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Area Mathematics - General concepts and linear algebra / Numbers

IEV ref102-02-09

en
complex number
element of a set containing the real numbers and other elements, which may be represented by an ordered pair of real numbers (a, b), with following properties:

  • the pair (a, 0) represents the real number a,
  • an addition is defined by ( a 1 , b 1 )+( a 2 , b 2 )=( a 1 + a 2 , b 1 + b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWGHbWaaS baaSqaaiaaigdaaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiabgUcaRiaacIcacaWGHbWaaSbaaSqaaiaaik daaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiabg2da9iaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaabYcacaaMe8UaamOy amaaBaaaleaacaaIXaaabeaakiabgUcaRiaadkgadaWgaaWcbaGaaG OmaaqabaGccaGGPaaaaa@55F8@ ,
  • a multiplication is defined by ( a 1 , b 1 )×( a 2 , b 2 )=( a 1 a 2 b 1 b 2 , a 1 b 2 + a 2 b 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWGHbWaaS baaSqaaiaaigdaaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiabgEna0kaacIcacaWGHbWaaSbaaSqaaiaaik daaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiabg2da9iaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaam yyamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadkgadaWgaaWcbaGa aGymaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaeilaiaays W7caWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamOyamaaBaaaleaacaaI YaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaWGIb WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@5E98@

Note 1 to entry: All properties of real numbers (operations and limits) are extended to complex numbers except the order relation.

Note 2 to entry: The complex number defined by the pair (a, b) is denoted by c=a+jb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadogacqGH9aqpca WGHbGaey4kaSIaaiOAaiaadkgaaaa@3E58@ where j is the imaginary unit (IEV 102-02-10) represented by the pair (0, 1), a is the real part and b the imaginary part. A complex number may also be expressed as c=|c|(cosφ+jsinφ)=|c| e jφ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadogacqGH9aqpda abdaqaaiaadogaaiaawEa7caGLiWoacaGGOaGaci4yaiaac+gacaGG ZbGaeqy1dyMaey4kaSIaaiOAaiaaysW7caaMc8Uaci4CaiaacMgaca GGUbGaeqy1dyMaaiykaiabg2da9maaemaabaGaam4yaaGaay5bSlaa wIa7aiaacwgadaahaaWcbeqaaiaacQgacqaHvpGzaaaaaa@571D@ where |c| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaam4yaa Gaay5bSlaawIa7aaaa@3CD7@ is a non-negative real number called modulus and φ a real number called argument.

Note 3 to entry: In electrotechnology, a complex number is usually denoted by an underlined letter symbol, for example c _ .

Note 4 to entry: The set of complex numbers is denoted by ℂ (C with a vertical bar in the left arc) or C. This set without zero is denoted by an asterisk to the symbol, for example ℂ*.


fr
nombre complexe, m
élément d'un ensemble contenant les nombres réels et d'autres éléments, dont chacun peut être représenté par un couple ordonné de nombres réels (a, b), avec les propriétés suivantes:

  • le couple (a, 0) représente le nombre réel a,
  • une addition est définie par ( a 1 , b 1 )+( a 2 , b 2 )=( a 1 + a 2 , b 1 + b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWGHbWaaS baaSqaaiaaigdaaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiabgUcaRiaacIcacaWGHbWaaSbaaSqaaiaaik daaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiabg2da9iaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaabYcacaaMe8UaamOy amaaBaaaleaacaaIXaaabeaakiabgUcaRiaadkgadaWgaaWcbaGaaG OmaaqabaGccaGGPaaaaa@55F8@ ,
  • une multiplication est définie par ( a 1 , b 1 )×( a 2 , b 2 )=( a 1 a 2 b 1 b 2 , a 1 b 2 + a 2 b 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWGHbWaaS baaSqaaiaaigdaaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiabgEna0kaacIcacaWGHbWaaSbaaSqaaiaaik daaeqaaOGaaeilaiaaysW7caWGIbWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiabg2da9iaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaam yyamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadkgadaWgaaWcbaGa aGymaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaeilaiaays W7caWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamOyamaaBaaaleaacaaI YaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaWGIb WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@5E98@

Note 1 à l'article: Toutes les propriétés des nombres réels (opérations et limites) s'étendent aux nombres complexes, sauf la relation d'ordre.

Note 2 à l'article: Le nombre complexe défini par le couple (a, b) est noté c=a+jb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadogacqGH9aqpca WGHbGaey4kaSIaaiOAaiaadkgaaaa@3E58@ où j est l'unité imaginaire (IEV 102-02-10) représentée par le couple (0, 1), a est la partie réelle et b la partie imaginaire. Un nombre complexe peut aussi être représenté par c=|c|(cosφ+jsinφ)=|c| e jφ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadogacqGH9aqpda abdaqaaiaadogaaiaawEa7caGLiWoacaGGOaGaci4yaiaac+gacaGG ZbGaeqy1dyMaey4kaSIaaiOAaiaaysW7caaMc8Uaci4CaiaacMgaca GGUbGaeqy1dyMaaiykaiabg2da9maaemaabaGaam4yaaGaay5bSlaa wIa7aiaacwgadaahaaWcbeqaaiaacQgacqaHvpGzaaaaaa@571D@ |c| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaam4yaa Gaay5bSlaawIa7aaaa@3CD7@ est un nombre réel positif ou nul appelé module et φ un nombre réel appelé argument.

Note 3 à l'article: En électrotechnique, un nombre complexe est généralement représenté par un symbole littéral souligné, par exemple c _ .

Note 4 à l'article: L'ensemble des nombres complexes est noté ℂ (C avec une barre verticale dans l'arc gauche) ou C. L'ensemble sans zéro est noté en ajoutant un astérisque au symbole, par exemple ℂ*.


de
komplexe Zahl, f

es
número complejo

ja
複素数

pl
liczba zespolona

pt
número complexo

sr
комплексни број, м јд

sv
komplext tal

zh
复数

Publication date: 2008-08
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