Komplexe Zahlen/Rechenregeln für Betrag/Aufgabe/en

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

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