Minkowski's question-mark function

In mathematics, the question mark function is a function denoted ?(x) whose definition is due to Hermann Minkowski. If $[a_{0};a_{1},a_{2},\cdots ]$ is the continued fraction representation of an irrational number x, then

${\rm {?}}(x)=a_{0}+2\sum _{n=1}^{\infty }(-1)^{n+1}2^{-a_{1}-\cdots -a_{n}}$

whereas if $[a_{0};a_{1},a_{2},\cdots a_{m}]$ is a continued fraction for a rational number, then

${\rm {?}}(x)=a_{0}+2\sum _{n=1}^{m}(-1)^{n+1}2^{-a_{1}-\cdots -a_{n}}$

It should be noted that if a_m is greater than one, then $[a_{0};a_{1},a_{2},\cdots a_{m}-1,1]$ is also a continued fraction for the same number, but the two expressions give identical values for ?(x).

For rational numbers 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

$?({\frac {p+r}{q+s}})={\frac {1}{2}}(?({\frac {p}{q}})+?({\frac {r}{s}}))$

Properties of ?(x)

The question mark function is an increasing and continuous function. It sends rational numbers to dyadic rational numbers, meaning those whose base two representation terminates. It sends quadratic irrationalities to non-dyadic rational numbers. If ?(x) is irrational, then x is either algebraic of degree greater than two, or transcendental. The function is invertible, and the inverse function has also attracted the attention of various mathematicians, in particular John Conway.