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Dirichlet's Approximation Theorem not working for n=8 and α= pi?

  1. Jun 4, 2013 #1
    Dirichlet's Approximation Theorem not working for n=8 and α= pi???

    I am reading a number theory text book that states Dirichlet's Approximation Theorem as follows:

    If α is a real number and n is a positive integer, then there exists integers a and b

    with 1≤ a ≤ n such that |aα-b|< 1/n .

    There is a proof of this theorem given in the text as well.

    My question is if this theorem is suppose to be true for all real numbers α and positive integers n, then how come I cannot find integers a and b satisfying this inequality for α= pi and n = 8?

    In this case since a is restricted to 1≤ a ≤ 8 it is easy to look for b by trying all 8 values for a:

    1* pi = 3.14159...
    2* pi = 6.28319...
    3* pi = 9.42478...
    4* pi = 12.5664...
    5* pi = 15.708...
    6* pi = 18.8496...
    7* pi = 21.9911...
    8* pi = 25.1327...

    since "b" must also be an integer it is clear that the possibilities for b are 3,6,9,12,15,18,21,25.
    However, none of these satisfy the inequality.

    Since 1/8 = .125 the decimal values of the numbers I listed above for all the possible values of a should be less than .125 in one of the cases...but this does not happen. How can this be? I figure I must be overlooking something or computing something incorrectly because the theorem clearly states that such a, b can be found for all positive n and real α.
     
  2. jcsd
  3. Jun 4, 2013 #2

    CompuChip

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    Note it says |aα-b|< 1/n instead of (aα-b) < 1/n.
    The difference is subtle but crucial. Now look again at 7*pi.
     
  4. Jun 4, 2013 #3
    I completely overlooked the absolute value sign. Thank you for pointing this out!
     
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