Complex numbers/Conjugation real part modulus/Fact/Proof/Exercise

Prove the following calculating rules for the complex numbers.

1. ${}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}$.
2. ${}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}$.
3. ${}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}$.
4. ${}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}$.
5. For ${}z\neq 0$ we have ${}z^{-1}={\frac {\overline {z}}{\vert {z}\vert ^{2}}}$.

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