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7.x, except for ro, when xo is replaced by o. sions obtained in this way are called the first, second, derived functions of f(x). If we denote these by fi(x), f2(x), . so that f(x) is obtained from f.-1(x) by the above process, we have

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f(x+h) =f(x)+f1(x).h+f2(x)h2/2!+...+fr(x)h*/r!+... This is a particular case of Taylor's theorem (see INFINITESIMAL CALCULUS).

46. Relation of Binomial Coefficients to Summation of Series.(i.) The sum of the first n terms of an ordinary arithmetical progression (a+b), (a+2b), ... (a+nb) is (§ 28 (i.)) }n ((a+b)+ (a+nb)} = na+n(n+1)b = m[1].a+m[2]. b. Comparing this with the table in §43 (iv.), and with formula (21), we see that the series expressing the sum may be regarded as consisting of two, viz. a+a+ ... and b+2b+3b+ . ; for the first series we multiply the table (i.e. each number in the table) by a, and for the second series we multiply it by b, and the terms and their successive sums are given for the first series by the first and the second columns, and for the second series by the second and the third columns.

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(ii.) In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the 'next following column. Thus we get a formula for the

sum of n terms of a series such as

2.4.6+4.6.8+..., or 6.8.10.12 +8.10.12.14+...

(iii.) Suppose we have such a series as 2.5+5.8+8.11+ ... This cannot be summed directly by the above method. But the nth term is (3n−1)(3n+2)=18[2]-6[1]-2. The sum of n terms is therefore (§ 43 (iv.))

18n[3]-6n[2]-2n[1]=3n3+6n2+n.

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(iv.) Generally, let N be any rational integral function of n of degree r. Then, since nr] is also a rational integral function of n of degree r, we can find a coefficient cr, not containing n, and such as to make N-cm] contain no power of n higher than n1. Proceeding in this way, we can express N in the form Gr.n[r]+ where Cr, Cr-1, Cr-2, do not contain n; and thence we can obtain the sum of the numbers found by putting n=1, 2, 3,... n successively in N. These numbers constitute an arithmetical progression of the rth order. (v.) A particular case is that of the sum 1+2+3+...+n', where r is a positive integer. It can be shown by the above reasoning that this can be expressed as a series of terms containing descending powers of n, the first term being '+1/(r+1). The most important cases are

1 +2 +3 +...+n = {n(n+1),
12+22+32+...+n2 = } n (n + 1)(2n+1),

13+23+33+...+n3 = {n2 (n+1)2 = (1+2+...+n)2.

The general formula (which is established by more advanced methods) is

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may be

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 taken as denoting a reversal of direction, so that, if + 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 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)1 = x2-2ax+a2, 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+(−)36, subject to the conditions (1) 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 (+p) (+q)=(− p) (−q) = (+pq), (+p) (−q)=(−p) (+q)=(−pg), and that + (−s) means that s is to be subtracted.

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These constructed symbols may be called positive and negative coefficients; or a symbol such as (−p) 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 {x+(-)a}" or {x+(−a)}", so that

(x-a)=x-n(1)x^~1a+...+(−)'n (,)x" ̄'a'+..., where + (−)" 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 |P; thus "|x|< I " 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:(1) (2) 2.10+1) 6.100 +5.10+ 1(3.10+1 2.10+1) 6.100+1.10-1 (3.10 - I 6.100 +3.10 6.100 +3.10

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In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9.

It may be necessary to introduce terms with zero coefficients. Thus, to divide by 1+x algebraically, we may write it in the form 1+o.x+o.x2+o.x3+o.x', and we then obtain I 1+0.x+0.x2+0.x3+0.xa ̧ x4 I+x = ! −x + x2 − x2 + 1 + x' I+x I+x' where the successive terms of the quotient are obtained by a process which is purely formal.

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(iii.) In solving a quadratic equation by the method of § 38 (viii.) we may be led to a result which is apparently absurd. 51. If we divide the sum of 2 and a2 by the sum of x and a, If, for instance, we inquire as to the time taken to reach a given we get a quotient x-a and remainder 202, or a quotient a-x height by a body thrown upwards with a given velocity, we and remainder 2x2, according to the order in which we work. find that the time increases as the height decreases. Graphical | Algebraical division therefore has no definite meaning unless

1. 20

II

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=0, then M is said to be a factor 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.)).

assumption of the identity of results obtained in different ways; for the expansions of (A+a)2, (A+a)3, ...are there supposed to be obtained in one way only, viz. by successive multiplications by A+a.

55. Algebraical Division.-In order to extend these laws so as to include division, we need a definition of algebraical division. The divisions in §§ 50-52 have been supposed to be performed by a process similar to the process of arithmetical division, viz. by a series of subtractions. This latter process, however, is itself based on a definition of division in terms of multiplication (§§ 15, 16). If, moreover, we examine the process of algebraical

52. Relation between Roots and Factors.-(i.) If we divide the division as illustrated in § 50, we shall find that, just as arithmultinomial

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by x-a, according to algebraical division, the remainder is R=Poa"+Pian1+...+Pn.

This is the remainder-theorem; it may be proved by induction. (ii.) If x=a satisfies the equation P=0, then Poa+Pian-1+ +=0; and therefore the remainder when P is divided by x-a is o, i.e. x-a is a factor of P.

+.

...

(iii.) Conversely, if x-a is a factor of P, then poa"+pian-1 +pn=0; i.e. x=a satisfies the equation P=0. (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=0, where P has the value in (i.), cannot have more than n 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 x 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 Indices and Logarithms.-(i.) Applying the general principles of §§ 47-49 to indices, we find that we can interpret X-m as being such that

Xm.X-m=X0=1; i.e. X-m = I/Xm.

In the same way we interpret X-P/ as meaning 1/XP/¶. (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 a+b and b+a, or ab and ba, are equal. The corresponding law of form regards a+b 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)2=x2+2ax+a2, we can deduce that (x+b+c)2= {x+(b+c)}2=x2+2(b+c)x+ | (b+c)2.

(iii.) Another result is that we can equate coefficients of like powers of x in two multinomials obtained from the same expression by different methods of expansion. For instance, by equating coefficients of x in the expansions of (1+x)m+n and of (1+x). (1+x)" we obtain (22) of § 44 (ii.).

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metical division is really the solution of an equation (§ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a multinomial really implies the use of undetermined coefficients (§ 42). When, for instance, we find that the quotient, when 6+5x+7x2+13x3+5x4 is divided by 2+3x+x2, is made up of three terms+3, -2x, and +5x2, we are really obtaining successively the values of Co, C1, and c2 which satisfy the identity 6+5x+7x+133+5xa = (co+c1x+C2x2) (2+3x+x2); and we could equally obtain the result by expanding the right-hand side of this identity and equating coefficients in the first three terms, the coefficients in the remaining terms being then compared to see that there is no remainder. We therefore define algebraical division by means of algebraical multiplication, and say that, if P and M are multinomials, the statement " P/M=Q" means that Q is a multinomial such that MQ (or QM) and P are identical. In this sense, the laws mentioned in § 54 apply also to algebraical

division.

56. Extensions of the Binomial Theorem.-It has been mentioned in § 41 (ix.) that the binomial theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account. There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true The argument involves the value, we never attain it exactly. theorem that, if is a positive quantity less than 1, 0 can be made as small as we please by taking t large enough; this follows from the fact that log can be made as large (numerically) as we please. (i.) By algebraical division, I I+0.x +0.x2+...+o.xr+1 I+x I+x

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= 1 − x + x2 -— ... + ( − )′′x′ +(-)+12 (24). − x+x2 ...+ (−)"x" r+2; If, therefore, we take 1/(1+x) as equal to 1−x+x2-...+ (-)"x", there is an error whose numerical magnitude is x+1/ (1+x); and, if |x| <1, this can be made as small as we please. This is the foundation of the use of recurring decimals; thus we can replace ==10%/(1-0) by 363636(=36/102 +36/104+36/106), with an error (in defect) of only 36/(106 .99). (ii.) Repeated divisions of (24) by 1+x, r being replaced by +1 before each division, will give (1+x)=1-2x+3x2-4x3+...+(−)" (+1)x"

86

+(-) +1 +1 {(+1) (1 +x)¬1+ (1+x)−2}, (1+x)-3= 1 − 3x+6x2 - 10x3 + ... + ( − ) ' . § (r + 1 ) (r+2) xr + ( − ) + 1 x + 1 } } (r+1) (r+2)(1 +x)−1+(r+1)(1+x)−2+(1+x) ̃3},&c. Comparison with the table of binomial coefficients in § 43 suggests that, if m is any positive integer,

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R‚=(−)r+1xr+1{m[r](I+x) ̄1+(m −1)[r](1+x)-2+...+1[r](I+x) ̄TM}(27). This can be verified by induction. The same result would (855) be obtained if we divided 1+o.x+o.x2+... at once by the expansion of (1+x)m.

(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 (A+a) (A+a)"−1 has been called ("); and it has then been shown that (") = ("-1) + (-). This does not involve any negative; and it can hence be shown that, if lx|< 1, R. can be

(iii.) From (21) of § 43 (iv.) we see that R, is less than mr+11x+1 if x is positive, or than m+m+(1+x) | if x is

r+1]

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(v.) The further extension to fractional values (positive or negative) of ʼn 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 √2, for instance, we proceed as if 2.0000... were the exact square of some number of the form co+c1/10+c2/102+....

In the same way, to find X1/%, where X=1+a1x+a2x2+ and q is a positive integer, we assume that X1/9=1+b1x+b2x2. . ., and we then (cf. § 55) determine b1, b2, . . . in succession so that (1+b1x+b2x2+...) shall be identical with X.

The application of the method to the calculation of (1+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 b1, b2,... and of the numerical value of 1+b1x+b2x2+..... +b,x".

(vi.) The definition of nor, 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, {1+(p!g)(1)x+(p/g) (2)x2+...+(p/q) (r)x"} ¶=1+Þ(1)x+Þ(2)x2+...+P(r) +terms in x+1, xr+2, 2,...x.

(vii.) The comparison of the numerical value of 1+
+n(2)x2+...+n(x, when n is fractional, with that of (1+x)",
involves advanced methods (§ 64). It is found that this expres-
sion can be used for approximating to the value of (1+x)",
provided that |x|<1; the results are as follows, where u,
denotes
n(r) x and S, denotes uo+u1+u2+...+u,.
(a) If n>-1, then, provided r>n,

(1) If 1>x>0, (1+x)” lies between S, and Sr+1;
(2) If o>x>-1, (1+x)" lies between S, and S,+ur+1/(1+x).

(b) If n<-1, the successive terms will either constantly decrease (numerically) from the beginning or else increase up to a greatest term (or two equal consecutive greatest terms) and then constantly decrease. If S, is taken so as to include the greatest term (or terms), then,

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57. Generating Functions.-The series 1-m1jx+m{2}x2obtained by dividing 1+o. x+o.x2+...by (1+x)", or the series (p/q)1)x+(p/q) (2x2+... obtained by taking the qth root of (1)x+Þ(2)x2+ .. is an infinite series, i.e. a series whose successive terms correspond to the numbers 1, 2, 3, ... It is often convenient, as in § 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to arithmetical calculation; i.e. to consider the relations between the coefficients of powers of x, rather than the values of the terms themselves. From this point of view, the function which, by algebraical operations on 1+o.x+o.x2+..., produces the series, is called its generating function. The generating functions of the two series, mentioned above, for example, are (1+x)" and (1+x). In the same way, the generating function of the series 1+2x+x2+0.x+o.x+... is (1+x)2.

Considered in this way, the relations between the coefficients of the powers of x in a series may sometimes be expressed by a formal equality involving the series as a whole. Thus (4) of § 41 (ii.) may be written in the form 1+n(1)x+n(2)x2+...+n(r)x2+..... 7 (1+x) {1+(n−1)(1)x+(n−1) (2)x2+... +(n−1)(r)x+...};

the symbol" "being used to indicate that the equality is only formal, not arithmetical.

This accounts for the fact that the same table of binomial coefficients serves for the expansions of positive powers of 1+x and of negative powers of 1-x. For (4) may (§ 43 (iv.)) be written (n − 1) [r] = N[r]—N[r−1],

and this leads to relations of the form

each set of coefficients being the numbers in a downward diagonal
1+2x+3x2+... 7 (1−x) (1+3x+6x2+10x3+...) (30),
of the table. In the same way (21) of § 43 (iv.) leads to such

relations as

1+3x+6x2+...7 (1+x+x2+...) (1+2x+3x2+.....) (31), the relation of which to (30) is obvious.

An application of the method is to the summation of a recurring series, i.e. a series co+c1x+C2x2+... whose coefficients are connected by a relation of the form poc,+pic,-1+...+PkCr-k=0, where po,p1, .pk are independent of x and of r.

58. Approach to a Limit.-There are two kinds of approach to a limit, which may be illustrated by the series forming the ex

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pansion of (x+h)", where n is a negative integer and 1>h/x>o.
(i.) Denote nrx-ht by ur, and uo+u1+...+u, by Sr.
Then (§ 56 (iii.)) (x+h) lies between S, and Sr+1; and provided
S, includes the numerically greatest term, |S+1-Sconstantly
decreases as r increases, and can be made as small as we please by
taking r large enough. Thus by taking r=0, 1, 2,
we have
a sequence So, S1, S2, . . . (i.e. a succession of numbers correspond-
ing to the numbers 1, 2, 3,...) which possesses the property
that, by starting far enough in the sequence, the range of varia-
tion of all subsequent terms can be made as small as we please,
but (x+h)" always lies between the two values determining the
range. This is expressed by saying that the sequence converges
to (x+h)" as its limit; it may be stated concisely in any of the
three ways,

(1) If I>x>0, (1+x)" lies between S, and Sr+1;
(2) Ifo>x>-1,(1+x)" lies between S, and S,+r+1/(I-Ur+1/Ur).
The results in (b) apply also if n is a negative integer.
(viii.) In applying the theorem to concrete cases, conversion
of a number into a continued fraction is often useful. Suppose,
for instance, that we require to calculate (23/13). We want to
express (23/13)3 in the form a2b, where b is nearly equal to 1. We
find that log10 (23/13)=3716767=log1u (2·3533) = log10 (40/17)
nearly; and thence that (23/13)=(40/17) (1+1063/3515200), (x+h)"=lim(x"+n (1),xn−1h+ ··· +n(r)x" ̄†h2+···),(x+h)" = lim Sr.
which can be calculated without difficulty to a large number of
significant figures.

(ix.) The extension of nr), and therefore of nr1, to negative and fractional values of n, enables us to extend the applicability of the binomial coefficients to the summation of series (§ 46 (ii.)). Thus the nth term of the series 2.5+5.8+8.11+... in § 46 (iii.) is 18(n-1) [2]; formula (20) of § 43 (iv.) holds for the extended coefficients, and therefore the sum of n terms of this series is 18.(n-1)[3]—18. (−})[3]=3n3+6n2+n. In this way we get the general rule that, to find the sum of n terms of a series, the rth term of which is (a+rb)(a+r+1.b)... (a+r+p−1·b), we

|

ST=(x+h)".

It will be noticed that, although the differences 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 increases, let us examine what happens as h/x decreases, r remaining unaltered. Denote h/x by 0, where 1>0>0; and suppose further that e<1/n, so that the first term of the series uo+u1tu2+.....is

the greatest (numerically). Then {(x+h) "—S,}/hr+1 lies between | is then more simply obtained by the differential calculus than

+ and n(r+1)x^~~1(1+0)"; and the difference between these can be made as small as we please by taking h small enough. Thus we can say that the limit of {(x+h)”—S,}/hr+1 is n(+1x1; but the approach to this limit is of a different kind from that considered in (i.), and its investigation involves the idea of continuity.

V. Continuity.

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 ax2+bx+c=o, we consider, not merely the value of x for which ax2+bx+c is o, but the value of ax2 +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 x 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 x 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 when is made indefinitely small; and this gives an interpretation of nx-1 as the derived function of x" (8 45).

This conception of a limit enables us to deal with algebraical expressions which assume such forms as for particular values of the variable (§ 39 (iii.)). We cannot, for instance, say that the x2-1 fraction is arithmetically equal to x+1 when x=1, as well as for other values of x; but we can say that the limit of the ratio of x2-1 to x-1 when x becomes indefinitely nearly equal

x-I

to I is the same as the limit of x+1.

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 x, though this is the case with the functions with which we are usually concerned.

62. The elementary idea of a differential coefficient is useful in reference to the logarithmic and exponential series. We know that log10N(1+0)= log10N+log10(1+0), and inspection of a table of logarithms shows that, when is small, log10(1+0) is approximately equal to X0, where X is a certain constant, whose value is 434... If we took logarithms to base a, we should

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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.

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convergents to a continued fraction of the form at art art

Po

Qo

where a,ɑ1,ɑ2, . . . are integers. Denoting these convergents by Po/Qo, P1/Q1, P2/Q2,... they may be regarded as obtained from Po a series Pe+ (PP)+(-)+...; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.

Another kind of sequence is that which is formed by introducing the successive factors of a continued product; e.g. the successive factors on the right-hand side of Wallis's theorem π 2.2 4.4 6.6

2

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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, S1, S2, ... 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).

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The word " sequence,' as defined in § 58 (i.), includes progressions 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 √(2n).(n/e)"; the approximation may be improved by Stirling's theorem

loge2+loge3+...+loge(n − 1)+}log,n = {log.(2π)+nlogen—n Bi B2 (-)1B, + I.2.n 3.4.n3 i + ... + (2 r = 1). 27. n2r-1+..., 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=x2; then to every rational value of x 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

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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 O, 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 x2+b2=o, for instance, has no real root; but we may treat the roots as being +b√-1, and b√-1, if I is treated as something which obeys the laws of arithmetic and emerges into reality under the condition √1.√-I—I. Expressions of the form b√-1 and a+b√−1, 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 trigonometry, which suggests the graphic representation of a+b√ -I by a line of length (a2+b2) drawn in a direction different from that of the line along which real numbers are represented.

théorie des fonctions d'une variable (1st ed. 1886, 2nd ed. 1904);
H. Weber, Lehrbuch der Algebra, 2 vols. (1st ed. 1895-1896, 2nd ed.
1898-1899; vol. i. of 2nd ed. transl. by Griess as Traité d'algèbre
supérieure, 1898). For a fuller bibliography, see Encyclopädie der
math. Wissenschaften (vol. i., 1898). A list of early works on algebra
is given in Encyclopaedia Britannica, 9th ed., vol. i. p. 518.
(W. F. SH.)

B. SPECIAL KINDS OF ALGEBRA

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1. 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 +, √, 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 (ab), 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 ing" 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.

66 mean

(a+b)+c=a+b+c) (1) (aXb) Xc = ax (bXc) (^^)

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a(b+c)=ab+ac
(a−b)+b=a (1) (a÷b)Xb=a

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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

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. Among school-books may be mentioned those of W. M. Baker and A. A. Bourne, W. G. Borchardt, W. D. Eggar, F. Gorse, H. S. Hall 3. Although the results of ordinary algebra will be taken for and S. R. Knight, A. E. F. Layng, R. B. Morgan. G. Chrystal, granted, it is convenient to give the principal rules upon which Introduction to Algebra (1898); H. B. Fine, A College Algebra (1905); | it is based. They are C. Smith, A Treatise on Algebra (1st ed. 1888, 3rd ed. 1892), are more suitable for revision purposes; the second of these deals rather fully with irrational numbers. For the algebraic theory of number, and the convergence of sequences and of series, see T. J. I'A. Bromwich, Introduction to the Theory of Infinite Series (1908); H. S. Carslaw, Introduction to the Theory of Fourier's Series (1906); H. B. Fine, The Number-System of Algebra (1891); H. P. Manning, Irrational Numbers (1906); J. Pierpont, Lectures on the Theory of Functions of Real Variables (1905). For general reference, G. Chrystal, TextBook of Algebra (pt. i. 5th ed. 1904, pt. ii. 2nd ed. 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 DIFFERENCES, CALCULUS OF; INFINITESIMAL CALCULUS; INTERPOLATION; VECTOR ANALYSIS. The following may also be consulted:--E. Borel and J. Drach, Introduction à l'étude de la théorie des nombres et de l'algèbre supérieure (1895); C. de Comberousse, Cours de mathématiques, vols. i. and iii. (1884-1887); H. Laurent, Traité d'analyse, vol. i. (1885); E. Netto, Vorlesungen über Algebra (vol. i. 1896, vol. ii. 1900); S. Pincherle, Algebra complementare (1893); G. Salmon, Lessons introductory to the Modern Higher Algebra (4th ed., 1885); J. A. Serret, Cours d'algèbre supérieure (4th ed., 2 vols., 1877); O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (pt. i. 1900, pt. ii. 1902) and Einleitung in die Funktionen-theorie (pt. i. 1904, pt. ii. 1905)— these being developments from O. Stolz, Vorlesungen über allgemeine Arithmetik (pt. i. 1885, pt. ii. 1886); J. Tannery, Introduction à la

to be unambiguous. The special symbols o and I 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+ẞ√-1, where a, ẞ 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+ẞ√ - I (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.

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