Difference between revisions of "Complex conjugate"
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− | The ''' | + | 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]]. |
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Revision as of 19:30, 4 September 2008
This is an AoPSWiki Word of the Week for August 29-September 4 |
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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