Modulus in Terms of Conjugate

Theorem

Let $z = a + i b$ be a complex number.

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

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

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

Proof

Let $z = a + i b$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z \overline z\)	\(=\)	\(\displaystyle \left({a + i b}\right) \left({a - i b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of conjugate
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({a^2 - \left({-b^2}\right)}\right) + i \left({a b - a b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of complex multiplication
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a^2 + b^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\vert{z}\right\vert^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of modulus

$\blacksquare$