Complex numbers/Rules for modulus/Exercise

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

1. ${}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}\,.$

2. For a real number ${}z$ its real absolute value and its complex absolute value coincide.
3. We have ${}\vert {z}\vert =0$ if and only if ${}z=0$.

4. ${}\vert {z}\vert =\vert {\overline {z}}\vert \,.$

5. ${}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert \,.$

6. For ${}z\neq 0$ we have ${}\vert {1/z}\vert =1/\vert {z}\vert$.

7. ${}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert \,.$

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