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Binary Decimals - Dual Expansion

  1. Jan 29, 2009 #1

    I'm trying to prove that there's a bijection between the open interval (0,1) and the set of all sequences whose elements are 0 or 1 in order to show cardinality continuum.

    So let C={a1, a2, a3,...|ai is either 0 or 1} which is the set of all sequences of 0's and 1's
    and let D={0.b1b2b3...|bi is either 0 or 1} which is the set of all binary decimals on the closed interval [0,1]
    I think it's pretty clear that there's a bijection between these two sets.

    Then the open interval (0,1)=D\{0.000...., binary decimals with tails of repeating 1's} which is the part that gives me problems.

    I'm trying to get rid of the dual expansions by getting rid of the tails of 1's. For example, I have both 0.1 and 0.0111..... in D which are the same number, and I want to get rid of 0.0111.....

    Can someone please explain to me the rational numbers that will have this dual expansion in binary? A denominator of what form will cause this? I think this will help me in explicitly defining (0,1) in terms of D. Any other thoughts on what I've already stated would be appreciated as well.
  2. jcsd
  3. Jan 29, 2009 #2


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    Repeating tails of 1 in the binary expansions correspond to numbers with denominators that are powers of two.
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