# Multiplication of Numbers Distributes over Addition

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## Contents

## 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
- Integer Multiplication Distributes over Addition
- Rational Multiplication Distributes over Addition
- Real Multiplication Distributes over Addition
- Complex 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**.

## Also see

- Modulo Multiplication Distributes over Modulo Addition
- Matrix Multiplication Distributes over Matrix Addition

- Euclid's proofs:

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems: $\text{VII}.$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 16$ - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...