Difference between revisions of "Complex conjugate"
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* <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 | + | * <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>. |
Revision as of 21:20, 1 January 2008
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
.
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