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Geometric representation (Argand diagram) of z and its conjugate z? in the complex plane. The complex conjugate is found by reflectingz across the real axis.
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. Given a complex number (where a and b are real numbers), the complex conjugate of , often denoted as , is equal to 
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, while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". For example, writing means .
The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proved by writing z and w in the form a + bi.
For any two complex numbers w,z, conjugation is distributive over addition, subtraction, multiplication and division.
Real numbers are the only fixed points of conjugation. A complex number is equal to its complex conjugate if its imaginary part is zero.
Composition of conjugation with the modulus is equivalent to the modulus alone.
Conjugation is an involution; the conjugate of the conjugate of a complex number z is z.
The product of a complex number with its conjugate is equal to the square of the number's modulus. This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates.
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 is 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 fieldautomorphism. 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 z-variable:
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 V 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. Note that on generic complex vector spaces, there is no canonical notion of complex conjugation.