Rational Addition is Associative

Theorem

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

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

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 $+$ 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