# Komplexe Zahlen/Real und Imaginärteil/Eigenschaften/Fakt/Beweis/Aufgabe/en

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

1. ${\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }}$.
2. ${\displaystyle {}\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}}$.
3. ${\displaystyle {}\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}}$.
4. For ${\displaystyle {}r\in \mathbb {R} }$ we have
${\displaystyle \operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ and }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.}$
5. ${\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}}$ if and only if ${\displaystyle {}z\in \mathbb {R} }$, and this is exactly the case when ${\displaystyle {}\operatorname {Im} \,{\left(z\right)}=0}$.