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Math Help - recurring digits

  1. #1
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    recurring digits

    In an irrational number what is the longest possible sequence of a single digit that can occur?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by anthp1234 View Post
    In an irrational number what is the longest possible sequence of a single digit that can occur?
    In almost all irrationals a string of 1's will occur that is longer than n for all n in \mathbb{N}.

    Google for normal numbers.

    CB
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  3. #3
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    There is no such limit

    Just by using the word "irrational" you automatically attach absolutely no general rules to the decimal expansion of the number. Here is a simple proof.

    Define x_n as follows:

    The first n digits of x_n are 1. After this string of 1's terminates, the next digits are exactly the digits of \pi.

    x_{20}=.111111111111111111113141592653589793238462  643383279502884197...

    This number is automatically irrational because it has an infinite non-repeating decimal expansion, and yet n can take on whatever value you choose, whether it be 20 or 20 trillion.
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  4. #4
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    OK just take

    Ok, just consider , what is the longest string of a single recurring digit in ?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by anthp1234 View Post
    Ok, just consider , what is the longest string of a single recurring digit in ?
    Already answered in 2nd post in this thread, \pi is believed to be normal (though not proven so), so there is no longest string of a single digit in its' decimal expansion.

    CB
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