Strictly Positive Real Numbers are Closed under Division
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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
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 $1$