In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if and are real numbers then the complex conjugate of is The complex conjugate of is often denoted as or .
The product of a complex number and its conjugate is a real number: (or in polar coordinates).
The complex conjugate of a complex number is written as or The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics.
The following properties apply for all complex numbers and unless stated otherwise, and can be proved by writing and in the form
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.
Conjugation does not change the modulus of a complex number:
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:
In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and and are defined, then
The map from to is a homeomorphism (where the topology on is taken to be the standard topology) and antilinear, if one considers as a complex vector space over itself. Even though it appears to be a 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 This Galois group has only two elements: and the identity on Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation.
Use as a variable
Once a complex number or is given, its conjugate is sufficient to reproduce the parts of the -variable:
Furthermore, can be used to specify lines in the plane: the set
These uses of the conjugate of as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.
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.
- where and is the identity map on
- for all and
- for all
Of course, is a -linear transformation of if one notes that every complex space has a real form obtained by taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space 
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.
- Absolute square – Product of a number by itself
- Complex conjugate line – Operation in complex geometry
- Complex conjugate representation
- Complex conjugate vector space – Mathematics concept
- Composition algebra – Type of algebras, possibly non associative
- Conjugate (square roots) – Change of the sign of a square root
- Hermitian function – Type of complex function
- Wirtinger derivatives – Concept in complex analysis
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
- "Lesson Explainer: Matrix Representation of Complex Numbers | Nagwa". www.nagwa.com. Retrieved 2023-01-04.
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29