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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

## Cantor Expansion

⮞ Go to: Cantor Expansion

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

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