Distributive Laws/Arithmetic

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

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


 * 's proofs:
 * Multiplication of Numbers is Left Distributive over Addition
 * Multiplication of Numbers is Right Distributive over Addition

Beware:
 * Addition of Numbers is not Distributive over Multiplication