Complex Number equals Conjugate iff Wholly Real

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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:

\(\displaystyle z\) \(=\) \(\displaystyle x + 0 i\)
\(\displaystyle \) \(=\) \(\displaystyle x\)
\(\displaystyle \) \(=\) \(\displaystyle x - 0 i\)
\(\displaystyle \) \(=\) \(\displaystyle \overline z\)

$\blacksquare$


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