Difference between revisions of "Complex conjugate"
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| − | The ''' | + | The '''conjugate''' of a [[complex number]] <math>z = a + bi</math> is <math>a - bi</math>, denoted by <math>\overline{z}</math>. Geometrically, <math>\overline z</math> is the [[reflect]]ion of <math>z</math> across the [[real axis]] if both points were plotted in the [[complex plane]]. |
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==Properties== | ==Properties== | ||
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| − | [[Category: | + | [[Category:Definition]] |
Revision as of 19:30, 4 September 2008
| This is an AoPSWiki Word of the Week for August 29-September 4 |
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.
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 .- If a complex number is a root of a polynomial with real coefficients, then so is .
.
.
for 
, 
. That is, 
where 
is the 
where 
is the This article is a stub. Help us out by expanding it.
