Rational Multiplication is Closed

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The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:

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

Proof 1

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 well-defined and closed on $\Q$.


Proof 2

From the definition of rational numbers, there exists four integers $p$, $q$, $r$, $s$, where:

$q \ne 0$
$s \ne 0$
$\dfrac p q = x$
$\dfrac r s = y$

We have that:

$p \times r \in \Z$
$q \times s \in \Z$

Since $q \ne 0$ and $s \ne 0$, we have that:

$q \times s \ne 0$

Therefore, by the definition of rational numbers:

$x \times y = \dfrac {p \times r} {q \times s} \in \Q$

Hence the result.