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Concept
Minkowski's question-mark function
Summary
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.
One way to define the question-mark function involves the correspondence between two different ways of representing fractional numbers using finite or infinite binary sequences. Most familiarly, a string of 0's and 1's with a single point mark ".", like "11.001001000011111..." can be interpreted as the binary representation of a number. In this case this number is
There is a different way of interpreting the same sequence, however, using continued fractions.
Interpreting the part before the point mark as a binary number in the same way, replace each consecutive block of 0's or 1's after the point by its run length, in this case generating the sequence . Then, use this sequence as the coefficients of a continued fraction:
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, . To define this formally, if an irrational number has the (non-terminating) continued-fraction representation
then the value of the question-mark function on is defined as the value of the infinite series
In the same way, if a rational number has the terminating continued-fraction representation then
the value of the question-mark function on is a finite sum,
Analogously to the way the question-mark function reinterprets continued fractions as binary numbers, the Cantor function can be understood as reinterpreting ternary numbers as binary numbers.
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