Difference between revisions of "Complex conjugate"
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The '''complex conjugate''' of a [[complex number]] <math>z = a + bi</math> is the complex number <math>\overline{z} = a - bi</math>. | The '''complex conjugate''' of a [[complex number]] <math>z = a + bi</math> is the complex number <math>\overline{z} = a - bi</math>. | ||
Revision as of 08:34, 31 August 2008
This is an AoPSWiki Word of the Week for August 29-September 4 |
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
)
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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