Recursive Definition For Rational Arguments
For rational numbers in the unit interval, the function may also be defined recursively; if p/q and r/s are reduced fractions such that | ps − rq | = 1 (so that they are adjacent elements of a row of the Farey sequence) then
Using the base cases
it is then possible to compute ?(x) for any rational x, starting with the Farey sequence of order 2, then 3, etc.
If and are two successive convergents of a continued fraction, then the matrix
has determinant ±1. Such a matrix is an element of, the group of two-by-two matrices with determinant ±1. This group is related to the modular group.
Read more about this topic: Minkowski's Question Mark Function
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