# Rational Numbers under Multiplication form Commutative Monoid

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## Theorem

The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a countably infinite commutative monoid.

## Proof

From Rational Numbers under Multiplication form Monoid, $\struct {\Q, \times}$ is a monoid.

Then:

- from Rational Multiplication is Commutative we have that $\times$ is commutative on $\Q$
- from Rational Numbers are Countably Infinite we have that $\Q$ is a countably infinite set.

$\blacksquare$

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups