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 field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $\times$ is commutative on $\Q$.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction