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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 (where a and b are real numbers) is defined to be z * = a - ib. It is also often denoted by a bar over the number, rather than a star, which often is used also for the conjugate transpose. If a complex number is treated as a 1×1 vector, the notations are identical.

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

These properties apply for all complex numbers z and w, unless stated otherwise.

(z + w) * = z * + w *
(zw) * = z * w *
$\left({\frac{z}{w}}\right)^* = \frac{z^*}{w^*}$ if w is non-zero
z * = z if and only if z is real
$\left| z^* \right| = \left| z \right|$
${\left| z \right|}^2 = zz^*$
$z^{-1} = \frac{z^*}{{\left| z \right|}^2}$ 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. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation.

## 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:

${\left(zw\right)}^* = w^* z^*.$

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

03-10-2013 05:06:04