\begin{definition}[Definition:Multiplication/Natural Numbers/Addition]
Let $\N$ be the natural numbers.
Let $+$ denote addition.
The binary operation $\times$ is recursively defined on $\N$ as follows:
:$\forall m, n \in \N: \begin{cases}
m \times 0 & = 0 \\
m \times \left({n + 1}\right) & = m \times n + m
\end{cases}$
This operation is called '''multiplication'''.
Equivalently, '''multiplication''' can be defined as:
:$\forall m, n \in \N: m \times n := +^n m$ 
where $+^n m$ denotes the $n$th power of $m$ under $+$.
\end{definition}