# 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:

\(\displaystyle z\) | \(=\) | \(\displaystyle \overline z\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x + i y\) | \(=\) | \(\displaystyle x - i y\) | Definition of Complex Conjugate | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle +y\) | \(=\) | \(\displaystyle -y\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle 0\) |

Hence by definition, $z$ is wholly real.

$\Box$

Now suppose $z$ is wholly real.

Then:

\(\displaystyle z\) | \(=\) | \(\displaystyle x + 0 i\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x - 0 i\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \overline z\) |

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.2$. The Algebraic Theory