MULTIPLICATION AND DIVISION OF ALGEBRAIC NUMBERS

69. Multiplication.

Multiplication is generally defined in Arithmetic as the process of finding the result when one number (the multiplicand) is taken as many times as there are units in another number (the multiplier). This definition fails when the multiplier is a fraction. In multiplying by a fraction, we divide the multiplicand into as many equal parts as there are units in the denominator, and take as many of these parts as there are units in the numerator.

If, for example, we multiply $6$ by $\frac{2}{3}$, we divide $6$ into three equal parts and take two of these parts, obtaining $4$ for the product. The multiplier, $\frac{2}{3}$, is $\frac{2}{3}$ of $1$, and the product, $4$, is $\frac{2}{3}$ of $6$, in other words, the product is obtained from the multiplicand precisely as the multiplier is obtained from $1$.

This statement is also true when the multiplier is a whole number. Thus in $5\times 7=35$, the multiplier, $5$, is equal to $1+1+1+1+1,$ and the product, $35$, is equal to $7+7+7+7+7.$

may be defined, therefore,

As the operation of finding from two given numbers, called multiplicand and multiplier, a third number called product, which is formed from the multiplicand as the multiplier is formed from unity.

According to this definition of multiplication,

From these four cases it follows that in finding the product of two algebraic numbers, if the two numbers have like signs, the product will have the plus sign, and if unlike signs, the product will have the minus sign.

Hence the Law of Signs in Multiplication is:

Like signs give $+$, and unlike signs give $-$.

If $a$ and $b$ stand for any two numbers, we have

73. The Index Law in Multiplication.

lint Since a^2 = aa, a^3 = aaa, a^2 × a^3 = aa × aaa = aaaaa = a^5 = a^2 + 3; a^4 × a = aaaa × a = aaaaa = a^5 = a^4 + 1.

If $a$ stands for any number, and $m$ and $n$ for any integers, $a^m\times a^n=a^{m+n}.\text{quadtext{Hence,}\}$

The index of the product of two powers of the same number is equal to the sum of the indices of the factors.