| Version 8 |
Version 7 |
| 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{2}{4!} + \frac{2}{5!} \\ |
| \frac{4}{5} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{3}{4!} + |
\frac{4}{5} &=& \frac{1}{2!} + \frac{1}{3!} + \frac{3}{4!} + |
| \frac{1}{5!} |
\frac{1}{5!} |
| \end{eqnarray*} |
\end{eqnarray*} |
|
|
| We can also employ a more concise notation as is used in |
We can also employ a more concise notation as is used in |
| representing fractions in other bases and simply list |
representing fractions in other bases and simply list |
| digits after a point. Since we would need an infinite |
digits after a point. Since we would need an infinite |
|
supply of digits, we make the same compromise as when
|
supply of digits, we make the same compropmise as when
|
| writing factorial base representations of integers. |
writing factorial base representations of integers. |
| With this convention, we than have the following table |
With this convention, we than have the following table |
| of factorial base representations of fractions. |
of factorial base representations of fractions. |
|
|
| \begin{tabular} |
\begin{tabular} |
| {| c | c |} |
{| c | c |} |
| 1/2 & 0 . 1 \\ |
1/2 & 0 . 1 \\ |
| 1/3 & 0 . 0 2 \\ |
1/3 & 0 . 0 2 \\ |
| 2/3 & 0 . 1 1 \\ |
2/3 & 0 . 1 1 \\ |
| 1/4 & 0 . 0 1 2 \\ |
1/4 & 0 . 0 1 2 \\ |
| 3/4 & 0 . 1 1 2 \\ |
3/4 & 0 . 1 1 2 \\ |
| 1/5 & 0 . 0 1 0 4 \\ |
1/5 & 0 . 0 1 0 4 \\ |
| 2/5 & 0 . 0 2 1 3 \\ |
2/5 & 0 . 0 2 1 3 \\ |
| 3/5 & 0 . 1 0 2 2 \\ |
3/5 & 0 . 1 0 2 2 \\ |
| 4/5 & 0 . 1 1 3 1 \\ |
4/5 & 0 . 1 1 3 1 \\ |
| 1/6 & 0 . 0 1 \\ |
1/6 & 0 . 0 1 \\ |
| 5/6 & 0 . 1 2 \\ |
5/6 & 0 . 1 2 \\ |
| 1/7 & 0 . 0 0 3 2 0 6 \\ |
1/7 & 0 . 0 0 3 2 0 6 \\ |
| 2/7 & 0 . 0 1 2 4 1 5 \\ |
2/7 & 0 . 0 1 2 4 1 5 \\ |
| 3/7 & 0 . 0 2 2 1 2 4 \\ |
3/7 & 0 . 0 2 2 1 2 4 \\ |
| 4/7 & 0 . 1 0 1 3 3 3 \\ |
4/7 & 0 . 1 0 1 3 3 3 \\ |
| 5/7 & 0 . 1 1 1 0 4 2 \\ |
5/7 & 0 . 1 1 1 0 4 2 \\ |
| 6/7 & 0 . 1 2 0 2 5 1 \\ |
6/7 & 0 . 1 2 0 2 5 1 \\ |
| 1/8 & 0 . 0 0 3 \\ |
1/8 & 0 . 0 0 3 \\ |
| 3/8 & 0 . 0 2 1 \\ |
3/8 & 0 . 0 2 1 \\ |
| 5/8 & 0 . 1 0 3 \\ |
5/8 & 0 . 1 0 3 \\ |
| 7/8 & 0 . 1 2 1 \\ |
7/8 & 0 . 1 2 1 \\ |
| 1/9 & 0 . 0 0 2 3 2 \\ |
1/9 & 0 . 0 0 2 3 2 \\ |
| 2/9 & 0 . 0 1 1 1 4 \\ |
2/9 & 0 . 0 1 1 1 4 \\ |
| 4/9 & 0 . 0 2 2 3 2 \\ |
4/9 & 0 . 0 2 2 3 2 \\ |
| 5/9 & 0 . 1 0 1 1 4 \\ |
5/9 & 0 . 1 0 1 1 4 \\ |
| 7/9 & 0 . 1 1 2 3 2 \\ |
7/9 & 0 . 1 1 2 3 2 \\ |
| 8/9 & 0 . 1 2 1 1 4 \\ |
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 |
|
| \end{tabular} |
\end{tabular} |
