Elements of Algebra

CHAP. XVI. OF ALGEBRA. 51 $\frac{a}{a}$, or $1$; and, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must be $a^0$; and hence we deduce this remarkable property, that $a^0$ is always equal to $1$, however great or small the value of the number $a$ may be, and even when $a$ is nothing; that is to say, $a^0$ is equal to $1$. 176. We may also continue our series of powers in a retrograde order, and that in two different ways; first, by dividing always by $a$; and secondly, by diminishing the exponent by unity; and it is evident that, whether we follow the one or the other, the terms are still perfectly equal. This decreasing series is represented in both forms in the following Table, which must be read backwards, or from right to left. $\frac{1}{aaaaaa}$ $\frac{1}{aaaaa}$ $\frac{1}{aaaa}$ $\frac{1}{aaa}$ $\frac{1}{aa}$ $\frac{1}{a}$ $1$ $a$ 1st. $\frac{1}{a^6}$ $\frac{1}{a^5}$ $\frac{1}{a^4}$ $\frac{1}{a^3}$ $\frac{1}{a^2}$ $\frac{1}{a^1}$ 2d. $a^{-6}$ $a^{-5}$ $a^{-4}$ $a^{-3}$ $a^{-2}$ $a^{-1}$ $a^0$ $a^1$ 177. We are now come to the knowledge of powers whose exponents are negative, and are enabled to assign the precise value of those powers. Thus from what has been said, it appears that $a^0$ $a^{-1}$ $a^{-2}$ $a^{-3}$ $a^{-4}$ is equal to $1$ $\frac{1}{a}$ $\frac{1}{aa}$ or $\frac{1}{a^2}$ $\frac{1}{a^3}$ $\frac{1}{a^4}, \ldots$ 178. It will also be easy, from the foregoing notation, to find the powers of a product, $ab$; for they must evidently be $ab$, or $a^1b^1$, $a^2b^2$, $a^3b^3$, $a^4b^4$, $a^5b^5, \ldots$ and the powers of fractions will be found in the same manner; for example, those of $\frac{a}{b}$ are $\frac{a^1}{b^1}$, $\frac{a^2}{b^2}$, $\frac{a^3}{b^3}$, $\frac{a^4}{b^4}$, $\frac{a^5}{b^5}$, $\frac{a^6}{b^6}$, $\frac{a^7}{b^7}, \ldots$ E 2