In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
Definition and intuition One way to define the question-mark function involves the correspondence between two different ways of representing real numbers using finite or infinite binary sequences. Most familiarly, a string of 0s and 1s with a single point mark ".", like "11.0010010000111111..." can be interpreted as the binary representation of a number. In this case this number is
2 + 1 + 1 8 + 1 64 + ⋯ = π . {\displaystyle 2+1+{\frac {1}{8}}+{\frac {1}{64}}+\cdots =\pi .}
There is a different way of interpreting the same sequence, however, using continued fractions. Interpreting the fractional part "0.00100100001111110..." as a binary number in the same way, replace each consecutive block of 0s or 1s by its run length (or, for the first block of zeroes, its run length plus one), in this case generating the sequence [3;3,1,2,1,4,6, … ] {\displaystyle \dots ]} . Then, use this sequence as the coefficients of a continued fraction:
3 + 1 3 + 1 1 + 1 2 + 1 1 + 1 4 + 1 6 + … ≈ 3.2676 {\displaystyle 3+{\frac {1}{\displaystyle 3+{\frac {1}{\displaystyle 1+{\frac {1}{\displaystyle 2+{\frac {1}{\displaystyle 1+{\frac {1}{\displaystyle 4+{\frac {1}{\displaystyle 6+\dots }}}}}}}}}}}}\approx 3.2676}
The question-mark function reverses this process: it translates the continued-fraction of a given real number into a run-length encoded binary sequence, and then reinterprets that sequence as a binary number. For instance, for the example above, ? ( 3.2676 ) ≈ π {\displaystyle \operatorname {?} (3.2676)\approx \pi } . To define this formally, if an irrational number x {\displaystyle x} has the (non-terminating) continued-fraction representation
x = a 0 + 1 a 1 + 1 a 2 + ⋯ = [ a 0 ; a 1 , a 2 , … ] {\displaystyle x=a_{0}+{\frac {1}{\displaystyle a_{1}+{\frac {1}{\displaystyle a_{2}+\cdots }}}}=[a_{0};a_{1},a_{2},\dots ]}
then the value of the question-mark function on x {\displaystyle x} is defined as the value of the infinite series
? ( x ) = a 0 + 2 ∑ n = 1 ∞ ( − 1 ) n + 1 2 a 1 + ⋯ + a n . {\displaystyle \operatorname {?} (x)=a_{0}+2\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n+1}}{2^{a_{1}+\cdots +a_{n}}}}.}
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