# Definition:Power (Algebra)/Natural Number

## Definition

Let $\N$ denote the natural numbers.

For each $m \in \N$, recursively define $e_m: \N \to \N$ to be the mapping:

$e_m \left({n}\right) = \begin{cases} 1 & : n = 0 \\ m \times e_m \left({x}\right) & : n = x + 1 \end{cases}$

where:

$+$ denotes natural number addition.
$\times$ denotes natural number multiplication.

$e_m \left({n}\right)$ is then expressed as a binary operation in the form:

$m^n := e_m \left({n}\right)$

and is called $m$ to the power of $n$.