# Champernowne’s constant

For a given base $b$, Champernowne’s constant ${C}_{b}$ is the result of concatenating the base $b$ digits of the positive integers in order after 0 and a decimal point, that is,

$$\sum _{i=1}^{\mathrm{\infty}}\frac{i}{{b}^{{\sum}_{j=1}^{i}k}}$$ |

(where $k$ is the number of digits of $j$ in base $b$).

Kurt Mahler proved that ${C}_{10}$ (approximately 0.123456789101112131415161718192021…) is a transcendental number^{}. Champernowne had earlier proved that ${C}_{10}$ is a normal number^{}.

Title | Champernowne’s constant |
---|---|

Canonical name | ChampernownesConstant |

Date of creation | 2013-03-22 17:04:09 |

Last modified on | 2013-03-22 17:04:09 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | Champernowne constant^{} |