Modulus in Terms of Conjugate

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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:

\(\displaystyle z \overline z\) \(=\) \(\displaystyle a^2 + b^2\) Product of Complex Number with Conjugate
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{z}\right\vert^2\) Definition of Complex Modulus

$\blacksquare$


Sources