Positive Real Numbers Closed under Division

Theorem

The set $\R_+^*$ of strictly positive real numbers is closed under division:

$\forall a, b \in \R_+^*: a \div b \in \R_+^*$

Proof

From the definition of division:

$a \div b := a \times \left({\dfrac 1 b}\right)$

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

As the algebraic structure $\left({\R_+^*, \times}\right)$ forms a group, it follows that:

$\forall a, b \in \R: a \times \left({\dfrac 1 b}\right) \in \R$

Therefore real number division is closed.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.2$: Example $1$