Rational Division is Closed

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Theorem

The set of rational numbers less zero is closed under division:

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


Proof

From the definition of division:

$a / b := a \times \paren {b^{-1} }$

where $b^{-1}$ is the inverse for rational multiplication.

From Non-Zero Rational Numbers under Multiplication form Abelian Group, the algebraic structure $\struct {\Q_{\ne 0}, \times}$ is a group.

From group axiom $G3$ it follows that every $b \in \Q_{\ne 0}$ has an inverse element $b^{-1} \in \Q$ under multiplication.

From group axiom $G0$ it follows that $\Q_{\ne 0}$ is closed under multiplication.

Hence the result:

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

$\blacksquare$


Sources