# Rational Multiplication Distributes over Addition

## Theorem

The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition:

$\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
$\forall x, y, z \in \Q: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$

## Proof

Follows directly from the definition of rational numbers as the quotient field of the integral domain $\struct {\Z, +, \times}$ of integers.

So $\struct {\Q, +, \times}$ is a field, and therefore a priori $\times$ is distributive over $+$ on $\Q$.

$\blacksquare$