Continued fraction
}</math> In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other  than the first, must be positive. The integers formula_3 are called the coefficients or terms of the continued fraction. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form. Continued fractions have a number of  remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to formula_6. The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number formula_7 is the value  of a "unique" infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values formula_7 and 1. This way of expressing real numbers (rational and irrational) is called their "continued fraction representation". The term "continued fraction" may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions. Motivation and notation. Consider, for example, the rational number , which is around 4.4624. As a first approximation, start with 4, which is the integer part; . The  fractional part is the reciprocal of which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of . The remaining fractional part, , is the reciprocal of , and is around 6.1429. Use 6 as an approximation for this to obtain as an approximation for and , about 4.4615, as the third approximation; . Finally, the fractional part, , is the reciprocal of 7, so its approximation in this scheme, 7, is exact () and produces the exact expression for . The expression is called the continued fraction representation of . This can be represented by the abbreviated notation  = [4; 2, 6, 7]. (It is customary to replace only the "first" comma by a semicolon.) Some older textbooks use all commas in the -tuple, for example, [4, 2, 6, 7]. If the starting number is rational, then this process exactly parallels the Euclidean algorithm applied to the numerator and denominator of the number. In particular, it must terminate and produce a finite continued fraction representation of the number. The sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm. If the starting number is irrational, then the process continues  indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are: Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties: Basic formula. A (generalized) continued fraction is an expression of the form where "a" and "b" can be any complex numbers. When "b" = 1 for all "i" the expression is  called a "simple" continued fraction. When the expression contains finitely many terms, it is called a "finite" continued fraction. When the expression contains infinitely many terms, it is called an "infinite" continued fraction. When the terms eventually repeat from some point onwards, the expression is called a periodic continued fraction. Thus, all of the following illustrate valid finite simple continued fractions: For simple continued fractions of the form formula_10 the formula_11 term can also be calculated using the following recursive formula: formula_12 where formula_13 and formula_14 From which it can be understood that the formula_11 sequence stops if formula_16. Calculating  continued fraction representations. Consider a real number . Let formula_17 be the integer part of and let formula_18 be the fractional part of . Then the continued fraction representation of is formula_19, where formula_20 is the continued fraction representation of formula_21. To calculate a continued fraction representation of a number , write down the integer part (technically the floor) of . Subtract this integer part from . If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if is rational. This process can be efficiently implemented using the Euclidean algorithm when the number  is rational. The table below shows an implementation of this procedure for the number 3.245, resulting in the continued fraction expansion [3; 4,12,4]. Notations. The integers formula_2, formula_23 etc., are called the "coefficients" or "terms" of the continued fraction. One can abbreviate the continued fraction in the notation of Carl Friedrich Gauss or as or in the notation of Pringsheim as or in another related notation as Sometimes angle brackets are used, like this: The semicolon in the square and angle bracket notations is sometimes replaced by a comma. One may also define "infinite simple continued fractions" as limits: This  limit exists for any choice of formula_2 and positive integers formula_32. Finite continued fractions. Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and the other coefficients are positive integers. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation  is therefore always greater than 1, if present. In symbols: Reciprocals. The continued fraction representations of a positive rational number and its reciprocal are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by formula_33 and formula_34 are reciprocals. For instance if formula_35 is an integer and formula_36 then If formula_39 then The last number that generates the remainder of the continued fraction is the same for both formula_42 and its reciprocal. For example, Infinite continued fractions and convergents. Every  infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction. An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like "e" have only  small terms early in their continued fraction, which makes them more difficult to approximate rationally. The golden ratio ϕ has terms equal to 1 everywhere—the smallest values possible—which makes ϕ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers. Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger. For a continued fraction , the first four convergents (numbered 0 through 3) are The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third coefficient , and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called "continuants". If successive convergents are found, with numerators , , ... and denominators , , ... then the relevant recursive relation is: The successive convergents are given by the formula Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are ⁄ and ⁄. For example, here are the convergents for [0;1,5,2,2]. When using the Babylonian  method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ... , , ... For example, the continued fraction expansion for is [1;1,2,1,2,1,2,1,2...]. Comparing the convergents with the approximants derived from the Babylonian method: 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 formula_2, formula_23, formula_47, formula_48 is an infinite sequence of positive integers, define the sequences formula_49 and formula_50 recursively: Theorem 1. For any positive real number formula_51 Theorem 2. The convergents of [formula_2; formula_23, formula_47, formula_48] are given by Theorem 3. If the formula_1th convergent to a continued fraction is formula_49/formula_50, then Corollary 1: Each convergent  is in its lowest terms (for if formula_49 and formula_50 had a nontrivial common divisor it would divide formula_64, 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 formula_68 unimodular matrices formula_69. Theorem 4. Each (formula_70th) convergent is nearer to a subsequent (formula_1th) convergent than any preceding (formula_72th) convergent is. In symbols, if the formula_1th convergent is taken to be formula_74, then  for all formula_76. Corollary 1: The even convergents (before the formula_1th) continually increase, but are always less than formula_78. Corollary 2: The odd convergents (before the formula_1th) continually decrease, but are always greater than formula_78. Theorem 5. Corollary 1: A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent. Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction. Semiconvergents. If are consecutive convergents, then any fractions of the form where  formula_84 is an integer such that formula_85, are called "semiconvergents", "secondary convergents", or "intermediate fractions". The formula_86-st semiconvergent equals the mediant of the formula_84-th one and the convergent formula_88. Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., formula_89), rather than that a convergent is a kind of semiconvergent. It follows that semiconvergents represent a monotonic sequence of fractions between the convergents formula_90 (corresponding to formula_91) and formula_92 (corresponding to formula_93). The consecutive semiconvergents formula_94 and formula_95 satisfy the property formula_96. If a rational approximation formula_97 to a  real number formula_42 is such that the value formula_99 is smaller than that of any approximation with a smaller denominator, then formula_97 is a semiconvergent of the continued fraction expansion of formula_42. The converse is not true, however. Best rational approximations. One can choose to define a "best rational approximation" to a real number as a rational number , , that is closer to than any approximation with a smaller or equal denominator. The simple continued fraction for can be used to generate "all" of the best rational approximations for by applying these three rules: For example, 0.84375 has continued fraction [0;1,5,2,2 ]. Here are all of its best rational approximations. The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation. The "half rule" mentioned above requires that when is even, the halved term /2 is admissible if and only if This is equivalent to: . The convergents to are "best approximations" in a much stronger sense than the one defined above. Namely, / is a convergent for if and only if has the smallest value among the analogous expressions for all rational approximations / with ; that is , we have so long as . (Note also that as .) Best rational within an interval. A rational that falls within the interval , for , can be found with the continued fractions for and . When both and are irrational and where and have identical continued fraction expansions up through , a rational that falls within the interval is given by the finite continued fraction, This rational will be best in the sense that no other rational in will have a smaller numerator or a smaller denominator. If is rational, it will have "two" continued fraction representations that are "finite", and , and similarly a rational   will have two representations, and . The coefficients beyond the last in any of these representations should be interpreted as ; and the best rational will be one of , , , or . For example, the decimal representation 3.1416 could be rounded from any number in the interval . The continued fraction representations of 3.14155 and 3.14165 are and the best rational between these two is Thus, is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator. Interval for a convergent . A rational number, which can be expressed as finite continued fraction in two ways, will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between (see this proof) The numbers and are formed by incrementing the last coefficient in the two representations for . It is the case that when is even, and when is odd. For example, the number has the continued fraction representations and thus is a convergent of any number strictly between Comparison. Consider and . If is the smallest index for which is unequal to then  if and otherwise. If there is no such , but one expansion is shorter than the other, say and with for , then if is even and if is odd. Continued fraction expansion of and its convergents. To calculate the convergents of we may set , define and , and , . Continuing like this, one can determine the infinite continued fraction of as The fourth convergent of is [3;7,15,1] = = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of . Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write  down at once the convergent fractions which result from these quotients without developing the continued fraction. The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, . Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, , which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction , we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator , and for our denominator, . The third convergent, therefore, is . We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113. In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions: To sum up, the pattern is These convergents are  alternately smaller and larger than the true value of , and approach nearer and nearer to . The difference between a given convergent and is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction is greater than , but − is less than  =  (in fact, − is just more than = ). The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is  always unity and the denominator the product of the two denominators. Thus the difference between and is , in excess; between and , , in deficit; between and , , in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series: The first term, as we  see, is the first fraction; the first and second together give the second fraction, ; the first, the second and the third give the third fraction , and so on with the rest; the result being that the series entire is equivalent to the original value. Generalized continued fraction. A generalized continued fraction is an expression of the form where the "a" ("n" > 0) are the partial numerators, the "b" are the partial denominators, and the leading term "b" is called the "integer" part of the continued fraction. To illustrate the use of generalized continued fractions, consider the following example. The  sequence of partial denominators of the simple continued fraction of does not show any obvious pattern: or However, several generalized continued fractions for have a perfectly regular structure, such as: The first two of these are special cases of the function with = 4 arctan (1) and the fourth and fifth one can be derived using the Wallis product. The continued fraction of formula_109 above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler. Other continued fraction expansions. Periodic continued fractions. The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations  with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden 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 formula_111 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 that the th root of the denominator of the th convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, approximately 3.27582, which is known as Lévy's constant. Lochs' theorem states that th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over decimal places. Applications. Square roots. Generalized continued fractions are used in a method for computing square roots. The identity leads via recursion to the generalized continued fraction for any square root: Pell's equation. Continued fractions play an essential  role in the solution of Pell's equation. For example, for positive integers and , and non-square , it is true that if , then is a convergent of the regular continued fraction for . The converse holds if the period of the regular continued fraction for is 1, and in general the period describes which convergents give solutions to Pell's equation. Dynamical systems. Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma. The  backwards shift operator for continued fractions is the map called the Gauss map, which lops off digits of a continued fraction expansion: . The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution. Eigenvalues and eigenvectors. The Lanczos algorithm uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix. Networking applications. Continued fractions have also been used in modelling optimization problems for wireless network virtualization to find  a route between a source and a destination.