# 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 $z$

$\blacksquare$