# 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 $z = \overline z$ if and only if $z$ is wholly real.

## Proof

Let $z = x + i y$.

Then:

 $\ds z$ $=$ $\ds \overline z$ $\ds \leadsto \ \$ $\ds x + i y$ $=$ $\ds x - i y$ Definition of Complex Conjugate $\ds \leadsto \ \$ $\ds +y$ $=$ $\ds -y$ $\ds \leadsto \ \$ $\ds y$ $=$ $\ds 0$

Hence by definition, $z$ is wholly real.

$\Box$

Now suppose $z$ is wholly real.

Then:

 $\ds z$ $=$ $\ds x + 0 i$ $\ds$ $=$ $\ds x$ $\ds$ $=$ $\ds x - 0 i$ $\ds$ $=$ $\ds \overline z$

$\blacksquare$