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47. Negative quantities will have arisen in various ways, e.g.
(i.) The logical result of the commutative law, applied to a succession of additions and subtractions, is to produce a negative quantity—3s. such that —5s.+3s.= o(§ 28 (vi.)).
(ii.) Simple equations, especially equations in which the unknown quantity is an interval of time, can often only be satisfied by a negative solution (§ 33).
(iii.) In solving a quadratic equation by the method of § 38 (Vlii-l we may be led to a result which is apparently absurd. If, for instance, we inquire as to the time taken to reach a given height by a body thrown upwards with a given velocity, we
find that the time increases as the height decreases. GraphiCal i. 20
representation shows that there are two solutions, and that an equation X’=oa2 may be taken to be satisfied not only by X=3a but also by X=——3a.
48. The occurrence of negative quantities does not, however, involve the conception of negative numbers. In (iii.) of § 47, for instance, “ —3a " does not mean that a is to be taken (—3-) times, but that a is to be taken 3 times, and the result treated as subtractive; i.e.—3a means —(3a), not (—3)a (cf. § 27 (i,)).
In the graphic method of representation the sign — may be taken as denoting a reversal of direction, so that, it + 3 represents a length of 3 units measured in one direction,—3 represents a length of 3 units measured in the other direction. But even so there are two distinct operations concerned in the—3,-viz. the multiplication by 3 and the reversal of direction. The graphic method, therefore, does not give any direct assistance 0 towards the conception of negative numbers as operators, though it is useful for interpreting negative quantities as results.
49. In algebraical transformations, however, such as (x—a)’ =x’—2ax+a’, the arithmetical rule of signs enables us to combine the sign—with a number and to treat the result as a whole, subject to its own laws of operation. We see first that any operation with 4a—3b can be regarded as an operation with (+)4a+(—)3b, subject to the conditions (i) that the signs (+) and (—) obey the laws (+)(+)=(+),(+)(—)=(—)(+)= (-), (—) (—)=(+), and (2) that, when processes of multiplication are completed, a quantity is to be added or subtracted according as it has the sign (+) or (—) prefixed. We are then able to combine any number with the + or the —- sign inside the bracket, and to deal with this constructed symbol according to special laws; i.e. we can replace pr or —pr by (+p)r or (—p)r, subject to the conditions that (+1)) (+q) =(—p) (—q) =(+pq), (+9) (~q)=(—P) (+q)=(—M), and that + (—8) means that s is to be subtracted.
These constructed symbols may be called Positive and negative coeflicients; or a symbol such as (—1:) may be called a negative number, in the same way that we call § a‘fractional number.
This increases the extent of the numbers with which we have to deal; but it enables us to reduce the number of formulae. The binomial theorem may, for instance, be stated for (x+a)" alone; the formula for (x—a)" being obtained by writing it as lx+(—)al" or lx+(—a)l", so that
(x—a)"=x"—n(,)x""a+...+(—)'n(r>x""a'+..., where + (—)r means — or + according as r is odd or even.
The result of the extension is that the number or quantity represented by any symbol, such as P, may be either positive or negative. The numerical value is then represented by IPI; thus “ |x|< r ” means that x is between --I and +1.
50. The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a binomial, as illustrated by the following examples:—
dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (§ 28 (iv.)), and are arranged in descending or ascending powers of x. If P and M are rational integral functions of x, arranged in descending powers of x, the division of P by M is complete when we obtain a remainder R whose degree (§ 45) is less than that of M. If R=o, then M is said to be afactor of P.
The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M. in arithmetic; numerical coefficients of the factor as a whole being ignored (cf. § 36 (iv.)).
52. Relation between Roots and Factors.—-—(i.) If we divide the multinomial
PEpox"+p1x“‘1+...+p,, by x—a, according to algebraical division, the remainder is ' REpoa"+p1a"_l—l-...+p,,. This is the remainder-theorem; it may be proved by induction.
(ii.) If x=a satisfies the equation P=o, then poa"+pla”_‘+ . . . +p..=o; and therefore the remainder when P is divided by 20—0 is o, i.e. x—a is a factor of P.
(iii.) Conversely, if x—a is a factor of P, then poa”+pla"'1 +. . . +p,.=o; i.e. x=a satisfies the equation P=o.
(iv.) Thus the problems of determining the roots of an equation P=o and of finding the factors of P, when P is a rational integral function of x, are the same.
(v.) In particular, the equation P=o, where P has the value in (i.), cannot have more than 11 different roots.
The consideration of cases where two roots are equal belongs to the theory of equations (see EQUATION).
(vi.) It follows that, if two multinomials of the nth degree in a: have equal values for more than n values of x, the corresponding coefficients are equal, so that the multinomials are equal for all values of x.
53. Negative I ndices and Logarithms.—(i.) Applying the general principles of §§ 47-49 to indices, we find that we can interpret X‘” as being such that
X'".X-'" =X° = 1; i.e. X‘" = I/X'".
(ii.) This leads to negative logarithms (see LOGARITHM).
54. Laws of Algebraic Form—(i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (§ 26 (i.)) but differ from them in relating, not to arithmetical value, but to algebraic form. The commutative law in arithmetic, for instance, states that 0-H: and b+a, or ab and ba, are equal. The corresponding law of form regards 0—H) and b+a, or ab and ba, as being not only equal but identical (cf. § 37 (ii.)), and then says that A+B and B+A, or AB and BA, are identical, where A and B are any multinomials. Thus a(b+c) and (b+c)a give the same result, though it may be written in various ways, such as ab+ac, ca+ab, &c. In the same way the associative law is that A(BC) and (AB)C give the same formal result.
These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in § 52 (vi.).
(ii.) One result of these laws is that, when we have obtained any formula involving a letter a, we can replace a by a multinomial. For instance, having found that (x+a)’=xz+2ax+a’, we can deduce that (x+b+c)2= lx+(b+c)}’=x’+2(b+c)x+ (b+c)’.
(iii.) Another result is that we can equate coefiicients of like powers of x in two multinomials obtained from the same expression by difierent methods of expansion. For instance, by equating coefficients of x’ in the expansions of (1+x)"+" and of (r+x)"'. (1+x)" we obtain (22) of § 44 (ii.).
(iv.) On the other hand, the method of equating coefficients often applies without the assumption of these laws. In § 41 (ii.), for instance, the coefficient of A"_'a’ in the expansion of
so that, if nE—m, ' 5,5i+n(i)x+n(,)x2+...+n(,)x' (29),
. (v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division. In calculating ~12, for instance, We proceed as if 2~oooo . . . were the exact square of some number of the form Qi+61/10+Cg/102+. . ..
In the same way, to find X1”, where XE t+a1x+aixfl+ . . . and q is a positive integer, we assume that X1"! = I+b1x+b,x'. . ., and we then (cf. § 55) determine bi, bi, . . . in succession so that (1+blx+b$x‘+ . . .)° shall be identical with X.
The application of the method to the calculation of (i-l—x)", when n=p/q, q being a positive integer and p a positive or negative integer, involves, as in the case where n is a negative integer, the separate consideration of the form of the coefficients bi, bi, . .. and of the numerical value of i+b1x+ng’+. . . +b,x'.
(vi.) The definition of nm, which has already been extended in (iv.) above, has to be further extended so as to cover fractional values of n, positive or negative. Certain relations still hold, the most important being (22) of § 44 (ii.), which holds whatever the values of m and of n may be; r, of course, being a positive integer. This may be proved either by induction or by the method of § 52 (vi.). The relation, when written in the form (23), is known as Vandermonde’s theorem. By means of this theorem it can be shown that, whatever the value of n may be,
It will be noticed that, although the difi'erences between successive terms of the sequence will ultimately become indefinitely small, there will always be intermediate numbers that do not occur in the sequence. The approach to the limit will therefore be by a series of jumps, each of which, however small, will be finite; i.e. the approach will be discontinuous.
(ii.) Instead of examining what happens as r increases, let us examine what happens as h/x decreases, r remaining unaltered. Denote h/x by 0, Where i>9>o; and suppose further that 0<| i/ri I, so that the first term of the series m+u1+u,+. . . is
59. The idea of continuity must in the first instance be introduced from the graphical point of view; arithmetical continuity being impossible without a considerable extension of the idea of number (§ 65). The idea is utilized in the elementary consideration of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.
60. The first step consists in the functional treatment of equations. .Thus, to solve the equation ax“+bx+c=o, we consider, not merely the value of x for which axz+bx+c is 0, but the value of ax’ +bx+c for every possible value of x. By graphical treatment we are able, not merely to see why the equation has usually two roots, and also to understand why there is in certain cases only one root (i.e. two equal roots) and in other cases no root, but also to see why there cannot be more than two roots.
Simultaneous equations in two unknowns x and y may be treated in the same way, except that each equation gives a functional relation between a; and y. (“ Indeterminate equations ” belong properly to the theory of numbers.)
‘61. From treating an expression involving x as a function of x which may change continuously when x changes continuously, we are led to regard two functions as and y as changing together, so that (subject to certain qualifications) to any succession of values of x or of y there corresponds a succession of values of y or of x; and thence, if (x, y) and (x+h, y+k) are pairs of corresponding values, we are led to consider the limit (§ 58 (ii.)) of the ratio k/h when h and k are made indefinitely small. Thus we arrive at the differential coefficient of f (x) as the limit of the ratio of f(x+0) —f(x) to 9 when 0 is made indefinitely small; and this gives an interpretation of nx"" as the derived function of x" (§ 45)
This conception of a limit enables us to deal with algebraical
as for other values of x; but we can say that the limit of the ratio of x’— r to x— I when 1: becomes indefinitely nearly equal to I is the same as the limit of x+r.
On the other hand, if f(y) has a definite and finite value for y=x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value as, though this is the case with the functions with which we are usually concerned.
62. The elementary idea of a difierential coefficient is useful in reference to the logarithmic and exponential series. We know that log10N(r +0) =logmN+logm(1+0), and inspection of a table of logarithms shows that, when 0 is small, log10(r+0) is approximately equal to )0, where h is a certain constant, whose value is -434. . . If we took logarithms to base a, we should have
is then more simply obtained by the difi'erential calculus than by ordinary algebraic methods.
63. The theory of inequalities is closely connected with that of maxima and minima, and therefore seems to come properly under this head. The more simple properties, however, only require the use of elementary methods. Thus to show that the arithmetic mean of n positive numbers is greater than their geometric mean (i.e. than the nth root of their product) we show that if any two are unequal their product may be increased, without altering their sum, by making them equal, and that if all the numbers are equal their arithmetic mean is equal to their geometric mean.
VI. Special Developments.
64. One case of convergence of a sequence has already been considered in § 58 (i.). The successive terms of the sequence in that case were formed by successive additions of terms of a series; the series is then also said to converge to the limit which is the limit of the sequence.
Another example of a sequence is afiorded by the successive
. . r r convergents to a continued fraction of the form 00+ g m. . . , where 00,111,111, . . . are integers. Denoting these convergents by Po/Qn, P, /Q1, P,/Q,, . . . they may be regarded as obtained from
a series d+ + +. . . ; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators 1 and denominators continually increasing.
Another kind of sequence is that which is formed by intro— ducing the successive factors of a continued product; e.g. the successive factors on the right-hand side of Wallis’: theorem
1-21 in 62
2 _ 1.3 ' 3.5 ' 5.7'“ A continued product of this kind can, by taking logarithms, be replaced by an infinite series.
In the particular case considered in § 58 (i.) we were able to examine the approach of the sequence So, 51, Sa, . . . to its limit X by direct examination of the value of X—S,. In most cases this is not possible; and we have first to consider the convergence of the sequence or of the series which it represents, and then to determine its limit by indirect methods. This constitutes the general theory of convergence of series (see SERIES).
The word “ sequence,” as defined in § 58 (i.), includes prm gressions such as the arithmetical and geometrical progressions, and, generally, the succession of terms of a series. It is usual, however, to confine it to those sequences (e.g. the sequence formed by taking successive sums of a series) which have to be considered in respect of their convergence or non-convergence.
In order that numerical results obtained by summing the first few terms of a series may be of any value, it is usually necessary that the series should converge to a limit; but there are exceptions to this rule. For instance, when n is large, n! is approximately equal to V(21rn).(n/e)"; the approximation may be improved by Stirling’s theorem
l 7 _' FlBr +m_3.4.n’+"'+l27—l l-zr.n""l+'" ' where B1, B2, . . . are Bernoulli’s numbers (§ 46 (v.)), although the series is not convergent.
65. Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis’s theorem (§ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e. a number which cannot be expressed as the ratio of two integers.
These are isolated cases of irrational numbers. Other cases arise when we consider the continuity of a function. Suppose, for instance, that y=x’; then to every rational value of as there corresponds a rational value of y, but the converse does not hold. Thus there appear to be discontinuities in the values of y.
The difficulty is due to the fact that number is naturally not continuous, so that continuity can only be achieved by an artificial development. The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages. Taking any number n to be represented by a point on a line at distance nL from a fixed point 0, where L is a unit of length, we start with a series of points representing the integers 1, 2, 3, . . . This series is of course discontinuous. The next step is to suppose that fractional numbers are represented in the same way. This extension produces a change of character in the series of numbers. In the original integral series each number had a definite number next to it, on each side, except 1, which began the series. But in the new series there is no first number, and no number can be said to be next to any other number, since, whatever two numbers we take, others can be inserted between them. On the other hand, this new series is not continuous; for we know that there are some points on the line which represent surds and other irrational numbers, and these numbers are not contained in our series. We therefore take a third step, and obtain theoretical continuity by considering that every point on the line, if it does not represent a rational number, represents something which may be called an irrational number.
This insertion of irrational numbers (with corresponding negative numbers) requires for its exact treatment certain special methods, which form part of the algebraic theory of number, and are dealt with under NUMBER.
66. The development of the theory of equations leads to the amplification of real numbers, rational and irrational, positive and negative, by imaginary and complex numbers. The quadratic equation x4+b’=o, for instance, has no real root; but we may treat the roots'as being +b~J —r, and —b\/ -1, if \I —1 is treated as something which obeys the laws of arithmetic and emerges into reality under the condition \I — 1.4 — 1 = — 1. Expressions of the form b4 —-1 and a+b\/ —r, where a and b are real numbers, are then described as imaginary and complex numbers respectively; the former being a particular case of the latter.
Complex numbers are conveniently treated in connexion not only with the theory of equations but also with analytical trigono
'metry, which suggests the graphic representation of a+b4 — 1 by a line of length (aH-b’)l drawn in a direction different from that of the line along which real numbers are represented.
REFERENCES.——w. K. Clifford, The Common Sense of the Exact Sciences (1885), chapters i. and iii., forms a good introduction to algebra. As to the teaching of algebra, see references under ARITHMETIC to works on the teaching of elementary mathematics. Amon school-books may be mentioned th0se of W. M. Baker and A. A. ourne, W. G. Borchardt, W. D. Biggar, F. Gorse, H. S. Hall and S. R. Knight, A. E. F. La ng, R. . Mor an. G. Chrystal, Introduction to Algebra (1898); . B. Fine, A Col ege Algebra (1905); C. Smith, A Treatise on Algebra (1st ed. 1888, rd ed. 1892), are more suitable for revision purposes; the second of t ese deals rather fully with irrational numbers. For the algebraic theo of number, and the convergence of sequences and of series, see T. . I'A. Bromwich, Introduction to the Theory of Infinite Series (1908 ; H. S. Carslaw, Introduction to the Theory 0; ourier's Series 1 06); H. B. Fine, The Number-System of Alge ra (1891); H. P. arming, Irrational Numbers (1906); Pierpont, Lectures on the Theor of Functions 0 Real Variables 1905). For general reference, G. h stal, Text
vote of Algebra (pt. i. 5th ed. 1904, pt. ii. 2nd e . 1900) is indispensable; unfortunately, like many of the works here mentioned, it lacks a proper index. Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Drrranancss, CALCULUS or; lNFlNrrESIMAL CALCULUS; INTERPOLATION; Vacrox ANALYSIS. The followin may also be consulted :—-E. Borel and Drach, Introduction l’e'tude de la théorie des nombres et de l‘alg bre supé< neure__(1895); C. de Comberousse, Caurs de mathématiques, vols. i. and in. (1884—1837); H. Laurent, Traite' d'analyse, vol. i. (1885); E. Netto, Vorlesungen fiber Algebra (vol. i. 1896, vol. ii. 1900); S. Pincherle, Algebra corn lementare (1893): G. Salmon. Lessons introductory to the Modern igher algebra (4th ed., 1885); J. A. Serret, Cour; d'algebre supérieure (4th ., 2 vols., 1877); O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (pt. i. 1900, pt. ii. 1902) and Emlertung in die Funktionen-theorie (pt. i. 1904, pt. ii. 1905)— these being developments from O. Stolz, Vorlesungen fiber allgerneine Arithmetik (pt. 1. 1885, pt. ii. 1886); J. Tannery, Introduction 6 la
B. SPECIAL Knms or ALGEBRA
r. A special algebra is one which differs from ordinary algebra in the laws of equivalence which its symbols obey. Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication. ~ ~
2. Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts. Although the distinction is one which cannot be ultimately maintained, it is convenient to classify the signs of algebra into symbols of quantity (usually figures or letters), symbols of operation, such as +, 4 , and symbols of distinction, such as brackets. Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings. It could not escape notice that one and the same symbol, such as \/ (a—b), or even (a—b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved. This led to a prolonged controversy on the nature of negative and imaginary quantities, which was ultimately settled in a very curious way. The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a “ meaning " or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained. It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction. When this fundamental truth had been fully grasped, mathematicians began to inquire whether algebras might not be discovered which obeyed laws different from those obtained by the generalization of arithmetic. The answer to this question has been so manifold as to be almost embarrassing. All that can be done here is to give a sketch of the more important and independent special algebras at present known to exist. .
3. Although the results of ordinary algebra will be taken for granted, it is convenient to give the principal rules upon which it is based. They are
These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the definitions of the inverse symbols — and +, which are assumed to be unambiguous. The special symbols 0 and r are used to denote a—a and a+a. They behave exactly like the corresponding symbols in arithmetic; and it follows from this that whatever “ meaning ” is attached to the symbols of quantity, ordinary algebra includes arithmetic, or at least an image of it. Every ordinary algebraic quantity may be regarded as of the form a+fiwl —-r, where a, fi are “ real ”; that is to say, every algebraic equivalence remains valid when its symbols of quantity are interpreted as complex numbers of the type a+fi~l —1 (cf. NUMBER). But the symbols of ordinary algebra do not necessarily denote numbers; they may, for instance, be interpreted as coplanar points or vectors. Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational algebraic quantities analogous to that of irrational numbers.