Inverse for Complex Multiplication

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

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

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

Proof
$$ $$ $$

Similarly for $$\frac {x - i y} {x^2 + y^2} \left({x + i y}\right)$$.

So the inverse of $$x + i y \in \left({\C^*, \times}\right)$$ is $$\frac {x - i y} {x^2 + y^2}$$.

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

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:
 * $$\left|{z}\right| = \sqrt {a^2 + b^2}$$

From Modulus in Terms of Conjugate, we have that:
 * $$\left|{z}\right|^2 = z \overline z$$

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