Distributive Laws/Arithmetic

Theorem
On all the number systems: the operation of multiplication is distributive over addition.
 * natural numbers $\N$
 * integers $\Z$
 * rational numbers $\Q$
 * real numbers $\R$
 * complex numbers $\C$


 * If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple of one of the magnitudes is of one, that multiple will also be of all.


 * If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.

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.

Also see

 * Modulo Multiplication Distributes over Modulo Addition
 * Matrix Multiplication Distributes over Matrix Addition