complex number:conjugate

Complex conjugate
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Geometric representation of z and its conjugate in the complex plane.
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

(where a and b are real numbers) is

The complex conjugate is also very commonly denoted by z * . Here is chosen to avoid confusion with the notation for the conjugate transpose of a matrix (which can be thought of as a generalization of complex conjugation). Notice that if a complex number is treated as a matrix, the notations are identical.
For example,



Complex numbers are often depicted as points in a plane with a cartesian coordinate system (see diagram). 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.
In polar form, however, the conjugate of reiφ is given by re − iφ. This can easily be verified by using Euler's formula.
Pairs of complex conjugates are significant because the imaginary unit i is qualitatively indistinct from its additive and multiplicative inverse − i, as they both satisfy the definition for the imaginary unit: x2 = − 1. Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.
Contents[hide]
1 Properties
2 Use as a variable
3 Generalizations
4 See also
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[edit] Properties
These properties apply for all complex numbers z and w, unless stated otherwise.



if w is non-zero
if and only if z is real
for any integer n


Idempotence; 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 p is a polynomial with real coefficients, and p(z) = 0, then as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs. (See the complex conjugate root theorem article.)
The function from to is a homeomorphism (where the topology on is taken to be the standard topology). 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 . This Galois group has only two elements: φ and the identity on . Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation.