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INDICES OR LOGARITHMS; AND MATHEMATICAL
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, +(a - c)= +a-c..
I. 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
II. I. and II. here represent equivalences of operation.
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 0 has no effect), the right hand of (I.) becomes – C; that of II. becomes + C. Hence (I.) and (II.) become
+c=-cand -ī= + c. In words :
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.
=0 and c=0; then ūd=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 c is used to denote X X X X X X to m factors.
Here x is called the Base; m, the Index; and 2cm, the mth Power
An mth Root of a given quantity means a quantity whose mth Power is equal to the given quantity.
The symbol mja denotes an mth Root of x; i.e. (m/w)” = 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 a negative, there is no positive and there is no negative mth root of w.
(4) There cannot be two different mth roots of a having the same sign. For, if possible, let y and be two such roots, so that y" = zm = x : then
y – z = (y-2) (3-1 + zy"-2 + z2ym-3 + + 2-1) = 0.
But, by the rules of signs, every term in the second factor has the same sign, .. this second factor cannot be zero. ..y-%= 0,
Hence the two roots supposed to be different are not different.
Thus, considering positive or negative values of mx, we have the following results * :
If m is odd, and a positive, m/s can have only one value, and this positive.
If m is odd, and a negative, m/x can have only one value, and this negative.
If m is even, and a positive, m/x can have only two values, one positive and the other negative.
If m is even, and x negative, mic can have no values, positive or negative.
i.e. y = %.
231. The symbol m/x may be used for the present to denote the positive value of the mth root of x, if u 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 mac, 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, 82197=13 and is, therefore, rational. But y2=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
(1) (2 x y)" = 20m x ym ;
(4)/(x + y) = "/C "/y.
= (3C x x x to m factors) * (y xyx to m factors)
...(1). Similarly (x + y)" = 20 - ym
.(2) In (1) write mac and my instead of x and y respectively. Thus (m/xm/y)" = (m/20) (m/y)" = 2 x y. .. taking the mth root of both sides m/ *m/y = m/(2 x y)
(3). Similarly m/c = m/y = m/(ac y)
* That is, incommensurable with unity as explained in Art. 31.
Law II. If m is greater than n, um = 2c= xm-n.
= . Law IV. If m is divisible by n, */(xcm) = xm+n. 234.
Law I. xm x xn = xm+n. For
20" means X X X X to m factors;
acmeans x x x x to n factors. .. 2cm x 2" = (2 x 2 x to m factors) x (x x x x to n factors)
to (m + n) factors = 2m+n.
= X X X X ...
235. Law II. If m>n, 20" - 20" = 3cm-n.
2cm-n x oc" = cm-nun Dividing both sides by ac", we have
2cm-n = 2cm : 2c".
237. Law IV. If m is divisible by n, */(2cm) = 2cm+n.
(acm+n)n = 2m+nxn = xcm. Taking the nth root of both sides
am+N = */(ccm).