Complex conjugate

If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:[1]

A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation.

The product of a complex number with its conjugate is equal to the square of the number's modulus:

Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:

The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.

Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.

All these generalizations are multiplicative only if the factors are reversed:

Since the multiplication of planar real algebras is commutative, this reversal is not needed there.

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no canonical notion of complex conjugation.