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Pigeonhole Principle & irrational numbers

  1. Jan 4, 2009 #1
    1. The problem statement, all variables and given/known data

    Let x be an irrational number. Show that the absolute value of the difference between jx and the nearest integer to jx is less than 1/n for some positive integer j not exceeding n.

    2. Relevant equations



    3. The attempt at a solution

    Ok, I know that it should be solved using pigeonhole Principle

    and there is the fact that

    for real numbers: 0 <= | jx - [jx] | < 1

    specifically for irrational numbers: 0 < | jx - [jx] | < 1

    that should make the difference but i cant exactly come up with the correct intervals that form the final conclusion.
     
    Last edited: Jan 4, 2009
  2. jcsd
  3. Jan 4, 2009 #2

    HallsofIvy

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    I think you are looking at it backwards. It is "n" that is 'given', not j. Given a positive integer, n, divide the interval from 0 to 1 into n equal sized (1/n of course) intervals.
     
  4. Jan 4, 2009 #3
    yes, but considering that 1 <= j <= n we have n different js.

    that leaves us with n pigeons (js) and n pigeonholes (intervals). :biggrin: and no useful conclusion.

    and why only irrational numbers?!
     
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