5 || 6 || 7 |- align="right" ! | || || 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1 |- align="right" ! | 0 || 1 || 1 || 2 || 5 || 7 || 19 || 26 || 71 || 97 |- align="right" ! | 1 || 0 || 1 || 1 || 3 || 4 || 11 || 15 || 41 || 56 |} Properties A 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 usual topology on the reals). The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals, 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 transformations 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. Some useful theorems If , , , is an infinite sequence of positive integers, define the sequences and recursively: Theorem 1. For any positive real number Theorem 2. The convergents of [; , , ] are given by Theorem 3. If the th convergent to a continued fraction is /, then Corollary 1: Each convergent is in its lowest terms (for if and had a nontrivial common divisor it would divide , which is impossible). Corollary 2: The difference between successive convergents is a fraction whose numerator is unity: Corollary 3: The continued fraction is equivalent to a series of alternating terms: Corollary 4: The matrix has determinant plus or minus one, and thus belongs to the group of unimodular matrices . Theorem 4. Each (th) convergent is nearer to a subsequent (th) convergent than any preceding (th) convergent is. In symbols, if the th convergent is taken to be , then for all . Corollary 1: The even convergents (before the th)