Tool to write / convert a number from factorial base/number system to base 10 and vice versa from decimal base to factorial base automatically.

Factorial Base - dCode

Tag(s) : Arithmetics

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⮞ Go to: Cantor Expansion

The **factorial base** (factoradic) is a writing of the numbers whose position of the figures is defined by the factorial of the position. The nth digit (from the right) of a number is written in base i, and its value in base 10 will be multiplied by $ (i - 1)! $

__Example:__ $ 3210_{(!)} = 3,2,1,0_{(!)} = 3_{(3)} 2_{(2)} 1_{(1)} 0_{(0)} = 23_{(10)} $

All the numbers in **factorial base** end with 0, thus, it is current to omit it on condition of specifying it clearly.

To write a number $ n $ in **factorial base**, noted $ n_{(!)} $ Or $ n_! $, perform the successive divisions of $ n $ by $ i $ ranging from 1 to infinity, as long as the result of the Euclidean division is not zero. The factorial notation consists of the remainders obtained in reverse order.

__Example:__ $ 1234_{(10)} = 1,4,1,1,2,0,0_{(!)}= 1411200_{(!)} $

n | i | calculation | remainder |
---|---|---|---|

1234 | 1 | 1234 ÷ 1 = 1234 | 0 |

1234 | 2 | 1234 ÷ 2 = 617 | 0 |

617 | 3 | 617 ÷ 3 = 205 | 2 |

205 | 4 | 205 ÷ 4 = 51 | 1 |

51 | 5 | 51 ÷ 5 = 10 | 1 |

10 | 6 | 10 ÷ 6 = 1 | 4 |

1 | 7 | 1 ÷ 7 = 0 | 1 |

Number conversion from $ n _ {(!)} $ (In factorial basis) to base/radix 10, carry out the successive multiplications of its digits (starting from the right) by $ i! $ ranging from 1 to infinity.

__Example:__ $ 1411200_{(!)} = 1 \times 6! + 4 \times 5! + 1 \times 4! + 1 \times 3! + 2 \times 2! + 0 \times 1! + 0 \times 0! = 1234 $

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