(Untitled) (Untitled)(Untitled) (Untitled)(Untitled) (Untitled)(Untitled) (Untitled)(Untitled)Examples




IEVref:102-02-09ID:
Language:enStatus: Standard
Term: complex number
Synonym1:
Synonym2:
Synonym3:
Symbol:
Definition: 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*.


Publication date:2008-08
Source:
Replaces:
Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO
CO remarks:
TC/SC remarks:
VT remarks:
Domain1:
Domain2:
Domain3:
Domain4:
Domain5: