Multiplication (natural numbers)

In Peano arithmetic, multiplication is defined by a recursion of addition of natural numbers.

Definition

Given an arbitrary $a \in \mathbb{N}$, we will define $ab$ recursively as follows: $a\cdot 0=0$ and $ab'=ab+a$, for all $b \in \mathbb{N}$.

Properties

Multiplication on the natural numbers has some important properties:

• The natural number $0'$ is the multiplicative identity (proof)
• Multiplication is distributive over addition (proof)
• Multiplication is commutative (proof) and associative (proof)

See also

• Peano arithmetic
• Recursion
• Addition (natural numbers)
• Associative property of addition on the natural numbers
• Commutative property of addition on the natural numbers
• Distributive property of multiplication over addition on the natural numbers