ratio φ = [1;1,1,1,1,1,...] and = [1;2,2,2,2,...], while = [3;1,2,1,6,1,2,1,6...] and = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for ) or 1,2,1 (for ), followed by the double of the leading integer. A property of the golden ratio φ Because the continued fraction expansion for φ doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. Hurwitz's theorem states that any irrational number can be approximated by infinitely many rational with While virtually all real numbers will eventually have infinitely many convergents whose distance from is significantly smaller than this limit, the convergents for φ (i.e., the numbers , , , , etc.) consistently "toe the boundary", keeping a distance of almost exactly away from φ, thus never producing an approximation nearly as impressive as, for example, for . It can also be shown that every real number of the form , where , , , and are integers such that , shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated. Regular patterns in continued fractions While there is no discernible pattern in the simple continued fraction expansion of , there is one for , the base of the natural logarithm: which is a special case of this general expression for positive integer : Another, more complex pattern appears in this continued fraction expansion for positive odd : with a special case for : Other continued fractions of this sort are where is a positive integer; also, for integer : with a special case for : If is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals by which is defined for all rational numbers, with and in lowest terms. Then for all nonnegative rationals, we have with similar formulas for negative rationals; in particular we have Many of the formulas can be proved using Gauss's continued fraction. Typical continued fractions Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, Khinchin proved that for almost all real numbers , the (for ) have an astonishing property: their geometric mean tends to a constant (known as Khinchin's constant, ) independent of the value of . Paul Lévy showed