Difference between revisions of "Complex conjugate"
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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{(w \cdot z)}</math> is the same as <math>\overline{(w \cdot \frac{1}{z})}) | + | * <math>\overline{(w \cdot z)} = \overline{w} \cdot \overline{z}</math> (<math>\overline{(w \cdot 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))}) |
| − | It also interacts in simple ways with other operations on <math>\mathbb{C}< | + | It also interacts in simple ways with other operations on </math>\mathbb{C}<math>: |
| − | * <math>|\overline{z}| = |z|< | + | * </math>|\overline{z}| = |z|<math> |
| − | * <math>\overline{z}\cdot z = |z|^2< | + | * </math>\overline{z}\cdot z = |z|^2<math> |
| − | * If <math>z = r\cdot e^{it}< | + | * 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>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} = 2i \mathrm{Im}(z)< | + | * </math>z - \overline{z} = 2i \mathrm{Im}(z)<math> where </math>\mathrm{Im}(z)<math> is the [[imaginary part]] of </math>z$. |
{{stub}} | {{stub}} | ||
[[Category:Number Theory]] | [[Category:Number Theory]] | ||
Revision as of 09:57, 4 December 2007
The complex conjugate of a complex number 
is the complex number 
.
Geometrically, if 
is a point in the complex plane, 
is the reflection of across the real axis.
Properties
Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:
(
is the same as 
(
is the same as 
\mathbb{C}
|\overline{z}| = |z|
\overline{z}\cdot z = |z|^2
z = r\cdot e^{it}
r, t \in \mathbb{R}
\overline z = r\cdot e^{-it}
\overline z![$is the complex number of same [[absolute value]] but opposite [[argument]] of$](//latex.artofproblemsolving.com/e/9/1/e91b25856ffcff7238d2e100654fbaac683af8df.png)
z
z + \overline z = 2 \mathrm{Re}(z)
\mathrm{Re}(z)![$is the [[real part]] of$](//latex.artofproblemsolving.com/1/3/0/130e002ddf8cbf9f4c487be86d057c16dc0a17b2.png)
zz - \overline{z} = 2i \mathrm{Im}(z)\mathrm{Im}(z)![$is the [[imaginary part]] of$](//latex.artofproblemsolving.com/5/6/3/5638ba3a6f76e07c6ccd890d6b0d411d1739590a.png)
z$.
(
is the same as 
(
is the same as 
\mathbb{C}
|\overline{z}| = |z|
\overline{z}\cdot z = |z|^2
z = r\cdot e^{it}
r, t \in \mathbb{R}
\overline z = r\cdot e^{-it}
\overline z![$is the complex number of same [[absolute value]] but opposite [[argument]] of$](http://latex.artofproblemsolving.com/e/9/1/e91b25856ffcff7238d2e100654fbaac683af8df.png)
z
z + \overline z = 2 \mathrm{Re}(z)
\mathrm{Re}(z)![$is the [[real part]] of$](http://latex.artofproblemsolving.com/1/3/0/130e002ddf8cbf9f4c487be86d057c16dc0a17b2.png)
z![$is the [[imaginary part]] of$](http://latex.artofproblemsolving.com/5/6/3/5638ba3a6f76e07c6ccd890d6b0d411d1739590a.png)
z$.
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