PlanetMath: factorial base representation of fractions

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factorial base representation of fractions

(Definition)

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

$\displaystyle x = \sum_{k=2}^N {d_k \over k!}$

where $0 \le d_k < k$ for some integer

. 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:

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

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

"factorial base representation of fractions" is owned by rspuzio.

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Attachments:: expressing fractions in factorial base (Algorithm) by rspuzio

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Cross-references: infinite, point, digits, bases, simple, rational numbers, factorial base, base, fixed, integer, rational number, representation, factorials, numbers, fractions, represent
There are 2 references to this entry.

This is version 10 of factorial base representation of fractions, born on 2007-02-26, modified 2007-02-26.
Object id is 8994, canonical name is FactorialBaseRepresentationOfFractions.
Accessed 889 times total.

Classification:

AMS MSC:

11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)

Pending Errata and Addenda

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