Question Mark Function

In Mathematics , the Minkowski question mark function, sometimes called the '''slippery devil's staircase''', is a Function , denoted ?(''x''), possessing various unusual Fractal properties. It was defined by Hermann Minkowski in 1904 by matching the Quadratic Irrational s to the Dyadic Rational s on the Unit Interval . The expression relating continued fractions to the dyadics (as commonly used, and defined below) was given by Arnaud Denjoy in 1938 .

DEFINITION

If

a_1, a_2, \ldots

is the Continued Fraction Representation of an Irrational Number ''x'', then

:

{
m ?}(x) = a_0 + 2 \sum_{n=1}^\infty (-1)^{n+1}2^{-a_1 - \cdots -a_n}

whereas:

If

a_1, a_2, \ldots, a_m

is a continued fraction representation of a Rational Number ''x'', then

:

{
m ?}(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>1

then

a_1, a_2, \ldots, a_m-1, 1

is also a continued fraction for the same number, but the two expressions give identical values for ?(''x'').

INTUITIVE EXPLANATION

To get some intuition for the definition above,
let's consider two different ways
of interpreting an infinite string of bits beginning with 0 as a real number in {Link without Title} .
One obvious way to interpret such a string is to place a binary point after the first 0 and read the string as a Binary expansion: thus, for instance, the string 001001001001001001001001...
represents the binary number 0.010010010010..., or 2/7. Another interpretation
views a string as the Continued Fraction {Link without Title} , where the integers ''a''_i are the run lengths in a Run-length Encoding of the string. The same example string 001001001001001001001001... then
corresponds to = (√3-1)/2. (If the string ends in an infinitely long run of the same bit, we ignore it and terminate the representation; this is suggested by the formal "identity" [0;''a''₁,...,''a''_n,∞ =[0;''a''₁,...,''a''_n+1/∞]=
{Link without Title} = {Link without Title} .)

The effect of the question mark function on {Link without Title} can then be understood as mapping the second interpretation of a string to the first interpretation of the same string. Our example string gives the equality
:

?\left(rac{\sqrt3-1}{2}
ight)=rac{2}{7}.

RECURSIVE DEFINITION FOR RATIONAL ARGUMENTS

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Question Mark Function