Minkowski's question mark function

In mathematics, the Minkowski question mark function (or the slippery devil's staircase), denoted by $?(x)$ , is a function possessing various unusual fractal properties, defined by Hermann Minkowski (1904, pages 171–172). It maps quadratic irrationals 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. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

Definition

If $[a_0; a_1, a_2, \ldots]$ is the continued fraction representation of an irrational number $x$ , then

whereas:

If $[a_0; a_1, a_2, \ldots, a_m]$ is a continued fraction representation of a rational number $x$ , then

Intuitive explanation

To get some intuition for the definition above, consider the different ways of interpreting an infinite string of bits beginning with 0 as a real number in $[0,1]$ . 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 $[0; a 1, a 2, \dots ]$ , 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 $[0; 2, 1, 2, 1, 2, 1, \dots] =$ √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":

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Pre-Chorus:
Hey, hey bitch, try this!
Guarantee turn a square to a bop bitch
You ain't down, bye bitch
I ain't got time for playin, I'm just saying man
Chorus: (x2)
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Trump orders a plan to dismantle the Education Department while keeping some core functions (copy)

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