Multiplication of natural numbers

Definition

Let $\mathbb{N}$ be the set of natural numbers, let $S$ be the successor function and let $+$ be the addition of $\mathbb{N}$. Multiplication is the function $\cdot \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(a, b) \mapsto a \cdot b$, with these two properties for all $a$ and $b$ in $\mathbb{N}$: $$a \cdot 0 = 0, \qquad a \cdot S(b) = a \cdot b + a.$$

Remarks

For each $a$, these equations are a definition by recursion on $b$. Thus exactly one such function exists. From the definition, $a \cdot 1 = a \cdot 0 + a = a$. We often write $ab$ for $a \cdot b$.

Explanations

L1 human

Multiplication is repeated addition: $3 \cdot 4$ is $3 + 3 + 3 + 3 = 12$. The definition says this in two rules: zero copies give $0$, and one more copy adds one more $a$.