Elements of Algebra

CHAP. XX. OF ALGEBRA. 61 tracting $3$ from $8$: and, in general, to find $b$, we must subtract $a$ from $c$, whence arises $b = c - a$; for, by adding $a$ to both sides again, we have $b + a = c - a + a$, that is to say, $= c$, as we supposed. 209. Subtraction therefore takes place, when we invert the question which gives rise to addition. But the number which it is required to subtract may happen to be greater than that from which it is to be subtracted; as, for example, if it were required to subtract $9$ from $5$: this instance therefore furnishes us with the idea of a new kind of numbers, which we call negative numbers, because $5 - 9 = -4$. 210. When several numbers are to be added together, which are all equal, their sum is found by multiplication, and is called a product. Thus, $ab$ means the product arising from the multiplication of $a$ by $b$, or from the addition of the number $a$, $b$ number of times; and if we represent this product by the letter $c$, we shall have $ab = c$; thus multiplication teaches us how to determine the number $c$, when the numbers $a$ and $b$ are known. 211. Let us now propose the following question: the numbers $a$ and $c$ being known, to find the number $b$. Suppose, for example, $a = 3$, and $c = 15$; so that $3b = 15$, and let us inquire by what number $3$ must be multiplied, in order that the product may be $15$; for the question proposed is reduced to this. This is a case of division; and the number required is found by dividing $15$ by $3$; and, in general, the number $b$ is found by dividing $c$ by $a$; from which results the equation $b = \frac{c}{a}$. 212. Now, as it frequently happens that the number $c$ cannot be really divided by the number $a$, while the letter $b$ must however have a determinate value, another new kind of numbers present themselves, which are called fractions. For example, suppose $a = 4$, and $c = 3$, so that $4b = 3$; then it is evident that $b$ cannot be an integer, but a fraction, and that we shall have $b = \frac{3}{4}$. 213. We have seen that multiplication arises from addition; that is to say, from the addition of several equal quantities: and if we now proceed farther, we shall perceive that, from the multiplication of several equal quantities together, powers are derived; which powers are represented in a general manner by the expression $a^b$. This signifies that the number $a$ must be multiplied as many times by itself, minus 1, as is indicated by the number $b$. And we know from what has been already said, that, in the present in-