Minkowski's Question Mark Function - Intuitive Explanation

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 a_i 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