Inverse for Rational Multiplication

Theorem
Each element $x$ of the set of non-zero rational numbers $\Q^*$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication:
 * $\displaystyle \forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$

Proof
From the definition, the field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

From Rational Multiplication Identity is One, the identity for $\struct {\Q, \times}$ is $1 = \dfrac 1 1 = \dfrac p p$ where $p \in \Z$ and $p \ne 0$.

From the definition of product inverses in field of quotients, we have that:
 * $\paren {\dfrac p q}^{-1} = \dfrac q p$

which can be demonstrated directly:

Note that this is defined only when $p \ne 0$ and $q \ne 0$.

Now let $x = \dfrac p q \in \Q$.

We define $\dfrac 1 x$ as $1 \times \paren {\dfrac p q}^{-1}$.

Hence: