Cantor set and Base 3 expansion

In summary, the conversation discusses how to do a ternary expansion of numbers and how to prove that a number is part of the 2^k iteration of the Cantor set if and only if each decimal expansion position is either a two or zero. The participants also discuss the relation between the base 3 expansion of a number and its inclusion in the Cantor set. The conversation concludes with the acknowledgement of a helpful resource.
  • #1
barksdalemc
55
0
How do I do a ternary expansion of numbers, and prove that if a number is part of the 2^k iteration of the cantor set if and only if each decimal expansion position is either a two or zero? If you guys can give me a hint, I would love to go from there.
 
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  • #2
I think you have that wrong. A number is in the set formed after the kth iteration of the process of removing middle thirds used to create the cantor set iff the kth digit in its base 3 expansion is not 1 (more specifically (to handle endpoints like 1/3=0.1=0.0222...), if it can be written in such a form). Since this is true, maybe it will be easier to prove.
 
  • #3
Yep

You are right. Thanks a lot. I didnt realize this resource was availabe and hopefully will be able to contribute both ways.
 

1. What is the Cantor set?

The Cantor set is a set of real numbers that is created by repeatedly removing the middle third of a line segment. This process is continued infinitely, resulting in a set of numbers that has no length or size, but is still considered to be uncountably infinite.

2. How is the Cantor set related to the Base 3 expansion?

The Cantor set is closely related to the Base 3 expansion because it can be represented using a ternary (base 3) number system. Each point in the Cantor set can be expressed as a ternary number, with the digits representing whether that point was removed in the first, second, or third iteration of the construction process.

3. How is the Base 3 expansion of a number related to its Cantor set representation?

The Base 3 expansion of a number can be used to determine its position in the Cantor set. For example, if we have the number 0.202 in Base 3, this represents a point in the Cantor set that was removed in the first iteration, then not removed in the second, and removed again in the third.

4. What is the relationship between the Cantor set and fractals?

The Cantor set is considered to be a fractal, as it exhibits self-similarity and has a non-integer dimension. It is also a precursor to other well-known fractals, such as the Sierpinski triangle.

5. Can the Cantor set be visualized?

Yes, the Cantor set can be visualized using a fractal plot or by drawing the construction process. It can also be represented geometrically as a set of points on a line segment that are infinitely close together, but never touching.

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