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Complex numbers/Conjugation real part modulus/Fact/Proof/Exercise
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Complex numbers
Prove the following calculating rules for the
complex numbers
.
|
z
|
=
z
z
¯
{\displaystyle {}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}}
.
Re
(
z
)
=
z
+
z
¯
2
{\displaystyle {}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}}
.
Im
(
z
)
=
z
−
z
¯
2
i
{\displaystyle {}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}}
.
z
¯
=
Re
(
z
)
−
i
Im
(
z
)
{\displaystyle {}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}}
.
For
z
≠
0
{\displaystyle {}z\neq 0}
we have
z
−
1
=
z
¯
|
z
|
2
{\displaystyle {}z^{-1}={\frac {\overline {z}}{\vert {z}\vert ^{2}}}}
.
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