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In formal, does there exist such that for all ?

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  • This better goes to http://math.stackexchange.com/ – a_guest Oct 15 '15 at 12:50
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    I'm voting to close this question as off-topic because this is a purely mathematical question (structure of real numbers) and thus belongs to math.se – Lutz Lehmann Oct 15 '15 at 12:51
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    I'm voting to close this question as off-topic because it is about mathematics, not programming or software development. – Pang Oct 16 '15 at 01:23

1 Answers1

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No, $\pi$ and thus $\pi/2$ are irrational, thus the (additive) equivalence classes of the integers modulo $2\pi$ are dense in $\Bbb R$ and thus approach infinitesimally, but never reach $\pi/2$.

The fundamental fact is that for any given number x the set of numbers {mx+n : m,n integer} is either

  • an arithmetic sequence {mr : r integer} which implies and is equivalent to x as a multiple of r being rational, or

  • dense in the real numbers, which by the first case happens for all irrational x.

Lutz Lehmann
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