Complex conjugate

Revision as of 11:05, 15 October 2006 by JBL (talk | contribs)

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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