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Complex Conjugation/Rules/Fact/Proof/Exercise
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Complex Conjugation/Rules/Fact
Prove the following properties of the
complex conjugation
.
z
+
w
¯
=
z
¯
+
w
¯
{\displaystyle {}{\overline {z+w}}={\overline {z}}+{\overline {w}}}
.
−
z
¯
=
−
z
¯
{\displaystyle {}{\overline {-z}}=-{\overline {z}}}
.
z
⋅
w
¯
=
z
¯
⋅
w
¯
{\displaystyle {}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}}
.
For
z
≠
0
{\displaystyle {}z\neq 0}
we have
1
/
z
¯
=
1
/
z
¯
{\displaystyle {}{\overline {1/z}}=1/{\overline {z}}}
.
z
¯
¯
=
z
{\displaystyle {}{\overline {\overline {z}}}=z}
.
z
¯
=
z
{\displaystyle {}{\overline {z}}=z}
if and only if
z
∈
R
{\displaystyle {}z\in \mathbb {R} }
.
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