Difference between revisions of "Complex conjugate"

 
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==Properties==
 
==Properties==
Conjugation is its own [[inverse]] and 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)} = \overline{w} \cdot \overline{z}</math>

Revision as of 11:05, 15 October 2006

The complex conjugate of a complex number $z = a + bi$ is the complex number $\overline{z} = a - bi$.

Geometrically, if $z$ is a point in the complex plane, $\overline z$ is the reflection of $z$ across the real axis.

Properties

Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:

  • $\overline{(\overline z)} = z$
  • $\overline{(w \cdot z)} = \overline{w} \cdot \overline{z}$
  • $\overline{(w + z)} = \overline{w} + \overline{z}$

It also interacts in simple ways with other operations on $\mathbb{C}$:


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