# Multiplication (natural numbers)

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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: