factorial base representation of fractions

One can represent fractions as well as whole numbers using factorials much in the same way that one has, say, a decimal representation of both whole numbers and fractions.

Suppose that $x$ is a rational number. For simplicity, let us assume that . Then we can write

$$x=\sum _{k=2}^N\frac{d_k}{k!}$$

where for some integer $N$. Unlike decimal representations of fractions and, more generally representations with any fixed base, factorial base representations of rational numbers all terminate.

Let us illustrate with some simple examples:

${\displaystyle \frac{1}{2}}$ $=$ ${\displaystyle \frac{1}{2!}}$

${\displaystyle \frac{1}{3}}$ $=$ ${\displaystyle \frac{2}{3!}}$

${\displaystyle \frac{2}{3}}$ $=$ ${\displaystyle \frac{1}{2!}}+{\displaystyle \frac{1}{3!}}$

${\displaystyle \frac{1}{4}}$ $=$ ${\displaystyle \frac{1}{3!}}+{\displaystyle \frac{2}{4!}}$

${\displaystyle \frac{3}{4}}$ $=$ ${\displaystyle \frac{1}{2!}}+{\displaystyle \frac{1}{3!}}+{\displaystyle \frac{2}{4!}}$

${\displaystyle \frac{1}{5}}$ $=$ ${\displaystyle \frac{1}{3!}}+{\displaystyle \frac{1}{4!}}+{\displaystyle \frac{1}{5!}}$

${\displaystyle \frac{2}{5}}$ $=$ ${\displaystyle \frac{2}{3!}}+{\displaystyle \frac{1}{4!}}+{\displaystyle \frac{3}{5!}}$

${\displaystyle \frac{3}{5}}$ $=$ ${\displaystyle \frac{1}{2!}}+{\displaystyle \frac{2}{4!}}+{\displaystyle \frac{2}{5!}}$

${\displaystyle \frac{4}{5}}$ $=$ ${\displaystyle \frac{1}{2!}}+{\displaystyle \frac{1}{3!}}+{\displaystyle \frac{3}{4!}}+{\displaystyle \frac{1}{5!}}$

We can also employ a more concise notation as is used in representing fractions in other bases and simply list digits after a point. Since we would need an infinite supply of digits, we make the same compromise as when writing factorial base representations of integers. With this convention, we than have the following table of factorial base representations of fractions.

1/2	0 . 1
1/3	0 . 0 2
2/3	0 . 1 1
1/4	0 . 0 1 2
3/4	0 . 1 1 2
1/5	0 . 0 1 0 4
2/5	0 . 0 2 1 3
3/5	0 . 1 0 2 2
4/5	0 . 1 1 3 1
1/6	0 . 0 1
5/6	0 . 1 2
1/7	0 . 0 0 3 2 0 6
2/7	0 . 0 1 2 4 1 5
3/7	0 . 0 2 2 1 2 4
4/7	0 . 1 0 1 3 3 3
5/7	0 . 1 1 1 0 4 2
6/7	0 . 1 2 0 2 5 1
1/8	0 . 0 0 3
3/8	0 . 0 2 1
5/8	0 . 1 0 3
7/8	0 . 1 2 1
1/9	0 . 0 0 2 3 2
2/9	0 . 0 1 1 1 4
4/9	0 . 0 2 2 3 2
5/9	0 . 1 0 1 1 4
7/9	0 . 1 1 2 3 2
8/9	0 . 1 2 1 1 4
1/10	0 . 0 0 2 2
3/10	0 . 0 1 3 1
7/10	0 . 1 1 0 4
9/10	0 . 1 2 1 3
1/11	0 . 0 0 2 0 5 3 1 4 0 10
2/11	0 . 0 1 0 1 4 6 2 8 1 9
3/11	0 . 0 1 2 2 4 2 4 3 2 8
4/11	0 . 0 2 0 3 3 5 5 7 3 7
5/11	0 . 0 2 2 4 3 1 7 2 4 6
6/11	0 . 1 0 1 0 2 5 0 6 5 5
7/11	0 . 1 0 3 1 2 1 2 1 6 4
8/11	0 . 1 1 1 2 1 4 3 5 7 3
9/11	0 . 1 1 3 3 1 0 5 0 8 2
10/11	0 . 1 2 1 4 0 3 6 4 9 1
1/12	0 . 0 0 2
5/12	0 . 0 2 2
7/12	0 . 1 0 2
11/12	0 . 1 2 2

Title	factorial base representation of fractions
Canonical name	FactorialBaseRepresentationOfFractions
Date of creation	2013-03-22 16:46:01
Last modified on	2013-03-22 16:46:01
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	13
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 11A63