Complex Conjugate - Properties

Properties

These properties apply for all complex numbers z and w, unless stated otherwise, and can be easily proven by writing z and w in the form a + ib.

if and only if z is real

for any integer n

, involution (i.e., the conjugate of the conjugate of a complex number z is again that number)

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 z is non-zero

In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and is defined, then

Consequently, 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).

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.