# Sum of Complex Conjugates

## Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

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

Then:

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

## Proof

Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.

Then:

 $\ds \overline {z_1 + z_2}$ $=$ $\ds \overline {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }$ $\ds$ $=$ $\ds \paren {x_1 + x_2} - i \paren {y_1 + y_2}$ Definition of Complex Conjugate $\ds$ $=$ $\ds \paren {x_1 - i y_1} + \paren {x_2 - i y_2}$ Definition of Complex Addition $\ds$ $=$ $\ds \overline {z_1} + \overline {z_2}$ Definition of Complex Conjugate

$\blacksquare$