Thread: Ternary Expansions and Cantor Set

1. 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=$\displaystyle (.002002...)_3$
2/13=2x1/13=(.004004...)=$\displaystyle (.011011...)_3$
3/13=3x1=$\displaystyle 3(.002002...)_3=(.02002...)_3$
4/13=2x2/13=$\displaystyle 2(.011011...)_3=(.022022...)_3$
5/13=4/13+1/13=$\displaystyle (.002002...)_3+(.022022...)_3=(.02110211)_3$
6/13=3x2/13=$\displaystyle 3(.011011...)_3=(.11011...)_3$
7/13=6/13+1/13=$\displaystyle (.11011...)_3+(.002002...)_3=(.112112...)_3$
8/13=7/13+1/13=$\displaystyle (.002002...)_3+(.112112...)_3=(.111111...)_3$
9/13=3x3/13=$\displaystyle 3(.02002...)_3=(.2002002...)_3$
10/13=9/13+1/13=$\displaystyle (.2002002...)_3+(.002002...)_3=(.2022022...)_3$
11/13=2/13+9/13=$\displaystyle (.011011...)_3+(.2002002...)_3=(.211211)_3$
12/13=3x4/13=$\displaystyle 3(.022022...)_3=(.22022)_3$

2. Originally Posted by eg37se
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=$\displaystyle (.002002...)_3$
2/13=2x1/13=(.004004...)=$\displaystyle (.011011...)_3$
3/13=3x1=$\displaystyle 3(.002002...)_3=(.02002...)_3$
4/13=2x2/13=$\displaystyle 2(.011011...)_3=(.022022...)_3$
5/13=4/13+1/13=$\displaystyle (.002002...)_3+(.022022...)_3=(.02110211)_3$ Should be $\displaystyle \color{red}(.101101...)_3$.
6/13=3x2/13=$\displaystyle 3(.011011...)_3=(.11011...)_3$
7/13=6/13+1/13=$\displaystyle (.11011...)_3+(.002002...)_3=(.112112...)_3$
8/13=7/13+1/13=$\displaystyle (.002002...)_3+(.112112...)_3=(.111111...)_3$ Should be $\displaystyle \color{red}(.121121...)_3$.
9/13=3x3/13=$\displaystyle 3(.02002...)_3=(.2002002...)_3$
10/13=9/13+1/13=$\displaystyle (.2002002...)_3+(.002002...)_3=(.2022022...)_3$
11/13=2/13+9/13=$\displaystyle (.011011...)_3+(.2002002...)_3=(.211211{\color{red }...})_3$
12/13=3x4/13=$\displaystyle 3(.022022...)_3=(.22022{\color{red}0...})_3$
The rest are correct.