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]].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. |
==Properties== | ==Properties== | ||
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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>. | * 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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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 .
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