made less than any assignable number whatsoever by taking /x great enough. It will be shown presently that the number defined by any regular sequence, a 1? a 2 , ••• stands in this same relation to its term a M . 28. Zero, Positive, Negative. In any regular sequence a 1} a 2 , "• a term a^ may always be found which itself, as well as each term which follows it, is either (1) numerically less than any assignable number, or (2) greater than some definite positive rational number, or (3) less than some definite negative rational number. In the first case the number a, which the sequence de- fines, is said to be zero, in the second positive, in the third negative. 29. The Four Fundamental Operations. Of the numbers defined by the two sequences : a 1? a 2 , a 3 , ••• a^, a^ + i, ••• a^ + v , •••, HD H2> Ps> ••• Phi fin + b •*• fin + vy ••• (1) The sum is the number defined by the sequence : a l + A? a 2 + /?2> • • • a /n + A> • • • O-fJL + v — P^ + vi • • • (3) The product is the number defined by the sequence : a iA a 2/fe ... a^A, a^ + i/3^ + 1, ... CLu + vP^ + y, ... (4) T%e quotient is the number defined by the sequence : o-i a 2 a, x a a + i a a+r For these definitions are consistent with the correspond- ing definitions for rational numbers ; they reduce to these elementary definitions, in fact, whenever the sequences 30 NUMBER-SYSTEM OF ALGEBBA. a 1? a 2 , ...; ft, ft, ... either reduce to the forms a, a, ...; ft ft ... or have rational limiting values. They conform to the fundamental laws I-IX. This is immediately obvious with respect to the commutative, asso- ciative, and distributive laws, the corresponding terms of the two sequences c^ft, a 2 ft, ... ; fta l7 fta 2 , •••? f° r instance, being identically equal, by the commutative law for ra- tionals. But again division as just defined is determinate. For division can be indeterminate only when a product may vanish without either factor vanishing (cf. § 24) ; whereas a ift? o. 2 /3 2 , •" can define 0, or its terms after the nth fall below any assignable number whatsoever, only when the same is true of one of the sequences a 1? a 2 , ... ; ft, ft, ... # It only remains to prove, therefore, that the sequences (1), (2), (3), (4) are qualified to define numbers (§ 27). (1) and (2) Since the sequences a 1? a 2 , ... ; ft, ft, ... are, by hypothesis, such as define numbers, corresponding terms in the two, a M , ft, may be found, such that a^ + v — a^ is numerically < 8, and ft* + v — ft is numerically < 8, and, therefore, (a^ + v ± ft + r ) — (ft By choosing a^ and ft as before the numerator of this fraction, and therefore the fraction itself, may be- made less than any assignable number ; and that for all values of v. Therefore the sequence —, — , ••• is regular, ft ft 30. Equality. Greater and Lesser Inequality. Of two numbers, a and b, defined by regular sequences a ly a 2 , ... ; ft, ft, ..., the first is greater than, equal to or less than the second, according as the number defined by a x — (3^ a 2 — ft, ... is greater than, equal to or less than 0. This definition is to be justified exactly as the definitions of the fundamental operations on numbers defined by regu- lar sequences were justified in § 29. From this definition, and the definition of in § 28, it immediately follows that Cor. Two numbers ivhich differ by less than any assignable number are equal. 31. The Number Denned by a Regular Sequence is its Limiting Value. The difference between a number a and the term a^ of the sequence by which it is defined may be made less than any assignable number by taking /x great enough.. 32 NUMBER-SYSTEM OF ALGEBRA. For it is only a restatement of the definition of a regular sequence a 1? a 2 , ... to say that the sequence which defines the difference a — a^ (§ 29, 2), is one whose terms after the /xth can be made less than any assignable number by choosing /x great enough, and which, therefore, becomes, as /x is indefinitely increased, a sequence which defines (§ 28). In other words, the limit of a — a^ as /x is indefinitely increased is 0, or a = limit ( <^3 — <*3< fa, etc. Then limit (a m — a m ) = (§§ 28, 31), or limit (a m ) = limit (a m ). This theorem justifies the use of regular sequences of irrationals for defining numbers, and so makes possible a simple expression of the results of some very complex THE IMAGINARY. COMPLEX NUMBERS. 35 operations. Thus a m , where m is irrational, is a number; the number, namely, which the sequence a**, a a z, ... defines, when a b a 2 , ... is any sequence of rationals defining m. But the importance of the theorem in the present discus- sion lies in its declaration that the number-system defined by regular sequences of rationals contains all numbers which result from the operations of regular sequence-building in general. iTls a closed system with respect to the four fundamental operations and this new operation, exactly as the rational numbers constitute a closed system with respect to the four fundamental operations only (cf. § 25). The number-system defined by regular sequences of " (rationals contains every number which lies between the extreme limits of the rational number-system ( — cc , + ooj and with respect to whose relation to each and every num- ber of that system it can be said that it is either greater than, equal to or less than fhat number: greater, equal or less in the sense in which one rational is greater than, equal to, or less than another (compare §§ 28, 30 and § 21). Y. THE IMAGINAEY. COMPLEX NUMBEES. 34. The Pure Imaginary. The other symbol which is needed to complete the number-system of algebra, unlike the irrational but like the negative and the fraction, admits of definition by a single equation of a very simple form, viz., x 2 + 1 = 0. It is the symbol whose square is — 1, the symbol V— 1, now commonly written t\* It is called the unit of imag- inaries. In contradistinction to i all the forms of number hitherto considered are called real. These names, "real" and "imagi- * Gauss introduced the use of i to represent V— 1. 36 NUMBER-SYSTEM OF ALGEBRA. nary/'' are unfortunate, for they suggest an opposition which does not exist. Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irra- tional, but in no other sense ; all are alike mere symbols devised for the sake of representing the results of opera- tions even when these results are not numbers (positive integers), i got the name imaginary from the difficulty once found in discovering some extra-arithmetical reality to correspond to it. As the only property attached to i by definition is that its square is — 1, nothing stands in the way of its being " multiplied " by any real number a; the product, ia, is called &pure imaginary. An entire new system of numbers is thus created, coex- tensive with the system of real numbers, but distinct from it. Except 0, there is no number in the one which is at the same time contained in the other. * Numbers in either system may be compared with each other by the definitions of equality and greater and lesser inequality (§ 30), ia being called = ib, as a — b : but a number in one system cannot be said to be either greater than, equal to or less than a number in the other system. 35. Complex Numbers. The sum a + ib is called a com- plex number. Its terms belong to two distinct systems, of which the fundamental units are 1 and i. The general complex number a + ib is defined by a com- plex sequence «i + ipi, a 2 + i(3 2 , . . ., cv + ^73 M , . . ., where a 1? a 2 , ... ; /? 1? /3 2 , ... are regular sequences. * Throughout this discussion is not regarded as belonging to the number- system, but as a limit of the system, lying without it, a sym- bol for something greater than any number of the system. THE IMAGINARY. COMPLEX NUMBERS. 37 ' Since a = a + i'O (§ 36, 3, Cor.) and ib = -f t'6, all real numbers, a, and pure imaginaries, ib, are contained in the system of complex numbers a + ib. a + ib can vanish only when both a = and 6 = 0. 36. The Four Fundamental Operations on Complex Num- bers. The assumption of the permanence of the fundamen- tal laws leads immediately to the following definitions of the addition, subtraction, multiplication, and division of complex numbers. 1. (a + ib) + (a' + ib 1 ) = a + a' + i(b + 6') . For (a + i'6) + (a' + #>') = a + i6 + a' + ib', Law II. = a + a' + i*6 + ib', Law I. = a+a' + i(6+6') % Laws II, Y. 2. (a + ib) - (a' + ib') = a- a[ + i(b - 6'). By definition of subtraction (VI) and § 36, 1. Cor. The necessary as well as the sufficient condition for the equality of tivo complex numbers a + ib, a' + ib' is that a = a' and b = b'. For if (a + ib) - (a' + ib 1 ) =a-a' + i(b -6') = 0,. a - a' = 0, & - 6' = (§ 35), or a = a', 6 = 6'. \ 3. (a + ib) (a' + i'6') =*= ana' - 66' + i(ab' + ba'). For (a + 16) (a' + iV) =?*(a + 16) a'+ (a -f ib)'ib', Law V. == aa'+ib • tx' + a • i'6' + i'6 • ib', Law V. = (aa' — 66') + i(ab'+ba'). Laws I-V. Cor. ijf either factor of a product vanish, the product vanishes. For ixO= i(b - 6) = ib - 16 (§ 10, 5), = (§ 14, 1). Hence (a + 16) = C6 x + i6 X = a x + i(& X 0) = 0. • Laws V, IV, § 28, § 29, 3. 38 NUMBER-SYSTEM OF ALGEBRA. a a + ib __ aa' -f- bb' .ba' — ab f ' a ' + ib'~~ a' 2 +b' 2 % a' 2 + b' 2 ' For let the quotient of a + ib by a' + *&' be x + %• By the definition of division (VIII), (x -f iy) (a'-f- #') = a + id. .-. xa' — yb' -\-i(xb' -\-ya')~ a -{-ib. §36,3 .-. xa' — yb' = a, xb' + ya f = 6. § 36, 2, Cor. Hence, solving for x and ?/ between these two equations, aa' 4- bb' ba' — ab' X '2 + ^2 > ^ a /2 + b ,2 Therefore, as in the case of real numbers, division is a determinate operation, except when the divisor is 0; it is then indeterminate. For x and y are determinate (by IX) unless a' 2 + b' 2 = 0, that is, unless a' = b'= 0, or a'-f- ib' = 0; for a' and V being real, a' 2 and b' 2 are both positive, and one cannot destroy the other. # Hence, 'lay the reasoning in § 24, Cor. If a product of tivo complex numbers vanish, one of the factors must vanish. 37. Numerical Comparison of Complex Numbers. Two complex numbers, a + ib, a' + ib', do not, generally speak- ing, admit of direct comparison with each other, as do two real numbers or two pure imaginaries ; for a may be greater than a', while b is less than b'. They are compared numerically, however, by means of their moduli Va 2 -f b 2 , Va' 2 + b' 2 ; a + ib being said to be * What is here proven is that in the system of complex numbers formed from the fundamental units 1 and i there is one, and hut one, number which is the quotient of a + ib by «' -f ib' ; this being a conse- quence of the determinateness of the division of real numbers and the peculiar relation (i 2 = — 1) holding between the fundamental units. For the sake of the permanence of IX we make the assumption, otherwise irrelevant, that this is the only value of the quotient whether within or without the system formed from the units 1 and u THE IMAGINARY. COMPLEX NUMBERS. 39 numerically greater than, equal to or less than a' ■+- ib ! according as Va 2 + b 2 is greater than, equal to or less than Va' 2 + b' 2 . Compare § 47. 38. The Complex System Adequate. The system a + ib is an adequate number-system for algebra. For, as will be shown (Chapter VII), all roots of algebraic equations are contained in this system. But more than this, the system a + ib is a closed system with respect to all existing mathematical operations, as are the rational system with respect to all finite combinations of the four fundamental operations and the real system with respect to these operations and regular sequence-building. For the results of the four fundamental operations on complex numbers are complex numbers (§ 36, 1, 2, 3, 4). Any other operation may be resolved into either a finite combination of additions, subtractions, multiplications, divis- ions or such combinations indefinitely repeated. In either case the* result, if determinate, is a complex number, as fol- lows from the definitions 1, 2, 3, 4 of § 36, and the nature of the real number-system as developed in the preceding chapter (see Chapter VIII). The most important class of these higher operations, and the class to which the rest may be reduced, consists of those operations which result in infinite series (Chapter VIII); among which are involution, evolution, and the taking of logarithms (Chapter IX), sometimes included among the fundamental operations of algebra. 39. Fundamental Characteristics of the Algebra of Num- ber. The algebra of number is completely characterized, formally considered, by the laws and definitions I-IX and the fact that its numbers are expressible linearly in terms of two fundamental units.* It is a linear, asso- ciative, distributive, commutative algebra. Moreover, the * That is, in terms of the first powers of these units. 40 NUMBER-SYSTEM OF ALGEBRA. most general linear, associative, distributive, commutative algebra, whose numbers are complex numbers of the form xfa -h x 2 e 2 + • • • + x n e n , built from n fundamental units e ly e 2 , • ••, e n , is reducible to the algebra of the complex number a -f- ib. For Weierstrass * has shown that any two complex numbers a and b of the form x 1 e 1 + x 2 e 2 -h Y x r&m whose sum, difference, product, and quotient are numbers of this same form, and for which the laws and definitions I-IX hold good, may by suitable transformations be resolved into com- ponents a 1? Og, • •• a^ ; &u b% • • • b r , such that a = a 1 -{- a 2 -\ -f- a r , 6=^ + ^+... + ^, a ± b = a x ± &J -f a 2 ± & 2 -\ h a,. ± b r , ab — «!&! + «AH — + a A, a __ % a 2 , , a r & &! ft 2 & r The components a,-, 5 f are constructed either from one fun- damental unit g t or from two fundamental units g i} Avt For components of the first kind the multiplication for- mula is * Zur Theorie der aus n Haupteinheiten gebildeten complexen Grossen. Gottinger Nachrichten Nr. 10, 1884. Weierstrass finds that these general complex numbers differ in only- one important respect from the complex number a + ib. If the num- ber of fundamental units be greater than 2, there always exist num- bers, different from 0, the product of which by certain other numbers is 0. Weierstrass calls 'them divisors of 0. The number of exceptions to the determinateness of division is infinite instead of one. t These units are, generally speaking, not e v e 2 , ..., e n , but linear combinations of them, as y^ + y 2 e 2 H J- y n e n , k^ + K 2 e 2 -\ f- K n e n . Any set of n independent linear combinations of the units e v e 2 , ... e n may be regarded as constituting a set of fundamental units, since all numbers of the form a^ + a 2 e 2 ^ !- a n e n may be expressed linearly in terms of them. GBAPHICAL REPRESENTATION OF NUMBERS. 41 For components of the second kind the multiplication formula is (a 9i + pk t ) (a' 9i + p%) = (aa'- PP^+W + &)K And these formulas are evidently identical with the mul- tiplication formulas (al)(j81) = (aj8)l, (al + pi) (a'l + P'i) = (aa'- pp')l + (a/3' + pj)% of common algebra. VI. GEAPHICAL EEPEESENTATION OF NUMBEES. THE VAEIABLE. 40. Correspondence between the Real Number-System and the Points of a Line. Let a right line be chosen, and on it a fixed point, to be called the null-point ; also a fixed unit for the measurement of lengths. Lengths may be measured on this line either from left to right or from right to left, and equal lengths measured in opposite directions, when added, annul each other ; opposite algebraic signs may, therefore, be properly attached to them. Let the sign -f- be attached to lengths measured to the right, the sign — to lengths measured to the left. The entire system of real numbers may be represented by the points of the line, by taking to correspond to each number that point whose distance from the null-point is represented by the number. For, as we proceed to demonstrate, the distance of every point of the line from the null-point, measured in terms of the fixed unit, is a real number; and there is no real number which may not represent such a distance. 1. The distance of any point on the line from the null-point is a real number. Let any point on the line be taken, and suppose the seg- ment of the line lying between this point and the null-point 42 NUMBEB-SYSTEM OF ALGEBBA. to contain the nnit line a times, with a remainder d x , this remainder to contain the tenth part of the unit line f$ times, with a remainder d 2 , d 2 to contain the hundredth part of the unit line y times, with a remainder d 3 , etc. The sequence of rational numbers thus constructed, viz., a, a . /?, a . /fy, . . . (adopting the decimal notation) is regular ; for the difference between its /xth term and each succeeding term is less than — — , a fraction which may be made less than any assignable number by taking p great enough ; and, by construction, this number represents the distance of the point under consideration from the null-point. By the convention made respecting the algebraic signs of lengths this number will be positive when the point lies to the right of the null-point, negative when it lies to the left. 2. Corresponding to every real number is a point on the line, the distance of which from the nidi-point is represented by the number. This is immediately evident for rational numbers ; a rational length may be actually measured off, and so the point be actually constructed. If the number be irrational, let a 1? a 2? ... be a sequence of rationals defining it. There is a point on the line which the point corresponding to the term a^ of this sequence approaches as limit as //. is indefinitely increased, and whose distance from the null-point the number a, defined by a 1? a 2 , ..., represents. For among the numbers to which points do correspond (by 1), one can be found which is equal to a. For, let b (defined by /3 1} f3 2 ...) be that one of these numbers which differs least from a. If this difference is not 0, in the sequence ^—(3^ a 2 —/3 27 ... can be found a term a^—fi^ which itself, as well as each term a IJi + v —/3 l j i+v following it, is either greater than some positive rational number 8 or less than some negative rational number — S\ The GRAPHICAL REPRESENTATION OF NUMBERS. 43 number b */■ 8 (or b -^ 8') differs from a less than 6 differs from a; and a point corresponds to it, namely, the point got by measuring off from B (which by hypothesis cor- responds to b) the rational length 8 (or —8'). Therefore, unless b is equal to a, among the numbers to which points correspond is one which differs less from a than b does, which is contrary to hypothesis. 41. The Real Number-System Continuous. The Variable. The theorem just demonstrated is of the highest impor- tance, for it establishes the right to represent geometric magnitudes by numbers and to discuss geometric relations algebraically. This right is evidently due to the presence of the irra- tional in the system of numbers. The geometric magnitudes are continuous ; that is to say, the boundary separating two contiguous parts of a geometric magnitude is common to both these parts. For instance, the point C, at which a given line AB is divided into the segments AC, CB, belongs to both of these segments. It is altogether different with the series of the rational numbers. This series belongs to the class of discrete magnitudes, or magnitudes consecutive parts of which have distinct boundaries ; for, between any two rational numbers, however nearly equal, may always be inserted an irrational. The entire system of real numbers, however, inasmuch as it contains an individual number to correspond to every individual point in the continuous series of points forming a right line, is continuous. If a point be made to move continuously along a line, its distance from any fixed point on the line will run through a portion of this continuous number series. Any quantity which is supposed to be changing is called a variable; and if, like the distance under consideration, its successive values form a continuous series, it is called a continuous variable. 44 NUMBER-SYSTEM OF ALGEBRA. B 42. Correspondence between the Complex Number-System and the Points of a Plane. The entire system of complex numbers may be represented by the points of a plane, as follows : In the plane let two right lines X'OX and Y'OY be drawn intersecting at right angles at the point 0. Make X'OX the "axis" of real numbers, using its points to represent real numbers, after the manner described in § 40, and make Y'OY the axis of pure imaginaries, represent- ing ib by the point of OY whose distance from is b when b is positive, and by the corresponding point of OY' when b is negative. X The point taken to represent the complex number a+ib is P, constructed by drawing through A and B, the points which rep- resent a and ib, parallels to Y'OY and X'OX, respectively. The correspondence between the complex numbers and the points of the plane is a one-to-one correspondence. To every point of the plane there is a complex number corre- sponding, and but one, while to each number there corre- sponds a single point of the plane.* * * A reality has thus been found to correspond to the hitherto unin- terpreted symbol a + ib. But this reality has no connection with the reality which gave rise to arithmetic, the number of things in a group of distinct things, and does not at all lessen the purely symbolic char- acter of a -f ib when regarded from the standpoint of that reality, the standpoint which must be taken in a purely arithmetical study of the origin and nature of the number concept. The connection between the numbers a-f ib and the points of a plane is purely artificial. The tangible geometrical pictures of the relations among complex numbers to which it leads are nevertheless a valuable aid in the study of these relations. Fig. 1. GRAPHICAL REPRESENTATION OF NUMBERS. 45 It follows, by the reasoning of § 41, that the system of the complex numbers is a continuous system. If the point P be made to move about in its plane, the corresponding number runs through a continuous series of complex values, and is called a complex variable. 43. M odulus. The length of the line OP (Fig. 1), i.e. V«- + b 2 , is called the modulus of a + ib. Let it be repre- sented by p. ~ — 44. Argument. The angle XOP made by OP with the positive half of the axis of real numbers is called the angle of a + ib, or its argument. Let its numerical measure be represented by 0. The angle is always to be measured " counter-clockwise " from the positive half of the axis of real numbers to the modulus line. 45. Sine. The ratio of PA, the perpendicular from P to the axis of real numbers, to OP, i.e. -, is called the sine of 0, written sin 0. P Sin is by this definition positive when P lies above the axis of real numbers, negative when P lies below this line. • 46. Cosine. The ratio of PB, the perpendicular from P to the axis of imaginaries, to OP, i.e. -, is called the cosine of 0, written cos 0. ? Cos is positive or negative according as P lies to the right or the left of the axis of imaginaries. 47. Theorem. The expression of a + ib in terms of its modulus and angle is p (cos + i sin 0) . For by § 46 - = cos 0, .-. a = p cos ; P and by § 45, - = sin 0, .-. b = p sin 0. P Therefore a + ib = p (cos -f- i sin 0). 46 NVMBEB-SYSTEM OF ALGEBBA. The factor cos + i sin has the same sort of geometrical meaning as the algebraic signs + and — , which are indeed but particular cases of it : it indicates the direction of the point which represents the number from the null-point. It is the other factor, the modulus p, the distance from the null-point of the point which corresponds to the number, which indicates the " absolute value " of the number, and may represent it when compared numerically with other numbers (§ 37), — that one of two numbers being numeri- cally the greater whose corresponding point is the more distant from the null-point. 48. Problem I. Given the points P and P f , representing a-j-ib and a' + ib' respectively; required the point represent- ing a + a f -f- i (p + &') • The point required is P", the intersection of the parallel to OP through P 1 with the parallel to OP f through P. For completing the construction indicated by the figure, we have OD 1 = PE = DD", and therefore OD" = OD+ OD 1 ; and similarly P"D" = PD + P'D\ Cor. I. To get the point corresponding to a — a'-t-i(b — b'), produce OP' to P'" y making OP'"=OP', and complete the parallelogram OP, OP 1 ". Cor. II. The modulus of the sum or difference of two complex numbers is less than {at greatest equal to) the sum of their moduli. For OP" is less than OP + PP" and, therefore, than OP FlG<2> +OP\ unless 0, P, P' are in the same • straight line, when 0P"= 0P+ OP 1 . Similarly, PP', which is equal to the modulus of the difference of the numbers represented by P and P', is less than, at greatest equal to, OP + OP 1 . GBAPHICAL REPRESENTATION OF NUMBERS. 47 49. Problem II. Given P and P', representing a + ib and a* + ib' respectively ; required the point representing (a + ib)(a' +ib'). Let a -M& = p (cos + i'sin0), §47 and a' +ib' = p'(cos & + * sin #') 5 then (a + ifyty' + iV) = pp'(cos + £ sin 0) (cos 0' + i sin 0') = pp'[_ (cos cos 6' — sin sin 0') + 1 (sin cos 0' + cos sin 0') ] . But • cos (9 cos (9' — sin 0sin0' = cos (0 + 0'),* and sin cos 0' + cos sin 0' = sin (0 + 0') * Therefore (a+t&) (a' + iV)=pp'[cos(0 + 0O+*'sin(0+0')] ; or, The modulus of the product of two complex numbers is the product of their moduli, its argument the sum of their arguments. The required construction is, therefore, made by drawing through a line making an angle + 0' with OX, and lay- ing off on this line the length pp'. Cor. I. Similarly the product of n numbers having moduli p, p, p", • •• p (n) respectively, and arguments 0, 0', 0", ... (n) , is the number pp'p" ... P <">[cos(0 + 0' + 0" + ... 0<">) + isin(0 + 0' + 0" + ...0< n) )]. In particular, therefore, by supposing the n numbers equal, we may infer the theorem [p (cos + i sin 0) ] n = p n (cos nO + i sin nO) , which is known as Demoivre's Theorem. * For the demonstration of these, the so-called addition theorems of trigonometry, see Wells' Trigonometry, § 65, or any other text-book of trigonometry. 48 NUMBER-SYSTEM OF ALGEBBA. Cor. II. From the definition of division and the preceding demonstration it follows that ^| f = £ [ cos(.-^) + -in(^^)] ; the construction for the point representing ^ is, there- fore, obvious. . a ' + ib ' 50. Circular Measure of Angle. Let a circle of unit radius be constructed with the vertex of any angle for centre. The length of the arc of this circle which is intercepted between the legs of the angle is called the circular measure of the angle. 51. Theorem. Any complex number may be expressed in the form pe ie ; ivhere p is its modulus and the circular meas- ure of its angle. It has already been proven that a complex number may be written in the form p(cos + ism #), where p and have the meanings just given them. The theorem will be demon- strated, therefore, when it shall have been shown that e ie = cos -f i sin 0. If n be any positive integer, we have, by § 36 and the binomial theorem, 1 | MY =1 | n M | n(n-l) (JO)' nj n 2 ! ri 1 , n(n-l)(n-2) (W) 3 , 3! 1 = 1 + 0+ * (*■*)« »8 1- ~2l i-m- 2 +± — ^ — ^w+. Let n be indefinitely increased ; the limit of the right side of this equation will be the same as that of the left. GRAPHICAL REPRESENTATION OF NUMBERS. 49 But the limit of the right side is (joy , (My , 1 + iO- 21 3! i.e. e 40 # 20\ n Therefore e 1 ' 9 is the limit of f 1 H — ) as n approaches oo. \ n J f i0\ n To construct the point representing ( 1 -\ — ) : V nJ On the axis of real numbers . lay off OA = 1. Draw AP equal to and par- allel to OB, and divide it into n equal parts. Let AA Y be one of these parts. Then A Y is the point 1 H n Through A x draw A Y A 2 right angles to OA Y and con- struct the triangle OA 1 A 2 simi- lar to OAA Y . A 2 is then the point [ 1 + n j For AOA 2 = 2AOA 1 ; and since OA 2 : 0A 1 : : 0A X : OA, and OA = 1, the length OA 2 = the square of length OA x . (see § 49) In like manner construct A 3 to represent tl-\ — j, A± for Let 7i be indefinitely increased. The broken line AA Y A 2 ... A n will approach as limit an arc of length of the circle of radius OA and, therefore, its extremity, A n , will approach as limit the point representing cos + i sin (§ 47). * This use of the symbol e** will be fully justified in § 73. 50 NUMBER-SYSTEM OF ALGEBRA. f i0\ n Therefore the limit of (l-\ — ] as n is indefinitely in- V n J creased is cos + i sin 0. Bnt this same limit has already been proved to be e ie . Hence e ie = cos -f- i sin Q* VII. THE FUNDAMENTAL THEOKEM OP ALGEBEA. 52. The General Theorem. If w = a^ n + a^ 1 + a 2 z n ~ 2 -{ 1- a n ^z + a n , where n is a positive integer, and a , a 1} . . ., a n any numbers, real or complex, independent of z, to each value of z corre- sponds a single value of w. We proceed to demonstrate that conversely to each value of iv corresponds a set of n values of z, i.e. that there are n numbers which, substituted for z in the polynomial a& n -f ctiZ 71 ' 1 -f- ••• + a n , will give this polynomial any value, to , which may be assigned. It will be sufficient to prove that there are n values of z which render a z n + a^z 71 " 1 -\ + a n equal to 0, inasmuch as from this it would immediately follow that the polynomial takes any other value, iv , for n values of z ; viz., for the values which render the polynomial of the same degree, a z n + a^ 1 -\ -f- (a n — w ), equal to 0. 53. Root of an Equation. A value of z for which a 2 n -f- a^* -1 -f- • • • + a n is is called a root of this poly- nomial, or more commonly a root of the algebraic equation aoZ n + c^z 11 - 1 -\ h a n = 0. * This demonstration is due to Dr. F. Franklin. See American Journal of Mathematics, Vol. VII, p. 376. FUNDAMENTAL THEOBEM OF ALGEBRA. 51 54. Theorem. Every algebraic equation has a root. Given the equation described in § 52, w = a$ n + a$ x ~ Y + a 2 z n ~ 2 -f- • • • -f- a n _ x z + a n - We are to demonstrate that in the system of complex numbers there is a value which, if assigned z, will render w = ; or for which the point representing w in the plane of complex numbers (the w-point we may call it) will coincide with the null-point. If not, let P be a point nearer to than any other with which the w-point can be made to coincide (or at least as near as any other). Through P draw a circle having its centre in the null- point 0. Then, by the hy- pothesis made, no value can be given z which will bring the corresponding w-point within this circle. But the w-point can be brought within this circle. For,'z and iv Q being the values of z and w which corre- spond to P, change z by adding to z a small increment 8, and let A represent the consequent change in w. A is defined by the equation (w + A) = a (z + 8)" + Oi(% + S)' 1 " 1 . + a 2 (z + 8) n ~ 2 + • • • + a n _ Y (z + 8) + a n . On applying the binomial theorem and arranging the terms with reference to powers of 8, the right member of this equation becomes '[eos(0 + 0') + *sin (0 + 0')] + terms involving p 2 , p 3 , etc. § 49. The point which represents pp' [cos (0 + 0') -\-isui(6 + 0')] for any particular value of p can be made to describe a circle of radius pp' about the null-point by causing to increase continuously from to 4 right angles. In the same circumstances the point representing w + pp' [cos (0 + 0') + i sin (0 + 0') ] will describe an equal circle about the point P and, there- fore, come within the circle OP. But by taking p small enough, A may be made to differ as little as we please from pp' [cos (0 + 0') + tsin(0 + 0')], # and, therefore, the curve traced out by P' (which represents w -f A, as runs through its cycle of values), to differ as little as we please from the circle of centre P and radius pp'. Therefore by assigning proper values to p and 0, the w- point (P') may be brought within the circle OP. * In the series Ap + Bp 2 + Cp 3 + etc. , the ratio of all the terms fol- lowing the first to the first, i.e. Bp 2 -f C p 3 + etc. _ v B + Cp + etc. . > — p x - , Ap A which by taking p small enough may evidently be made as small as we please. FUNDAMENTAL THEOREM OF ALGEBRA. 53 The to-point nearest the null-point must therefore be the null-point itself.* 55. Theorem. If a be a root of a z n -f a^ -1 -\ h a n , this polynomial is divisible by z — a. For divide a^z n -f- a^ 71 " 1 + • • • -f- a n by z — a, continuing the division until z disappears from the remainder, and call this remainder B, the quotient Q, and, for convenience, the poly- nomial f(z) . Then we have immediately f(z) = (z-a)Q + B, holding for all values of z. Let z take the value a ; then f(z) vanishes, as also the product (z — a) Q. Therefore when z = a, R = 0, and being independent of z it is hence always 0. 56. The Fundamental Theorem. The number of the roots of the polynomial a Q z n -f- a^z 1 ^ 1 -f- ... -f- a n is n. For, by § 54, it has at least one root ; call this a ; then, by § 55, it is divisible by z — a, the degree of the quotient being n — 1. Therefore we have o^+a^-M \-a n =(z— a)(a » n " 1 +&i» n-2 H h&»~i)> Again, by § 54, the polynomial a^ 71 - 1 + b x z n - 2 -\ h b n _ Y has a root ; call this /?, and dividing as before, we have a^ n + a x z n ~ l + . . . +a„= (*r^») (z-0) (a^ n ~ 2 + c^' 3 + . . . + c n _ 2 ) . * In the above demonstration it is assumed that the coefficient of 8, i.e. na z n ~ l -f (n — l)^ 71 " 2 + ••• + «n-i, is not 0. If it be 0, it is only necessary to take instead of P some other point on the circle OP ; na^- 1 -f etc., will evidently not vanish for all points of this circle, since the number of its roots would then be infinite (see § 56) . 54 NUMBER-SYSTEM OF ALGEBRA. Since the degree of the quotient is lowered by 1 by each repetition of this process, n — 1 repetitions reduce it to the first degree, or we have ao^+a^-M \-a n = a (z-a)(z-P)(z-y)--'(z-v), a product of n factors, each of the first degree. Now a product vanishes when one of its factors vanishes (§ 36, 3, Cor.), and the factor z — a vanishes when z — a, z — ft when z—ft, • • •, z — v when z — v. Therefore a z n +a 1 2f~ 1 + ••• + a> n vanishes for the n values, a, ft, y, ••• v, of z. Furthermore, a product cannot vanish unless one of its factors vanishes (§ 36, 4, Cor.), and not one of the factors z — a, z — ft, •••, z — .v, vanishes unless z equals one of the numbers a, ft, • • • v. The polynomial has therefore n and but n roots. The theorem that the number of roots of an algebraic equation is the same as its degree is called the fundamental theorem of algebra. VIII. INFINITE SEEIES. 57. Definition. Any operation which is the limit of ad- ditions indefinitely repeated produces an infinite series. We are to determine the conditions which an infinite series must fulfil to represent a number. If the terms of a series are real numbers, it is called a real series; if complex, a complex seizes. I. REAL SERIES. 58. Sum. Convergence. Divergence. An infinite series % + ^2 + ^sH M»H represents a number or not, according as the sequence S l) S 2) S 3) '" S /«? S m+1? •" S m+n) "*) where s x = a ly s 2 — «i + a 2 , • • •, s t = a, + a 2 -f • • • <%i, is regular or not. INFINITE SERIES. 55 If Sa s 2 ---, be a regular sequence, the number which it defines, or lim (s n ), is called the s?m of the infinite series a-i + a 2 + a 3 + h«„H j and the series is said to be convergent. If s l7 s 2 , be not a regular sequence, s n either transcends any finite value whatsoever, as n is indefinitely increased, or while remaining finite becomes altogether indeterminate. The infinite series then has no r sum, and is said to be diver- gent. The series 1 + 1 + 1 + ••• and 1 — 1 + 1 — 1 + ••• are examples of these two classes of divergent series. A divergent series cannot represent a number. 59. General Test of Convergence. From these definitions and § 27 it immediately follows that : The infinite series a x + a 2 + ••• + a m -\ is convergent when m may be so taken that the differences s m+n — s m are numeri- cally less than any assignable number 8 for cdl values of n, where s m and s m+n are the sum of the first m and of the first m-\-n terms of the series respectively. If these conditions be not fulfilled, the series is divergent. The limit of the last term of a convergent series is ; for the condition of convergence requires that by taking m great enough, s m+1 — s m , i.e. a m+1 , may be found less than any assignable number. But it is not to be assumed con- versely that a series is convergent, if the limit of its last term is ; other conditions have also to be fulfilled, s m+n — s m must be less than 8 for all values of n. *\ 1 1 Thus the limit of the last term of the series \-\ 1 1- ••• is ; but, as will presently be shown, this is a divergent series. 60. Absolute Convergence. It is important to distin- guish between convergent series which remain convergent when all the terms are given the same algebraic signs and »s 56 X UMBER-SYSTEM OF ALGEBRA. convergent series which become divergent on this change of signs. Series of the first class are said to be absolutely con- vergent ; those of the second class, only conditionally con- vergent. Absolutely convergent series have the character of ordinary sums; i.e. the order of the terms may be changed without altering the sum of the series. For consider the series u x + a 2 -f a 3 + ••• supposed to be absolutely convergent and to have the sum S, when the terms are in the normal order of the indices. It is immediately obvious that no change can be made in the sum of the series by interchanging terms with finite indices ; for n may be taken greater than the index of any of the interchanged terms. Then S n has not been affected by the change, since it is a finite sum and it is immaterial in what order the terms of a finite sum are added ; and as for the rest of the series, no change has been made in the order of its terms. But a x -f a 2 -f a 3 -f- • • • may be separated into a number of infinite series, as, for instance, into the series a x + a 3 + a 5 + .•• and a 2 + cr 4 + a 6 + • ••, and these series summed separately. Let it be separated into I such series, the sums of which — they must all be absolutely convergent, as being parts of an absolutely convergent series — are S^\ S&\ ••• SV\ respectively ; it is to be proven that s= m + #(2) + sw + - + s®. Let 8^\ S^\ ••• be the sums of the first m terms of the series # (1 \ S( 2 \ • ••, respectively. Then, by the hypothesis that the series a x -\- a 2 -\- • •• is absolutely convergent, m may be taken so large that the sum S m+n + S m+n + '" + ^ +n shall differ from S by less than any assignable number 5 for all values of n ; therefore the limit of this sum is S. But again, n may be so taken that S^ shall differ from SW by less than -, S^ 2 ] from S (2) by less than -, • • • ; and therefore the sum Cn +C + - + S% +n from S {1) + S {2) + - + S {1) by less than (^ ) I ; i.e. by less than 8. Hence the limit of this sum is SW + # (2) + ••• + SM. Therefore S and #0) + SW + ••• + S® are limits of the same finite sum and hence equal. INFINITE SERIES. 57 61. Conditional Convergence. On the other hand, the terms of a conditionally convergent series can be so arranged that the sum of the series may take any real value whatsoever. In a conditionally convergent series the positive and the negative terms each constitute a divergent series having for the limit of its last term. If, therefore, C be any positive number, and S n be constructed by first adding positive terms (beginning with the first) until their sum is greater than (7, to these negative terms until their sum is again less than O, then positive terms till the sum is again greater than (7, and so on indefinitely ; the limit of S n , as n is indefinitely increased, is C. 62. Special Tests of Convergence. 1. If each of the terms of a series a x + a 2 + • • • be numerically less than (at greatest equal to) the corresponding term of an absolutely convergent series, or if the ratio of each term of a 1 -f- a 2 + • • * to the corre- sponding term of an absolutely convergent series never exceed some finite number C, the series a 1 + a 2 + • • • is absolutely convergent. If on the other hand, each term of a 1 + a 2 + • • • be numeri- cally greater than (at the lowest equal to) the corresponding term of a divergent series, or if the ratio of each term of d\-\- a 2 -\ to the corresponding term of a divergent series be never numerically less than some finite number C ! , different from 0, the series a 1 -f- a 2 -f- ••• is divergent. 2. The series a x — a 2 + a 3 — a 4 H , the terms of which are alternately positive and negative, is convergent, if after some term a { each term be numerically less or, at least, not greater than the term which immediately precedes it, and the limit of a n , as n is indefinitely increased, be 0. For here S m +n — S m =: (— l) m [a m+ i — « m+2 + ... (— l)"-^^]. The expression within brackets may be written in either of the forms (a m +i — a m+2 ) + (a m+3 — a m+4 ) + ••• (1) or a m+ i — (a m+2 — a m+3 ) . (2) + 2A-lj' 58 NUMBER-SYSTEM OF ALGEBRA. It is therefore positive, (1), and less than a m+h (2); and hence by- taking m large enough, may be made numerically less than any assign- able number whatsoever. The series 1 j f- ... is, by this theorem, con- , 2 3 4 ' J ' vergent. 3. The series 1-J \-~-\ h .■••• is divergent. Tor the first 2* terms after the first may be written 2 + \2^1 + T+2j + U 2 + 1 + 22 + 2 + 2* + 3 + 2* + & ) + + f 1 + —*— + ■■■ \2A-1 + 1 2A-1 + 2 2A-1 + 2A- where, obviously, each of the expressions within parentheses is greater than — 2 The sum of the first 2* terms after the first is therefore greater than -, and may be made to exceed any finite quantity whatsoever by tak- 2 ing \ great enough. This series is commonly called the harmonic series. By a similar method of |3roof it may be shown that the series 1 + — -f- — -\ is convergent if p >1. Here, 1 + J- 6p I* 4p \2p ) and the sum of the series is, therefore, less than that of the decreasing 2 / 2 \ 2 geometric series 1 + J- ( — ) +•••. The series 1 H — -f- — - -| is divergent if p < 1, the terms being then greater than the corresponding terms of 2 3 4. The series a 1 + a 2 + a 3 +". is absolutely convergent if after some term of finite index, a t , the ratio of each term to that which immediately precedes it be numerically less than 1 and, as the index of the term is indefinitely increased, approach INFINITE SERIES. 59 a limit which is less than 1 ; but divergent, if this ratio and its limit be greater than 1. For — to consider the first hypothesis first — let a be the 'greatest value which this ratio has after the term a t . By the hypothesis a is a fraction. Then, 5i±l < a, .\ a i+l < a ia ; -^ < a, .*. a i+ 2 S a i+ia ^ a*** 1 ,2 *i+l ■ a *' + * < a, /. a i+k < a i+Ck -i )a < — < a ia *. tf< + (ft_l) The given series is therefore < Si + 0<[a + a 2 + a 3 + — a k + — ]• And this is an absolutely convergent series. For a + a 2 + ~. a* + - = fS^ (a + a 2 + ... + a-) _ limit { a — a n+i \ ~n = cc [ i_ a ) = — - — , since a is a fraction. The given series is therefore absolutely convergent, § 62, 1. The same course of reasoning would prove that the series is diver- gent when after some term a* the ratio of each term to that which precedes it is never less than some quantity, a, which is itself greater than 1. When the limit of the ratio of each term of the series to the term immediately preceding it is 1, the series is some- times convergent, sometimes divergent. The series con- sidered in § 62 ? 3 are illustrations of this statement. 63. Limits of Convergence. An important application of the theorem just demonstrated is in determining what are called the limits of convergence of infinite series of the form Oq -f- a x x + a 2 x? -f- a 3 x s + • • •, 60 NUMBEB-SYSTEM OF ALGEBBA. where x is supposed variable, but the coefficients a , a 1? etc., constants as in the preceding discussion. Such a series will be convergent for very small values of x, if the coefficients be all finite, as will be supposed, and generally divergent for very great values of X) and by the limits of conver- gence of the series are meant the values of x for which it ceases to be convergent and becomes divergent. By the preceding theorem the series will be convergent if the limit of the ratio of any term to that which precedes it be numerically less than 1 ; i.e. if limit n = oo M>n+l X or limit f a '^x\ <1: that is, if x be numerically < limit ( — — ) ; and divergent, if x be numerically > limit I — - 1. Thus the infinite series a m + ma m ^x -\ * ) -a m ~ 2 x L -f- • • •, which is the expansion, by the binomial theorem, of (a-\-x) m for other than positive integral values of m, is convergent for values of x numerically less than a, diver- gent for values of x numerically greater than a. For in this case m(m — l)---(m — n -f 1)~ limit / a n \ _ limi aX- in) l m(m — ])-"(m — n) limit / w n + 1 ' imit \ n J limit ' n = oo -1 + - INFINITE SERIES. 61 2. Again, the expansion of e x , i.e. 1 -f- x -\ 1- ... ; is con- vergent for all finite values of x. For here limit f-M= limit -il The same is true for the series which is the expansion of a x . 64. Operations on Infinite Series. 1. The sum of two convergent series, % + % -]-••• and b l -j-b 2 -\- • •-, is the series (<2j +&!) + (a 2 + b 2 ) + ••• ; and their difference is the series (a 1 — b 1 )-\-(a 2 — b 2 )-{ . The sum of the series a x + a 2 + • •• is the number defined by s 1$ s 2 , ..., and the sum of the series b l -\-b 2 -] — is the number defined by t v t 2 , ..., where s^ = a x + a 2 -\ \-a t and t i =b l -\-b 2 -\ \-b L . The sum of the two series is therefore the number defined by s Y + t v s 2 + t v • «•, § 29, (1). But if Si = («! + &0 + (« 2 + b 2 ) H f- (a t -f bt), we have St = s t + U for all values of i. This is immediately obvious for finite values of r, and there can be no difference between Si and *< + t t as s approaches co, since it would be a difference having for its limit. Therefore the number defined by s x + t v s 2 -f t 2 , • ••, is the sum of the series (a x + b{) -f (a 2 + 6 2 ) + • ••. 2. TAe product of two absolutely convergent series &i + a 2H °m^ &i + b 2 -\ is the series a ± b x + (ajj 2 + aj^i) + ( a A + <^2 + a 3^i) H + O A + «A-1 H h a»-l& 3 + « A) H • Each set of terms within parentheses is to be regarded as constitut- ing a single term of the product ; and it will be noticed that the first of them consists of the one partial product in which the sum of the indices is 2, the second of all in which the sum of the indices is 3, etc. By § 29, (3), the product of a, + a 2 + ... by \ + b 2 + ... is J 1 ™^ «*.)'• where s n and t n represent the sums of the first n terms of a x -f a 2 + •••, L\ + fyj + m "t respectively. Suppose first that the terms of a l + a 2 +"> and b x + b 2 H are all positive. Then if S n be the sum of the first n terms of 62 NUMBER-SYSTEM OF ALGEBRA. a A + ( a A + a 2^i) + ••*» an d m represent - when n is even and n ~~ when n is odd, evidently s n ^ n > & > «**«. -o , limit , . v limit / . x But n ± ^ (S n t n ) = n ± ao (Smtm). Therefore XW;-^M>- If the terms of o^ + a 2 + •••, 6j + 6 2 + ••• be not all of the same sign, call the sums of the first n terms of the series got by making all the signs plus, s n f and t n ' respectively ; also S n ', the sum of the first n terms of the series which is their product. Then by the demonstration just given limit (,S"„)= limit 0V'»); n = go v nj n — go v . nn but S n always differs from s n t n by ress than (at greatest by as much as) S' n from s'j'n ; therefore, as before, limit («„)= limit ( SA ). n = go v nj n = co K nnj 3. The quotient of a 2 + a 2 -\ by b l -\-b 2 -\ does not admit of simple expression in terms of the a/s and 6/s. It is an absolutely convergent series when a l -\-a 2 -\ and bi -f- b 2 -I- • • • are absolutely convergent and the sum of b\ + b 2 H is not 0. II. COMPLEX SERIES. The terms sum, convergent, divergent, have the same mean- ings in connection with complex as in connection with real series. 65. General Test of Convergence. A complex series, ** convergent wlien the modulus of s m+n — s TO ma?/ 6e made ?ess £7ia?i amy assignable number 8 5?/ taking m great enough, and that for all values of n; divergent, when this condition is not satisfied. See § 48, Cor. II ; § 59. 66. Of Absolute Convergence. Let a t Z\ :. m{~ ]>a^). But this is an abso- lutely convergent series (§62,4). Hence the series a + a Y z -\ is absolutely convergent for all values of z within the circle through Z (§62, 1). 64 NUMBER-SYSTEM OF ALGEBRA. 2. For a point on the circumference of this circle the series may be convergent or it may be divergent. Thus the circle of convergence of the series 1 + | -f — -] — is of radius unity, and the series is convergent for the point — 1, divergent for -f 1. 68. Theorem. The following is a theorem on which many of the properties of functions defined by series depend. If the series a + a x z + a 2 z 2 H \- a n z n -\ have a circle of convergence greater than the null-point itself and z run through a regular sequence of values z Y , z 2 , ... de- fining 0, the sum of all terms following the first, viz., a x z + a 2 z 2 + — h a n z n + • • • ivill run through a sequence of values likeivise regular and defining ; or, the entire series may be made to differ as little as one chooses from its first term a . The numbers z v z 2 , ••• are, of course, all supposed to lie within the circle of convergence, and for convenience, to be real. It will be con- venient also to suppose «i>*{>Zn etc.; i.e. that each is greater than the one following it. Since a + a L z + a 2 z 2 + ••• + a n z n -{ converges absolutely for z = z v so also does a Y z + a 2 z 2 + M^ n +-, and, therefore, a x + a 2 z -\ — + a n z n '-\- ••♦. Hence A x + A 2 z Y -\ \- A n z x n H — (where Ai — modulus ai) is convergent, and a number M can be found greater than its sum. And since for z = z 2 , z 3 , • •• the individual terms of A l + Aji+ — + A f &+ — are less than the corresponding terms of A x + A 2 z x + ••• -f A n z™ + •••, this series and, therefore, modulus (a x -f a 2 z + •••) remain always less than M as z runs through the sequence of values z 2 , z 3 , •••. Hence the values of modulus (a x z + a 2 z 2 + •••) which correspond to z = z v z 2 ••• constitute a regular sequence denning 0, each term being numerically less than the corresponding term of the regular sequence z x M, z 2 M, • • • which defines 0. INFINITE SERIES. 65 Cor. The same argument proves that if or z m (a m + be the sum of all terms of the series from the (m -f- l)th on, the series a m + a m+1 z -\ can be made to differ as little as one may please from its first term a m . 69. Operations on Complex Series. The definitions of sum, difference, and product of two convergent complex series are the same as those already given for real series, viz. : 1. The sum of two convergent series, a^ a 2 -\ and °i 4- b 2 + • • •, is the series (a x + b x ) + (a 2 + b 2 ) + • • • ; their difference, the series (c^ — & x ) + (a 2 — b 2 ) -f- • ••. For if s . = a Y + a 2 + • •• + a { and U = b x + b 2 -f • •• + &<, modulus [(5 TO+n ± * m+n ) - (s m ± t m )~\ < modulus (s m+n - s TO ) + modulus (f m+n - f m ), and may, therefore, be made less than any assignable number by tak- ing m great enough. The theorem therefore follows by the reasoning of § 64, 1. 2. The product of two absolutely convergent series, &i + &2 + % + • • • and £\ + b 2 + b 3 H , is the series a^ + {a$ 2 + a 2 &i) + (afis + a 2 £> 2 + a s bi) • ••. For, letting 5< = A Y + A 2 + ••• + A { and Ti = j8, + B 2 + ••• + B h where A it Bt, are the moduli of a t -, 6;, respectively, and representing by A n =(-iy-^,.... A n As in the case of the exponential function, a part of the equations among the coefficients are sufficient to determine them all in terms of the one coefficient A Y . But as in that case (by assuming the truth of the binomial theorem for negative integral values of the exponent) it can be readily shown that these values will satisfy the remaining equa- tions also. The series *--+- +(- l) n ~ 1 - + ••• 2 3 n converges for all values of z whose moduli are less than 1 (§ 62, f). For such values, therefore, the function ^-| 2 +-+(-l) n - 1 f+-) ( 9 ) satisfies the functional equation /[(! + *)(! +0]=/(l+z)+/(l + 0- 74 NUMBEB-SYSTEM .OF ALGEBBA. And since z = 1 — (1 — z) and t = 1 — (1 — t) f the function - aIi-z ^ 1 ^ ' + ... + Q^¥+...\ satisfies this equation when written in the simpler form /(at) =f(z) +f(t), . for values of 1 — z and 1 — t whose moduli are both less than 1. 1. Log e b. To identify the general function /(l + 2) with the particular function log e (l+3) it is only necessary to give the undetermined coefficient A the value 1. For since log e (l+z) belongs to the class of functions which satisfy the equation (8), log e (l + z) = A(z-^+..X Therefore = i+ ^H+-) + 2 V 2 ( z -l 2+ - Y+ - But e l0 s e (i+*) = l+z. Hence z* l + , = l + ^-|+...j + ^-| + ...J + ...; or ? equating the coefficients of the first power of z, A = 1. The coefficients of the higher powers of z in the right number are then identically 0. It has thus been demonstrated that log e & is a number (real or complex), if when b is written in the form 1+2, the absolute value of z is less than 1. To prove that it is a number for other than such values of b, let b = pe ie (§ 51), where p, as being the modulus of b, is positive. Then log e 6 = log e/ o + i0, and it only remains to prove that log e p is a number. Let p be written in the form e n — (e n — p), where e n is the first integral power of e greater than p. THE LOGARITHMIC FUNCTION. 75 Then since § ■ e n - (e n — p) = e n /l — e n p \ loge/o = log e e n + log/l - e -^~J 1 ^ j is a number since " is less than 1. 2. Log a b. It having now been fully demonstrated that a z is a number satisfying the equation a z a T == a z+T for all finite values of a, Z, T\ let a z = z, a T = t, and call Z the logarithm of z to the base a, or log a 2, and in like manner T, log.*. Then, since zt = a z a T = a z+r , • log a (^) = log a s+log a *, or log a z belongs, like log e z, to the class of functions which satisfy the functional equation (8). Pursuing the method followed in the case of log e b, it will be found that log a (l -f z) is equal to the series A(z — ^--\ ) when A = . This number is called the modulus of log e a the system of logarithms of which a is base. 77. Indeterminateness of loga. Since any complex num- ber a may be thrown into the form pe ie , log e a = log e/ o + i<9. (10) This, however, is only one of an infinite series of possible values of log e a. For, since e ie = e*(*±2»») (§ 75), log e a = log eP e*(0 ±2 ™O= log e/ o + i(0 ±2mr), where n may be any positive integer. Log e a n-i> '•' a o are a ^l numbers less than 10 ; and then repre- sented by the mere sequence of numbers a n a n _! • • • a — it being left to the position of any number a { in this sequence to indicate the power of 10 with which it is to be associ- ated. For a system of this sort to be complete — to be capable of representing all numbers unambiguously — a sym- bol (0), which will indicate the absence of any particular power of 10 from the sum a n 10 n + a^lO*"" 1 -\ J-%10 + a ', is indispensable. Thus without 0, 101 and 11 must both be written 11. But this symbol at hand, any number may be expressed unambiguously in terms of it and symbols for 12 ... 9 The positional idea is very old. The ancient Babylonians commonly employed a decimal notation similar to that of the Egyptians ; but their astronomers had besides this a very remarkable notation, a sexagesimal positional system. HISTORIC SYSTEMS OF NOTATION. bi In 1854 a brick tablet was found near Senkereh on the Euphrates, certainly older than 1600 B.C., on one face of which is impressed a table of the squares, on the other, a table of the cubes of the numbers from 1 to 60. The squares of 1, 2, ••• 7 are written in the ordinary decimal notation, but 8 2 , or 64, the first number in the table greater than 60, is written 1, 4 (1 X 60 + 4) ; similarly 9 2 , and so on to 59 2 , which is written 5S, 1 (58 X 60 + 1) ; while 60 2 is written 1. The same notation is followed in the table of cubes, and on other tablets which have since been found. This is a posi- tional system, and it only lacks a symbol for of being a perfect positional system. The inventors of the 0-symbol and the modern complete decimal positional system of notation were the Indians, a race of the finest arithmetical gifts. The earlier Indian notation is decimal but not positional. It has characters for 10, 100, etc., as well as for 1, 2, ••• 9, and, on the other hand, no 0. Most of the Indian characters have been traced back to an old alphabet # in use in Northern India 200 b.c The original of each numeral symbol 4, 5, 6, 7, 8 (?), 9, is the initial letter in this alphabet of the corresponding numeral word (see table on page 89, f colunfti 1). The characters first occur as numeral signs in certain inscriptions which are assigned to the 1st and 2d centuries a.d. (column 2 of table). Later they took the forms given in column 3 of the table. When was invented and the positional notation replaced the old notation cannot be exactly determined. It was * Dr. Isaac Taylor, in his book " The Alphabet," names this alpha- bet the Indo-Bactrian. Its earliest and most important monument is the version of the edicts of King Asoka at Kapur-di-giri. In this inscription, it may be added, numerals are denoted by strokes, as I, II, III, MM, Mill. t Columns 1-5, 7, 8 of the table on page 89 are taken from Tay- lor's Alphabet, II, p. 266 ; column 6, from Cantor's Geschichte der Mathematik. 88 NUMBER-SYSTEM OF ALGEBRA. certainly later than 400 a.d., and there is no evidence that it was earlier than 500 a.d. The earliest known instance of a date written in the new notation is 738 a.d. By the time that came in, the other characters had developed into the so-called Devanagari numerals (table, column 4), the classical numerals of the Indians. The perfected Indian system probably passed over to the Arabians in 773 a.d., along with certain astronomical writings. However that may be, it was expounded in the early part of the 9th century by Alkhwarizml, and from that time on spread gradually throughout the Arabian world, the numerals tak- ing different forms in the East and in the West. Europe in turn derived the system from the Arabians in the 12th century, the " Gobar " numerals (table, column 5) of the Arabians of Spain being the pattern forms of the Euro- pean numerals (table, column 7). The arithmetic founded on the new system was at first called algorithm (after Alkhwarizmi), to distinguish it from the arithmetic of the abacus which it came to replace. A word must be said with reference to this arithmetic on the abacus. In the primitive abacus, or reckoning table, unit counters were used, and a number represented by the appropriate number of these counters in the appropriate columns of the instrument ; e.g. 321 by 3 counters in the column of 100' s, 2 in the column of 10' s, and 1 in the column of units. The Eomans employed such an abacus in all but the most elementary reckonings, it was in use in Greece, and is in use to-day in China. Before the introduction of algorithm, however, reckon- ing on the abacus had been improved by the use in its columns of separate characters (called apices) for each of the numbers 1, 2, • • • 9, instead of the primitive unit counters. This improved abacus reckoning was probably invented by Gerbert (Pope Sylvester II.), and certainly used by him at Kheims about 970-980, and became generally known in the following century. HISTORIC SYSTEMS OF DOTATION. 89 TABLE SHOWING THE EVOLUTION OF THE ARABIC CIPHERS. z E 2 H i- a r. INDIAN ARABIC EURO PEA N A. D. GOBAR APICES LETTERS OF INDO-BACT ALPHABE Sec. I. Sec. V. Sec.X. Sec.X. SecX. Sec.XII. Sec.XIV. r*\ 1 i 1 V 1 — 0^-* \ I IT I 7* B. C. Sec. II. — - *-~*~* £ a ? 5 ? V ¥ $ f^ A 4 h h Y <\ i H < 7 y 4 c, s i Eb 3 .p™ :~«4-«„^ «,v:»v» ;« „i 4-~ 22 30 60 60 2 ' this notation be written 22 30. Thus the ability to represent fractions by a single integer or a sequence of integers, which the Egyptians secured by the use of fractions having a com- mon numerator, 1, the Babylonians found in fractions having common denominators and the principle of position. The Egyptian system is superior in that it gives an exact expres- sion of every quotient, which the Babylonian can in general do only approximately. As regards practical usefulness, however, the Babylonian is beyond comparison the better system. Supply the 0-symbol and substitute 10 for 60, and * The Khind papyrus of the British Museum ; translated by A. Eisenlohr, Leipzig, 1877. fraction -, for instance, which is equal to 1 , would in THE FRACTION. 93 this notation becomes that of the modern decimal fraction, in whose distinctive merits it thus shares. As in their origin, so also in their subsequent history,* the sexagesimal fractions are intimately associated with astronomy. The astronomers of Greece, India, and Arabia all employ them in reckonings of any complexity, in those involving the lengths of lines as well as in those involving the measures of angles. So the Greek astronomer, Ptolemy (150 a.d.), in the Almagest (/meydXr} owra&s) measures chords as well as arcs in degrees, minutes, and seconds — the de- gree of chord being the 60th part of the radius as the degree of arc is the 60th part of the arc subtended by a chord equal to the radius. The sexagesimal fraction held its own as the fraction par excellence for scientific computation until the 16th century, when it was displaced by the decimal fraction in all uses except the measurement of angles. 93. Greek Fractions. Fractions occur in Greek writings — both mathematical and non-mathematical — much earlier than Ptolemy, but not in arithmetic* The Greeks drew as sharp a distinction between pure arithmetic, dptO^TiKr), and the art of reckoning, XoyiaTtKr), as between pure and metrical geom- etry. The fraction was relegated to Xoyio-TLKr}. There is no place in a pure science for artificial concepts, no place, there- fore, for the fraction in aptO/jLVTiKr} ; such was the Greek posi- tion. Thus, while the metrical geometers — as Archimedes (250 B.C.), in his "Measure of the Circle" (kvkXov /xerp^o-ts), and Hero (120 b.c.) — employ fractions, neither of the trea- tises on Greek arithmetic before Diophantus (300 a.d.) which * The usual method of expressing fractions was to write the numer- ator with an accent, and after it the denominator twice with a double 17 accent : e.q. iC ku" tea" — — -. Before sexagesimal fractions came into j % 21 ° vogue actual reckonings with fractions were effected by unit fractions, of which only the denominators (doubly accented) were written. 94 NUMBER-SYSTEM OF ALGEBRA. have come down to us — the 7th, 8th, 9th books of Euclid's "Elements" (300 B.C.), and the " Introduction to Arithmetic '^ (etcraycoy^ apiOfjLrjTLKrj) of Mcomachus (100 a.d.) — recognizes the fraction. They do, it is true, recognize the fractional relation. Euclid, for instance, expressly declares that any number is either a multiple, a part, or parts (/xep^), i.e. multiple of a part, of every other number (Euc. VII, 4), and he demonstrates such theorems as these : If A be the same parts of B that C is of D, then the sum or difference of A and C is the same parts of the sum or difference of B and D that A is of B (VII, 6 and 8). If Abe the same parts of B that C is of D, then, alternately, A is the same parts of C that B is of ' D (VII, 10). But the relation is expressed by two integers, that which indicates the part and that which indicates the multiple. It is a ratio, and Euclid has no more thought of expressing it except by two numbers than he has of expressing the ratio of two geometric magnitudes except by two magni- tudes. There is no conception of a single number, the fraction proper, the quotient of one of these integers by the other. In the apiOfxyrtKOL of Diophantus, on the other hand, the last and transcendently the greatest achievement of the Greeks in the science of number, the fraction is granted the position in elementary arithmetic which it has held ever since. IV. OEIGIrT OP THE IEEATIOrTAL. 94. The Discovery of Irrational Lines. The Greeks attrib- uted the discovery of the Irrational to the mathematician and philosopher Pythagoras # (525 B.C.). * This is the explicit declaration of the most reliable document extant on the history of geometry before Euclid, a chronicle of the ancient ORIGIN OF THE IRRATIONAL. 95 If, as is altogether probable, # the most famous theorem of Pythagoras — that the square on the hypothenuse of a right triangle is equal to the sum of the squares on the geometers which Proems (a.d. 450) gives in his commentary on Euclid, deriving it from a history written by Eudemus about 330 b.c. This chronicle credits the Egyptians with the discovery of geometry and Thales (600 b.c.) with having first introduced this study into Greece. Thales and Pythagoras are the founders of the Greek mathematics. But while Thales should doubtless be credited with the first conception of an abstract deductive geometry in contradistinction to the practical empirical geometry of Egypt, the glory of realizing this conception belongs chiefly to Pythagoras and his disciples in the Greek cities of Italy (Magna Grsecia) ; for they established the principal theorems respecting rectilineal figures. To the Pythagoreans the discovery of many of the elementary properties of numbers is due, as well as the geometric form which characterized the Greek theory of numbers throughout its history. In the middle of the fifth century before Christ Athens became the principal centre of mathematical activity. There Hippocrates of Chios (430 b.c.) made his contributions to the geometry of the circle, Plato (380 b.c) to geometric method, Theaetetus (380 b.c) to the doctrine of incommensurable magnitudes, and Eudoxus (360 b.c) to the theory of proportion! There also was begun the study of the conies. About 300 b.c the mathematical centre of the Greeks shifted to Alexandria, where it remained. The third century before Christ is the most brilliant period in Greek mathematics. At its beginning — in Alexandria — Euclid lived and taught and wrote his Elements, collecting, systematizing, and per- fecting the work of his predecessors. Later (about 250) Archimedes of Syracuse flourished, the greatest mathematician of antiquity and founder of the science of mechanics ; and later still (about 230) Apol- lonius of Perga, "the great geometer," whose Conies marks the cul- mination of Greek geometry. Of the later Greek mathematicians, besides Hero and Diophantus, of whom an account is given in the text, and the great summarizer of the ancient mathematics, Pappus (300 a.d.), only the famous astrono- mers Hipparchus (130 b.c) and Ptolemy (150 a.d.) call for mention here. To them belongs the invention of trigonometry and the first trigonometric tables, tables of chords. The dates in this summary are from Gow's Hist, of Greek Math. * Compare Cantor, Geschichte der Mathematik, p. 153. > 96 NUMBER-SYSTEM OF ALGEBRA. other two sides — was suggested to him by the fact that 32 _j_ 42 _ 52^ i n connection with the fact that the triangle whose sides are 3, 4, 5, is right-angled, — for both almost certainly fell within the knowledge of the Egyptians, -7 he would naturally have sought, after he had succeeded In demonstrating the geometric theorem generally, for number triplets corresponding to the sides of any right triangle as do 3, 4, 5 to the sides of the particular triangle. The search of course proved fruitless, fruitless even in the case which is geometrically the simplest, that of the isosceles right triangle. To discover that it was necessarily fruitless ; in the face of preconceived ideas and the appar- ent testimony of the senses, to conceive that lines may exist which have no common unit of measure, however small that unit be taken ; to demonstrate that the hypothenuse and side of the isosceles right triangle actually are such a pair of lines, was the great achievement of Pythagoras. # 95. Consequences of this Discovery in Greek Mathematics. One must know the antecedents and follow the consequences of this discovery to realize its great significance. It was * His demonstration may easily have been the following, which was old enough in Aristotle's time (340,fe.c.). to be made the subject of a popular reference, and which is to be found at the end of the 10th book in all old editions of Euclid's Elements : If there be any line which the side and diagonal of a square both contain an exact number of times, let their lengths in terms of this line be a and b respectively ; then b 2 = 2 a 2 . The numbers a and b may have a common factor, 7 ; so that a— ay and 6 = #7, where a and /3 are prime to each other. The equation b 2 = 2 a 2 then reduces, on the removal of the factor 7 2 common to both its members, to fi 2 = 2 a 2 . From this equation it follows that # 2 , and therefore £, is an even number, and hence that a which is prime to # is odd. But set j8 = 2£', where /3' is integral, in the equation & 2 = 2a 2 ; it becomes 4 &' 2 — 2 a 2 , or 2 &' 2 = a 2 , whence a 2 , and therefore a, is even. a has thus been proven to be both odd and even, and is therefore not a number. ORIGIN OF THE IRRATIONAL. 97 the first recognition of the fundamental difference between the geometric magnitudes and number, which Aristotle for- mulated brilliantly 200 years later in his famous distinc- tion between the continuous and the discrete, and as such was potent in bringing about that complete banishment of numerical reckoning from geometry which is so character- istic of this department of Greek mathematics in its best, its creative period. No one before Pythagoras had questioned the possibility of expressing all size relations among lines and surfaces in terms of number, — rational number of course. Indeed, except that it recorded a few facts regarding congruence of figures gathered by observation, the Egyptian geometry was nothing else than a meagre collection of formulas for com- puting areas. The earliest geometry was metrical. But to the severely logical Greek no alternative seemed possible, when once it was known that lines exist whose lengths — whatever unit be chosen for measuring them — cannot both be integers, than to have done with number and measurement in geometry altogether. Congruence be- came not only the final but the sole test of equality. For the study of size relations among unequal magnitudes a pure geometric theory of proportion was created, in which proportion, not ratio, was the primary idea, the method of exhaustions making the theory available for figures bounded by curved lines and surfaces. The outcome was the system of geometry which Euclid expounds in his Elements and of which Apollonius makes splendid use in his Conies, a system absolutely free from extraneous concepts or methods, yet, within its limits, of great power. It need hardly be added that it never occurred to the Greeks to meet the difficulty which Pythagoras' discovery had brought to light by inventing an irrational number, itself incommensurable with rational numbers. For arti- ficial concepts such as that they had neither talent nor liking. 98 NUMBER-SYSTEM OF ALGEBRA, On the other hand, they did develop the theory of irra- tional magnitudes as a department of their geometry, the irrational line, surface, or solid being one incommensurable with some chosen (rational) line, surface, solid. Such a theory forms the content of the most elaborate book of Euclid's Elements, the 10th. 96. Approximate Values of Irrationals. In the practical or metrical geometry which grew up after the pure geometry had reached its culmination, and which attained in the works of Hero the Surveyor almost the proportions of our modern elementary mensuration,* approximate values of irrational numbers played a very important r6le. Nor do such approx- imations appear for the first time in Hero. In Archimedes' " Measure of the Circle " a number of excellent approxima- - . 22 tions occur, among them the famous approximation — for 7r, the ratio of the circumference of a circle to its diam- eter. The approximation - for V2 is reputed to be as old as Plato. It is not certain how these approximations were effected. f They involve the use of some method for extracting square roots. The earliest explicit statement of the method in common use to-day for extracting square roots of numbers (whether exactly or approximately) occurs in the com- mentary of Theon of Alexandria (380 a.d.) on Ptolemy's * The formula Vs (s — a)(s — b) (s — c) for the area of a triangle in terms of its sides is due to Hero. t Many attempts have been made to discover the methods of approximation used by Archimedes and Hero from an examination of their results, but with little success. The formula Va 2 ± b = a ± — ■ 2a will account for some of the simpler approximations, but no single method or set of methods have been found which will account for the more difficult. See Gunther: Die quadratischen Irrationalitaten der Alten und deren Entwicldungsmethoden. Leipzig, 1882. Also in Handbuch der klassischen Altertums-Wissenschaft, liter. Halbband. GREEK ALGEBRA. 99 Almagest. Theon, who like Ptolemy employs sexagesimal fractions, thus finds the length of the side of a square con- taining 4500° to be 67° 1' 55". 97. The Later History of the Irrational is deferred to the chapters which follow (§§ 106, 108, 112, 121, 129). It will be found that the Indians permitted the simplest forms of irrational numbers, surds, in their algebra, and that they were followed in this by the Arabians and the mathema- ticians of the Kenaissance, but that the general irrational did not make its way into algebra until after Descartes. V. OKIGIN OF THE NEGATIVE AND THE IMAGINAKY. THE EQUATION. 98. The Equation in Egyptian Mathematics. While the irrational originated in geometry, the negative and the imaginary are of purely algebraic origin. They sprang directly from the algebraic equation. The authentic history of the equation, like that of geome- try and arithmetic, begins in the book of the old Egyptian scribe Ahmes. For Ahmes, quite after the present method, solves numerical problems which admit of statement in an equation of the first degree involving one unknown quantity.* 99. In the Earlier Greek Mathematics. The equation was slow in arousing the interest of Greek mathematicians. They were absorbed in geometry, in a geometry whose methods were essentially non-algebraic. To be sure, there are occasional signs of a concealed algebra under the closely drawn geometric cloak. Euclid * His symbol for the unknown quantity is the word hau, meaning heap. 100 NUMBER-SYSTEM OF ALGEBRA. solves three geometric problems which, stated algebraically, are but the three forms of the quadratic ; or + ax = b 2 , x 2 = ax + b 2 , x 2 + b 2 = ax* And the Conies of Apollonius, so astonishing if regarded as a product of the pure geo- metric method used in its demonstrations, when stated in the language of algebra, as recently it has been stated by Zeuthen,f almost convicts its author of the use of algebra as his instrument of investigation. 100. Hero. But in the writings of Hero of Alexandria (120 b.c.) the equation first comes clearly into the light again. Hero was a man of practical genius whose aim was to make the rich pure geometry of his predecessors available for the surveyor. With him the rigor of the old geometric method is relaxed ; proportions, even equations, among the measures of magnitudes are permitted where the earlier geometers allow only proportions among the magnitudes themselves ; the theorems of geometry are stated metrically, in formulas ; and more than all this, the equation becomes a recognized geometric instrument. Hero gives for the diameter of a circle in terms of s, the sum of diameter, circumference, and area, the formula : t ^_ Vl54s-f-841-^-29 a ~ ' IT He could have reached this formula only by solving a quadratic equation, and that not geometrically, — the nature of the oddly constituted quantity s precludes that suppo- sition, — but by a purely algebraic reckoning like the following : 7rd 2 The area of a circle in terms of its diameter being — — , * Elements, VI, 29, 28 ; Data, 84, 85. t Die Lehre von den Kegelschnitten im Altertum. Copenhagen, 1886. \ See Cantor ; Geschichte der Mathematik, p. 341. GREEK ALGEBRA. 101 the length of its circumference 71-d, and tt according to 22 Archimedes' approximation — , we have the equation : , , wd 2 , , 11 # , 29 , s = d -\ h 7ra, or -—d~-\ d = s. 4 14 7 Clearing of fractions, multiplying by 11, and completing the square, 121 d 2 + 638 d + 841 = 154 s + 841, whence 11 d + 29 = Vlo4 s + 841, Vl54s + 841-29 or d = - 11 Except that he lacked an algebraic symbolism, therefore, Hero was an algebraist, an algebraist of power enough to solve an affected quadratic equation. 101. Diophantus (300 a.d. ?). The last of the Greek mathematicians, Diophantus of Alexandria, was a great algebraist. The period between him and Hero was not rich in cre- ative mathematicians, but it must have witnessed a grad- ual development of algebraic ideas and of an algebraic symbolism. At all events, in the apiOix-qriKa of Diophantus the alge- braic equation has been supplied with a symbol for the unknown quantity, its powers and the powers of its recip- rocal to the 6th, and a symbol for equality. Addition is represented by mere juxtaposition, but there is a special symbol, /p, for subtraction. On the other hand, there are no general symbols for known quantities, — symbols to serve the purpose which the first letters of the alphabet are made to serve in elementary algebra nowadays, — there- fore no literal coefficients and no general formulas. With the symbolism had grown up many of the formal rules of algebraic reckoning also. Diophantus prefaces the 102 NUMBER-SYSTEM OF ALGEBRA. apL$fjLr)TiKd with, rules for trie addition, subtraction, and mul- tiplication of polynomials. He states expressly that the product of two subtractive terms is additive. The apiOfjLrjTLKOL itself is a collection of problems concern- ing numbers, some of which are solved by determinate algebraic equations, some by indeterminate. Determinate equations are solved which have given posi- tive integers as coefficients, and are of any of the forms ax m = bx n , ax 2 -f- bx — c, ax 2 -f- c = bx, ax 2 = bx-\-c 9 also a single cubic equation, x s -f x = Ax 2 + 4. In reducing equa- tions to these forms, equal quantities in opposite members are cancelled and subtractive terms in either member are rendered additive by transposition to the other member. The indeterminate equations are of the form y 2 = ax 2 + bx -f c, Diophantus regarding any pair of positive rational numbers (integers or fractions) as a solution which, substi- tuted for y and x, satisfy the equation.* These equations are handled with marvellous dexterity in the apiB^TiKo.. No effort is made to develop general comprehensive methods, but each exercise is solved by some clever device suggested by its individual peculiarities. Moreover, the discussion is never exhaustive, one solution sufficing when the possible number is infinite. Yet until some trace of indeterminate equations earlier than the apiOfxrjriKa is discovered, Diophan- tus must rank as the originator of this department of mathematics. The determinate quadratic is solved by the method which we have already seen used by Hero. The equation is first multiplied throughout by a number which renders the co- efficient of x 2 a perfect square, the " square is completed," the square root of both members of the equation taken, and * The designation " Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for inte- gral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree. INDIAN ALGEBRA. 103 the value of x reckoned out from the result. Thus from ax 2 -f- c = bx is derived first the equation a 2 x 2 -f ac = abx, then a 2 x 2 — abx + (-) = f x J —etc, then ax — - = + l( - J — ac, V(IT- and finally, x = *■* The solution is regarded as possible only when the num- ber under the radical is a perfect square (it must, of course, be positive), and only one root — that belonging to the positive value of the radical — is ever recognized. Thus the number system of Diophantus contained only the positive integer and fraction ; the irrational is excluded ; and as for the negative, there is no evidence that a Greek mathematician ever conceived of such a thing, — certainly not Diophantus with his three classes and one root of affected quadratics. The position of Diophantus is the more interesting in that in the dpitf/x^TiKa the Greek science of number culminates. 102. The Indian Mathematics. The pre-eminence in math- ematics passed from the Greeks to the Indians. Three mathematicians of India stand out above the rest : Arya- bhatta (born 476 a.d.), Brahmagupta (born 598 a.d.), Bhds- kara (born 1114 a.d.). While all are in the first instance astronomers, their treatises also contain full expositions of the mathematics auxiliary to astronomy, their reckoning, algebra, geometry, and trigonometry. # * The mathematical chapters of Brahmagupta and Bhaskara have been translated into English by Colebrooke : "Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahmagupta and Bhaskara," 1817 ; those of Aryabhatta into French by L. Kodet (Journal Asiatique, 1879). 104 NUMBER-SYSTEM OF ALGEBRA. An examination of the writings of these mathematicians and of the remaining mathematical literature of India leaves little room for doubt that the Indian geometry was taken bodily from Hero, and the algebra — whatever there may have been of it before Aryabhatta — at least powerfully affected by Diophantus. Nor is there occasion for surprise in this. Aryabhatta lived two centuries after Diophantus and six after Hero, and during those centuries the East had frequent communication with the West through various channels. In particular, from Trajan's reign till later than 300 a.d. an active commerce was kept up between India and the east coast of Egypt by way of the Indian Ocean. Greek geometry and Greek algebra met very different fates in India. The Indians lacked the endowments of the geometer. So far from enriching the science with new discoveries, they seem with difficulty to have kept alive even a proper understanding of Hero's metrical formulas. But algebra flourished among them wonderfully. Here the fine talent for reckoning which had created a perfect nu- meral notation, supported by a talent equally fine for sym- bolical reasoning, found a great opportunity and made great achievements. With Diophantus algebra is no more than an art by which disconnected numerical problems are solved ; in India it rises to the dignity of a science, with general methods and concepts of its own. 103. Its Algebraic Symbolism. Eirst of all, the Indians devised a complete, and in most respects adequate, sym- bolism. Addition was represented, as by Diophantus, by mere juxtaposition ; subtraction, exactly as addition, except that a dot was written over the coefficient of the subtra- hend. The syllable bha written after the factors indicated a product ; the divisor written under the dividend, a quo- tient ; a syllable, ka, written before a number, its (irrational) square root ; one member of an equation placed over the other, their equality. The equation was also provided with INDIAN ALGEBRA. 105 symbols for any number of unknown quantities and their powers. 104. Its Invention of the Negative. The most note- worthy feature of this symbolism is its representation of subtraction. To remove the subtractive symbol from be- tween minuend and subtrahend (where Diophantus had placed his symbol 41), to attach it wholly to the subtrahend and thus connect this modified subtrahend with the minuend additively, is, formally considered, to transform the sub- traction of a positive quantity into the addition of the corresponding negative. It suggests what other evidence makes certain, that algebra owes to India the immensely useful concept of the absolute negative. Thus one of these dotted numbers is allowed to stand by itself as a member of an equation. Bhaskara recognizes the double sign of the square root, as well as the impossi- bility of the square root of a negative number (which is very interesting, as being the first dictum regarding the imag- inary), and no longer ignores either root of the quadratic. More than this, recourse is had to the same expedients for interpreting the negative, for attaching a concrete phys- ical idea to it, which persist to this day. The primary meaning of the very name, given the negative was debt, as that given the positive was means. The opposition between the two was also pictured by lines described in opposite directions. 105. Its Use of Zero. But the contributions of the Ind- ians to the fund of algebraic concepts did not stop with the absolute negative. They made a number of 0^ and though some of their reckonings with it are childish, Bhaskara, at least, had sufficient understanding of the nature of the " quotient " - (infinity) to say "it suffers no change, however much it is increased or diminished." He associates it with Deity. 106 NUMBEB-SYSTEM OF ALGEBBA. 106. Its Use of Irrational Numbers. Again, the Indians were the first to reckon with irrational square roots as with numbers; Bhaskara extracting square roots of binomial surds and rationalizing irrational denominators of fractions even when these are polynomial. Of course they were as little able rigorously to justify such a procedure as the Greeks ; less able, in fact, since they had no equivalent of the method of exhaustions. But it probably never occurred to them that justification was necessary; they seem to have been unconscious of the gulf fixed between the discrete and continuous. And here, as in the case of and the negative, with the confidence of apt and successful reckoners, they were ready to pass immediately from numerical to purely symbolical reasoning, ready to trust their processes even where formal demonstration of the right to apply them ceased to be attainable. Their skill was too great, their instinct too true, to allow them to go far wrong. 107. Determinate and Indeterminate Equations in Indian Algebra. As regards equations — the only changes which the Indian algebraists made in the treatment of determinate equations were such as grew out of the use of the negative. This brought the triple classification of the quadratic to an end and secured recognition for^both roots of the quadratic. Brahmagupta solves the quadratic by the rule of Hero and Diophantus, of which he gives an explicit and gen- eral statement. Cridhara, a mathematician of some dis- tinction belonging to the period between Brahmagupta and Bhaskara, made the improvement of this method which consists in first multiplying the equation throughout by four times the coefficient of the square of the unknown quantity and so preventing the occurrence of fractions under the radical sign.* Bhaskara also solves a few cubic and biquadratic equa- tions by special devices. * This method still goes under the name " Hindoo method." ARABIAJS ALGEBRA. 107 The theory of indeterminate equations, on the other hand, made great progress in India. The achievements of the Indian mathematicians in this beautiful but difficult depart- ment of the science are as brilliant as those of the Greeks in geometry. They created the doctrine of the indetermi- nate equation of the first degree, ax -{-by = c, which they treated for integral solutions by the method of continued fractions in use to-day. They worked also with equations of the second degree of the forms ax 2 -\-b = cy 2 , xy = ax -{-by -f- c, originating general and comprehensive methods where Dio- phantus had been content with clever jugglery. 108. The Arabian Mathematics. The Arabians were the instructors of modern Europe in the ancient mathematics. The service which they rendered in the case of the numeral notation and reckoning of India they rendered also in the case of the geometry, algebra, and astronomy of the Greeks and Indians. Their own contributions to mathematics are unimportant. Their receptiveness for mathematical ideas was extraordinary, but they had little originality. The history of Arabian mathematics begins with the reign of Almansur (754-775), # the second of the Abbasid caliphs. It is related (by Ibn-al-Adami, about 900) that in this reign, in the year 773, an Indian brought to Bagdad certain astronomical writings of his country, which contained a method called u Sindhind," for computing the motions of the stars, — probably portions of the Siddhanta of Brahma- gupta, — and that Alfazari was commissioned by the caliph to translate them into Arabic. t Inasmuch as the Indian * It was Almansur who transferred the throne of the caliphs from Damascus to Bagdad which immediately became not only the capital city of Islam, but its commercial and intellectual centre. t This translation remained the guide of the Arabian astronomers until the reign of Almamun (813-833), for whom Alkhwarizmi pre- pared his famous astronomical tables (820). Even these were based chiefly on the "Sindhind," though some of the determinations were made by methods of the Persians and Ptolemy. 108 NUMBER-SYSl^ip^OF ALGEBRA. astronomers put full expositions of their reckoning, algebra, and geometry into their treatises, Alfazarfs translation laid open to his countrymen a rich treasure of mathematical ideas and methods. It is impossible to set a date to the entrance of Greek ideas. They must have made themselves felt at Damascus, the residence of the later Omayyad caliphs, for that city had numerous inhabitants of Greek origin and culture. But the first translations of Greek mathematical writings were made in the reign of Harun Arraschid (786-809), when Euclid's Elements and Ptolemy's Almagest were put into Arabic. Later on, translations were made of Archimedes, Apollo- nius, Hero, and last of all, of Diophantus (by Abu'l Wafa, 940-998). The earliest mathematical author of the Arabians is Alkhwarizmi, who flourished in the first quarter of the 9th century. Besides astronomical tables, he wrote a treatise on algebra and one on reckoning (elementary arithmetic). The latter has already been mentioned. It is an exposition of the positional reckoning of India, the reckoning which mediaeval Europe named after him Algorithm. a J ^he treatise on algebra bears a title in which the word Mlgebra^ appears for the first time : viz., Ald/jebr walmu- p Jcdbala. Aldjehr {i.e. reduction) signifies the making of all terms of an equation positive by transferring negative terms to the opposite member of the equation; almukabala (i.e. opposition), the cancelling of equal terms in opposite mem- bers of an equation. Alkhwarizmfs classification of equations of the 1st and 2d degrees is that to which these processes would naturally lead, viz. : ax 2 = bx, bx 2 = c, bx = c, x 2 + bx — c, x 2 -f c = bx, x 2 =bx-\- c. These equations he solves separately, following up the solution in each case with a geometric demonstration of its ARABIAN ALGEBRA. 109 correctness. He recognizes both roots of the quadratic when they are positive. In this respect he is Indian ; in all others — the avoidance of negatives, the nse of geometric demonstration — he is Greek. Besides Alkhwarizml, the most famous algebraists of the Arabians were Alkarchi and Alchayydmt, both of whom lived in the 11th century. r Alkarchi gave the solution of equations of the forms : ax 2p + bx p = c, ax 2p + c = bx p , bx p + c = ax 2p . ■\ He also reckoned with irrationals, the equations V8 + Vl8 = V50, 2 , and for o, p 2 r. Then x s = p 2 (r — x). Now this equation is the result of eliminating y from between the two equations, x 2 = py, y 2 = x (r — x) ; the first of which is the equa- tion of a parabola, the second, of a circle. Let these two curves be constructed ; they will intersect in one real point distinct from the origin, and the abscissa of this point is a root of x z -f- bx = a. See Hankel, Geschichte der Mathematik, p. 279. This method is of greater interest in the history of geometry than in that of algebra. It involves an anticipation of some of the most important ideas of Descartes' Geometric (see p. 118). 110 NUMBER-SYSTEM OF ALGEBRA. frequent enough to exercise a controlling influence on their aesthetic or scientific development. Their mathematical productions are of a later date than those of the East and almost exclusively arithmetico-algebraic. They con- structed a formal algebraic notation which went over into the Latin translations of their writings and rendered the path of the Europeans to a knowledge of the doctrine of equations easier than it would have been, had the Arabians of the East been their only instructors. The best known of their mathematicians are Ibn Aflah (end of 11th century), Ibn Albanna (end of 13th century), Alkasadl (15th century). 109. Arabian Algebra Greek rather than Indian. Thus, of the three greater departments of the Arabian mathematics, the Indian influence gained the mastery in re ckoning^o nly. The Arabian geometry is Greek through and through. While the algebra contains both elements, the Greek pre- dominates. Indeed, except that both roots of the quadratic are recognized, the doctrine of the determinate equation is altogether Greek. It avoids the negative almost as care- fully as Diophantus does ; and in its use of the geometric method of demonstration it is actuated by a spirit less modern still — the spirit in which Euclid may have con- ceived of algebra when he solved his geometric quadratics. The theory of indeterminate equations seldom goes beyond Diophantus ; where it does, it is Indian. The Arabian trigonometry is based on Ptolemy's, but is its superior in two important particulars. It employs the sine where Ptolemy employs the chord (being in this re- spect Indian), and has an algebraic instead of a geometric form. Some of the methods of approximation used in reckoning out trigonometric tables show great cleverness. Indeed, the Arabians make some amends for their ill-advised return to geometric algebra by this excellent achievement in algebraic geometry. The preference of the Arabians for Greek algebra was EARLY EUROPEAN ALGEBRA. Ill especially unfortunate in respect to the negative, which was in consequence forced to repeat in Europe the fight for recognition which it had already won in India. 110. Mathematics in Europe before the Twelfth Century. The Arabian mathematics found entrance to Christian Eu- rope in the 12th century. During this century and the first half of the next a good part of its literature was translated into Latin. Till then the plight of mathematics in Europe had been miserable enough. She had no better representatives than the Romans, the most deficient in the sense for mathematics of all cultured peoples, ancient or modern ; no better lit- erature than the collection of writings on surveying known as the Codex Arcerianus, and the childish arithmetic and geometry of Boetius. Prior to the 10th century, however, Northern Europe had not sufficiently emerged from barbarism to call even this paltry mathematics into requisition. What learning there was was confined to the cloisters. Beckoning (computus) was needed for the Church calendar and was taught in the cloister schools established by Alcuin (735-804) under the patronage of Charlemagne. Reckoning was commonly done on the fingers. Not even was the multiplication table gen- erally learned. Reference would be made to a written copy of it, as nowadays reference is made to a table of loga- rithms. The Church did not need geometry, and geometry in any proper sense did not exist. 111. Gerbert. But in the 10th century there lived a man of true scientific interests and gifts, Gerbert,* Bishop of Rheims, Archbishop of Ravenna, and finally Pope Sylvester II. In him are the first signs of a new life for mathematics. His achievements, it is true, do not extend beyond the revival of Roman mathematics, the authorship of a geom- * See § 88. 112 NUMBEBSYSTEM OF ALGEBBA. etry based on the Codex Arcerianus, and a method for effect- ing division on the abacus with apices. Yet these achieve- ments are enough to place him far above his contemporaries. His influence gave a strong impulse to mathematical studies where interest in them had long been dead. He is the fore- runner of the intellectual activity ushered in by the trans- lations from the Arabic, for he brought to life the feeling of the need for mathematics which these translations were made to satisfy. 112. Entrance of the Arabian Mathematics. Leonardo. It was the elementary branch of the Arabian mathematics which took root quickest in Christendom — reckoning with 9 digits and 0. Leonardo of Pisa — Fibonacci, as he was also called — did great service in the diffusion of the new learning through his Liber Abaci (1202 and 1228), a remarkable presentation of the arithmetic and algebra of the Arabians, which remained for centuries the fund from which reckoners and algebraists drew and is indeed the foundation of the modern science. The four fundamental operations on integers and frac- tions are taught after the Arabian method ; the extraction of the square root and the doctrine of irrationals are presented in their pure algebraic form ; quadratic equations are solved and applied to quite complicated" problems ; negatives are accepted when they admit of interpretation as debt. The last fact illustrates excellently the character of the Liber Abaci. It is not a mere translation, but an inde- pendent and masterly treatise in one department of the new mathematics. Besides the Liber Abaci, Leonardo wrote the Practica Geometriae, which contains much that is best of Euclid, Archimedes, Hero, and the elements of trigonometry ; also the Liber Quadratorum, a collection of original algebraic problems most skilfully handled. EARLY EUROPEAN ALGEBRA. 113 113. Mathematics during the Age of Scholasticism. Leo- nardo was a great mathematician,* but tine as his work was, it bore no fruit until the end of the 15th century. In him there had been a brilliant response to the Arabian impulse. But the awakening was only momentary ; it quickly yielded to the heavy lethargy of the " dark " ages. The age of scholasticism, the age of devotion to the forms of thought, logic and dialectics, is the age of greatest dul- ness and confusion in mathematical thinking.! Algebra owes the entire period but a single contribution; the concept of the fractional power. Its author was Nicole Oresme (died 1382), who also gave a symbol for it and the rules by which reckoning with it are governed. * Besides Leonardo there flourished in the first quarter of the 13th century an able German mathematician, Jordanus Nemorarius. He was the author of a treatise entitled Be numeris datis, in which known quantities are for the first time represented by letters, and of one De trangulis which is a rich though rather systemless collection of theorems and problems principally of Greek and Arabian origin. See Giinther : Geschichte des mathemathischen Unterrichts im deutschen Mittelalter, p. 156. t Compare Hankel, Geschichte der Mathematik, pp. 349-352. To the unfruitfulness of these centuries the Summa of Luca Pacioli bears witness. This book^ which has the distinction of being the earliest book on algebra printed, appeared in 1494, and embodies the arith- metic, algebra, and geometry of the time just preceding the Renais- sance. It contains not an idea or method not already presented by Leonardo. Even in respect to algebraic symbolism it surpasses the Liber Abaci only to the extent of using abbreviations for a few fre- quently recurring words, as p. for "plus," and R. for "res" (the unknown quantity) . And this is not to be regarded as original with Pacioli for the Arabians of Leonardo's time made a similar use of abbreviations. In a translation made by Gerhard of Cremona (12th century) from an unknown Arabic original the letters r (radix), c (census) , d (dragma) are used to represent the unknown quantity, its square, and the absolute term respectively. Pacioli' s demonstration that "minus times minus is plus" is per- haps worth inserting here, not, unfortunately, because it has gone altogether out of vogue, but for the sake of the scholastic principle on 114 NUMBER-SYSTEM OF ALGEBRA. 114. The Renaissance. Solution of the Cubic and Bi- quadratic Equations. The first achievement in algebra by the mathematicians of the Renaissance was the algebraic solntion of the cubic equation : a fine beginning of a new era in the history of the science. The cubic x s -f- «W3 = n was solved by Ferro of Bologna in 1505, and a second time and independently, in 1535, by Ferro's countryman, Tartaglia, who by help of a transfor- mation made his method apply to x s ± mx 2 = ± n also. But Cardan of Milan was the first to publish the solution, in his Ars Magna,* 1545. The Ars Magna records another brilliant discovery : the solution — after a general method — of the biquadratic x 4 + 6x 2 + 36 = 60 x by Ferrari, a pupil of Cardan. Thus in Italy, within fifty years of the new birth of algebra, after a pause of sixteen centuries at the quad- ratic, the limits of possible attainment in the algebraic solution of equations were reached; for the algebraic solution of the general equation of a degree higher than 4 is impossible, as was first demonstrated by Abel.f The general solution of higher equations proving an obstinate problem, nothing was left the searchers for the which he bases it. He reasons thus : Since 8 • 8 = (10 — 2) (10 — 2) = 64, and 10 • 10 = 100, and - 2 . 10 = - 20 ; therefore, - 2 • - 2 = + 4 — and adds that this method of reasoning is well-known to philosophers, being " a disjunctiva plurium partium a destructione multarum supra unam semper tenet consequentia. " It should be added that the loth century produced a mathemati- cian who deserves a distinguished place in the general history of mathematics on account of his contributions to trigonometry, the astronomer Regiomontanus (1436-1476). Like Jordanus, he was a German. * The proper title of this work is : " Artis magnae sive de regulis Algebraicis liber unus." It has stolen the title of Cardan's "Ars magna Arithmeticae," published at Basel, 1570. t Memoire sur les Equations Algebriques : Christiania, 1826. Also in Crelle's Journal, I, p. 65. EABLY EUROPEAN ALGEBEA. 115 roots of equations but to devise a method of working them out approximately. In this the French mathematician Vieta (1540-1603) was successful, his method being essen- tially the same as that now known as Newton's. 115. The Negative in the Algebra of this Period. First Appearance of the Imaginary. But the general equation presented other problems than the discovery of rules for obtaining • its roots; the nature of these roots and the relations between them and the coefficients of the equation invited inquiry. We witness another phase of the struggle of the negative for recognition. The imaginary is now ready to make com- mon cause with it. Already in the Ars Magna Cardan distinguishes between numeri veri — the positive integer, fraction, and irrational, — and numeri ficti, or falsi — the negative and the square root of the negative. Like Leonardo, he tolerates negative roots of equations when they admit of interpretation as " debitum," not otherwise. While he has no thought of accepting im- aginary roots, he shows that if 5 + V— 15 be substituted for x in x (10 — x) = 40, that equation is satisfied ; which, of course, is all that is meant nowadays when 5 4- V— 15 is called a root. His declaration that 5 ± V— 15 are " vere sophistica " does not detract from the significance of this, the earliest recorded instance of reckoning with the imaginary. It ought perhaps to be added that Cardan is not always so successful in these reckonings ; for in another place he sets 4\ \ AJ \64 8* 4\ Following Cardan, Bombelli* reckoned with imaginaries to good purpose, explaining by their aid the irreducible case in Cardan's solution of the cubic. * L' Algebra, 1579. He also formally states rules for reckoning with ± V — 1 and a + b V — 1. 116 NUMBER-SYSTEM OF ALGEBRA. On the other hand, neither Yieta nor his distinguished follower, the Englishman Harriot (1560-1621), accept even negative roots ; though Harriot does not hesitate to perform algebraic reckonings on negatives, and even allows a nega- tive to constitute one member of an equation. 116. Algebraic Symbolism. Vieta and Harriot. Yieta and Harriot, however, did distinguished service in perfect- ing the symbolism of algebra ; Yieta, by the systematic use of letters to represent known quantities, — algebra first became "literal" or "universal arithmetic" in his hands, * — Harriot, by ridding algebraic statements of every non- symbolic element, of everything but the letters which rep- resent quantities known as well as unknown, symbols of operation, and symbols of relation. Harriot's Artis Analy- ticae Praxis (1631) has quite the appearance of a modern algebra, f * There are isolated instances of this use of letters much earlier than Vieta in the De numeris datis of Jordanus Nemorarius, and in the AJgorithnus demonstrates of Regiomontanus. But the credit of making it the general practice of algebraists belongs to Vieta. t One has only to reflect how much of the power of algebra is due to its admirable symbolism to appreciate the importance of the Artis Analyticae Praxis, in which this symbolism is finally established. But one addition of consequence has since been made to it, integral and fractional exponents introduced by Descartes (1637) and Wallis (1659). Harriot substituted small letters for the capitals used by Vieta, but followed Vieta in representing known quantities by consonants and unknown by vowels. The present convention of representing known quantities by the earlier letters of the alphabet, unknown by the later, is due to Descartes. Vieta' s notation is unwieldy and ill adapted to purposes of alge- braic reckoning. Instead of restricting itself, as Harriot's does, to the use of brief and easily apprehended conventional symbols, it also employs words subject to the rules of syntax. Thus for A s — 3 B 1 A = Z (or aaa — 3 bba = z, as Harriot would have written it) , Vieta writes A cubus — B quad 3 in A aeqnatur Z solido. In this respect Vieta is inferior not only to Harriot, but to several of his predecessors and EARLY EUROPEAN ALGEBRA. 117 117. Fundamental Theorem of Algebra. Harriot and Girard. Harriot has been credited with the discovery of the "fundamental theorem" of algebra — the theorem that the number of roots of an algebraic equation is the same as its degree. The Artis Analyticae Praxis contains no mention of this theorem — indeed, by ignoring negative and imagi- nary roots, leaves no place for it; yet Harriot develops systematically a method which, if carried far enough, leads to the discovery of this theorem as well as to the relations holding between the roots of an equation and its coefficients. By multiplying together binomial factors which involve the unknown quantity, and setting their product equal to 0, he builds "canonical" equations, and shows that the roots of these equations — the only roots, he says — are the positive values of the unknown quantity which render these binomial factors 0. Thus he builds act — ba — ca= — be, in which a is the unknown quantity, out of the factors a — b, a + c, and proves that & is a root of this equation and the only root, the negative root c being totally ignored. While no attempt is made to show that if the terms of a "common" equation be collected in one member, this can notably to his contemporary, the Dutch mathematician Stevinus (1548-1620), who would, for instance, have written x 2 + 3x — 8 as l® + 3® — 8®. The geometric affiliations of Vieta's notation are obvious. It suggests the Greek arithmetic. It is surprising that algebraic symbolism should owe so little to the great Italian algebraists of the 16th century. Like Pacioli (see note, p. 113) they were content with a few abbreviations for words, a "syncopated" notation, as it has been called, and an incomplete one at that. The current symbols of operation and relation are chiefly of English and German origin, having been invented or adopted as follows : viz. =, by Recorde in 1540 ; ^/, by Rudolf in 1526 ; the vinculum, by Vieta in 1591 ; brackets, by Girard in 1629 ; -f-, by Pell in 1630 ; X, >, <, by Harriot in 1631. The signs -f and — occur in a 15th century manu- script discovered by Gerhardt at Vienna. The notations a — b and - for the fraction were adopted from the Arabians. 118 NUMBER-SYSTEM OF ALGEBRA. be separated into binomial factors, the case of canonical equations raised a strong presumption for the soundness of this view of the structure of an equation. The first statement of the fundamental theorem and of the relations between coefficients and roots occurs in a remarkably clever and modern little book, the Inven- tion Nouvelle en VAlyebre, of Albert Girard, published in Amsterdam in 1629, two years earlier, therefore, than the Artis Analyticae Praxis. Girard stands in no fear of imag- inary roots, but rather insists on the wisdom of recognizing them. They never occur, he says, except when real roots are lacking, and then in number just sufficient to fill out the entire number of roots to equality with the degree of the equation. Girard also anticipated Descartes in the geometrical in- terpretation of negatives. But the Invention Nouvelle does not seem to have attracted much notice, and the genius and authority of Descartes were needed to give the interpreta- tion general currency. VI. ACCEPTANCE OF THE NEGATIVE, THE GENEKAL IEKATIONAL, AND THE IMAGINAKY AS NUMBEKS. 118. Descartes' Geometrie and the Negative. The Geome- trie of Descartes appeared in 1637. This famous little trea- tise enriched geometry with a general and at the same time simple and natural method of investigation : the method of representing a geometric curve by an equation, which, as Descartes puts it, expresses generally the relation of its points to those of some chosen line of reference.* To form such equations Descartes represents line segments by letters, — the known by a, b, c, etc., the unknown by x and y. He * See Geometrie, Livre II. In Cousin's edition of Descartes' works, Vol. V, p. 337. THE NEGATIVE. 119 supposes a perpendicular, y, to be dropped from any point of the curve to the line of reference, and then the equation to be found from the known properties of the curve which connects y with x, the distance of y from a fixed point of the line of reference. This is the equation of the curve in that it is satisfied by the x and y of each and every curve-point. # To meet the difficulty that the mere length of the perpen- dicular (y) from a curve-point will not indicate to which side of the line of reference the point lies, Descartes makes the convention that perpendiculars on opposite sides of this line (and similarly intercepts (x) on opposite sides of the point of reference) shall have opposite algebraic signs. This convention gave the negative a new position in mathematics. Not only was a "real" interpretation here found for it, the lack of which had made its position so dif- ficult hitherto, but it was made indispensable, placed on a footing of equality with the positive. The acceptance of the negative in algebra kept pace with the spread of Descar- tes' analytical method in geometry. 119. Descartes' Geometric Algebra. But the Geometrie has another and perhaps more important claim on the atten- tion of the historian of algebra. The entire method of the book rests on the assumption — made only tacitly, to be sure, and without knowledge of its significance — that two algebras are formally identical whose fundamental opera- tions are formally the same ; i.e. subject to the same laws of combination. For the algebra of the Geome'trie is not, as is commonly said, mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; but * Descartes fails to recognize a number of the conventions of our modern Cartesian geometry. He makes no formal choice of two axes of reference, calls abscissas y and ordinates x, and as frequently regards as positive ordinates below the axis of abscissas as ordinates above it. 120 NUMBER-SYSTEM OF ALGEBRA. symbols which there represent numbers here represent line segments. Not only is this the case with the letters a, b, x, y, etc., which are mere names (noms) of line segments, not their numerical measures, but with the algebraic combina- tions of these letters, a + b and a — b are respectively the sum and difference of the line segments a and b; ab, the fourth proportional to an assumed unit line, a, and b ; -, the b fourth proportional to b, a, and the unit line ; and Va, Va, etc., the first, second, etc., mean proportionals to the unit line and a.* Descartes' justification of this use of the symbols of numerical algebra is that the geometric constructions of which he makes a + b, a — b, etc., represent the results are " the same " as numerical addition, subtraction, multiplica- tion, division, and evolution, respectively. Moreover, since all geometric constructions which determine line segments may be resolved into combinations of these constructions as the operations of numerical algebra into the fundamental operations, the correspondence which holds between these fundamental constructions and operations holds equally between the more complex constructions and operations. The entire system of the geometric constructions under consideration may therefore be regarded as formally iden- tical with the system of algebraic operations, and be represented by the same symbolism. In what sense his fundamental constructions are "the same " as the fundamental operations of arithmetic, Des- cartes does not explain. The true reason of their formal identity is that both are controlled by the commutative, associative, and distributive laws. Thus in the case of the former as of the latter, ab = ba, and a (be) =abc, for the fourth proportional to the unit line, a, and b is the same as the fourth proportional to the unit line, b, and a ; and the fourth proportional to the unit line, a, and be is the same as * Geometrie, Livre I. Ibid. pp. 313-314. THE GENERAL IRRATIONAL. 121 the fourth proportional to the unit line, ab, and c. But this reason was not within the reach of Descartes, in whose day the fundamental laws of numerical algebra had not yet been discovered. 120. The Continuous Variable. Newton. Euler. It is customary to credit the Geometrie with having introduced the continuous variable into mathematics, but without suffi- cient reason. Descartes prepared the way for this con- cept, but he makes no use of it in the Geometrie. The x and y which enter in the equation of a curve he regards not as variables but as indeterminate constants, a pair of whose values correspond to each curve-point.* The real author of this concept is Newton (1642-1727), of whose great inven- tion, the method of fluxions, continuous variation, "flow," is the fundamental idea. But Newton's calculus, like Descartes' algebra, is geo- metric rather than purely numerical, and his followers in England, as also, to a less extent, the followers of his great rival, Leibnitz, on the continent, in employing the calculus, for the most part conceive of variables as lines, not num- bers. The geometric form again threatened to become para- mount in mathematics, and geometry to enchain the new " analysis " as it had formerly enchained the Greek arith- metic. It is the great service of Euler (1707-1783) to have broken these fetters once for all, to have accepted the con- tinuously variable number in its purity, and therewith to have created the pure analysis. For the relations of con- tinuously variable numbers constitute the field of the pure analysis ; its central concept, the function, being but a device for representing their interdependence. 121. The General Irrational. While its concern with variables puts analysis in a certain opposition to elementary algebra, concerned as this is with constants, its establish- * Geometrie, Livre II. Ibid. pp. 337-338. 122 NUMBEB-SYSTEM OF ALGEBRA. ment of the continuously variable number in mathematics brought about a rich addition to the number-system of alge- bra — the general irrational. Hitherto the only irrational numbers had been "surds," impossible roots of rational numbers ; henceforth their domain is as wide as that of all possible lines incommensurable with any assumed unit line. 122. The Imaginary, a Recognized Analytical Instrument. Out of the excellent results of the use of the negative grew a spirit of toleration for the imaginary. Increased atten- tion was paid to its properties. Leibnitz, noticed the real sum of conjugate imaginaries (1676-7) ; Demoivre dis- covered (1730) the famous theorem (cos 6 -\- i sin 0) n = cos nO + i sin nO ; and Euler (1748) the equation cos & + i sin == e <9 , which plays so great a r61e in the modern theory of functions. Euler also, practising the method of expressing complex numbers in terms of modulus and angle, formed their prod- ucts, quotients, powers, roots, and logarithms, and by many brilliant discoveries multiplied proofs of the power of the imaginary as an analytical instrument. 123. Argand's Geometric Representation of the Imaginary. But the imaginary was never regarded as anything better than an algebraic fiction — to be avoided, where possible, by the mathematician who prized purity of method — until the discovery of a geometric picture for it such as that with which Descartes had supplied the negative. The first to render it this service was a French mathematician, Argand, in 1806.* * Essai sur une maniere de representer les quantit€s imaginaires dans les constructions geometriques. THE COMPLEX NUMBER. 123 As -f- 1 and — 1 may be represented by unit lines drawn in opposite directions from any point, 0, and as t (i.e. V — 1) is a mean proportional to + 1 and — 1, it occurred to Argand to represent this symbol by the line whose direction with respect to the line + 1 is the same as the direction of the line —1 with respect to it; viz., the unit perpendicular through to the 1 — line. Let only the direction of the 1 — line be fixed, the position of the point in the plane is altogether indifferent. Between the segments of a given line, whether taken in the same or opposite directions, the equation holds ; AB + BC=AC It means nothing more, however, when the directions of AB and BG are opposite, than that the result of carrying a moving point from A first to B, and thence back to <7, is the same as carrying it from A direct to C. But in this sense the equation holds equally when A, B, C are not in the same right line. Given, therefore, a complex number, a + ib ; choose any point A in the plane ; from it draw a line AB, of length a, in the direction of the 1 — line, and from B a line BO, of length b, in the direction of the % — line. The line AC, thus fixed in length and direction, but situated anywhere in the plane, is Argand's picture of a -f- ib. ■ Argand's skill in the use of his new device was equal to the discovery of the demonstration given in § 54, that every algebraic equation has a root. 124. Gauss. The Complex Number. The method of rep- resenting complex numbers in common use to-day, that described in § 42, is due to Gauss. He was already in pos- session of it in 1811, though he published no account of it until 1831. To Gauss belongs the conception of i as an independent unit co-ordinate with 1, and of a -f- ib as a complex number, / 124 NUMBER-SYSTEM OF ALGEBRA. a sum of multiples of the units 1 and %\ his also is the name " complex number " and the concept of complex num- bers in general, whereby a + ib secures a footing in the theory of numbers as well as in algebra. He too, and not Argand, must be credited with really breaking down the opposition of mathematicians to the imaginary. Argand' s Essai was little noticed when it ap- peared, and soon forgotten ; but there was no withstanding the great authority of Gauss, and his precise and masterly presentation of this doctrine.* VII. EECOGNITION OP THE PUKELY SYMBOLIC CHAEACTEE OF ALGEBEA. QUATERNIONS. AUSDEHNUNGSLEHRE. 125. The Principle of Permanence. Thus, one after another, the fraction, irrational, negative, and imaginary, gained entrance to the number-system of algebra. iNot one of them was accepted until its correspondence to some actually existing thing had been shown, the fraction and irrational, which originated in relations among actually ex- isting things, naturally making good their position earlier than the negative and imaginary, which grew immediately out of the equation, and for which a " real " interpretation had to be sought. Inasmuch as this correspondence of the artificial numbers to things extra-arithmetical, though most interesting and the reason of the practical usefulness of these numbers, has not the least bearing on the nature of their position in pure arithmetic or algebra ; after all of them had been accepted as numbers, the necessity remained of justifying this * See Gauss, Complete Works, II, p. 174. SYMBOLIC CHARACTER OF ALGEBRA. 125 acceptance by purely algebraic considerations. This was first accomplished, though incompletely, by the English, mathematician, Peacock* Peacock begins with, a valuable distinction between arith- metical and symbolical algebra. Letters are employed in the former, but only to represent positive integers and frac- tions, subtraction being limited, as in ordinary arithmetic, to the case where subtrahend is less than minuend. In the latter, on the other hand, the symbols are left altogether general, untrammelled at the outset with any particular meanings whatsoever. It is then assumed that the rules of operation applying to the symbols of arithmetical algebra apply without altera- tion in symbolical algebra ; the meanings of the operations themselves and their results being derived from these rules of operation. This assumption Peacock names the Principle of Perma- nence of Equivalent Forms, and illustrates its use as follows : t In arithmetical algebra, when a > b, c > d, it may readily be demonstrated that (a — b) (c — d) = ac — ad — bc-{- bd. By the principle of permanence, it follows that (0 _ b) (0 - d) = x - x d - b x + bd, or (-&)(-<*) = &<£ Or again. In arithmetical algebra a m a n = a m+n , when m and n are positive integers. Applying the principle of permanence, p p p (aty = a? - a* ... to q factors -= a |+|+ -t° J terms = a p , whence afl — i/a p . * Arithmetical and Symbolical Algebra, 1830 and 1845 ; especially the later edition. Also British Association Reports, 1833. t Algebra, edition of 1845, §§ 631, 569, 639. 126 NUMBER-SYSTEM OF ALGEBRA. Here the meanings of the product ( — b) ( — d) and of the p symbol a q are both derived from certain rules of operation in arithmetical algebra. Peacock notices that the symbol = also has a wider mean- ing in symbolical than in arithmetical algebra ; for in the former = means that " the expression which exists on one side of it is the result of an operation which is indicated on the other side of it and not performed." # He also points out that the terms "real" and "imagi- nary" or '-impossible" are relative, depending solely on the meanings attaching to the symbols in any particular application of algebra. For a quantity is real when it can be shown to correspond to any real or possible existence ; otherwise it is imaginary. f The solution of the problem : to divide a group of 5 men into 3 equal groups, is imaginary though a positive fraction, while in Argand's geometry the so-called imaginary is real. The principle of permanence is a fine statement of the assumption on which the reckoning with artificial numbers depends, and the statement of the nature of this dependence is excellent. Eegarded as an attempt at a complete presen- tation of the doctrine of artificial numbers, however, Pea- cock's Algebra is at fault in classing the positive fraction with the positive integer and not with the negative and imaginary, where it belongs, in ignoring the most difficult of all artificial numbers, the irrational, in not defining arti- ficial numbers as symbolic results of operations, but princi- pally in not subjecting the operations themselves to a final analysis. 126. The Fundamental Laws of Algebra. "Symbolical Algebras." Of the fundamental laws to which this analysis leads, two, the commutative and distributive, had been noticed years before Peacock by the inventors of symbolic * Algebra, Appendix, § 631. t Ibid. § 557. SYMBOLIC CHABACTER OF ALGEBBA. 127 methods in the differential and integral calculus as being common to number and the operation of differentiation. In . fact, one of these mathematicians, Servois,* introduced the names commutative and distributive. Moreover, Peacock's contemporary, Gregory, in a paper "On the Eeal Nature of Symbolical Algebra," which ap- peared in the interim between the two editions of Peacock's Algebra,f had restated these two laws, and had made their significance very clear. To Gregory the formal identity of complex operations with the differential operator and the operations of numer- ical algebra suggested the comprehensive notion of algebra embodied in his fine definition : " symbolical algebra is the science which treats of the combination of operations de- fined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject." This definition recognizes the possibility of an entire class of algebras, each characterized primarily not by its subject- matter, but by its operations and the formed laws to which they are subject; and in which the algebra of the complex num- ber a -f ib and the system of operations with the differential operator are included, the two (so far as their laws are identical) as one and the same particular case. So long, however, as no " algebras " existed whose laws differed from those of the algebra of number, this definition had only a speculative value, and the general acceptance of * Gergonne's Annates, 1813. One mnst go back to Enclid for the earliest known recognition of any of these laws. Euclid demonstrated, of integers (Elements, VII, 16), that ab=ba. t In 1838. See The Mathematical Writings of D. F. Gregory, p. 2. Among other writings of this period, which promoted a correct under- standing of the artificial numbers, should be mentioned Gregory's interesting paper, " On a Difficulty in the Theory of Algebra,"' Writ- ings, p. 235, and De Morgan's papers "On the Foundation of Algebra " (1839, 1841 ; Cambridge Philosophical Transactions, VII). 128 NUMBEB-SYSTEM OF ALGEBBA. the dictum that the laws regulating its operations consti- tuted the essential character of algebra might have been long delayed had not Gregory's paper been quickly followed by the discovery of two "algebras/' the quaternions of Hamilton and the Ausclelinungslelire of Grassmann, in which one of the laws of the algebra of number, the commutative law for multiplication, had lost its validity. 127. Quaternions. According to his own account of the discovery,* Hamilton came upon quaternions in a search for a second imaginary unit to correspond to the perpendicu- lar which may be drawn in space to the lines 1 and i. In pursuance of this idea he formed the expressions, a + ib +jc, x + iy +jz, in which a, b, c, x, y, z were sup- posed to be real numbers, and j the new imaginary unit sought, and set their product (a + ib +jc) (x -f- iy -±-jz) = ax — by — cz-\-i (ay -J- bx) +j (az + ex) + ij (bz + cy) . The question then was, what interpretation to give ij. It would not do to set it equal to a' + ib r -\- jc' , for then the theorem that the modulus of a product is equal to the product of the moduli of its factors, which it seemed indis- pensable to maintain, would lose its validity ; unless, in- •deed, a' = b' = c' = 0, and therefore ij = 0, a very unnatural supposition, inasmuch as 1 i is different from 0. No course was left for destroying the ij term, therefore, but to make its coefficient, bz -f- cy, vanish, which was tanta- mount to supposing, since b, c, y, z are perfectly general, that ji = — ij. Accepting this hypothesis, denial of the commutative law as it was, Hamilton was driven to the conclusion that the system upon which he had fallen contained at least three imaginary units, the third being the product ij. He called * Philosophical Magazine, II, Vol. 25, 1844. SYMBOLIC CHABACTEB OF ALGEBRA. 129 this ft, took as general complex numbers of the system, a + ib + jc + kd, x -\-iy -\-jz -\-kw, quaternions, built their products, and assuming jk = — kj = i ft£ = — ik=j, found that the modulus law was fulfilled. A geometrical interpretation was found for the "imag- inary triplet" ib +Jc -f- kd, by making its coefficients, b, c, d, the rectangular co-ordinates of a point in space ; the line drawn to this point from the origin picturing the triplet by its length and direction. Such directed lines Hamilton named vectors. To interpret geometrically the multiplication of i into j, it was then only necessary to conceive of the j axis as rigidly connected with the i axis, and turned by it through a right angle in the jk plane, into coincidence with the k axis. The geometrical meanings of other operations fol- lowed readily. In a second paper, published in the same volume of the Philosophical Magazine, Hamilton compares in detail the laws of operation in quaternions and the algebra of number, for the first time explicitly stating and naming the asso- ciative law. 128. Grassmann's Ausdehnungslehre. In the Ausdeh- nungslehre, as Grassmann first presented it, the elementary magnitudes are vectors. The fact that the equation AB + BC= AC always holds among the segments of a line, when account is taken of their directions as well as their lengths, suggested the probable usefulness of directed lengths in general, and led Grassmann, like Argand, to make trial of this definition of 130 NUMBER-SYSTEM OF ALGEBRA. addition for the general case of three points, A, B, C, not in the same right line. But the outcome was not great until he added to this his definition of the product of two vectors. He took as the product ab, of two vectors, a and &,the parallelogram gen- erated by a when its initial point is carried along b from initial to final extremity. This definition makes a product vanish not only when one of the vector factors vanishes, but also when the two are parallel. It clearly conforms to the distributive law. On the other hand, since (a + b) (a -f- b) = aa -f ab + ba -f bb, and (a + b) (a + b) = aa = bb = 0, ab -f- ba = 0, or ba = — ab, the commutative law for multiplication has lost its validity, and, as in quaternions, an interchange of factors brings about a change in the sign of the product. The opening chapter of Grassmann's first treatise on the Ausdehnungslehre (1844) presents with admirable clear- ness and from the general standpoint of what he calls " "Formenlehre " (the doctrine of forms), the fundamental laws to which operations are subject as well in the Aus- dehnungslehre as in common algebra. y 129. The Doctrine of the Artificial Numbers fully Devel- oped. The discovery of quaternions and the Ausdehnungs- lehre made the algebra of number in reality what Gregory's definition had made it in theory, no longer the sole algebra, but merely one of a class of algebras. A higher standpoint was created, from which the laws of this algebra could be seen in proper perspective. Which of these laws were distinctive, and what was the significance of each, came out clearly enough when numerical algebra could be com- pared with other algebras whose characteristic laws were not the same as its characteristic laws. SYMBOLIC CHARACTER OF ALGEBRA. 131 The doctrine of the artificial numbers regarded from this point of view — as symbolic results of the operations which the fundamental laws of algebra define — was fully presented for the negative, fraction, and imaginary, by Hankelf in his Complexe Zahlensystemen (1867). Hankel re-anhounced Peacock's principle of permanence. The doctrine of the irrational now accepted by mathematicians is due to Weierstrass and G. Cantor.* A number of interesting contributions to the literature of the subject have been made recently ; among them a paper f by Kronecker in which methods are proposed for avoiding the artificial numbers by the use of congruences and " inde- terminates," and papers t by Weierstrass, Dedekind, Holder, Study, Scheffer, and Schur, all relating to the theory of general complex numbers built from n fundamental units (see page 40). * See Cantor in Mathematische Annalen, V, p. 123, XXI, p. 567. The first paper was written in 1871. In the second, Cantor compares his theory with that of Weierstrass, and also with the theory proposed by Dedekind in his Stetigkeit und irrationale Zahlen (1872). The theory of the irrational, set forth in Chapter IV of the first part of this book, is Cantor's. t Journal fur die reine und angewandte Mathematik, Vol. 101, p. 337. | Gottinger Nachrichten for 1884, p. 395 ; 1885, p. 141 ; 1889, p. 34, p. 237. Leipziger Berichte for 1889, p. 177, p. 290, p. 400. Mathe- mathische Annalen, XXXIII, p. 49. ^ 4* M -** &V >rf^ ' . NOV 1 q 19£ fi , , UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DD6, BERKELEY, CA 94720 EMHH^HHBHHIBSBfilH^HBglHIHHnBHflMVnHnHtfHHHHBBHHHiHBHHHSHlHB\;':if? » B MlM«t-M-" ,BH UNIVERSITY OF CALIFORNIA LIBRARY