1) Here is a recipe for a fractal: You start with a simple line segment. Then erase the middle third of that segment, leaving two separate segments. At the next stage, take each of those smaller line segments and erase the middle third of them. Use that rule to draw the next two stages of this:
2) How many line segments will you have at stage n?
3) Look back at stage 0. You can think about that as being a shaded-in number line, from 0 to 1. Then stage one consists of two disjoint subintervals: [0 to 1/3] and [2/3 to 1]. The open interval (1/3 to 2/3) has been removed. Name the intervals that are included in stage two. Use interval notation.
4) Were this process to go on forever, what, if anything, is left at the end? (Whatever remains is known as the Cantor Set.)
5) We can also use base-3 decimals to look at the Cantor Set. In fact they're not even "decimals" at all, because they're based on powers of three instead of powers of ten. Remember "tressimals"? This string, for example, would represent the number 5/9:
312
6) What fraction does this tressimal represent? 30201
7) The tressimals can be infinite in length, too. What number does this string represent?
3111111111111111111111111111111111111...
8) You may be wondering how this connects to the fractal you drew. Look back at your original sketch. At each stage, a line segment gets broken into three parts: left, middle, and right. We can assign the digit 0 to the left section, 1 to the middle, and 2 to the right. Then consider a tressimal to be like a set of directions to go to a particular address. For example, 30121 is left, middle, right, middle. It's telling you: go to the lefthand section at stage one. Then at stage two, go to the middle of that lefthand section that you are already in. Next, go to the righthand part of that middle. Finally, go to the middle of that righthand part. You are now in the middle of the right of the middle of the left. Try it! Are you in an empty area or are you in a shaded interval?
9) Now think of 32221 as an address. Where do you land if you follow the directions? Make sure you are comfortable with this address idea before you continue. Call me over if you need help.
10) From the address point of view, is 322 the same as 3220?
11) Think again about 3111111111111111111.... Use the address idea to argue that .1 repeating must be one half.
12) Use the address idea to argue that 30222222222222222... must be one third.
13) You saw above that 30121 lands you at an address that can't possibly be in the Cantor Set, because the interval is empty. Make up a bunch of tressimals that you believe will protect you from landing in an empty interval.
14) Is there any kind of pattern or rule to which tressimals will work? Or does it seem to be random?
15) Remember that these tressimals can be infinite in length. How is it possible to write an infinite string that is guaranteed to keep you within the Cantor Set? Does it have to have a repeating pattern?
16) There are infinitely many strings you can make that will work to keep you in the Cantor set. In fact, there are so many of them that they are uncountably infinite. Prove it.
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