Difference between revisions of "Complex conjugate"
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==Properties== | ==Properties== | ||
Conjugation is its own [[Function/Introduction#The_Inverse_of_a_Function | functional inverse]] and [[commutative property | commutes]] with the usual [[operation]]s on complex numbers: | Conjugation is its own [[Function/Introduction#The_Inverse_of_a_Function | 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>) |
| − | * <math>\overline{(w + z)} = \overline{w} + \overline{z}</math> (<math>\overline{(w + z)}</math> is the same as <math>\overline{(w + (-z))}</math>) | + | * <math>\overline{(w + z)} = \overline{w} + \overline{z}</math>. (<math>\overline{(w + z)}</math> is the same as <math>\overline{(w + (-z))}</math>) |
It also interacts in simple ways with other operations on <math>\mathbb{C}</math>: | It also interacts in simple ways with other operations on <math>\mathbb{C}</math>: | ||
| − | * <math>|\overline{z}| = |z|</math> | + | * <math>|\overline{z}| = |z|</math>. |
| − | * <math>\overline{z}\cdot z = |z|^2</math> | + | * <math>\overline{z}\cdot z = |z|^2</math>. |
* If <math>z = r\cdot e^{it}</math> for <math>r, t \in \mathbb{R}</math>, <math>\overline z = r\cdot e^{-it}</math>. That is, <math>\overline z</math> is the complex number of same [[absolute value]] but opposite [[argument]] of <math>z</math>. | * If <math>z = r\cdot e^{it}</math> for <math>r, t \in \mathbb{R}</math>, <math>\overline z = r\cdot e^{-it}</math>. That is, <math>\overline z</math> is the complex number of same [[absolute value]] but opposite [[argument]] of <math>z</math>. | ||
* <math>z + \overline z = 2 \mathrm{Re}(z)</math> where <math>\mathrm{Re}(z)</math> is the [[real part]] of <math>z</math>. | * <math>z + \overline z = 2 \mathrm{Re}(z)</math> where <math>\mathrm{Re}(z)</math> is the [[real part]] of <math>z</math>. | ||
Revision as of 10:59, 4 December 2007
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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