In mathematics, the

**complex conjugate**of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number

*z*=

*a*+

*ib*is defined to be

*z*=

^{*}*a*-

*ib*. It is also often denoted by a bar over the number, rather than a star.

For example, (3-2*i*)^{*} = 3 + 2*i*, *i*^{*} = -*i* and 7^{*} = 7.

One usually thinks of complex numbers as points in a plane with a cartesian coordinate system. The *x*-axis contains the real numbers and the *y*-axis contains the multiples of *i*. In this view, complex conjugation corresponds to reflection at the *x*-axis.

## Properties

The following are valid for all complex numbers *z* and *w*, unless stated otherwise.

- (
*z*+*w*)^{*}=*z*+^{*}*w*^{*} - (
*zw*)^{*}=*z*^{*}*w*^{*} - (
*z/w*)^{*}=*z*/^{*}*w*if^{*}*w*is non-zero -
*z*^{*}=*z*if and only if*z*is real - |
*z*^{*}| = |*z*| - |
*z*|^{2}=*z**z*^{*} -
*z*^{-1}=*z*^{*}/ |z|^{2}if*z*is non-zero

If *p* is a polynomial with real coefficients, and *p*(*z*) = 0, then *p*(*z*^{*}) = 0 as well. Thus the roots of real polynomials outside of the real line occur in complex conjugate pairs.

The function φ(*z*) = *z*^{*} from **C** to **C** is continuous. Even though it appears to be a "tame" well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension **C** / **R**. This Galois group has only two elements: φ and the identity on **C**. Thus the only two field automorphisms of **C** that leave the real numbers fixed are the identity map and complex conjugation.

## Generalizations

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-star algebras.

One may also define a conjugation for quaternions: the conjugate of *a* + *bi* + *cj* + *dk* is *a* - *bi* - *cj* - *dk*.

Note that all these generalizations are multiplicative only if the factors are reversed:

- (
*zw*)^{*}=*w*^{*}*z*^{*}