Definition:Multiplication/Natural Numbers/Addition

Definition

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 $+$.

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 4$: Number systems $\text{I}$: Peano's Axioms
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 8$ Definition by finite recursion: Exercise $8.4 \ \text {(b)}$