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From the Fun Fact files, here is a Fun Fact at the Medium level:

Cantor Set

Figure 1
Figure 1

Start with the interval [0,1]. Remove the (open) middle third of it, i.e. get (1/3, 2/3). Now remove the middle thirds of each of the remaining intervals, i.e. get (1/9, 2/9) and (7/9, 8/9). Continue this process ad infinitum. The points left over form a fractal called the standard Cantor Set. It is an infinite set since a lot of points, including the endpoints of the removed intervals, are never removed. Can you list the endpoints?

Now let's think about lengths. The length of the original interval is 1. Now how much "length" do we remove during the process?

At the first step, we remove an interval of length 1/3. At the second step, we remove two intervals of length 1/9. At the third step, we remove 4 intervals with total length 1/27, etc. What is the total length removed during the entire process? A geometric series!

SUM1 to infinity (2n-1/3n) = 1.

Wait a minute ... you mean we have an infinite set left over with 0 length??

Yes, and it's worse than that: the set is uncountable (i.e. has the same cardinality as the real numbers) which means that there are "more" Cantor set points than there are rationals!

Presentation Suggestions:
Use this Fun Fact after students learn to sum geometric series. You may wish to assign the sum of the lengths "removed" as an exercise before presenting this Fun Fact. (Of course, you can also do the computation in terms of how much remains after each stage and get a limiting sequence instead.) Follow with the Fun Fact on the Devil's Staircase.

The Math Behind the Fact:
The real numbers are far stranger than any of your students might have suspected! The standard Cantor Set is quite interesting: it is an uncountable, totally disconnected, perfect (every point is a limit point) set of "Lebesgue measure zero". Another way to describe the standard Cantor Set is the set of all real numbers in [0,1] expressible without 1's in its base 3 expansion! Though first constructed as a pathological example, it arises naturally these days in the study of dynamical systems.

How to Cite this Page:
Su, Francis E., et al. "Cantor Set." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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References:
    Any text on real analysis.

Keywords:    geometric series, measure theory
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Joshua Sabloff
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