- #1
tylerc1991
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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!