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| Title of object: |
factorial base representation of fractions |
| Canonical Name: |
FactorialBaseRepresentationOfFractions |
| Type: |
Definition |
| Created on: |
2007-02-26 10:58:14 |
| Modified on: |
2007-02-26 15:00:52 |
| Classification: |
msc:11A63 |
Preamble:
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Content:
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 $0 < x < 1$. Then we can write
\[x = \sum_{k=2}^N {d_k \over k!}\]
where $0 \le d_k < k$ 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:
\begin{eqnarray*}
\frac{1}{2} &=& \frac{1}{2!} \\
\frac{1}{3} &=& \frac{2}{3!} \\
\frac{2}{3} &=& \frac{1}{2!} + \frac{1}{3!} \\
\frac{1}{4} &=& \frac{1}{3!} + \frac{2}{4!} \\
\frac{3}{4} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{2}{4!} \\
\frac{1}{5} &=& \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} \\
\frac{2}{5} &=& \frac{2}{3!} + \frac{1}{4!} + \frac{3}{5!} \\
\frac{3}{5} &=& \frac{1}{2!} + \frac{2}{4!} + \frac{2}{5!} \\
\frac{4}{5} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{3}{4!} +
\frac{1}{5!}
\end{eqnarray*}
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 compropmise as when
writing factorial base representations of integers.
With this convention, we than have the following table
of factorial base representations of fractions.
\begin{tabular}
{| c | c |}
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
\end{tabular} |
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