Editing Factorial number system (section)

== Definition ==

The factorial number system is a [[mixed radix]] [[numeral system]]: the ''i''-th digit from the right has base&nbsp;''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 is to be multiplied by {{math|(''i''&nbsp;−&thinsp;1)}}! (its place value).

{| class="wikitable" style="text-align:right;"
|-
! {{rh}} | Radix/Base                     
|    8  ||   7  ||   6 ||  5  ||  4 ||  3 ||  2 || 1
|-
! {{rh}} | Place value               
|   7!  ||  6!  ||  5! ||  4! || 3! || 2! || 1! || 0!
|-
! {{rh}} | Place value in decimal   
| 5040 || 720 || 120 || 24 || 6 ||  2 ||  1 || 1
|-
! {{rh}} | 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 {{OEIS|id=A124252}}. The factorial number system is sometimes defined with the 0! place omitted because it is always zero {{OEIS|id=A007623}}.

In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0<sub>!</sub> stands for

: {{=}} 3×5!&nbsp;+&nbsp;4×4!&nbsp;+&nbsp;1×3!&nbsp;+&nbsp;0×2!&nbsp;+&nbsp;1×1!&nbsp;+&nbsp;0×0!&nbsp;
: {{=}} ((((3×5&nbsp;+&nbsp;4)×4&nbsp;+&nbsp;1)×3&nbsp;+&nbsp;0)×2&nbsp;+&nbsp;1)×1&nbsp;+&nbsp;0
: {{=}} &nbsp;463<sub>10</sub>.

(The place value is the factorial of one less than the radix position, which is why the equation begins with 5! for a 6-digit factoradic number.)

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 integer [[quotient]], until this quotient becomes 0.

For example, 463<sub>10</sub> can be transformed into a factorial representation by these successive divisions:
{| style="text-align:right;"
|
: 463 ÷ 1 = 463, remainder 0
: 463 ÷ 2 = 231, remainder 1
: 231 ÷ 3 = 77, remainder 0
: 77 ÷ 4 = 19, remainder 1
: 19 ÷ 5 = 3, remainder 4
: 3 ÷ 6 = 0, remainder 3
|}
The process terminates when the quotient reaches zero. Reading the remainders backward gives 3:4:1:0:1:0<sub>!</sub>.

In principle, this system may be extended to represent [[rational number]]s, 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.