Complex conjugate
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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