Difference of Complex Number with Conjugate

Theorem

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

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

Let $\Im \left({z}\right)$ be the imaginary part of $z$.

Then

$z - \overline z = 2 i \Im \left({z}\right)$

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
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x + i y - x + i y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 i y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 i \Im \left({z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of imaginary part

$\blacksquare$