Multiplication of natural numbers

Multiplication $\cdot : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is defined by recursion on the second argument, using addition (Addition of natural numbers): for each fixed $m \in \mathbb{N}$,

$m \cdot 0 = 0, \qquad m \cdot \sigma(n) = m \cdot n + m.$

We write $mn$ for $m \cdot n$ when no confusion arises.

As with addition, this is a recursion and is legitimate by the recursion theorem (The recursion theorem): for each $m$ the map $n \mapsto m \cdot n$ is the unique function with $m \cdot 0 = 0$ and $m \cdot \sigma(n) = m \cdot n + m$, so $\cdot$ is a well-defined total operation (Multiplication is a well-defined total operation).

The base clause $m \cdot 0 = 0$ builds in absorption on the right; that $0 \cdot n = 0$ on the left, and $1 \cdot n = n = n \cdot 1$, are proved by induction (Zero and one under multiplication). Multiplication distributes over addition (Distributivity and the successor law for multiplication) and is associative and commutative (Multiplication is associative, Multiplication is commutative); with The natural numbers have no zero divisors and Cancellation for multiplication by a nonzero factor this makes $(\mathbb{N}, +, \cdot)$ a commutative semiring in which cancellation holds.