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

## Definition

Given an arbitrary , we will define recursively as follows: and , for all .

## Properties

Multiplication on the natural numbers has some important properties:

- The natural number is the multiplicative identity (proof)
- Multiplication is distributive over addition (proof)
- Multiplication is commutative (proof) and associative (proof)