The '''Minkowski question-mark function''' converts the continued fraction representation {{math|[a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ...]}} of a number into a binary decimal representation in which the integer part {{math|a<sub>0</sub>}} is unchanged and the {{math|a<sub>1</sub>, a<sub>2</sub>, ...}} become alternating runs of binary zeroes and ones of those lengths. The decimal point takes the place of the first zero. 

Thus, {{math|?}}(31/7) = 71/16 because 31/7 has the continued fraction representation {{math|[4;2,3]}} giving the binary expansion {{math|4 + 0.0111<sub>2</sub>}}.

Among its interesting properties is that it maps roots of quadratic equations, which have repeating continued fractions, to rational numbers, which have repeating binary digits.

The question-mark function is continuous and monotonically increasing, so it has an inverse.

* Produce a function for {{math|?(x)}}. &nbsp; Be careful: rational numbers have two possible continued fraction representations:
:::* &nbsp; {{math|[a<sub>0</sub>;a<sub>1</sub>,... a<sub>n−1</sub>,a<sub>n</sub>]}}  &nbsp; &nbsp; and 
:::* &nbsp; {{math|[a<sub>0</sub>;a<sub>1</sub>,... a<sub>n−1</sub>,a<sub>n</sub>−1,1]}} 
* Choose one of the above that will give a binary expansion ending with a &nbsp; '''1'''.
* Produce the inverse function {{math|?<sup>-1</sup>(x)}}
* Verify that {{math|?(φ)}} = 5/3, where {{math|φ}} is the Greek golden ratio.
* Verify that {{math|?<sup>-1</sup>(-5/9)}} = (√13 - 7)/6
* Verify that the two functions are inverses of each other by showing that {{math|?<sup>-1</sup>(?(x))}}={{math|x}} and {{math|?(?<sup>-1</sup>(y))}}={{math|y}} for {{math|x, y}} of your choice 


Don't worry about precision error in the last few digits. 


;See also:
* Wikipedia entry: [[wp:Minkowski%27s_question-mark_function|Minkowski's question-mark function]]

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