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 $\Re \left({z}\right)$ be the real part of $z$.

Then:

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

Proof

Let $z = x + i y$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z + \overline z\)	\(=\)	\(\displaystyle \left({x + i y}\right) + \left({x - i y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of complex conjugate
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \Re \left({z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of real part

$\blacksquare$