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Factorial Base

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

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Factorial Base -

Tag(s) : Arithmetics

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Factorial Base

Factorial Base to Decimal Converter



Decimal to Factorial base Converter



Cantor Expansion

⮞ Go to: Cantor Expansion

Answers to Questions (FAQ)

What is factorial base? (Definition)

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.

How to write/convert a number to factorial base?

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_{(!)} $

nicalculationremainder
123411234 ÷ 1 = 12340
123421234 ÷ 2 = 6170
6173617 ÷ 3 = 2052
2054205 ÷ 4 = 511
51551 ÷ 5 = 101
10610 ÷ 6 = 14
171 ÷ 7 = 01

How to convert a number written in factorial base?

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 $

How to convert a negative number or a decimal number?

The factorial operator is only defined for positive integers, the factorial of any other number is not defined, therefore cannot be calculated in factorial basis.However, it is possible to extend to decimal numbers using (1/1!), 1/2!, 1/3!, …, 1/n! as a place value after the decimal point.

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Factorial Base on dCode.fr [online website], retrieved on 2024-05-27, https://www.dcode.fr/factorial-base

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