# Multiplication of Numbers Distributes over Addition

## Theorem

On all the number systems:

natural numbers $\N$
integers $\Z$
rational numbers $\Q$
real numbers $\R$
complex numbers $\C$

the operation of multiplication is distributive over addition:

$m \paren {n + p} = m n + m p$
$\paren {m + n} p = m p + n p$

## Proof

This is demonstrated in these pages:

Natural Number Multiplication Distributes over Addition

$\blacksquare$

## Also known as

This result is known as the Distributive Property.

As such, it typically refers to the various results contributing towards this.

At elementary-school level, this law is often referred to as (the principle of) multiplying out brackets.