# The Cantor Sequence with Bits

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The Cantor sequence is similar to the Cantor set but deals with integers. To construct it, start with the natural numbers , writing them in ternary notation as . Construct a binary sequence : if in ternary has only 0s or 2s, let ; otherwise let . This gives .

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Contributed by: Michael Schreiber (September 2007)

Open content licensed under CC BY-NC-SA

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The Cantor set is constructed iteratively; starting with the closed unit interval , the open middle third is taken out, leaving the two closed intervals and . Then the middle thirds of those two intervals are taken out, leaving four intervals of length , and so on. The Cantor set is the limit (or intersection) of all such sets.

Here is a way to look at the Cantor set in terms of the base 3 (ternary) representation of numbers. In the unit interval numbers in base 3 have the form , where the are all possible combinations of the digits 0, 1 or 2. The numbers in the middle third of start with 0.1; the numbers in the middle thirds of and start with 0.01 and 0.21. In general, numbers in a middle third have the digit 1 somewhere in their ternary expansion. In other words, the Cantor set consists of numbers that only have 0 or 2 as ternary digits. Replacing the 2s with 1s in those expansions gives a representation of the Cantor set using binary numbers.

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