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, then) the complex conjugate of is equal to The complex conjugate of is often denoted as or .
In polar form, the conjugate of is This can be shown using Euler's formula.
The product of a complex number and its conjugate is a real number: (or in polar coordinates).
If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.
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.
If a complex number is represented as a matrix, the notations are identical, and the complex conjugate corresponds to a flip along the diagonal.
The following properties apply for all complex numbers and unless stated otherwise, and can be proved by writing and in the form
For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:[ref 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.
Conjugation does not change the modulus of a complex number:
Conjugation is an involution, that is, the conjugate of the conjugate of a complex number is In symbols, [ref 1]
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:
If is a polynomial with real coefficients and then as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).
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.
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.
The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.
For matrices of complex numbers, where represents the element-by-element conjugation of [ref 2] Contrast this to the property where represents the conjugate transpose of
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.
One may also define a conjugation for quaternions and split-quaternions: the conjugate of is
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.
There is also an abstract notion of conjugation for vector spaces over the complex numbers. In this context, any antilinear map that satisfies
is called a complex conjugation, or a real structure. As the involution is antilinear, it cannot be the identity map on
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.