Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.are real numbers, then the complex conjugate ofThe first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate.The second is preferred in physics, where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics.matrix, the notations are identical, and the complex conjugate corresponds to the matrix transpose, which is a flip along the diagonal.[1] The following properties apply for all complex numbersFor any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:[ref 1]In other words, real numbers are the only fixed points of conjugation.Conjugation does not change the modulus of a complex number:This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates:Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).is a holomorphic function whose restriction to the real numbers is real-valued, andis taken to be the standard topology) and antilinear, if one considersEven 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 extensionthat leave the real numbers fixed are the identity map and complex conjugation.is given, its conjugate is sufficient to reproduce the parts of theSimilarly, for a fixed complex unitas a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces.All these generalizations are multiplicative only if the factors are reversed:Since the multiplication of planar real algebras is commutative, this reversal is not needed there.There is also an abstract notion of conjugation for vector spacesthat satisfies is called a complex conjugation, or a real structure.has a real form obtained by taking the same vectors as in the original space and restricting the scalars to be real.The above properties actually define a real structure on the complex vector space[2] One example of this notion is the conjugate transpose operation of complex matrices defined above.