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

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

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

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