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Math Help - I am hope that my question belongs here.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    I am hope that my question belongs here.

    Let x be a rational number in (113/191 ,184/311). Find x=a/b when is known that a<400.

    { 113/191 < a/b < 184/311 }
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let x be a rational number in (113/191 ,184/311). Find x=a/b when is known that a<400.

    { 113/191 < a/b < 184/311 }
    I have no elegant mathematical answer to your question. But an answer can relatively easily be found with the help of a little program: since there are only finitely many candiadates for a/b to consider.
    For example, I have hacked together (rather hastily) the following Python script:
    Code:
    from math import floor, ceil
    lb = 113.0/191.0 # lower bound of interval
    ub = 184.0/311.0 # upper bound of interval
    maxnum = 399.0 # max for numerator
    checked = 0 # number of pairs a/b checked
    found = 0 # number of a/b with lb < a/b < ub found
    # we are not too fussy about the lower limit for the denominator,
    # but because the numerator is limited we need not consider that
    # many numerators a:
    for b in range(1,ceil(maxnum/lb)):
        for a in range(ceil(lb*b),max(ceil(lb*b)+1, ceil(ub*b))):
            checked += 1
            if (lb < float(a)/b and float(a)/b < ub):
                found += 1
                print "found: a/b = %d/%d" % (a,b)
    print "total: checked %d, found %d" % (checked, found)
    The final output is "total: checked 674, found 1".
    So if this program does not contain a major flaw, the only a/b that satisfies your requirements is 297/502
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    You did a wonderful job!!! But... I looking for a mathematical solution...
    Another question(that comes with your answer...) is: to prove that there is only one number such this.


    Thank you again for your effort!
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    Super Member Failure's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    You did a wonderful job!!! But... I looking for a mathematical solution...
    Of course, we are all looking for more elegant and general, in a word "mathematical" ways of answering such questions. On the other hand, for this purpose one would want to state the problem in a more general form, for otherwise the effort that goes into finding a general answer might not get amortized properly...

    Another question(that comes with your answer...) is: to prove that there is only one number such this.
    It comes down to a question about the correctness of this program. There is a small dark cloud on the horizon for example, because this program uses ordinary floating point arithmetic.
    That aside, I think that the program does an exhaustive search (there are only finitely many possibilities in any case), and I am thus quite confident that there are no other solutions to your problem.
    Btw: Initially I considered the possibility of searching the Stern?Brocot tree - Wikipedia, the free encyclopedia , but came to the conclusion that even if that could be done, the difference to the much simpler approch that I finally used would mainly be to arrive at a much more complex program (with correspondingly higher probability of such a program containing a bug).
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let x be a rational number in (113/191 ,184/311). Find x=a/b when is known that a<400.

    { 113/191 < a/b < 184/311 }
    This is one of those cases where the simplest method works best. If p/q < r/s then (p+r)/(q+s) lies in the interval ( p/q , r/s). So this problem is solved by \frac{113+184}{191+311} = \frac{297}{502}, as the computer analysis confirms.

    Of course, this method does not show that the answer is unique (subject to the condition a<400).
    Last edited by Opalg; April 15th 2010 at 11:27 AM.
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