ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2950 on D702: Involution

Complex conjugation operation is an involution

Formulation 0

Let $z = (x, y) \in \mathbb{C}$ be a D1207: Complex number.
Let $w \mapsto \overline{w}$ be the D734: Complex conjugation operation.

Then \begin{equation} \overline{\overline{z}} = z \end{equation}

Proofs

Proof 0

Let $z = (x, y) \in \mathbb{C}$ be a D1207: Complex number.
Let $w \mapsto \overline{w}$ be the D734: Complex conjugation operation.

By definition, $\overline{z} = \overline{(x, y)} = (x, - y)$. Thus \begin{equation} \overline{\overline{z}} = \overline{\overline{(x, y)}} = \overline{(x, - y)} = (x, y) \end{equation} $\square$