Rational Division is Closed

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 Infinite Abelian Group, the algebraic structure $\struct {\Q_{\ne 0}, \times}$ is a group.

From Group Axiom $\text G 3$: Existence of Inverse Element it follows that every $b \in \Q_{\ne 0}$ has an inverse element $b^{-1} \in \Q$ under multiplication.

From Group Axiom $\text G 0$: Closure 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

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $3$