Factorial number system
The factorial number system is a
|Place value in decimal||5040||720||120||24||6||2||1||1|
|Highest digit allowed||7||6||5||4||3||2||1||0|
From this it follows that the rightmost digit is always 0, the second can be 0 or 1, the third 0, 1 or 2, and so on (sequence A124252 in the
In this article, a factorial number representation will be flagged by a subscript "!", so for instance 341010! stands for 354413021100, whose value is
(Note that the place value is one less than the radix position, which is why these equations begin with 5!.)
General properties of mixed radix number systems also apply to the factorial number system. For instance, one can convert a number into factorial representation producing digits from right to left, by repeatedly dividing the number by the radix (1, 2, 3, ...), taking the remainder as digits, and continuing with the
For example, 46310 can be transformed into a factorial representation by these successive divisions:
The process terminates when the quotient reaches zero. Reading the remainders backward gives 341010!.
In principle, this system may be extended to represent fractional numbers, though rather than the natural extension of place values (−1)!, (−2)!, etc., which are undefined, the symmetric choice of radix values n = 0, 1, 2, 3, 4, etc. after the point may be used instead. Again, the 0 and 1 places may be omitted as these are always zero. The corresponding place values are therefore 1/1, 1/1, 1/2, 1/6, 1/24, ..., 1/n!, etc.