Complex conjugate

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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, (3-2''i'')^* = 3 + 2''i'', ''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 properties apply 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'''. 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: :(''zw'')^* = ''w''^* ''z''^* Since the multiplication of complex numbers is commutative, this reversal is "invisible" there. de:Konjugation (Mathematik) fr:Conjugu� ja:共役複素数 pl:Liczba sprzężona