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 + ib.
A significant property of the complex conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the complex number is real.
For any two complex numbers w,z:
The penultimate relation is involution; i.e., the conjugate of the conjugate of a complex number z is z. The ultimate relation is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
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.
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. It should be remarked that on generic complex vector spaces there is no canonical notion of complex conjugation.