Elements of Algebra

14 ELEMENTS SECT. I. gives the dividend $ab$. It is clear also, that if we had to divide $ab$ by $b$, the quotient would be $a$. And in all examples of division that can be proposed, if we divide the dividend by the quotient, we shall again obtain the divisor; for as $24$ divided by $4$ gives $6$, so $24$ divided by $6$ will give $4$. 50. As the whole operation consists in representing the dividend by two factors, of which one may be equal to the divisor, and the other to the quotient, the following examples will be easily understood. I say first that the dividend $abc$, divided by $a$, gives $bc$; for $a$, multiplied by $bc$, produces $abc$: in the same manner $abc$, being divided by $b$, we shall have $ac$; and $abc$, divided by $ac$, gives $b$. It is also plain, that $12mn$, divided by $3m$, gives $4n$; for $3m$, multiplied by $4n$, makes $12mn$. But if this same number $12mn$ had been divided by $12$, we should have obtained the quotient $mn$. 51. Since every number $a$ may be expressed by $1a$, or $a$, it is evident that if we had to divide $a$, or $1a$, by $1$, the quotient would be the same number $a$. And, on the contrary, if the same number $a$, or $1a$, is to be divided by $a$, the quotient will be $1$. 52. It often happens that we cannot represent the dividend as the product of two factors, of which one is equal to the divisor; hence, in this case, the division cannot be performed in the manner we have described. When we have, for example, $24$ to divide by $7$, it is at first sight obvious, that the number $7$ is not a factor of $24$; for the product of $7 \times 3$ is only $21$, and consequently too small; and $7 \times 4$ makes $28$, which is greater than $24$. We discover, however, from this, that the quotient must be greater than $3$, and less than $4$. In order therefore to determine it exactly, we employ another species of numbers, which are called *fractions*, and which we shall consider in one of the following chapters. 53. Before we proceed to the use of fractions, it is usual to be satisfied with the whole number which approaches nearest to the true quotient, but at the same time paying attention to the *remainder* which is left; thus we say, $7$ in $24$ goes $3$ times, and the remainder is $3$, because $3$ times $7$ produces only $21$, which is $3$ less than $24$. We may also consider the following examples in the same manner: $6)34(5$, that is to say, the divisor is $6$, the dividend $34$, the quotient $5$, and the remainder $4$. $30$ --- $4$