Convert repeating decimals to fractions in any base

My previous post ended with a discussion of repeating binary decimals such as

0.00110011…_two = 1/5.

For this post I’ll explain how calculations like that are done, how to convert a repeating decimal in any base to a fraction.

First of all, we only need to consider repeating decimals of the form

0.1_b, 0.01_b, 0.001_b, etc.

because every repeating decimal is an integer times an expression like one above. For example,

0.424242… = 42 × 0.010101…

You can think of that example as base 10, but it’s equally true in any base that has a 4, i.e. any base greater than 4.

Now suppose we have an expression

$0.\underbrace{000\ldots1}_{n \text{ terms}}\underbrace{000\ldots1}_{n \text{ terms}}\cdots$

in base b.

We can see that this expressions is

$\sum_{k=1}^\infty \left(b^{-n} \right )^k = \frac{1}{b^n - 1}$

by summing a geometric series.

So going back to our example above,

$0.010101\ldots_b = \frac{1}{b^2 -1}$

If we’re working in base 10, this equals 1/99. If we’re working in hexadecimal, this is 1/FF_hex = 1/255.

I’ll finish with a duodecimal example. Suppose we have

0.7AB7AB7AB…_twelve

and want to convert it to a fraction. We have

$0.7\text{AB}7\text{AB}\ldots_{\text{twelve}} = 7\text{AB}_{\text{twelve}} \times 0.001001\ldots_{\text{twelve}} = \frac{7\text{AB}_{\text{twelve}}}{BBB_{\text{twelve}}}$

Or 1139/1727 in base 10.