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
Famous quotes containing the words definition, rational and/or arguments:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“If we did not have rational souls, we would not be able to believe.”
—St. Augustine (354–430)
“Children are intensely invested in getting their way. They will devote more emotional and intellectual energy to winning arguments than parents ever will, and are almost always better rested.”
—Jean Callahan (20th century)