factorial base representation of fractions

One can represent fractions as well as whole numbers using factorialsmuch in the same way that one has, say, a decimal representation ofboth whole numbers and fractions.

Suppose that $x$ is a rational number. For simplicity, let us assumethat . Then we can write

$$x=\sum _{k=2}^N\frac{d_k}{k!}$$

where for some integer $N$. Unlike decimal representationsof 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 \frac{1}{2!}}$

${\displaystyle \frac{1}{3}}$$=$${\displaystyle \frac{2}{3!}}$

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

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

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

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

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

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

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

We can also employ a more concise notation as is used inrepresenting fractions in other bases and simply listdigits after a point. Since we would need an infinitesupply of digits, we make the same compromise as whenwriting factorial base representations of integers.With this convention, we than have the following tableof factorial base representations of fractions.