MinkowskiQuestionMark

MinkowskiQuestionMark[x]

gives Minkowski's question mark function .

Details

  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • For a real number with continued fraction representation , the Minkowski's question mark function equals .
  • For certain arguments, MinkowskiQuestionMark automatically evaluates to exact values.
  • MinkowskiQuestionMark can be evaluated to arbitrary numerical precision.
  • MinkowskiQuestionMark automatically threads over lists.

Examples

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Basic Examples  (2)

Scope  (4)

Evaluate at integers or rationals:

Evaluate at quadratic irrational numbers:

Evaluate at inexact real numbers:

TraditionalForm formatting:

Applications  (3)

Enumeration of rational numbers from the unit interval, based on the binary representation of the ordinal number:

First few rational numbers, according to the chosen enumeration strategy:

Plot the cumulative distribution function of so constructed rationals:

Compare it with the Minkowski's question mark function:

An unstable fixed point of the Minkowski's question mark function:

Build a continued fraction expansion of a rational number:

Compare to the builtin function:

Properties & Relations  (2)

The Minkowski's question mark function satisfies reflection identity TemplateBox[{{1, -, x}}, MinkowskiQuestionMark]=1-TemplateBox[{x}, MinkowskiQuestionMark]:

For the function also satisfies TemplateBox[{{x, /, {(, {x, +, 1}, )}}}, MinkowskiQuestionMark]=1/2TemplateBox[{x}, MinkowskiQuestionMark]:

MinkowskiQuestionMark satisfies TemplateBox[{{{(, {{p, _, 1}, +, {p, _, 2}}, )}, /, {(, {{q, _, 1}, +, {q, _, 2}}, )}}}, MinkowskiQuestionMark]=1/2 (TemplateBox[{{{(, {p, _, 1}, )}, /, {(, {q, _, 1}, )}}}, MinkowskiQuestionMark]+TemplateBox[{{{(, {p, _, 2}, )}, /, {(, {q, _, 2}, )}}}, MinkowskiQuestionMark]) for adjacent elements and of a Farey sequence:

Introduced in 2014
 (10.0)