Elements of Algebra

CHAP. V. OF ALGEBRA. 15 $9)41(4$, $36$ --- $5$ here the divisor is $9$, the dividend $41$, the quotient $4$, and the remainder $5$. The following rule is to be observed in examples where there is a remainder. 54. Multiply the divisor by the quotient, and to the product add the remainder, and the result will be the dividend. This is the method of proving the division, and of discovering whether the calculation is right or not. Thus, in the first of the two last examples, if we multiply $6$ by $5$, and to the product $30$ add the remainder $4$, we obtain $34$, or the dividend. And in the last example, if we multiply the divisor $9$ by the quotient $4$, and to the product $36$ add the remainder $5$, we obtain the dividend $41$. 55. Lastly, it is necessary to remark here, with regard to the signs $+$ plus and $-$ minus, that if we divide $+ab$ by $+a$, the quotient will be $+b$, which is evident. But if we divide $+ab$ by $-a$, the quotient will be $-b$; because $-a \times -b$ gives $+ab$. If the dividend is $-ab$, and is to be divided by the divisor $+a$, the quotient will be $-b$; because it is $-b$ which, multiplied by $+a$, makes $-ab$. Lastly, if we have to divide the dividend $-ab$ by the divisor $-a$, the quotient will be $+b$; for the dividend $-ab$ is the product of $-a$ by $+b$. 56. With regard, therefore, to the signs $+$ and $-$, division requires the same rules to be observed that we have seen take place in multiplication; viz. $+$ by $+$ makes $+$; $+$ by $-$ makes $-$; $-$ by $+$ makes $-$; $-$ by $-$ makes $+$. or, in few words, like signs give plus, and unlike signs give minus. 57. Thus when we divide $18pq$ by $-3p$, the quotient is $-6q$. Farther; $-30xy$ divided by $+6y$ gives $-5x$, and $-54abc$ divided by $-9b$ gives $+6ac$; for, in this last example, $-9b$ multiplied by $+6ac$ makes $-6 \times 9abc$, or $-54abc$. But enough has been said on the division of simple quantities; we shall therefore hasten to the explanation of fractions, after having added some further remarks on the nature of numbers, with respect to their divisors. b