# Komplexe Konjugation/Rechenregeln/Fakt/Beweis/Aufgabe/en

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Prove the following properties of the complex conjugation.

1. ${\displaystyle {}{\overline {z+w}}={\overline {z}}+{\overline {w}}}$.
2. ${\displaystyle {}{\overline {-z}}=-{\overline {z}}}$.
3. ${\displaystyle {}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}}$.
4. For ${\displaystyle {}z\neq 0}$ we have ${\displaystyle {}{\overline {1/z}}=1/{\overline {z}}}$.
5. ${\displaystyle {}{\overline {\overline {z}}}=z}$.
6. ${\displaystyle {}{\overline {z}}=z}$ if and only if ${\displaystyle {}z\in \mathbb {R} }$.