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 $n$.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic