Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number z = a + ib is defined to be z^* = a - ib. It is also often denoted by a bar over the number, rather than a star.

For example, (3-2i)^* = 3 + 2i, i^* = -i and 7^* = 7.

One usually thinks of complex numbers as points in a plane with a cartesian coordinate system. The x-axis contains the real numbers and the y-axis contains the multiples of i. In this view, complex conjugation corresponds to reflection at the x-axis.

Properties

The following are valid for all complex numbers z and w, unless stated otherwise.

(z + w)^* = z^* + w^*

(zw)^* = z^* w^*

(z/w)^* = z^* / w^* if w is non-zero

z^* = z if and only if z is real

|z^*| = |z|

|z|² = z z^*

z^-1 = z^* / |z|²    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 p is a polynomial with real coefficients, and p(z) = 0, then p(z^*) = 0 as well. Thus the roots of real polynomials outside of the real line occur in complex conjugate pairs.

The function φ(z) = z^* from C to C is continuous. Even though it appears to be a "tame" 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 C / R. This Galois group has only two elements: φ and the identity on C.

Generalizations

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-star algebras.

One may also define a conjugation for quaternions: the conjugate of a + bi + cj + dk is a - bi - cj - dk.

Note that all these generalizations are multiplicative only if the factors are reversed:

(zw)^* = w^* z^*

Since the multiplication of complex numbers is commutative, this reversal is "invisible" there.