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'factorial base representation of fractions'
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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 14:23:41 |
| 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*} |
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