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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)= ta-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 -ċ= +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.
.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.
O and d=0; then bc=0-bc-0+0=bc.
Put a=0 and c=0; then bă=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.
229. If m is any positive integer, the symbol " is used to denote X X X X X X to m factors. Here ac is called the Base; m, the Index; and 2cm, the mth
An mth Root of a given quantity means a quantity whose mth Power is equal to the given quantity.
The symbol m/x denotes an mth Root of x; i.e. ("/x)" = 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 a positive, there is no negative mth root of ac. (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 mth root of x.
(4) There cannot be two different mth roots of a having the same sign. For, if possible, let y and z be two such roots, so that y" =zm = : then yM – 2* = (y-2) (3–1 + zym–+ zym-3 + ...... +2m-1) = 0. zm
xm But, by the rules of signs, every term in the second factor has the same sign, .-. this second factor cannot be zero. :: * -z = 0,
Hence the two roots supposed to be different are not different.
Thus, considering positive or negative values of m/x, we have the following results*:
If m is odd, and a positive, ma can have only one value, and this positive.
If m is odd, and x negative, mæ can have only one value, and this negative.
If m is even, and a positive, m/xc can have only two values, one positive and the other negative.
If m is even, and x negative, mc can have no values, positive or negative.
i.e. y =?
231. The symbol m/s 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/2, when m and a 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, 32197=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) (x * y) = xm x YM ;
(4) m/(x + y) = "/w = /y.
= (x x x x to m factors) ~ (y xyx to m factors)
....(1). Similarly (x + y)" = 2cm - ym....
...(2). In (1) write mac and my instead of x and y respectively.
(3). Similarly m/= m/y="/(x + y)... ......... .(4)
* That is, incommensurable with unity as explained in Art. 31.
Laws involving different indices but the same bases.
30m x 2" = xm+N. Law II. If m is greater than n, 2cm = = xm-n. Law III.
= . Law IV. If m is divisible by n, */(2cm) = = 2min.
Law II. If m>n, 2cm - 2" -= 2cm-n. For, since m>n, m n is positive. .. in Law I., writing m-n instead of m, we have
cm-n x och = am-n+n= cm. Dividing both sides by ac", we have
acm-n = m = 2c".
237. Law IV. If m is divisible by n, "(ac") = xm+.
= 2". Taking the nth root of both sides