Complex conjugate
The complex conjugate of a complex number is the complex number
.
Geometrically, if is a point in the complex plane,
is the reflection of
across the real axis.
Properties
Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:
(
is the same as
\overline{(w + z)} = \overline{w} + \overline{z}
\overline{(w + z)}
\overline{(w + (-z))})
It also interacts in simple ways with other operations on :
- If
for
,
. That is,
is the complex number of same absolute value but opposite argument of
.
where
is the real part of
.
where
is the imaginary part of
.
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