Definition:Multiplication/Natural Numbers

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

It is defined as:


 * $\forall m, n \in \N: m \times n = \begin{cases}

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

This can be interpreted as:


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

The definition can equivalently be structured:
 * $\forall m, n \in \N: \begin{cases}

0 \times n & = 0 \\ s \left({m}\right) \times n & = m \times n + n \end{cases}$

Also defined as

 * $\forall m, n \in \N: m \times n = \begin{cases}

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

or:


 * $\forall m, n \in \N: \begin{cases}

1 \times n & = n \\ s \left({m}\right) \times n & = m \times n + n \end{cases}$