# Complex numbers/Real and imaginary part/Properties/Fact

The real part and the imaginary part of complex numbers fulfill the following properties (for ${\displaystyle {}z}$ and ${\displaystyle {}w}$ in ${\displaystyle {}\mathbb {C} }$).
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. ${\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}}$ holds if and only if ${\displaystyle {}z\in \mathbb {R} }$ holds, and this holds if and only if ${\displaystyle {}\operatorname {Im} \,{\left(z\right)}=0}$ holds.