Determining if Numbers are in the Cantor Set

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SUMMARY

This discussion focuses on determining the membership of the fractions 509/729 and 511/729 in the Cantor set. The fraction 509/729, represented as 0.200212 in base 3, is excluded from the Cantor set because it contains the digit '1'. In contrast, 511/729 can be expressed as 0.200221, which can be expanded to 0.200220222..., using only the digits '0' and '2', confirming its inclusion in the Cantor set. The endpoints of the removed intervals in the Cantor set remain included, further validating the membership of 511/729.

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  • Familiarity with decimal to base conversion techniques
  • Concept of endpoints in interval removal
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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?
 
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Indeed, both look good to me!
 
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.
 

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