Definition:Multiplication/Natural Numbers/Addition

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Let $\N$ be the natural numbers.

Let $+$ denote addition.

The binary operation $\times$ is recursively defined on $\N$ as follows:

$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \paren {n + 1} & = m \times n + m \end{cases}$

This operation is called multiplication.

Equivalently, multiplication can be defined as:

$\forall m, n \in \N: m \times n := \mathop {+^n} m$

where $\mathop {+^n} m$ denotes the $n$th power of $m$ under $+$.