# Difference between revisions of "Complex conjugate"

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The '''conjugate''' of a [[complex number]] <math>z = a + bi</math> is <math>a - bi</math>, denoted by <math>\overline{z}</math>. Geometrically, <math>\overline z</math> is the [[reflect]]ion of <math>z</math> across the [[real axis]] if both points were plotted in the [[complex plane]]. | The '''conjugate''' of a [[complex number]] <math>z = a + bi</math> is <math>a - bi</math>, denoted by <math>\overline{z}</math>. Geometrically, <math>\overline z</math> is the [[reflect]]ion of <math>z</math> across the [[real axis]] if both points were plotted in the [[complex plane]]. | ||

## Revision as of 17:52, 6 September 2008

The **conjugate** of a complex number is , denoted by . Geometrically, is the reflection of 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:

- .
- . ( 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 .
- If a complex number is a root of a polynomial with real coefficients, then so is .

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