Rational Multiplication is Closed

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 fortiori $\times$ 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:

$q \ne 0$

$s \ne 0$

$\dfrac p q = x$

$\dfrac r s = y$

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.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $3$