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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.

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.