Complex conjugation: Difference between revisions
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In [[mathematics]], '''complex conjugation''' is an operation on [[complex number]]s which reverses the sign of the imaginary part, that is, it sends <math>z = x + iy</math> to the '''complex conjugate''' <math>\bar z = x-iy</math>. | In [[mathematics]], '''complex conjugation''' is an operation on [[complex number]]s which reverses the sign of the imaginary part, that is, it sends <math>z = x + iy</math> to the '''complex conjugate''' <math>\bar z = x-iy</math>. | ||
In the [[Complex number#Geometric interpretation|geometrical interpretation]] in terms of the [[Argand diagram]], complex conjugation is represented by [[reflection]] in the x-axis. The complex numbers left fixed by conjugation are precisely the [[real number]]s. | In the [[Complex number#Geometric interpretation|geometrical interpretation]] in terms of the [[Argand diagram]], complex conjugation is represented by [[reflection]] in the x-axis. The complex numbers left fixed by conjugation are precisely the [[real number]]s. | ||
Conjugation respects the algebraic operations of the complex numbers: <math>\overline{z+w} = \bar z + \bar w</math> and <math>\overline{zw} = \bar z \bar w</math>. Hence conjugation represents an [[automorphism]] of the field of complex numbers over the field of real numbers, and is the only non-trivial automorphism. | Conjugation respects the algebraic operations of the complex numbers: <math>\overline{z+w} = \bar z + \bar w</math> and <math>\overline{zw} = \bar z \bar w</math>. Hence conjugation represents an [[automorphism]] of the field of complex numbers over the field of real numbers, and is the only non-trivial automorphism. One can say it is impossible to tell which is ''i'' and which is -''i''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:01, 31 July 2024
In mathematics, complex conjugation is an operation on complex numbers which reverses the sign of the imaginary part, that is, it sends to the complex conjugate .
In the geometrical interpretation in terms of the Argand diagram, complex conjugation is represented by reflection in the x-axis. The complex numbers left fixed by conjugation are precisely the real numbers.
Conjugation respects the algebraic operations of the complex numbers: and . Hence conjugation represents an automorphism of the field of complex numbers over the field of real numbers, and is the only non-trivial automorphism. One can say it is impossible to tell which is i and which is -i.
