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Thread: Ternary Expansions and Cantor Set

  1. #1
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    Ternary Expansions and Cantor Set

    The problem is to find out for what values of p (for integers between 0 and 13), is p/13 in the cantor set. I know that they are only in the cantor set if they can be expressed in the ternary expansion to base 3 using only 2's and 0's. I have a lot of problems with these ternary expansions so I was wondering if you guys could check my work quick, and let me know where I have gone wrong.

    THANKS

    1/13= (.002002...)_3
    2/13=2x1/13=(.004004...)= (.011011...)_3
    3/13=3x1= 3(.002002...)_3=(.02002...)_3
    4/13=2x2/13= 2(.011011...)_3=(.022022...)_3
    5/13=4/13+1/13= (.002002...)_3+(.022022...)_3=(.02110211)_3
    6/13=3x2/13= 3(.011011...)_3=(.11011...)_3
    7/13=6/13+1/13= (.11011...)_3+(.002002...)_3=(.112112...)_3
    8/13=7/13+1/13= (.002002...)_3+(.112112...)_3=(.111111...)_3
    9/13=3x3/13= 3(.02002...)_3=(.2002002...)_3
    10/13=9/13+1/13= (.2002002...)_3+(.002002...)_3=(.2022022...)_3
    11/13=2/13+9/13= (.011011...)_3+(.2002002...)_3=(.211211)_3
    12/13=3x4/13= 3(.022022...)_3=(.22022)_3
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  2. #2
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    Quote Originally Posted by eg37se View Post
    The problem is to find out for what values of p (for integers between 0 and 13), is p/13 in the cantor set. I know that they are only in the cantor set if they can be expressed in the ternary expansion to base 3 using only 2's and 0's. I have a lot of problems with these ternary expansions so I was wondering if you guys could check my work quick, and let me know where I have gone wrong.

    THANKS

    1/13= (.002002...)_3
    2/13=2x1/13=(.004004...)= (.011011...)_3
    3/13=3x1= 3(.002002...)_3=(.02002...)_3
    4/13=2x2/13= 2(.011011...)_3=(.022022...)_3
    5/13=4/13+1/13= (.002002...)_3+(.022022...)_3=(.02110211)_3 Should be \color{red}(.101101...)_3.
    6/13=3x2/13= 3(.011011...)_3=(.11011...)_3
    7/13=6/13+1/13= (.11011...)_3+(.002002...)_3=(.112112...)_3
    8/13=7/13+1/13= (.002002...)_3+(.112112...)_3=(.111111...)_3 Should be \color{red}(.121121...)_3.
    9/13=3x3/13= 3(.02002...)_3=(.2002002...)_3
    10/13=9/13+1/13= (.2002002...)_3+(.002002...)_3=(.2022022...)_3
    11/13=2/13+9/13= (.011011...)_3+(.2002002...)_3=(.211211{\color{red  }...})_3
    12/13=3x4/13= 3(.022022...)_3=(.22022{\color{red}0...})_3
    The rest are correct.
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