AoPSWiki
Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
Personal tools

Complex conjugate

From AoPSWiki

The conjugate of a complex number z = a + bi is a - bi, denoted by \overline{z}. Geometrically, \overline z is the reflection of z across the real axis if both points were plotted in the complex plane.

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{(\frac{w}{z})} is the same as \overline{(w \cdot \frac{1}{z})})
  • \overline{(w + z)} = \overline{w} + \overline{z}. (\overline{(w - z)} is the same as \overline{(w + (-z))})

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

  • |\overline{z}| = |z|.
  • \overline{z}\cdot z = |z|^2.
  • If z = r\cdot e^{it} for r, t \in \mathbb{R}, \overline z = r\cdot e^{-it}. That is, \overline z is the complex number of same absolute value but opposite argument of z.
  • z + \overline z = 2 \mathrm{Re}(z) where \mathrm{Re}(z) is the real part of z.
  • z - \overline{z} = 2i \mathrm{Im}(z) where \mathrm{Im}(z) is the imaginary part of z.
  • If a complex number z is a root of a polynomial with real coefficients, then so is \overline z.

This article is a stub. Help us out by expanding it.

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Complex_conjugate"
Preparing for MATHCOUNTS or the AMC contests, and having a tough time with number theory problems? Read Art of Problem Solving's Introduction to Number Theory by Mathew Crawford.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us