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 
)
. (
is the same as 
)
.
. (
is the same as 
)
. (
is the same as 
)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 .
.
.
for 
, 
. That is, 
where 
is the 
where 
is the
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