# digital root

Given an integer $m$ consisting of $k$ digits ${d}_{1},\mathrm{\dots},{d}_{k}$ in base $b$, let

$$j=\sum _{i=1}^{k}{d}_{i},$$ |

then repeat this operation^{} on the digits of $j$ until $$. This stores in $j$ the digital root of $m$. The number of iterations of the sum operation is called the additive persistence of $m$.

The digital root of ${b}^{x}$ is always 1 for any natural $x$, while the digital root of $y{b}^{n}$ (where $y$ is another natural number^{}) is the same as the digital root of $y$. This should not be taken to imply that the digital root is necessarily a multiplicative function^{}.

The digital root of an integer of the form $n(b-1)$ is always $b-1$.

Another way to calculate the digital root for $m>b$ is with the formula^{} $m-(b-1)\lfloor \frac{m-1}{b-1}\rfloor $.

Title | digital root |
---|---|

Canonical name | DigitalRoot |

Date of creation | 2013-03-22 15:59:34 |

Last modified on | 2013-03-22 15:59:34 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 13 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | repeated digit sum |

Synonym | repeated digital sum |

Defines | additive persistence |