# Complex Conjugation/Rules/Fact

For the complex conjugation the following rules hold (for arbitrary ${\displaystyle {}z,w\in \mathbb {C} }$).
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. We have ${\displaystyle {}{\overline {z}}=z}$ if and only if ${\displaystyle {}z\in \mathbb {R} }$ holds.