Complex Number Equals Conjugate iff Wholly Real

Theorem

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

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

Then $z = \overline z$ iff $z$ is wholly real.

Proof

Let $z = x + i y$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z\)	\(=\)	\(\displaystyle \overline z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x + i y\)	\(=\)	\(\displaystyle x - i y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of complex conjugate
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle +y\)	\(=\)	\(\displaystyle -y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle y\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Hence by definition, $z$ is wholly real.

$\Box$

Now suppose $z$ is wholly real.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z\)	\(=\)	\(\displaystyle x + 0 i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x - 0 i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \overline z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$