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
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)