Editing Simple continued fraction (section)

===Properties===
The [[Baire space (set theory)|Baire space]] is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a [[homeomorphism]] from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the [[Euclidean topology|usual topology]] on the reals). The infinite continued fraction also provides a map between the [[quadratic irrational]]s and the [[dyadic rational]]s, and from other irrationals to the set of infinite strings of binary numbers (i.e. the [[Cantor set]]); this map is called the [[Minkowski question-mark function]]. The mapping has interesting self-similar [[fractal]] properties; these are given by the [[modular group]], which is the subgroup of [[Möbius transformation]]s having integer values in the transform. Roughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) [[upper half-plane]]; this is what leads to the fractal self-symmetry.

The limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1) is the [[Gauss–Kuzmin distribution]].