Complex Conjugation is Involution

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Theorem

Let $z = x + i y$ be a complex number.

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


Then the operation of complex conjugation is an involution:

$\overline {\paren {\overline z} } = z$


Proof

\(\displaystyle \overline {\paren {\overline z} }\) \(=\) \(\displaystyle \overline {\paren {\overline {x + i y} } }\) Definition of $z$
\(\displaystyle \) \(=\) \(\displaystyle \overline {x - i y}\) Definition of Complex Conjugate
\(\displaystyle \) \(=\) \(\displaystyle x + i y\) Definition of Complex Conjugate
\(\displaystyle \) \(=\) \(\displaystyle z\) Definition of $z$

$\blacksquare$