Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can I choose an $x\in[a,b)\subseteq[0,1)$, where $a,b\in\mathbb{Q}$, such that $x$ has a non-repeating fractional part in some chosen base?

For example, say I'm looking at $[\frac{1}{2},\frac{3}{4})$ and I'm working in base 10. I could pick $x=\frac{2}{3}$, but that has a repeating fractional part in base 10, so I'd choose $x=\frac{1}{2}$. Is there an algorithmic method that works in general? (If it helps, the problem I'm trying to solve is in base 2.)

As a bonus question, is there a way of choosing $x$ such that its fractional part has a minimal length? So, following the above example, $\frac{1}{2}=0.5$ would be "better" than $\frac{5}{8}=0.625$.

share|cite|improve this question
up vote 4 down vote accepted

Consider numbers of the form $x = \frac{p}{q^n}$ where $q$ is your base. Surely with $n$ big enough you can pick $p$ to be inside $[aq^n,bq^n)$.

If you want an algorithmic approach, just multiply $a$ and $b$ by $q^n$ with increasing $n$ until you will find an integer between them. In base two it is even simpler, just take the binary representation of $a$ and $b$ and find the first place where they differ, and then find the first number $x_1 \geq a$ or $x_2 < b$.

a  = 0100011011010010111111010110110(110)...
b  = 0100011011010011001100110011001(1001)...
                    ^ first difference
x1 = 01000110110100101111111
                           ^ difference from a
x2 = 0100011011010011000
                       ^  difference from b
                         (of course remove the trailing zeros)


share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.