Intuitive Explanation
To get some intuition for the definition above, consider the different ways of interpreting an infinite string of bits beginning with 0 as a real number in . One obvious way to interpret such a string is to place a binary point after the first 0 and read the string as a binary expansion: thus, for instance, the string 001001001001001001001001... represents the binary number 0.010010010010..., or 2/7. Another interpretation views a string as the continued fraction, where the integers ai are the run lengths in a run-length encoding of the string. The same example string 001001001001001001001001... then corresponds to = √3 − 1/2. If the string ends in an infinitely long run of the same bit, we ignore it and terminate the representation; this is suggested by the formal "identity":
- = = = .
The effect of the question mark function on can then be understood as mapping the second interpretation of a string to the first interpretation of the same string, just as the Cantor function can be understood as mapping a triadic base 3 representation to a base 2 representation. Our example string gives the equality
Read more about this topic: Minkowski's Question Mark Function
Famous quotes containing the words intuitive and/or explanation:
“If mothers are told to do this or that or the other,... they lose touch with their own ability to act.... Only too easily they feel incompetent. If they must look up everything in a book, they are always too late even when they do the right things, because the right things have to be done immediately. It is only possible to act at exactly the right point when the action is intuitive or by instinct, as we say. The mind can be brought to bear on the problem afterwards.”
—D.W. Winnicott (20th century)
“To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.... It must involve throughout a resolute rejection of the demand for an explanation of the regularities in the observable course of nature, by means of truths concerning a reality beyond what is actual and observable, as a demand which plays no role in the scientific enterprise.”
—Bas Van Fraassen (b. 1941)