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Math Help - Prove that...

  1. #1
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    Prove that...

    Prove that, among the numbers \sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, ..., one of them gets arbitrarily close to an integer.
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  2. #2
    mfb
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    Re: Prove that...

    Here is a hint: If n \sqrt{2}=m+\epsilon with a small \epsilon, how is that related to \frac{m}{n}?
    Which property of the rational numbers can you use?
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  3. #3
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    Re: Prove that...

    Haha I already know the solution Hence the "that you can solve yourself" part in the title of this forum.

    There's a more elegant solution involving the Pigeonhole principle.
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  4. #4
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    Re: Prove that...

    Wait, does it say that n must be an integer? Or do we just infer that from the number sequence given? Because if not we can say that there are more combinations of n times root 2 than there are integers so one must be arbitrarily close to an integer? I'm not a calcmaster as my user name implies but I do like challenging problems! Hope I didn't say something too ridiculous...
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  5. #5
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    Re: Prove that...

    According to mfb's notation, n is an integer.

    There's a fairly simple solution to this...
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  6. #6
    mfb
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    Re: Prove that...

    Quote Originally Posted by calcmaster View Post
    Because if not we can say that there are more combinations of n times root 2 than there are integers so one must be arbitrarily close to an integer?
    [0.4 , 0.6] as interval in the real numbers has more numbers than there are integers, but it does not have any number less than 0.4 apart from an integer.
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