# 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:

 $\displaystyle \overline z$ $=$ $\displaystyle 0 - i y$ $\displaystyle$ $=$ $\displaystyle -i y$ $\displaystyle$ $=$ $\displaystyle -\left({0 + i y}\right)$ $\displaystyle$ $=$ $\displaystyle -z$

$\blacksquare$