Difference between revisions of "Distance to integers"

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(Created page with "Define the function $\mathrm{dist}_{\mathbb{Z}} \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$ where $\inf$ deno...")
 
 
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$$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$
 
$$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$
 
where $\inf$ denotes the [[infimum]]. This function can be computed using the [[floor]] and [[ceiling]] functions:
 
where $\inf$ denotes the [[infimum]]. This function can be computed using the [[floor]] and [[ceiling]] functions:
$$\mathrm{dist}_{\mathbb{Z}}(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right).$$
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$$\mathrm{dist}_{\mathbb{Z}}(x)=\min \left( x - \lfloor x \rfloor, \lceil x \rceil - x \right).$$
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[[Category:SpecialFunction]]

Latest revision as of 03:12, 6 July 2016

Define the function $\mathrm{dist}_{\mathbb{Z}} \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$ where $\inf$ denotes the infimum. This function can be computed using the floor and ceiling functions: $$\mathrm{dist}_{\mathbb{Z}}(x)=\min \left( x - \lfloor x \rfloor, \lceil x \rceil - x \right).$$