Inverse for Complex Multiplication

Theorem
Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $z^{-1}$ under the operation of complex multiplication:
 * $\forall z \in \C_{\ne 0}: \exists z^{-1} \in \C_{\ne 0}: z \times z^{-1} = 1 + 0 i = z^{-1} \times z$

This inverse can be expressed as:
 * $\dfrac 1 z := \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$

where $\overline z$ is the complex conjugate of $z$.

Proof
Similarly for $\dfrac {x - i y} {x^2 + y^2} \paren {x + i y}$.

So the inverse of $x + i y \in \struct {\C_{\ne 0}, \times}$ is $\dfrac {x - i y} {x^2 + y^2}$.

As $x^2 + y^2 > 0 \iff x, y \ne 0$ the inverse is defined for all $z \in \C: z \ne 0 + 0 i$.

From the definition, the complex conjugate $\overline z$ of $z = x + i y$ is $x - i y$.

From the definition of the modulus of a complex number, we have:
 * $\cmod z = \sqrt {a^2 + b^2}$

From Modulus in Terms of Conjugate, we have that:
 * $\cmod z^2 = z \overline z$

Hence the result:
 * $\dfrac 1 z = \dfrac {\overline z} {z \overline z}$