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Complex numbers/Real and imaginary part/Properties/Fact/Proof/Exercise
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Complex numbers
|
Real and imaginary part/Properties/Fact
Prove the following statements concerning the real and imaginary parts of a
complex number
.
z
=
Re
(
z
)
+
Im
(
z
)
i
{\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }}
.
Re
(
z
+
w
)
=
Re
(
z
)
+
Re
(
w
)
{\displaystyle {}\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}}
.
Im
(
z
+
w
)
=
Im
(
z
)
+
Im
(
w
)
{\displaystyle {}\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}}
.
For
r
∈
R
{\displaystyle {}r\in \mathbb {R} }
we have
Re
(
r
z
)
=
r
Re
(
z
)
und
Im
(
r
z
)
=
r
Im
(
z
)
.
{\displaystyle \operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ und }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.}
The equation
z
=
Re
(
z
)
{\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}}
holds if and only if
z
∈
R
{\displaystyle {}z\in \mathbb {R} }
and this holds if and only if
Im
(
z
)
=
0
{\displaystyle {}\operatorname {Im} \,{\left(z\right)}=0}
.
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