Difference between revisions of "Complex conjugate"
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==Properties== | ==Properties== | ||
| − | Conjugation is its own [[Function | functional inverse]] and [[commutative property | commutes]] with the usual [[operation]]s on complex numbers: | + | Conjugation is its own [[Function#Inverses | functional inverse]] and [[commutative property | commutes]] with the usual [[operation]]s on complex numbers: |
* <math>\overline{(\overline z)} = z</math>. | * <math>\overline{(\overline z)} = z</math>. | ||
* <math>\overline{(w \cdot z)} = \overline{w} \cdot \overline{z}</math>. (<math>\overline{(\frac{w}{z})}</math> is the same as <math>\overline{(w \cdot \frac{1}{z})}</math>) | * <math>\overline{(w \cdot z)} = \overline{w} \cdot \overline{z}</math>. (<math>\overline{(\frac{w}{z})}</math> is the same as <math>\overline{(w \cdot \frac{1}{z})}</math>) | ||
Revision as of 11:10, 29 June 2013
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.
