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Rationals dense on interval?

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Is the Set X Dense on any interval between (0,1)
    X= [itex] \{ \frac{p}{q} \} [/itex] where p and q are odd positive integers with
    p<q
    3. The attempt at a solution
    so we know that q is always bigger than p so it will always be less than 1.
    and since p and q are odd we will not have the rationals that have even factors.
    So I do not think it will be dense anywhere. The rationals are dense in the reals but we only have odd numbers divided by odd numbers.
     
  2. jcsd
  3. Mar 9, 2012 #2

    Dick

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    If you pick a rational p/q then I think (2^n*p+1)/(2^n*q+1) is awfully close to p/q for n large.
     
  4. Mar 9, 2012 #3
    so should I look at the limit of that.
     
  5. Mar 9, 2012 #4

    Dick

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    Suppose you did, what would that tell you?
     
  6. Mar 9, 2012 #5
    well it would go to zero because the bottom will grow faster than the top. But it seems like it would have a chance of maybe being dense close to zero.
     
  7. Mar 9, 2012 #6

    Dick

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    I don't think it goes to zero.
     
  8. Mar 9, 2012 #7
    ok, but with p<q I could make q as large as I want and keep p small.
     
  9. Mar 9, 2012 #8

    Dick

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    No! Fix p and q. Show there is a rational number of the form odd/odd that is as close to p/q as you want.
     
  10. Mar 9, 2012 #9
    if p and q are fixed then that limit should go to 1.
     
  11. Mar 9, 2012 #10

    Dick

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    What limit goes to 1? I really don't know what you are talking about.
     
  12. Mar 9, 2012 #11
    your post # 2 , as n goes to infinity , that whole formula should go to 1.
     
  13. Mar 10, 2012 #12

    Dick

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    No, it does not. It goes to p/q. That was my whole point!
     
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