| Version 3 |
Version 2 |
| One can represent fractions as well as whole numbers using factorials |
One can represent fractions as well as whole numbers using factorials |
| much in the same way that one has, say, a decimal representation of |
much in the same way that one has, say, a decimal representation of |
| both whole numbers and fractions. |
both whole numbers and fractions. |
|
|
| Suppose that $x$ is a rational number. For simplicity, let us assume |
Suppose that $x$ is a rational number. For simplicity, let us assume |
| that $0 < x < 1$. Then we can write |
that $0 < x < 1$. Then we can write |
| \[x = \sum_{k=2}^N {d_k \over k!}\] |
\[x = \sum_{k=2}^N {d_k \over k!}\] |
| where $0 \le d_k < k$ for some integer $N$. Unlike decimal representations |
where $0 \le d_k < k$ for some integer $N$. Unlike decimal representations |
| of fractions and, more generally representations with any fixed base, |
of fractions and, more generally representations with any fixed base, |
| factorial base representations of rational numbers all terminate. |
factorial base representations of rational numbers all terminate. |
|
|
| Let us illustrate with some simple examples: |
Let us illustrate with some simple examples: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \frac{1}{2} &=& \frac{1}{2!} \\ |
\frac{1}{2} &=& \frac{1}{2!} \\ |
| \frac{1}{3} &=& \frac{2}{3!} \\ |
\frac{1}{3} &=& \frac{2}{3!} \\ |
| \frac{2}{3} &=& \frac{1}{2!} + \frac{1}{3!} \\ |
\frac{2}{3} &=& \frac{1}{2!} + \frac{1}{3!} \\ |
| \frac{1}{4} &=& \frac{1}{3!} + \frac{2}{4!} \\ |
\frac{1}{4} &=& \frac{1}{3!} + \frac{2}{4!} \\ |
| \frac{3}{4} &=& \frac{1}{2!} + \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{1}{5} &=& \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} \\ |
| \frac{2}{5} &=& \frac{2}{3!} + \frac{1}{4!} + \frac{3}{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{3}{5} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{3}{4!} + \frac{1}{5!} |
| \frac{4}{5} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{3}{4!) + |
|
| \frac{1}{5!} |
|
| \end{eqnarray*} |
\end{eqnarray*} |
