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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,

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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

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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 a=0 and d=0; then bc=0-bc−0+0=bc.

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

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For (xxy) means (x × y) × (x × y) × to m factors

Similarly

...

= (x x x x to m factors) × (yxy x to m factors)

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...

(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

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* 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

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.. in Law I., writing m -n instead of m, we have

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237. Law IV. If m is divisible by n, /(x) = xm÷n ̧
For, since m is divisible by n, m÷n is an integer.
.. in Law III., writing m÷n instead of m, we have
(xm÷n)n = xm÷n×n = xm ̧

Taking the nth root of both sides

xm÷n = n/(xm).

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