# Rational Multiplication is Commutative

## Theorem

The operation of multiplication on the set of rational numbers $\Q$ is commutative:

$\forall x, y \in \Q: x \times y = y \times x$

## Proof

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

So $\left({\Q, +, \times}\right)$ is a field, and therefore a priori $\times$ is commutative on $\Q$.

$\blacksquare$