Definition:Multiplication/Natural Numbers

Definition
The multiplication operation in the domain of natural numbers $\N$ is written $\times$.

As the Natural Numbers are a Naturally Ordered Semigroup, it is defined using the Principle of Recursive Definition as follows:


 * $\forall m, n \in \N: n \times m = g_m \left({n}\right)$

where $g_m: \N \to \N$ is the unique mapping that satisfies:


 * $\forall m \in \N: g_m \left({n}\right) =

\begin{cases} 0 & : n = 0 \\ g_m \left({r}\right) + m & : n = r + 1 \end{cases}$

... and this can be interpreted as:


 * $n \times m = +^n m = \underbrace{m + m + \cdots + m}_{\text{$n$ copies of $m$}}$