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An element of the Cantor set
Show that 3/4 is an element of Cantor's ternary set that is not an endpoint of any removed interval.
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Show that 3/4 is an element of Cantor's ternary set that is not an endpoint of any removed interval.
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$3/4$ can be expressed as $0.\overline{20}$ in ternary expansion. Since it can be expressed solely in 0s and 2s it is a member of Cantor's ternary set. Since the "tail end" of the ternary expansion is not all 0s or all 2s, it is not an endpoint of a removed interval.
The algorithm I used to write $3/4$ as a ternary expansion is as follows:
1. Multiply it by $3$. So $3*\frac{3}{4}=\frac{9}{4}=2+\frac{1}{4}$.
2. Write down the whole number portion as the first value in the ternary expansion, so $0.2$.
3. Subtract the previous whole number from the fraction. $\frac{9}{4}-2=\frac{1}{4}$.
4. Repeat this process.
I dished out a couple of likes to Cornelius's comments. I particularly like the way that you found the expansion. Typically, though, we'd like to write this as
[dmath]\frac{2}{3} \sum_{k=0}^{\infty} \frac{1}{9^k} = \frac{2}{3}\frac{1}{1-1/9} = \frac{3}{4}.[/dmath]