tremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2x. Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms. In the same way we regard logma, not as a new kind of number, but as an approximation. . (vii) The use of fractional indiccs follows directly from this parallelism. We find that the product a'”><a"‘><a"‘ is equal to a”; and, by definition, the product daXfiaXi/a is equal to a, which is al. This suggests that we should write {/0 as all“; 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, aP/q mean (oi/q)”, not (a')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; Vscron 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. 3r. 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 difierent 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 IL, 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. ‘ read temperature from 60° instead of from 0°. ' form the conception, not only of a zero, but also of the arbitrariness of position of this zero (cf. §27 (i.)); and we are assisted a. definite meaning as the number corresponding to the ex- ~ 33. The importance of the bounding line of the graph lies in the fact that we can keep it unaltered while We alter the graph as a whole by moving OX up or down. We might, for instance, Thus we 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 I 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 0. (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/1)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=Io, 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 difierent 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 06 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-t-Q in the form of a continued fraction, 1 (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. ' 3 5. In relation to algebra, the graphic method is mainly useful in connexion with the theory of limits (§§ 58, 6:) 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 diflerent 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. 36. M onomials.——(i.) An expression such as a.2.a.a.b.c.3.a.o.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors. An expression denoting that two or more monomials are to be added or subtracted 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 oa‘bc’, which is, of course, equal to other expressions, such as 6basc‘. The numerical factor 6 is called the coeflicient of a'5bc2 (§ 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. vThus 605bc1+5a1bc=£aiq since b°= 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 a" as a factor may be divisible by a monomial containing a" as a factor, it is necessary that 1: should be not greater than m. (iv.) In algebra we have a theory of highest common factor and 10100:! 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 6a“’bc2 and 12a‘b’cd we mean the H.C.F. or L.C.M. of a‘bc' and a‘b’cd. The H.C.F. is then an expression of the form an°c'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 o'b'c'd'. Similarly the L.C.M. is of the form a'b‘lc'd', where p, q, r, s have the least possible values consistent with the condition that an'c’d' shall be divisible by each of the given expressions. In the particular case it is clear that the H.C.F. is a‘bc and the L.C.M. is a‘b’c’d. The extension to multinomials forms part of the theory of factors (5 51). 37. Products of M ultinomials.—(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 (iii.) By writing (A+a)’=A'+2Aa+a’ in the form (A+a)2= A’+(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 coefiicient. Thus 2bx—4x’+6ab+3ax would be written -4x’+(3a+2b)x+6ab. It is not necessary to regard -4 here as a negative number; all that is meant is that 41’ has to be subtracted. (v.) When we have to multiply twd multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication. If any power is absent, we treat it as present, but with coefficient 0. Thus, to multiply xa—zx-l-I by 2x2+4, we write the process + 1 +0 — 2 + I +2+0+4 +2+o—4+2 +0 +o—o+o +4+0—8+4 +2+0+0+2—8+4 giving 2x5-i- 2xL8x+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 ex . pressions 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 an, and A is arithmetically equal to B for the particular value of x which we require), and if B=C is an identity (i.e. if B and C are expressions involving 2: 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 lransPosilion. 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 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. §2r (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. (v.) An alternative method of obtaining the rule of transposition is to change the zero from which we measure. Thus 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 that 6(x+r)=2o(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). ' (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 at to the other. An equation of the form ax=b, where a and b do not contain at, is the standard form of simple equation. (viii.) The quadratic equation is the equation of two expressions, monomial or multinomialpnone of the terms involving any power of x except x and x”. The standard form is usually taken to be ax2+bx+c=o, from which we find, by transformation, , (2 ax+lg>22~ = b’ —4ac, and thence x = W. 41. The binomial theorem gives a formula for writing down the coefiicient 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 coefiicients 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—l-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+e) (D+d) (E+e) and find their‘product. The coefficient of A’aa in the expansion of (A+a)ls is then the number of terms su¢ as ABcde, AbcDe, Adee, . . . , 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 coeflicient of No8 is therefore the number of terms which can be obtained fromrABCDE 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 bee. Any one of these 3 might have appeared first, any one of the remaining 2 second, and the remaining I 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)-:— (r. 2. 3); and thisis therefore the coefficient of A’aa in the expansion of (A+a)‘. The reasoning is quite general; and, in the same way, the coefficient of A'Ha' in the expansion of (A+a)” is {(n—r+r) (n—r—l—z) . . . (n—r)nl+ [1.2.3 . . . rl. It is usual to write this as a fraction, inverting the order of the factors in the Hence the formula (2) is also true for the nth power of A+a. But it is true for the rst 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 . . . ris denoted by [Lor rl, 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 n"), then m.,-inm/rl. (iv.) We can write nm in the more symmetrical form "(r)=n(n-4) 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 nu) is an integer, n") is divisible by r!; i.e. the product of any r consecutive integers is divisible by 1! (see § 42 (ii.)). (vi.) The product r! arose in (i.) by the successive multiplication of r, r— r, r— 2, . . . 1.' In practice the successive factorials 1!, 2!, 3! . . . are supposed to be obtained successively by introduction of new factors, so that r!==r. (r—r)! (7)_ Thus in defining r! as r. 2. 3 . . . r we regard the multiplications as taking place from left to right; and similarly in n“). A product in which multiplications are taken in this order is called a continued product. (vii.) In order to make the formula (5) hold for the extreme values nm and n(,., we must adppt the convention that o . = r This is consistent with (7), which gives rl= r.o!. It should be observed that, for r=o, (4) is replaced by (s), which shows that Here we have introduced a number 0(0) given by 0(0) = I (16)) which is consistent with the relations in (i.). In this table any number is equal to the sum of the numbers which lie horizontally above it in the preceding column, and the difference of any two numbers in a column is equal to the sum of the numbers horizontally between them in the preceding column. The coeflicients in the expansion of (A-l-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 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 (r 5). Constructed in this way, the successive diagonal rows, commencing with the first, give the figurale numbers of the first, second, third, . . . order. The (r+1)th figurate number of the nth order, i.e. the (r+r)th number in the nth diagonal row, is n(n+r) . . . (n+r—r)/rl=nl']/r!; this may, by analogy with the notation of §4r, be denoted by nm. We then have ("+l)[']=('+1)[n]=("+')!/("! Y!) =("+')(*)=("+')<n\ (17). (iv.) By means of (17) the relations between the binomial coefficients in the form pm may be replaced by others with the coefficients expressed in the form pm. The table in (ii.) may be written Xioi The most important relations are "l'l=”['—l]+(""l)(r) (i8); °{r1=° ('ig); "[']"("_5)[r]="lr—ll+(""I)['-l1+--~+("_3+I)[r—1] (20); "('1="lr—11+("“lllr—ll+-'-+Ilr—l] (21). (v.) It should be mentioned that the notation of the binomial coefficients, and of the continued products such as n(n—i) . . . (n—r-l-r), is not settled. Some writers, for instance, use the symbol n, in place, in some cases, of nm, and, in other cases, of n"). It is convenient to retain an, to denote x’lrl, so that we have the consistent notation x,=x'/r!. n(,)=n(')/rl, n[,]=nl'1/r!. The binomial theorem for positive integral index may then be written (x+y)n =xnyo+xHyi +... +x.._.yr+---+xoy,.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 individuals 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 ("1%) 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~ viduals out of a stock of n has in efiect been proved in § 41 (i.) to be up). 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 isany one of the n individuals, the “C. groups of r consist of .._1C,_1 which contain A and kiC, which do not contain A. Similarly, considering the various ways in which a group of r may be obtained from two stocks, one containing m and the other containing n, we find that nu+nCr =Itcr‘nc0 +nGr-V1'ncl+~- +nc0'ncr' which gives (m+n)(r)=m(').n(o)+m(,_|).n(1)+...+m(o).n(r) This may also be written (m+n)<')=mf').n(°)+r(1).m('-".n(1>+...+r(.).m(°).n(') (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 r individuals out of the total stock is nm. 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', ray’, M’, . . . 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 terrr'rs or elements, the first of which a is always either A or not-A, the second b is B or not-B, the third 0 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 (i—p)N cases, where p 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 I—p; or we may consider that there are N individuals, for pN of which the attribute a is A, while for (r—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 (t-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 0, instead of being A and not-A, may be A1, A1, . . each of these having its own frequency; and similarly for b, c, . . . If the frequency of each A is independent of the frequency of each B, then the attributes a and b are independent; otherwise they are correlated. 45. Application of Binomial Theorem 10 Rational Integral Functions—An expression of the form cox"+c,x""+ . . . +6“, where co, oi, . . . do not involve x, and the indices of the powers of a: are all positive integers, is calledaralional inlegralfunclion of x of degree n. If we represent this expression by f (x), the expression obtained by changing so into x-l-h is [(x-i-h); and each term of this may be expanded by the binomial theorem. Thus we have (22). |