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* + *i**b* (where *a* and *b* are real numbers) is defined to be *z* ^{*} = *a* - *i**b*. It is also often denoted by a bar over the number, rather than a star, which often is used also for the conjugate transpose. If a complex number is treated as a 1×1 vector, the notations are identical.

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

These properties apply for all complex numbers *z* and *w*, unless stated otherwise.

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

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

- if
*w* is non-zero

*z* ^{*} = *z* if and only if *z* is real

- if
*z* is non-zero

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

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.

See complex conjugate vector space

## 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* + *b**i* + *c**j* + *d**k* is *a* - *b**i* - *c**j* - *d**k*.

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

Since the multiplication of complex numbers is commutative, this reversal is "invisible" there.