Determining if Numbers are in the Cantor Set

[spitz]

Homework Statement

I have two numbers: 509/729 and 511/729. I want to determine if they are in the Cantor set.

The Attempt at a Solution

I have:

509/729 in base 3 is: 0.200212

So this is not part of the cantor set because it can't be expanded in base 3 using only 0 and 2.

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511/729 in base 3 is: 0.200221

For this one, am I correct in saying that 0.200221 = 0.200220222...

So, it can be expanded using only 0 and 2, and so is part of the cantor set?

 

[micromass]

Indeed, both look good to me!

 

[HallsofIvy]

Since we form the Cantor set by removing open intervals, the endpoins of those intervals remain in the Cantor set. .0200221₃ is such an endpoint.