Rational Multiplication is Closed

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Theorem

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 priori $\times$ is well-defined and closed on $\Q$.

$\blacksquare$


Proof 2

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

$\displaystyle q \ne 0$
$\displaystyle s \ne 0$
$\displaystyle \frac p q = x$
$\displaystyle \frac r s = y$

We have that:

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

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

Therefore, by the definition of rational numbers:

$\displaystyle x \times y = \frac {p \times r} {q \times s} \in \Q$.

Hence the result.

$\blacksquare$


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