Strictly Positive Rational Numbers are Closed under Multiplication

Theorem

$\forall a, b \in \Q_{>0}: a b \in \Q_{>0}$

Proof

Let $a$ and $b$ be expressed in canonical form:

$a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$

where $p_1, p_2 \in \Z$ and $q_1, q_2 \in \Z_{>0}$.

As $\forall a, b \in \Q_{>0}$ it follows that $p_1, p_2 \in \Z_{>0}$.

By definition of rational multiplication:

$\dfrac {p_1} {q_1} \times \dfrac {p_2} {q_2} = \dfrac {p_1 \times p_2} {q_1 \times q_2}$

From Integers form Ordered Integral Domain, it follows that:

\(\ds p_1 \times p_2\)

\(>\)

\(\ds 0\)

\(\ds q_1 \times q_2\)

\(>\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {p_1} {q_1} \times \dfrac {p_2} {q_2}\)

\(>\)

\(\ds 0\)

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction