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- Complex_conjugate abstract "In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, 3 + 4i and 3 − 4i are complex conjugates.The conjugate of the complex number ,where and are real numbers, isFor example, An alternative notation for the complex conjugate is . However, the notation avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of complex conjugation. The star-notation is preferred in physics, where dagger is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical.Complex numbers are considered points in the complex plane, a variation of the Cartesian coordinate system where both axes are real number lines that cross at the origin, however, the y-axis is a product of real numbers multiplied by . On the illustration, the x-axis is called the real axis, labeled Re, while the y-axis is called the imaginary axis, labeled Im. The plane defined by the Re and Im axes represents the space of all possible complex numbers. In this view, complex conjugation corresponds to reflection of a complex number at the x-axis, equivalent to a 180 degree rotation of the complex plane about the Re axis.In polar form, the conjugate of is . This can be shown using Euler's formula.Pairs of complex conjugates are significant because the imaginary unit is qualitatively indistinct from its additive and multiplicative inverse , as they both satisfy the definition for the imaginary unit: . 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.In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". For example, writing means".
- Complex_conjugate thumbnail Complex_conjugate_picture.svg?width=300.
- Complex_conjugate wikiPageID "173918".
- Complex_conjugate wikiPageRevisionID "596790254".
- Complex_conjugate hasPhotoCollection Complex_conjugate.
- Complex_conjugate subject Category:Complex_numbers.
- Complex_conjugate comment "In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, 3 + 4i and 3 − 4i are complex conjugates.The conjugate of the complex number ,where and are real numbers, isFor example, An alternative notation for the complex conjugate is .".
- Complex_conjugate label "Complesso coniugato".
- Complex_conjugate label "Complex conjugate".
- Complex_conjugate label "Complex geconjugeerde".
- Complex_conjugate label "Conjugado (matemática)".
- Complex_conjugate label "Conjugado de um número complexo".
- Complex_conjugate label "Conjugué".
- Complex_conjugate label "Konjugation (Mathematik)".
- Complex_conjugate label "Sprzężenie zespolone".
- Complex_conjugate label "مرافق عدد مركب".
- Complex_conjugate label "共轭复数".
- Complex_conjugate label "複素共役".
- Complex_conjugate sameAs Komplexně_sdružené_číslo.
- Complex_conjugate sameAs Konjugation_(Mathematik).
- Complex_conjugate sameAs Conjugado_(matemática).
- Complex_conjugate sameAs Konjugatu_(matematika).
- Complex_conjugate sameAs Conjugué.
- Complex_conjugate sameAs Complesso_coniugato.
- Complex_conjugate sameAs 複素共役.
- Complex_conjugate sameAs 복소켤레.
- Complex_conjugate sameAs Complex_geconjugeerde.
- Complex_conjugate sameAs Sprzężenie_zespolone.
- Complex_conjugate sameAs Conjugado_de_um_número_complexo.
- Complex_conjugate sameAs m.017ggt.
- Complex_conjugate sameAs Q381040.
- Complex_conjugate sameAs Q381040.
- Complex_conjugate wasDerivedFrom Complex_conjugate?oldid=596790254.
- Complex_conjugate depiction Complex_conjugate_picture.svg.
- Complex_conjugate isPrimaryTopicOf Complex_conjugate.
