Complex numbers/Modulus/Rules/Fact

The modulus of complex numbers fulfils the following properties.

1. ${}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}$.
2. For real ${}z$ the real and the complex modulus are the same.
3. We have ${}\vert {z}\vert =0$ if and only if ${}z=0$.
4. ${}\vert {z}\vert =\vert {\overline {z}}\vert$.
5. ${}\vert {zw}\vert =\vert {z}\vert \cdot \vert {w}\vert$.
6. For ${}z\neq 0$ we have ${}\vert {1/z}\vert =1/\vert {z}\vert$.
7. ${}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert$.
8. ${}\vert {z+w}\vert \leq \vert {z}\vert +\vert {w}\vert$ (triangle inequality).

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