Search for a tool
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.

Results

Factorial Base -

Tag(s) : Arithmetics

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developed the best 'Factorial Base' tool, so feel free to write! Thank you!

# Factorial Base

## Cantor Expansion

⮞ Go to: Cantor Expansion

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

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

## Source code

dCode retains ownership of the online 'Factorial Base' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Factorial Base download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!