Factorial Number System - Definition

Definition

The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value to be multiplied by (i − 1)! (its place value).

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. The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in OEIS). Conversely, a further unchanging zero digit may be added in the rightmost position for the 0! place.

In this article, a factorial number representation will be flagged by a subscript "!", so for instance 341010_! stands for 3₆4₅1₄0₃1₂0₁, whose value is ((((3×5 + 4)×4 + 1)×3 + 0)×2 + 1)×1 + 0 = 463₁₀.

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 place values (1, 2, 3, ...), taking the remainder as digits, and continuing with the integer quotient, until this quotient becomes 0.

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.