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Problem: Multiples of pi

  1. Sep 16, 2012 #1
    Hi everyone,

    About 15 minutes ago I came up with a problem... What whole number multiple of pi would result in a number closest to a whole number?

    Does a single whole number multiple exist, and can we... prove it?


    Thanks for help in advance!


    -Daniel
     
  2. jcsd
  3. Sep 16, 2012 #2

    micromass

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    There is no solution to that problem. The thing is that we can get [itex]n\pi[/itex] as close to an integer as we like. This is basically Kroneckers density theorem. Of course, a nonzero multiple of [itex]\pi[/itex] can never actually equal an integer (since that would imply that [itex]\pi[/itex] is rational), but it can be arbitrary close.

    The number [itex]\pi[/itex] is not special here, it works for any irrational number.
     
  4. Sep 16, 2012 #3
    Thank you for the reply, micromass! I was not familiar with Kronecker's density theorem, but its logic clarifies this problem. I wonder if there's any pattern in what integers [itex]n[/itex] would bring us closer to a whole number...

    Ah, well there goes my bedtime tonight! Thanks for the direction :D
     
  5. Sep 25, 2012 #4
    That is a much more interesting problem (to me). For certain types of irrational numbers, there is indeed a pattern (you can check out Pell's equation and Continued Fractions to find ways to very closely approximate square roots).

    However, here is how you would find such integers for pi. We know the close approximation of 22/7 for pi. Then we have:

    22/7≈pi
    22≈7pi

    And verifying, we have 7*pi≈21.99114858

    Another close approximation is 333/106:

    333/106≈pi
    333≈106pi

    and 106pi≈333.0088213...

    I hope this proves useful!
     
  6. Oct 3, 2012 #5
    You might also be interested in the following thread from the wu riddle site;

    "Say I am given a number X = A*[sqrt]2 + B*[pi], where A and B are integers.
    Given X, how can you find A and B, without using brute force?"

    It comes with a long discussion.

    see
    http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi
     
  7. Oct 8, 2012 #6
    I ran a quick computer program just for interest sake.

    78256779
    103767361
    129277943
    131002976
    156513558
    180299107
    182024140
    183749173
    205809689
    207534722
    209259755
    233045304
    234770337
    236495370
    258555886
    260280919
    262005952

    Those numbers if multiplied with pi will give you a number so close to a integer that the decimal part can't fit in a double precision floating point. As stated above you can't actually get a integer from multiplying a integer with pi (except 0)
     
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