# Komplexe Zahlen/Rechenregeln für Betrag/Aufgabe/en

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Prove the following properties of the absolute value of a complex number.

1. For a real number ${\displaystyle {}z}$ its real absolute value and its complex absolute value coincide.
2. We have ${\displaystyle {}\vert {z}\vert =0}$ if and only if ${\displaystyle {}z=0}$.
3. ${\displaystyle {}\vert {z}\vert =\vert {\overline {z}}\vert }$.
4. ${\displaystyle {}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert }$.
5. ${\displaystyle {}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert }$.
6. For ${\displaystyle {}z\neq 0}$ we have ${\displaystyle {}\vert {1/z}\vert =1/\vert {z}\vert }$.