Complex conjugate
The conjugate of a complex number is
, denoted by
. Geometrically,
is the reflection of
across the real axis if both points were plotted in the complex plane.For all polynomials with real coefficients, if a complex number
is a root of the polynomial its conjugate
will be a root as well.
Properties
Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:
.
. (
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
.
- If a complex number
is a root of a polynomial with real coefficients, then so is
.
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