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 $\frac{5}{8}=0.625$.