Definition
See figure. Formally, the Cantor function c : → is defined as follows:
- Express x in base 3.
- If x contains a 1, replace every digit after the first 1 by 0.
- Replace all 2s with 1s.
- Interpret the result as a binary number. The result is c(x).
For example:
- 1/4 becomes 0.02020202... base 3; there are no 1s so the next stage is still 0.02020202...; this is rewritten as 0.01010101...; when read in base 2, this is 1/3 so c(1/4) = 1/3.
- 1/5 becomes 0.01210121... base 3; the digits after the first 1 are replaced by 0s to produce 0.01000000...; this is not rewritten since there are no 2s; when read in base 2, this is 1/4 so c(1/5) = 1/4.
- 200/243 becomes 0.21102 (or 0.211012222...) base 3; the digits after the first 1 are replaced by 0s to produce 0.21; this is rewritten as 0.11; when read in base 2, this is 3/4 so c(200/243) = 3/4.
Read more about this topic: Cantor Function
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