Integer Powers

Why

Definition

Let $a \in \Z $ and let $p \in \N $. Define the first power of $a$ to be $a$. Define the second power of $a$ to be $a^2$. Define the $p$th power of a for $p \geq 2$ to be $a^p = aa^{p-1}$.

Negative powers

Let $a \in \Z $ and let $p \in \Z $ with $p < 0$. Then define $a^{p}$ to be $1/a^{-p}$. Since $p$ is negative, $-p$ is positive and so we have defined $a^{-p}$.

Zero

Define $a^0 = 1$.

Future editions will include. This sheet include only a very basic outline of a few definitions. ↩︎