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Is there (for each fixed base system with digits $0,1,\dots,m$) and then for each real number $r\in\mathbb R$, an integer $n\in\mathbb Z$ and a sequence $(a_i)_{i\in\mathbb{N}}$ with $a_i\in\{0,1,\dots,m\}$, such that

$$r=n\cdot 0.a_1a_2a_3\dots$$

and $|n|$ is minimal among all such representations? How to compute it (up to arbitrary precision)?

Intuitively I'd say the answer is yes (apart from maybe pathologies involving signs, zero or one, which I forget) but I'm not acquainted with methods to properly show this. The question came up when thinking about definable and computable real numbers and what it really means to work with non-compact manifolds on a computer or even on paper.

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Yes. If $r=0$ let $n=0$; then you can take any digit sequence $(a_i)_{i\in\mathbb N}$. Otherwise, if $r\ne 0$ let $k=\bigl\lceil \strut|r|\bigr\rceil$ and the $n=\operatorname{sgn} (r) k$. Then the number $\frac{r}{n}$ is in $(0,1]$ and can be represented as requested.

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  • $\begingroup$ What is $||r||$? $\endgroup$ – TonyK Feb 5 '14 at 18:07
  • $\begingroup$ Why this $n$ is of minimal absolute value? $\endgroup$ – Bach Feb 5 '14 at 18:07

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