Modulus in Terms of Conjugate

Theorem
Let $$z = a + \imath b$$ be a complex number.

Let $$\left|{z}\right|$$ be the modulus of $$z$$.

Let $$\overline z$$ be the conjugate of $$z$$.

Then $$\left|{z}\right|^2 = z \overline z$$.

Proof
Let $$z = a + \imath b$$.

Then:

$$ $$ $$ $$