Complex numbers/Real and imaginary part/Properties/Fact/Proof/Exercise

Prove the following statements concerning the real and imaginary parts of a complex number.

1. $z=\mathrm{Re}\left(z\right)+\mathrm{Im}\left(z\right)\mathrm{i}$.
2. $\mathrm{Re}(z+w)=\mathrm{Re}\left(z\right)+\mathrm{Re}\left(w\right)$.
3. $\mathrm{Im}(z+w)=\mathrm{Im}\left(z\right)+\mathrm{Im}\left(w\right)$.
4. For $r\in ℝ$ we have

   $\mathrm{Re}\left(rz\right)=r\mathrm{Re}\left(z\right)\text{ und }\mathrm{Im}\left(rz\right)=r\mathrm{Im}\left(z\right).$

5. The equation $z=\mathrm{Re}\left(z\right)$ holds if and only if $z\in ℝ$ and this holds if and only if $\mathrm{Im}\left(z\right)=0$.

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