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Difference between revisions of "Complex conjugate"

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The '''complex conjugate''' of a [[complex number]] <math>z = a + bi</math> is the complex number <math>\overline{z} = a - bi</math>.
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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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Geometrically, if <math>z</math> is a point in the [[complex plane]], <math>\overline z</math> is the [[reflect]]ion of <math>z</math> across the [[real axis]].
 
  
 
==Properties==
 
==Properties==
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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:
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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>\overline{(w + z)} = \overline{w} + \overline{z}</math>. (<math>\overline{(w + z)}</math> is the same as <math>\overline{(w + (-z))}</math>)
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* <math>\overline{(w + z)} = \overline{w} + \overline{z}</math>. (<math>\overline{(w - z)}</math> is the same as <math>\overline{(w + (-z))}</math>)
 
It also interacts in simple ways with other operations on <math>\mathbb{C}</math>:
 
It also interacts in simple ways with other operations on <math>\mathbb{C}</math>:
 
* <math>|\overline{z}| = |z|</math>.
 
* <math>|\overline{z}| = |z|</math>.
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* <math>z + \overline z = 2 \mathrm{Re}(z)</math> where <math>\mathrm{Re}(z)</math> is the [[real part]] of <math>z</math>.
 
* <math>z + \overline z = 2 \mathrm{Re}(z)</math> where <math>\mathrm{Re}(z)</math> is the [[real 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>.
 
* <math>z - \overline{z} = 2i \mathrm{Im}(z)</math> where <math>\mathrm{Im}(z)</math> is the [[imaginary part]] of <math>z</math>.
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* 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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[[Category:Number Theory]]
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[[Category:Complex numbers]]
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[[Category:Definition]]
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[[category:Mathematics]]

Latest revision as of 17:40, 28 September 2024

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.For all polynomials with real coefficients, if a complex number $z$ is a root of the polynomial its conjugate $\overline{z}$ will be a root as well.

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 $\overline z$.


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