Editing Factorial number system (section)

== Fractional values ==

Unlike single radix systems whose place values are ''base''<sup>''n''</sup> for both positive and negative integral ''n'', the factorial number base cannot be extended to negative place values as these would be (−1)!, (−2)! and so on, and these values are undefined (see [[factorial]]).

One possible extension is therefore to use 1/0!, 1/1!, 1/2!, 1/3!, ..., 1/''n''! etc. instead, possibly omitting the 1/0! and 1/1! places which are always zero.

With this method, all rational numbers have a terminating expansion, whose length in 'digits' is less than or equal to the denominator of the rational number represented. This may be proven by considering that there exists a factorial for any integer and therefore the denominator divides into its own factorial even if it does not divide into any smaller factorial.

By necessity, therefore, the factoradic expansion of the reciprocal of a [[prime number|prime]] has a length of exactly that prime (less one if the 1/1! place is omitted). Other terms are given as the sequence [https://oeis.org/A046021 A046021] on the OEIS. It can also be proven that the last 'digit' or term of the representation of a rational with prime denominator is equal to the difference between the numerator and the prime denominator.

Similar to how checking the divisibility of 4 in base 10 requires looking at only the last two digits, checking the divisibility of any number in factorial number system requires looking at only a finite number of digits. That is, it has a [[divisibility rule]] for each number.

There is also a non-terminating equivalent for every rational number akin to the fact that in decimal 0.24999... = 0.25 = 1/4 and [[0.999...|0.999... = 1]], etc., which can be created by reducing the final term by 1 and then filling in the remaining infinite number of terms with the highest value possible for the radix of that position.

In the following selection of examples, spaces are used to separate the place values, otherwise represented in decimal. The rational numbers on the left are also in decimal:

* <math>1/2 = 0.0\ 1_!</math>
* <math>1/3 = 0.0\ 0\ 2_!</math>
* <math>2/3 = 0.0\ 1\ 1_!</math>
* <math>1/4 = 0.0\ 0\ 1\ 2_!</math>
* <math>3/4 = 0.0\ 1\ 1\ 2_!</math>
* <math>1/5 = 0.0\ 0\ 1\ 0\ 4_!</math>
* <math>1/6 = 0.0\ 0\ 1_!</math>
* <math>5/6 = 0.0\ 1\ 2_!</math>
* <math>1/7 = 0.0\ 0\ 0\ 3\ 2\ 0\ 6_!</math>
* <math>1/8 = 0.0\ 0\ 0\ 3_!</math>
* <math>1/9 = 0.0\ 0\ 0\ 2\ 3\ 2_!</math>
* <math>1/10 = 0.0\ 0\ 0\ 2\ 2_!</math>
* <math>1/11 \ \ = 0.0\ 0\ 0\ 2\ 0\ 5\ 3\ 1\ 4\ 0\ A_!</math>
* <math>2/11 \ \ = 0.0\ 0\ 1\ 0\ 1\ 4\ 6\ 2\ 8\ 1\ 9_!</math>
* <math>9/11 \ \ = 0.0\ 1\ 1\ 3\ 3\ 1\ 0\ 5\ 0\ 8\ 2_!</math>
* <math>10/11 = 0.0\ 1\ 2\ 1\ 4\ 0\ 3\ 6\ 4\ 9 \ 1_!</math>
* <math>1/12 \ \ = 0.0\ 0\ 0\ 2_!</math>
* <math>5/12 \ \ = 0.0\ 0\ 2\ 2_!</math>
* <math>7/12 \ \ = 0.0\ 1\ 0\ 2_!</math>
* <math>11/12 = 0.0\ 1\ 2\ 2_!</math>
* <math>1/15 = 0.0\ 0\ 0\ 1\ 3_!</math>
* <math>1/16 = 0.0\ 0\ 0\ 1\ 2\ 3_!</math>
* <math>1/18 = 0.0\ 0\ 0\ 1\ 1\ 4_!</math>
* <math>1/20 = 0.0\ 0\ 0\ 1\ 1_!</math>
* <math>1/24 = 0.0\ 0\ 0\ 1_!</math>
* <math>1/30 = 0.0\ 0\ 0\ 0\ 4_!</math>
* <math>1/36 = 0.0\ 0\ 0\ 0\ 3\ 2_!</math>
* <math>1/60 = 0.0\ 0\ 0\ 0\ 2_!</math>
* <math>1/72 = 0.0\ 0\ 0\ 0\ 1\ 4_!</math>
* <math>1/120 = 0.0\ 0\ 0\ 0\ 1_!</math>
* <math>1/144 = 0.0\ 0\ 0\ 0\ 0\ 5_!</math>
* <math>1/240 = 0.0\ 0\ 0\ 0\ 0\ 3_!</math>
* <math>1/360 = 0.0\ 0\ 0\ 0\ 0\ 2_!</math>
* <math>1/720 = 0.0\ 0\ 0\ 0\ 0\ 1_!</math>

There are also a small number of constants that have patterned representations with this method:

* <math>e = 1\ 0.0\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1..._!</math>
* <math>e^{-1} = 0.0\ 0\ 2\ 0\ 4\ 0\ 6\ 0\ 8\ 0\ A\ 0\ C\ 0\ E..._!</math>
* <math>\sin(1) = 0.0\ 1\ 2\ 0\ 0\ 5\ 6\ 0\ 0\ 9\ A\ 0\ 0\ D\ E..._!</math>
* <math>\cos(1) = 0.0\ 1\ 0\ 0\ 4\ 5\ 0\ 0\ 8\ 9\ 0\ 0\ C\ D\ 0..._!</math>
* <math>\sinh(1) = 1.0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0..._!</math>
* <math>\cosh(1) = 1.0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1..._!</math>