# complex number

The ring of complex numbers^{} $\u2102$ is defined to be the quotient ring^{} of the polynomial ring^{} $\mathbb{R}[X]$ in one variable over the reals by the principal ideal^{} $({X}^{2}+1)$. For $a,b\in \mathbb{R}$, the equivalence class^{} of $a+bX$ in $\u2102$ is usually denoted $a+bi$, and one has ${i}^{2}=-1$.

The complex numbers form an algebraically closed field. There is a standard metric on the complex numbers, defined by

$$d({a}_{1}+{b}_{1}i,{a}_{2}+{b}_{2}i):=\sqrt{{({a}_{2}-{a}_{1})}^{2}+{({b}_{2}-{b}_{1})}^{2}}.$$ |

Title | complex number |
---|---|

Canonical name | ComplexNumber |

Date of creation | 2013-03-22 11:52:35 |

Last modified on | 2013-03-22 11:52:35 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12D99 |

Classification | msc 30-00 |

Classification | msc 32-00 |

Classification | msc 46L05 |

Classification | msc 18B40 |

Classification | msc 46M20 |

Classification | msc 17B37 |

Classification | msc 22A22 |

Classification | msc 81R50 |

Classification | msc 22D25 |

Synonym | $\u2102$ |

Related topic | Complex |