# Imaginary number

An **imaginary number** is a complex number that can be written as a real number multiplied by the imaginary unit i,[note 1] which is defined by its property *i*^{2} = −1.[1][2] The square of an imaginary number bi is −*b*^{2}. For example, 5*i* is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.[3]

... Exponents repeat the pattern from blue area |

i^{−3} = i |

i^{−2} = −1 |

i^{−1} = −i |

i^{0} = 1 |

i^{1} = i |

i^{2} = −1 |

i^{3} = −i |

i^{4} = 1 |

i^{5} = i |

i^{6} = −1 |

i^{n} = i^{m } where m ≡ n mod 4 |

Originally coined in the 17th century by René Descartes[4] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).

An imaginary number *bi* can be added to a real number a to form a complex number of the form *a* + *bi*, where the real numbers a and b are called, respectively, the *real part* and the *imaginary part* of the complex number.[5][note 2]