Square of Complex Conjugate is Complex Conjugate of Square

Theorem

Let $z \in \C$ be a complex number.

Let $\overline z$ denote the complex conjugate of $z$.

Then:

$\overline {z^2} = \left({\overline z}\right)^2$

Proof

A direct consequence of Product of Complex Conjugates:

$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$

for two complex numbers $z_1, z_2 \in \C$.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory