# 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]].For all polynomials with real coefficients, if a complex number <math>z</math> is a root of the polynomial its conjugate <math>\overline{z}</math> will be a root as well. | |

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− | The ''' | ||

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==Properties== | ==Properties== | ||

− | Conjugation is its own [[Function | + | Conjugation is its own [[Function#Inverses | 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{(\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>) | ||

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* <math>z - \overline{z} = 2i \mathrm{Im}(z)</math> where <math>\mathrm{Im}(z)</math> is the [[imaginary part]] of <math>z</math>. | * <math>z - \overline{z} = 2i \mathrm{Im}(z)</math> where <math>\mathrm{Im}(z)</math> is the [[imaginary part]] of <math>z</math>. | ||

* If a complex number <math>z</math> is a root of a polynomial with real coefficients, then so is <math>\overline z</math>. | * If a complex number <math>z</math> is a root of a polynomial with real coefficients, then so is <math>\overline z</math>. | ||

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{{stub}} | {{stub}} | ||

− | [[Category: | + | [[Category:Complex numbers]] |

+ | [[Category:Definition]] |

## Latest revision as of 18:26, 25 January 2020

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.For all polynomials with real coefficients, if a complex number is a root of the polynomial its conjugate will be a root as well.

## 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 .

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