CHAPTER XI. INDICES OR LOGARITHMS; AND MATHEMATICAL TABLES. Negative Symbols. If a>c, the symbol a-c has an intelligible meaning. In this case we may easily show that I. The addition of a-c (to any number whatever) is equivalent to the addition of a followed by the subtraction of c. Or, in symbols, II. The subtraction of a- c (from any number>a) is equivalent to the subtraction of a followed by the addition of c. Or, in symbols, -(a−c)=a+c I. and II. here represent equivalences of operation. .II. But if a<c, the symbol a-c has no intelligible meaning by itself. It is convenient, however, to be able to use the above fundamental equivalences in all cases whatever. Putting, then, a=0; (a−c) becomes (0-c). Such a quantity is called a negative quantity. It may be written for brevity -c, or still better c. The sign is here called a sign of affection. It is useful to write it over a number to distinguish it from the sign - used to denote the operation of subtraction. Putting a=0 on the right-hand side of I. and II. (since the addition or subtraction of O has no effect), the right hand of (I.) becomes — c; that of II. becomes + c. Hence (I.) and (II.) become The addition or subtraction of a negative symbol is interpreted to mean the subtraction or addition of the corresponding positive. It remains to interpret multiplication involving a negative quantity. If a>b and c>d, we may prove that (a - b) (c-d)=ac-bc - ad+bd ......... .III. It is convenient to use this equation for all cases. Hence if a<b or c<d we so interpret multiplication of negatives that this shall always hold. Thus Put b=0 and d=0; then ac=ac-0-0+0=ac. Put b=0 and c=0; then ad=0—0—ad+0=ad. Put a=0 and c=0; then бd=0—0—0+bd=bd. Thus the multiplication of a negative by a positive is negative; and the multiplication of a negative by a negative is positive. § 1. INDEX NOTATION. 229. If m is any positive integer, the symbol am is used to denote x x x x x x ...... to m factors. of x. Here x is called the Base; m, the Index; and xm, the mth Power An mth Root of a given quantity means a quantity whose mth Power is equal to the given quantity. The symbol / denotes an mth Root of x; i.e. (m/x)m = x. 230. To determine how many positive or negative roots a given positive or negative quantity has. The rules for multiplying positive or negative quantities are I. The product of positive quantities is positive. II. A change in sign of one factor changes the sign of the product. Hence a product containing an even number of negative factors is positive; and a product containing an odd number of negative factors is negative. From this it follows conversely that (1) If m is odd, and x positive, there is no negative mth root of x. (2) If m is odd, and x negative, there is no positive mth root of x. (3) If m is even, and x negative, there is no positive and there is no negative mt th root of x. (4) There cannot be two different mth roots of x having the same sign. For, if possible, let y and z be two such roots, so that ym=zm=x: then ym -zm=(y-z) (ym−1 + zym−2 + z2ym−3 + ...... + m−1) = 0. But, by the rules of signs, every term in the second factor has the same sign, .. this second factor cannot be zero. ..y-z = 0, Hence the two roots supposed to be different are not i.e. y = %. different. Thus, considering positive or negative values of m/x, we have the following results*:— If m is odd, and x positive, "/x can have only one value, and this positive. If m is odd, and x negative, m/x can have only one value, and this negative. If m is even, and x positive, m/x can have only two values, one positive and the other negative. If m is even, and x negative, m/x can have no values, positive or negative. 231. The symbol m/x may be used for the present to denote the positive value of the mth root of x, if x is positive; and the negative value of the mth root of x, if x is negative (and m odd). * Whether there are roots which are neither positive nor negative, is not here discussed. 232. The method of finding the arithmetical value of m/x, when m and x have any particular values, is not explained here. But it is important to observe that its value cannot be exactly, but only approximately, determined in most cases. A number which cannot be arithmetically expressed by a fraction having a finite integral numerator and denominator is called an Irrational or Incommensurable* number. Thus, 3/2197-13 and is, therefore, rational. But 2=1·414 &c., cannot be exactly evaluated, and is therefore irrational. Laws involving the same indices but different bases. 233. The power [or root] of a product [or quotient] of two quantities is equal to the product [or quotient] of the corresponding powers [or roots] of the two quantities. Thus For (xxy) means (x × y) × (x × y) × to m factors Similarly ... = (x x x x to m factors) × (yxy x to m factors) ... (x ÷ y)m = xm ÷ ym. ... ..(1). ..(2). In (1) write / and /y instead of x and y respectively. Thus (m/xxm/y)m = (m/x)m × (m/y)m = x × y. .. taking the mth root of both sides * That is, incommensurable with unity as explained in Art. 31. .(3). .(4). Laws involving different indices but the same bases. Law I. хт х хот = xm+n Law II. If m is greater than n, x1 ÷ x2 = xm¬n .. in Law I., writing m -n instead of m, we have 237. Law IV. If m is divisible by n, /(x) = xm÷n ̧ Taking the nth root of both sides xm÷n = n/(xm). |