# Sum of Complex Number with Conjugate

## Theorem

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

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

Let $\map \Re z$ be the real part of $z$.

Then:

$z + \overline z = 2 \, \map \Re z$

## Proof

Let $z = x + i y$.

Then:

 $\ds z + \overline z$ $=$ $\ds \paren {x + i y} + \paren {x - i y}$ Definition of Complex Conjugate $\ds$ $=$ $\ds 2 x$ $\ds$ $=$ $\ds 2 \, \map \Re z$ Definition of Real Part

$\blacksquare$

## Also defined as

This result is also reported as:

$\map \Re z = \dfrac {z + \overline z} 2$