Rational Multiplication is Associative

Theorem

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

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

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 associative on $\Q$.

$\blacksquare$

Sources

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