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:

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

$\blacksquare$


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