Cantor Set: 7/12, 1/3, 1/4, 11/12

[tylerc1991]

Homework Statement

Which of the following are in the Cantor set: 7/12, 1/3, 1/4, 11/12? Give the ternary expansion of each.

The Attempt at a Solution

I see that 1/3 is in the Cantor set and has a ternary expansion:

1/3 = 0/3 + 2/3^2 + 2/3^3 + 2/3^4 + ...

I am fairly certain that 1/4 as well as 11/12 are in the Cantor set as well, although I am not exactly sure how to go about showing this. When deciding if 1/3 was in the Cantor set or not I noticed that the endpoints of the intervals would always stay in the set (like 0 and 1 for instance), and after removing the middle 1/3 the first time, 1/3 became an endpoint. As for 1/4 and 11/12 I am not certain and would appreciate any direction and/or intuition. Thank you!

 

[Dick]

They want you to get ternary expansions of all those numbers anyway, so why don't you do that part first. Numbers in the cantor set have ternary expansions consisting of only the digits 0 and 2. Like you have found 1/3=0.022222... You could also have written 1/3=0.1 in ternary, but that's the only sort of ambiguity. To systematically get ternary expansion use 'long division'. If you remember how to do it in base 10 you just have to make a few adjustments.