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a definite meaning as the number corresponding to the ex- 33. The importance of the bounding line of the graph lies in tremity of a length x, on the logarithmic scale, such that 5 the fact that we can keep it unaltered while we alter the graph corresponds to the extremity of 2x. Thus the concrete fact as a whole by moving OX up or down. We might, for instance, required to enable us to pass arithmetically from the concep-read temperature from 60° instead of from o°. Thus we tion of a fractional number to the conception of a surd is the form the conception, not only of a zero, but also of the arbitrarifact of performing calculations by means of logarithms.

In the same way we regard log102, not as a new kind of number, but as an approximation.

(vii.) The use of fractional indices follows directly from this parallelism. We find that the product amXamXam is equal to asm; and, by definition, the product a X va X va is equal to a, which is a1. This suggests that we should write va as al; and we find that the use of fractional indices in this way satisfies the laws of integral indices. It should be observed that, by analogy with the definition of a fraction, a/ mean (a1/), not (ap)1/.

II. Graphical Introduction to Algebra.

29. The science of graphics is closely related to that of mensuration. While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths. An important development, covering such diverse matters as the equilibrium of forces and the algebraic theory of complex numbers (§ 66), has relation to cases where the numerical quantity has direction as well as magnitude. There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of mensuration are then applied to determine related numerical quantities. General aspects of the subject are considered under MENSURATION; VECTOR ANALYSIS; INFINITEsimal Calculus.

30. The elementary use of graphic methods is qualitative rather than quantitative; i.e. it is for purposes of illustration and suggestion rather than for purposes of deduction and exact calculation. We start with related facts, and adopt a particular method of visualizing the relation. One of the relations most commonly illustrated in this way is the time-relation; the passage of time being associated with the passage of a point along a straight line, so that equal intervals of time are represented by equal lengths.

31. It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function. Simple examples of the time-relation are the number of scholars present in a class, the height of the barometer, and the reading of the thermometer, on successive days. Another useful set of graphs comprises those which give the relation between the expressions of a length, volume, &c., on different systems of measurement. Mechanical, commercial, economic and statistical facts (the latter usually involving the time-relation) afford numerous examples.

32. The ordinary method of representation is as follows. Let X and Y be the related quantities, their expressions in terms of selected units A and B being x and y, so that X=x.A, Y=y. B. For graphical representation we select units of length L and M, not necessarily identical. We take a fixed line OX, usually drawn horizontally; for each value of X we measure a length or abscissa ON equal to x.L, and draw an ordinate NP at right angles to OX and equal to the corresponding value of y. M. The assemblage of ordinates NP is then the graph of Y. The series of values of X will in general be discontinuous, and the graph will then be made up of a succession of parallel and (usually) equidistant ordinates. When the series is theoretically continuous, the theoretical graph will be a continuous figure of which the lines actually drawn are ordinates. The upper boundary of this figure will be a line of some sort; it is this line, rather than the figure, that is sometimes called the "graph." It is better, however, to treat this as a secondary meaning. In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines.

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ness of position of this zero (cf. § 27 (i.)); and we are assisted to the conception of negative quantities. On the other hand. the alteration in the direction of the bounding line, due to alteration in the unit of measurement of Y, is useful in relation to geometrical projection.

This, however, applies mainly to the representation of values of Y. Y is represented by the length of the ordinate NP, so that the representation is cardinal; but this ordinate really corresponds to the point N, so that the representation of X is ordinal. It is therefore only in certain special cases, such as those of simple time-relations (e.g. " J is aged 40, and K is aged 26; when will J be twice as old as K?"), that the graphic method leads without arithmetical reasoning to the properties of negative values. In other cases the continuation of the graph may constitute a dangerous extrapolation.

34. Graphic representation thus rests on the principle that equal numerical quantities may be represented by equal lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse property that the quantity represented by pL is pA, i.e. that pA is determined by finding the number of times that L is contained in pL. The graphic method may therefore be used in arithmetic for comparing two particular magnitudes of the same kind by comparing the corresponding lengths P and Q measured along a single line OX from the same point O.

(i.) To divide P by Q, we cut off from P successive portions each equal to Q, till we have a piece R left which is less than Q. Thus P=kQ+R, where k is an integer.

(ii.) To continue the division we may take as our new unit a submultiple of Q, such as Q/r, where r is an integer, and repeat the process. We thus get P=kQ+m.Q/r+S=(k+m/r)Q+S, where S is less than Q/r. Proceeding in this way, we may be able to express P÷Q as the sum of a finite number of terms | k+m/r+n/r2+ ; or, if r is not suitably chosen, we may not. If, e.g. r=10, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.

...

(iii.) In the above method the choice of r is arbitrary. We can avoid this arbitrariness by a different procedure. Having obtained R, which is less than Q, we now repeat with Q and R the process that we adopted with P and Q; i.e. we cut off from Q successive portions each equal to R. Suppose we find Q=sR+T, then we repeat the process with R and T; and so on. We thus express P÷Q in the form of a continued fraction, k+ which is usually written, for conciseness, k+1&c.,

,

I S+ I t+&c. or k+ &c.

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(iv.) If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p-kq)L, which is both less than pL and less than qL. Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t, . . . will ultimately come to an end. Moreover, if the last divisor is uL, then it follows from the theory of numbers (§ 26 (ii.)) that (a) u is a factor of p and of q, and (b) any number which is a factor of p and q is also a factor of u. Hence u is the greatest common measure of p and q.

35. In relation to algebra, the graphic method is mainly useful in connexion with the theory of limits (§§ 58, 61) and the functional treatment of equations (§ 60). As regards the latter, there are two classes of cases. In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced,

and the solution of the equation is the determination of the points where the ordinates of the graph are zero. The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the equation is the determination of their common ordinates. Graphic methods also enter into the consideration of irrational numbers (§ 65).

III. Elementary Algebra of Positive Numbers.

(iii.) By writing (A+a)2= A2+2Aa+a2 in the form (A+a)2= A2+(2A+a)a, we obtain the rule for extracting the square root in arithmetic.

*

(iv.) When the terms of a multinomial contain various powers of x, and we are specially concerned with x, the terms are usually arranged in descending (or ascending) order of the indices; terms which contain the same power being grouped so as to give a single coefficient. Thus 2bx-4x2+6ab+3ax would be written -4x2+(3a+2b)x+6ab. It is not necessary to regard here as a negative number; all that is meant is that 4x2 has to be subtracted.

36. Monomials.—(i.) An expression such as a.2.a.a.b.c.3.a.a.c,-4 denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) (v.) When we have to multiply two multinomials arranged which are multiplied together being its factors. An expression according to powers of x, the method of detached coefficients denoting that two or more monomials are to be added or sub-enables us to omit the powers of x during the multiplication. If tracted is a multinomial or polynomial, each of the monomials being a term of it. A multinomial consisting of two or of three

terms is a binomial or a trinomial.

(ii.) By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms. Thus the above expression is equal to 6abc2, which is, of course, equal to other expressions, such as 6ba5c2. The numerical factor. 6 is called the coefficient of a5bc2 (§ 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.

(iii.) The multiplication and division of monomials is effected by means of the law of indices. Thus 6abc5a2bc=u3c, since bo 1. It must, of course, be remembered (§ 23) that this is a statement of arithmetical equality; we call the statement an "identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.

In order that a monomial containing am as a factor may be divisible by a monomial containing ao as a factor, it is necessary that p should be not greater than m.

(iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple. We disregard numerical coefficients, so that by the H.C.F. or L.C.M. of 6a5bc2 and 12a4b2cd we mean the H.C.F. or L.C.M. of abc2 and a1b2cd. The H.C.F. is then an expression of the form abc'd, where p, q, r, s have the greatest possible values consistent with the condition that each of the given expressions shall be divisible by abec'd. Similarly the L.C.M. is of the form abc'd, where p, q, r, s have the least possible values consistent with the condition that ab'c'd' shall be divisible by each of the given expressions. In the particular case it is clear that the H.C.F. is a1bc and the L.C.M. is a5b2c2d.

The extension to multinomials forms part of the theory of factors (§ 51).

37. Products of Multinomials.—(i.) Special arithmetical results may often be used to lead up to algebraical formulae. Thus a comparison of numbers occurring in a table of squares

12 = 1 22 = 4 32=9

112=121
122 = 144
132 = 169

suggests the formula (A+a)2= A2+2Aa+a2.
equalities

99 × 101=9999 = 10000-I
98 X 102 = 9996 = 10000 — -4
97X103=9991 = 10000—9

any power is absent, we treat it as present, but with coefficient o. Thus, to multiply x3-2x+1 by 2x2+4, we write the process.

+I+0-2+1
+2+0+4

+2+0-4+2
+o+o-oto
+4+0-8+4
+2+0+0+2-8+4

giving 2x5+2x2-8x+4 as the result.

38. Construction and Transformation of Equations.-(i.) The statement of problems in equational form should precede the solution of equations.

(ii.) The solution of equations is effected by transformation, which may be either arithmetical or algebraical. The principles of arithmetical transformation follow from those stated in §§ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities. The principle of algebraic transformation has been stated in § 22; it is that, if A= B is an equation (i.e. if either or both of the expressions A and B involves x, and A is arithmetically equal to B for the particular value of x which we require), and if BC is an identity (i.e. if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A=C is an equation which is true for the same value of x for which A=B is true.

(iii.) A special rule of transformation is that any expression may be transposed from one side of an equation to the other, provided its sign is changed. This is the rule of transposition. Suppose, for instance, that P+Q-R+S=T. This may be written (P+Q-R)+S=T; and this statement, by definition of the sign, is the same as the statement that (P+Q-R)= T-S. Similarly the statements P+Q-R-S-T and P+ Q-R=T+S are the same. These transpositions are purely arithmetical. To transpose a term which is not the last term on either side we must first use the commutative law, which involves an algebraical transformation. Thus from the equation P+Q-R+S=T and the identity P+Q-R+S=P-R+S+Q we have the equation P-R+S+Q=T, which is the same statement as P-R+S=T-Q.

(iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general Similarly the theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation. The objection to this (cf. § 21 (ii.)) is that we do not need the general theorem, and that it is unwise to cultivate the habit of laying down a general law as a justification for an isolated action.

lead up to (A-a) (A+a)= A2—a2. These, with (A-a)2= A2-2Aa+a2, are the most important in elementary work.

(ii.) These algebraical formulae involve not only the distributive law and the law of signs, but also the commutative law. Thus (A+a)2 = (A+a)(A+a)=A(A+a) +a(A+a)=AA+Aa+aA+aa; and the grouping of the second and third terms as 2Aa involves treating Aa and aA as identical. This is important when we come to the binomial theorem (§ 41, and cf. § 54 (i.)).

Thus

(v.) An alternative method of obtaining the rule of transposition is to change the zero from which we measure. from P+Q-R+S-T we deduce P+(Q-R+S)=P+(T−P). If instead of measuring from zero we measure from P, we find Q-R+S-T-P. The difference between this and (iii.) is that we transpose the first term instead of the last; the two methods corresponding to the two cases under (i.) of § 15 (2).

(vi.) In the same way, we do not lay down a general rule

4(x-2)

5

3

that an equation is not altered by multiplying both members by the same number. Suppose, for instance, that (x+1)=(x-2). | Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+1)_4(x-2). This means that, whatever unit A we take, 2(x+1) A and A are equal. We therefore take A to be 15, and find that 6(x+1)=20(x-2). Thus, if we have an equation P-Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit. If the P and Q of our equation were quantities expressed in terms of a unit A, we should restate the equation in terms of a unit A/m, as explained in §§ 18 and 21 (i.) (a).

3

(vii.) One result of the rule of transposition is that we can transpose all the terms in x to one side of equation, and all the terms not containing x to the other. An equation of the form ax=b, where a and b do not contain x, is the standard form of simple equation.

(viii.) The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x2. The standard form is usually taken to be

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PU

Р

(ii.) In an equation of the form, the expressions P, Q, U, V are usually numerical. We then have QV.=V. QV, or PV=UQ, as in § 38 (vi.). This is the rule of cross-multiplication. x2-I (iii.) The restriction in (i.) is important. Thus x2+x-2 except when x=1. For this (x-1) (x+2) latter value it becomes %, which has no direct meaning, and requires interpretation (§ 61).

(x-1) (x+1) is equal to x+1

x+2'

=

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where the first line stands for (A+a)= 1. Aoao, and the successive numbers in the (n+1)th line are the coefficients of Ana, An-la1,... A°a" in the n+1 terms of the multinomial equivalent to (A+a)".

41. The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.

(i.) For the general formula, we need only consider (A+a)”. It is clear that, since the numerical coefficients of A and of a are

each 1, the coefficients in the expansions arise from the grouping and addition of like terms (§ 37 (ii.)). We therefore determine the coefficients by counting the grouped terms individually, instead of adding them. To individualize the terms, we replace (A+a) (A+a) (A+a)... by (A+a) (B+b) (C+c). . ., so that no two terms are the same; the "like" -ness which determines the placing of two terms in one group being the fact that they become equal (by the commutative law) when B, C, . . . and b, c,.. are each replaced by A and a respectively.

....

Suppose, for instance, that n=5, so that we take five factors (A+a) (B+b) (C+c) (D+d) (E+e) and find their product. The coefficient of A2a3 in the expansion of (A+a) is then the number of terms such as ABcde, AbcDe, AbCde, . . ., in each of which there are two large and three small letters. The first term is ABCDE, in which all the letters are large; and the coefficient of A2a3 is therefore the number of terms which can be obtained from ABCDE by changing three, and three only, of the large letters into small ones.

We can begin with any one of the 5 letters, so that the first change can be made in 5 ways. There are then 4 letters left, and we can change any one of these. Then 3 letters are left, and we can change any one of these. Hence the change can be made in 3.4.5 ways.

If, however, the 3.4.5 results of making changes like this are written down, it will be seen that any one term in the required product is written down several times. Consider, for instance, the term AbcDe, in which the small letters are bce. Any one of these 3 might have appeared first, any one of the remaining 2 second, and the remaining 1 last. The term therefore occurs 1. 2. 3 times. This applies to each of the terms in which there are two large and three small letters. The total number of such terms in the multinomial equivalent to (A+a) (B+b) (C+c) (D+d) (E+e) is therefore (3. 4. 5)+ (1. 2. 3); and this is therefore the coefficient of A2a3 in the expansion of (A+a)5.

the

The reasoning is quite general; and, in the same way, coefficient of Ana" in the expansion of (A+a)" is {(n-r+1) (n-r+2)... (n−1)n}÷{1.2.3. r). It is usual to write this as a fraction, inverting the order of the factors in the numerator. Then, if we denote it by n(,), so that _n (n − 1)... (n −r+1) 1. 2. 3...

we have

n(r):

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(1),

(2),

(A+a)" =n©0A*+n(1)A” ̃1a+...+n«)An¬a*+...+n(n)an where no), introduced for consistency of notation, is defined by (3).

n (0) I

This is the binomial theorem for a positive integral index.

(ii.) To verify this, let us denote the true coefficient of Ana” by ("), so that we have to prove that (") =n(,), where n(,) is defined by (1); and let us inspect the actual process of multiplying the expansion of (A+a)" by A+a in order to obtain that of (A+a)". Using detached coefficients (§ 37 (v.)), the multiplication is represented by the following:

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so that In the same way we have (A-a)2=A2-2 Aa+a2, (A-a)3 =A3-3A2a+3Aa-a3,..., so that the multinomial equivalent Now suppose that the formula (2) has been established for to (A-a)" has the same coefficients as the multinomial equivalent every power of A+a up to the (n-1)th inclusive, so that to (A+a)", but with signs alternately + and -.

n

The multinomial which is equivalent to (A)", and has its | ("') = (n-1) (+), (−1) = (n − 1) (-1). Then ("), the coefficient

terms arranged in ascending powers of a, is called the expansion of Ana" in the expansion of (A+a)", is equal to (n−1)(1)+ of (Aa)". (n-1)(-1). But it may be shown that (r being >o)

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Hence the formula (2) is also true for the nth power of A+a. But it is true for the 1st and the 2nd powers; therefore it is true for the 3rd; therefore for the 4th; and so on. Hence it is true for all positive integral powers of n.

(iii.) The product 1. 2. 3. . . r is denoted by or r!, and is called factorial r. The form r! is better for printing, but the form is more convenient for ordinary use. If we denote n(n−1). . . (n−r+1) (r factors) by nr), then nr)=n&r)/r!. (iv.) We can write n.) in the more symmetrical form

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n(r) = N(n-r)

(5),

(6). We should have arrived at this form in (i.) by considering the selection of terms in which there are to be two large and three small letters, the large letters being written down first. The terms can be built up in 5! ways; but each will appear 2! 3! times.

(v.) Since nr) is an integer, nr is divisible by r!; ie. the product of any r consecutive integers is divisible by r! (see § 42 (ii.)). (vi.) The product r! arose in (i.) by the successive multiplication of r, r-1, 1-2, . . . 1. In practice the successive factorials 1, 2, 3!... are supposed to be obtained successively by introduction of new factors, so that

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and similarly, for the final terms, we should note that P(a) = 0 if q>p

(9),

(10). (viii.) If u, denotes the term involving a' in the expansion of (A+a)", then ur/Ur_1={(n−r+1)/r} .a/A. This decreases as r increases; its value ranging from na/A to a/(nA). If na<A, the terms will decrease from the beginning; if n A<a, the terms will increase up to the end; if na >A and nA > a, the terms will first increase up to a greatest term (or two consecutive equal greatest terms) and then decrease.

(ix.) The position of the greatest term will depend on the relative values of A and a; if a/A is small, it will be near the beginning. Advantage can be taken of this, when n is large, to make approximate calculations, by omitting terms that are negligible.

(a) Let S, denote the sum uo+u1+

=

...

+u,, this sum being taken so as to include the greatest term (or terms); and let ur+1/u, 0, so that <1. Then the sum of the remaining terms Ur+i+ur+2+ ..+u, is less than (1+0+0+ . . . +0~~~~1)U,+1, which is less than ur+1/(1-0); and therefore (A+a)" lies between S, and S,+u,+1/(1-0). We can therefore stop as soon as ur+1/(1-0) becomes negligible.

methods of procedure. We know that (A+a)" is equal tɔ a multinomial of n+1 terms with unknown coefficients, and we require to find these coefficients. We therefore represent them by separate symbols, in the same way that we represent the unknown quantity in an equation by a symbol. This is the method of undetermined coefficients. We then obtain a set of equations, and by means of these equations we establish the required result by a process known as mathematical induction. This process consists in proving that a property involving p is true when p is any positive integer by proving (1) that it is true when p=1, and (2) that if it is true when p=n, where n is any positive integer, then it is true when p=n+1. The following are some further examples of mathematical induction.

(i.) By adding successively 1, 3, 5... we obtain 1, 4, 9, ... This suggests that, if u, is the sum of the first n odd numbers, then un=n2. Assume this true for u1, U2,..., Un. Then Un+1=Un+(2n+1)=n2+(2n+1)=(n+1)2, so that it is true for un+1. But it is true for u1. Therefore it is true generally. (ii.) We can prove the theorem of § 41 (v.) by a double application of the method.

(a) It is clear that every integer is divisible by 1!. (b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+1 consecutive integers is divisible by (p+1). Denote the product n(n+1). . . (n+r-1) by nr). Then the assumption is that, whatever positive integral value n may have, np is divisible by p!.

H

(1) n'p+1)—(n-1) ip+i)= n(n+1)... (n+p−1)(n+p)—(n−1)n ... (n+p−1) = (p+1). np. But, by hypothesis, n'p is divisible by p!. Therefore n'p+(n-1)+1) is divisible by p!. Therefore, if (n-1)+1l is divisible by (p+1)!, np+1 is divisible by (p+1)!.

(2) But 1p+1(p+1)!, which is divisible by (p+1)!.

=

(3) Therefore n+1l is divisible by (p+1)!, whatever positive integral value n may have.

(c) Thus, if the theorem of § 41 (v.) is true for r=p, it is true for rp+1. But it is true for r=1. Therefore it is true generally.

(iii.) Another application of the method is to proving the law of formation of consecutive convergents to a continued fraction (see CONTINUED FRACTIONS).

43. Binomial Coefficients.-The numbers denoted by n, in § 41 are the binomial coefficients shown in the table in § 40; nr) being the (r+1)th number in the (n+1)th row. They have arisen as the coefficients in the expansion of (A+a)"; but they may be considered independently as a system of numbers defined by (1) of § 41. The individual numbers are connected by various relations, some of which are considered in this section.

(i.) From (4) of § 41 we have n(r) − (n − 1)(r) = (n − 1)(r−1) (11). Changing n into n-1, n-2, and adding the results, n(7) − (n − s) (r) = (n − 1) (r−1) +· (n − 2) (r−1) + ... + (n − s) (r−1) (12). In particular,

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(14),

changing n and

n(r) — (n − 1) (r−1) = (n − 1)(r) into n-1 and r-1, repeating the process, and adding, we find, taking account of (9),

N(r) = (n − 1)(r) + (n − 2 ((r−1) + ... + (n − r − 1)o (15). (ii.) It is therefore more convenient to rearrange the table

(b) In the same way, for the expansion of (A -- a)", let σ, denote uo-u1+...±u,. Then, provided σ, includes the greatest term, it will be found that (A-a)" lies between σ, and σ,+1. For actual calculation it is most convenient to write the of § 40 as shown below, on the left; the table on the right giving

theorem in the form

n

2

..An±...

(A ±a)" = A" (1 ±x)" = A" ± 7" x . A" +"> 1x. 7x. A" ± where x=a/A; thus the successive terms are obtained by successive multiplication. To apply the method to the calculation of N", it is necessary that we should be able to express N in the form A+a or A-a, where a is small in comparison with A, A" is easy to calculate and a/A is convenient as a multiplier.

42. The reasoning adopted in § 41 (ii.) illustrates two general

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viduals out of a stock of n has in effect been proved in § 41 (i.) to be r). This property enables us to establish, by simple reasoning, certain relations between binomial coefficients. Thus (4) of § 41 (ii.) follows from the fact that, if A is any one of the n individuals, the C, groups of consist of -1C-1 which contain A and -1C, which do not contain A. Similarly, considering the various ways in which a group of ʼn may be obtained from two stocks, one containing m and the other containing ʼn, we find that m+nCr=mCrin Co+m Gr-1'n C1+.. +m Co'n Cr,

This

(m+n)(r)=m(r).N(0)+m(r−1). N(1) +...+m(0).n(r) (22). may also be written

The coefficients in the expansion of (A+a)" for any particular value of n are obtained by reading diagonally upwards from left to right from the (n+1)th number in the first column. (iii.) The table might be regarded as constructed by successive which gives applications of (9) and (4); the initial data being (16) and (10). Alternatively, we might consider that we start with the first diagonal row (downwards from the left) and construct the remaining diagonal rows by successive applications of (15). Constructed in this way, the successive diagonal rows, commencing with the first, give the figurate numbers of the first, second, third, . . . order. The (r+1)th figurate number of the nth order, i.e. the (r+1)th number in the nth diagonal row, is n(n+1) (n+r-1)/r!=n1r}/r!; this may, by analogy

...

with the notation of §41, be denoted by nr. We then have

(n+1)[r]=(r+1) [n] = (n+r)!! (n! r!) = (n+r) (r) = (n+r)(n) (17). (iv.) By means of (17) the relations between the binomial coefficients in the form p) may be replaced by others with the coefficients expressed in the form Pl. The table in (ii.) may be written

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n[r] (n―s)[r]=N[r-1]+(n − 1) [r-1]+...+(n−s+1)[r-1] (20); N[r]=n[r-1]+(n − 1) [r-1]+...+I [-1] (21), (v.) It should be mentioned that the notation of the binomial coefficients, and of the continued products such as n(n-1) . . . (n-r+1), is not settled. Some writers, for instance, use the symbol n, in place, in some cases, of nr), and, in other cases, of nr). It is convenient to retain x, to denote x/r!, so that we have the consistent notation

xr = x2/r!, n(r) = n(r)/r!, n[r]=n[r]/r!.

(m+n) (r) =m(r), n(0) +r(1). m(~1), n(1)+...+r(r).m°).n(r) (23). If r is greater than m or n (though of course not greater than m+n), some of the terms in (22) and (23) will be zero.

(iii.) If there are n types, the number of individuals in each type being unlimited (or at any rate not less than r), the number (H) of distinguishable groups of individuals out of the total stock is m. This is sometimes called the number of homogeneous products of r dimensions formed out of n letters; i.e. the number of products such as x, x-3μ3, x-22, ... that can be formed with positive integral indices out of n letters x, y, z, .. the sum of the indices in each product being r. (iv.) Other developments of the theory deal with distributions, partitions, &c. (see COMBINATORIAL ANALYSIS).

(v.) The theory of probability (q.v.) also comes under this head. Suppose that there are a number of arrangements of r terms or elements, the first of which a is always either A or not-A, the second b is B or not-B, the third c is C or not-C, and so on. If, out of every N cases, where N may be a very large number, a is A in pN cases and not-A in (1-p)N cases, where is a fraction such that pN is an integer, then p is the probability or frequency of occurrence of A. We may consider that we are dealing always with a single arrangement abc . . .. and that the number of times that a is made A bears to the number of times that a is made not-A the ratio of p to 1-p; or we may consider that there are N individuals, for pN of which the attribute a is A, while for (1-p)N it is not-A. If, in this latter case, the proportion of cases in which b is B to cases in which b is not-B is the same for the group of pN individuals in which a is A as for the group of (1-p)N in which a is not-A, then the frequencies of A and of B are said to be independent; if this is not the case they are said to be correlated. The possibilities of a, instead of being A and not-A, may be A1, A2, each of these having its own frequency; and similarly for frequency of each B, then the attributes a and b are independent; otherwise they are correlated.

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The binomial theorem for positive integral index may then be b, c, . . . If the frequency of each A is independent of the

written

(x+y)n =XnYo+Xn-1Y1 +...+Xn-r¥r+...+X0Yn•

This must not be confused with the use of suffixes to denote particular terms of a series or a progression (as in § 41 (viii.) and (ix.)).

44. Permutations and Combinations. The discussion, in § 41 (i.), of the number of terms of a particular kind in a particular product, forms part of the theory of combinatorial analysis (q.v.), which deals with the grouping and arrangement of indi

viduals taken from a defined stock. The following are some particular cases; the proof usually follows the lines already indicated. Certain of the individuals may be distinguishable from the remainder of the stock, but not from each other; these may be called a type.

(i.) A permutation is a linear arrangement, read in a definite direction of the line. The number (P,) of permutations of r individuals out of a stock of n, all being distinguishable, is n'). In particular, the number of permutations of the whole stock is n!.

If a of the stock are of one type, b of another, c of another, . . . the number of distinguishable permutations of the whole stock is n÷(a!b!c! . . .).

(ii.) A combination is a group of individuals without regard to arrangement. The number (C,) of combinations of r indi

-1

Functions. An expression of the form cox+x-1+... 45. Application of Binomial Theorem to Rational Integral +CA, where co, C1, . do not involve x, and the indices of the powers of x of degree n. of x are all positive integers, is called a rational integral function

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If we represent this expression by f(x), the expression obtained by changing x into x+h is f(x+h); and each term of this may be expanded by the binomial theorem. Thus we have h2 f(x+)h=cox"+ncox"~1/4+n (n − 1) cox?+...

=

h

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+c1xn−1 + (n − 1) c1xn−2 1 1 + (n − 1) (n − 2) c1xn−2+... it..

h

h2

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+ &c.

= {Cox"+c1xn−1 +C2xn−2+...}

+ {ncoxn¬2+(n − 1) c1xn−2+ (n − 2) c2xn−3 +...

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