Strictly Positive Real Numbers are Closed under Division

Theorem

The set $\R_{>0}$ of strictly positive real numbers is closed under division:

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

Proof

From the definition of division:

$a \div b := a \times \paren {\dfrac 1 b}$

where $\dfrac 1 b$ is the inverse for real number multiplication.

From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, the algebraic structure $\struct {\R_{>0}, \times}$ forms a group.

Thus it follows that:

$\forall a, b \in \R_{>0}: a \times \paren {\dfrac 1 b} \in \R$

Therefore real number division is closed in $\R_{>0}$.

$\blacksquare$

Also see

• Strictly Positive Real Numbers are Closed under Multiplication

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$