# Complex Number equals Negative of Conjugate iff Wholly Imaginary

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## Theorem

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

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

Then $\overline z = -z$ if and only if $z$ is wholly imaginary.

## Proof

Let $z = x + i y$.

Then:

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

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

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

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

Hence by definition, $z$ is wholly imaginary.

$\Box$

Now suppose $z$ is wholly imaginary.

Then:

\(\displaystyle \overline z\) | \(=\) | \(\displaystyle 0 - i y\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -i y\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\left({0 + i y}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -z\) |

$\blacksquare$

## Sources

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