{"history":[],"article":{"id":"bdc1f3e6-660e-4811-a8c8-8d49972a849e","title":"Simple continued fraction","slug":"simple-continued-fraction","content":":::summary\n:::\n\n# \nA **simple** or **regular [continued fraction](https://bliptext.com/articles/continued-fraction)** is a [continued fraction](https://bliptext.com/articles/continued-fraction) with numerators all equal one, and denominators built from a sequence \\{a_i\\} of integer numbers.\nThe sequence can be **finite** or in**finite**, resulting in a **finite** (or **terminated**) continued fraction like\nor an **infinite** continued fraction like\nTypically, such a continued fraction is obtained through an [iterative](https://bliptext.com/articles/iterative) process of representing a number as the sum of its [integer part](https://bliptext.com/articles/integer-part) and the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of another number, then writing this other number as the sum of its [integer part](https://bliptext.com/articles/integer-part) and another [reciprocal](https://bliptext.com/articles/multiplicative-inverse), and so on.\nIn the finite case, the iteration/[recursion](https://bliptext.com/articles/recursion) is stopped after finitely many steps by using an integer in lieu of another continued fraction.\nIn contrast, an infinite continued fraction is an [infinite expression](https://bliptext.com/articles/infinite-expression-mathematics).\nIn either case, all integers in the sequence, other than the first, must be [positive](https://bliptext.com/articles/positive-number).\nThe integers a_i are called the [coefficients](https://bliptext.com/articles/coefficient) or terms of the continued fraction.\nSimple continued fractions have a number of remarkable properties related to the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) for integers or [real numbers](https://bliptext.com/articles/real-number).\nEvery [rational number](https://bliptext.com/articles/rational-number) p⁄q has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to (p,q).\nThe numerical value of an infinite continued fraction is [irrational](https://bliptext.com/articles/irrational-number); it is defined from its infinite sequence of integers as the [limit](https://bliptext.com/articles/limit-mathematics) of a sequence of values for finite continued fractions.\nEach finite continued fraction of the sequence is obtained by using a finite [prefix](https://bliptext.com/articles/prefix-computer-science) of the infinite continued fraction's defining sequence of integers.\nMoreover, every irrational number \\alpha 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](https://bliptext.com/articles/commensurability-mathematics) values \\alpha and 1.\nThis way of expressing real numbers (rational and irrational) is called their continued fraction representation.\n\n# Motivation and notation\nConsider, for example, the [rational number](https://bliptext.com/articles/rational-number) 415⁄93, which is around 4.4624.\nAs a first [approximation](https://bliptext.com/articles/approximation), start with 4, which is the [integer part](https://bliptext.com/articles/floor-and-ceiling-functions); 415⁄93 = 4 + 43⁄93.\nThe fractional part is the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of 93⁄43 which is about 2.1628.\nUse the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of 4 + 1⁄2 = 4.5.\nNow, 93⁄43 = 2 + 7⁄43;\nthe remaining fractional part, 7⁄43, is the reciprocal of 43⁄7, and 43⁄7 is around 6.1429.\nUse 6 as an approximation for this to obtain 2 + 1⁄6 as an approximation for 93⁄43 and 4 + 1⁄2 + 1⁄6, about 4.4615, as the third approximation.\nFurther, 43⁄7 = 6 + 1⁄7.\nFinally, the fractional part, 1⁄7, is the reciprocal of 7, so its approximation in this scheme, 7, is exact (7⁄1 = 7 + 0⁄1) and produces the exact expression for 415⁄93.\nThat expression is called the continued fraction representation of 415⁄93.\nThis can be represented by the abbreviated notation 415⁄93 = [4; 2, 6, 7].\n(It is customary to replace only the first comma by a semicolon to indicate that the preceding number is the whole part.) Some older textbooks use all commas in the\n(n + 1)\n-tuple, for example, [4, 2, 6, 7].\nIf the starting number is rational, then this process exactly parallels the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) applied to the numerator and denominator of the number.\nIn particular, it must terminate and produce a finite continued fraction representation of the number.\nThe sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm.\nIf the starting number is [irrational](https://bliptext.com/articles/irrational-number), then the process continues indefinitely.\nThis produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit.\nThis is the (infinite) continued fraction representation of the number.\nExamples of continued fraction representations of irrational numbers are:\n√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...]\n. The pattern repeats indefinitely with a period of 6.\ne = [2;1,2,1,1,4,1,1,6,1,1,8,...]\n. The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.\nπ = [3;7,15,1,292,1,1,1,2,1,3,1,...]\n. No pattern has ever been found in this representation.\nΦ = [1;1,1,1,1,1,1,1,1,1,1,1,...]\n. The [golden ratio](https://bliptext.com/articles/golden-ratio), the irrational number that is the \"most difficult\" to approximate rationally.\nγ = [0;1,1,2,1,2,1,4,3,13,5,1,...]\n. The [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant), which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.\nContinued fractions are, in some ways, more \"mathematically natural\" representations of a [real number](https://bliptext.com/articles/real-number) than other representations such as [decimal representations](https://bliptext.com/articles/decimal-representation), and they have several desirable properties:\n[a0;a1,... an−1,an] = [a0;a1,...\nan−1,(an−1),1]\n. Usually the first, shorter one is chosen as the [canonical representation](https://bliptext.com/articles/canonical-form).\n\n# Formulation\nA continued fraction in canonical form is an expression of the form\nwhere ai are integer numbers, called the coefficients or terms of the continued fraction.\nWhen the expression contains finitely many terms, it is called a finite continued fraction.\nWhen the expression contains infinitely many terms, it is called an infinite continued fraction.\nWhen the terms eventually repeat from some point onwards, the continued fraction is called [periodic](https://bliptext.com/articles/periodic-continued-fraction).\nThus, all of the following illustrate valid finite simple continued fractions:\nFor simple continued fractions of the form\nthe a_n term can be calculated using the following recursive formula:\nwhere and\nfrom which it can be understood that the a_n sequence stops if N_{n+1} = 0.\n\n## Notations\nConsider a continued fraction expressed as\nBecause such a continued fraction expression may take a significant amount of vertical space, a number of methods have been tried to shrink it.\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz) sometimes used the notation\nand later the same idea was taken even further with the nested fraction bars drawn aligned, for example by [Alfred Pringsheim](https://bliptext.com/articles/alfred-pringsheim) as\nor in more common related notations as\nor\n[Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) used a notation reminiscent of [summation notation](https://bliptext.com/articles/summation),\nor in cases where the numerator is always 1, eliminated the fraction bars altogether, writing a list-style\nSometimes list-style notation uses angle brackets instead,\nThe semicolon in the square and angle bracket notations is sometimes replaced by a comma.\nOne may also define infinite simple continued fractions as [limits](https://bliptext.com/articles/limit-of-a-sequence):\nThis limit exists for any choice of a_0 and positive integers.\n\n# Calculating continued fraction representations\nConsider a real number r. Let and let f = r - i. When f\\neq 0, the continued fraction representation of r is [i;a_1,a_2,\\ldots], where is the continued fraction representation of 1/f.\nWhen r\\geq 0, then i is the [integer part](https://bliptext.com/articles/integer-part) of r, and f is the [fractional part](https://bliptext.com/articles/fractional-part) of r.\nIn order to calculate a continued fraction representation of a number r, write down the [floor](https://bliptext.com/articles/floor-function) of r. Subtract this value from r. If the difference is 0, stop; otherwise find the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the difference and repeat.\nThe procedure will halt if and only if r is rational.\nThis process can be efficiently implemented using the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) when the number is rational.\nThe table below shows an implementation of this procedure for the number 3.245 = 649/200:\n!Step\n!Real Number\n!Integer part\n!Fractional part\n!Simplified\n!Reciprocal of f !1\n!2 !3 !4\nThe continued fraction for 3.245 is thus or, expanded:\n\n# Reciprocals\nThe continued fraction representations of a positive rational number and its [reciprocal](https://bliptext.com/articles/multiplicative-inverse) are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively.\nIn other words, the numbers represented by and are reciprocals.\nFor instance if a is an integer and x < 1 then\nIf x>1 then\nThe last number that generates the remainder of the continued fraction is the same for both x and its reciprocal.\nFor example,\n\n# Finite continued fractions\nEvery finite continued fraction represents a [rational number](https://bliptext.com/articles/rational-number), and every [rational number](https://bliptext.com/articles/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.\nThese two representations agree except in their final terms.\nIn 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.\nThe final element in the short representation is therefore always greater than 1, if present.\nIn symbols:\n[a0; a1, a2, ..., an − 1, an, 1] = [a0; a1, a2, ..., an − 1, an + 1]\n.\n[a0; 1] = [a0 + 1]\n.\n\n# Infinite continued fractions and convergents\nEvery infinite continued fraction is [irrational](https://bliptext.com/articles/irrational-number), and every [irrational](https://bliptext.com/articles/irrational-number) number can be represented in precisely one way as an infinite continued fraction.\nAn infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number.\nThese rational numbers are called the **convergents** of the continued fraction.\nThe larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated.\nNumbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers.\nOther numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally.\nThe [golden ratio](https://bliptext.com/articles/golden-ratio) Φ has terms equal to 1 everywhere—the smallest values possible—which makes Φ the most difficult number to approximate rationally.\nIn this sense, therefore, it is the \"most irrational\" of all irrational numbers.\nEven-numbered convergents are smaller than the original number, while odd-numbered ones are larger.\nFor a continued fraction\n[a0; a1, a2, ...]\n, the first four convergents (numbered 0 through 3) are\nThe 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.\nThe denominators are formed similarly.\nTherefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain [multivariate polynomials](https://bliptext.com/articles/multivariate-polynomial) called [continuants](https://bliptext.com/articles/continuant-mathematics).\nIf successive convergents are found, with numerators h1, h2, ... and denominators k1, k2, ...\nthen the relevant recursive relation is that of [Gaussian brackets](https://bliptext.com/articles/gaussian-brackets):\nThe successive convergents are given by the formula\nThus to incorporate a new term into a rational approximation, only the two previous convergents are necessary.\nThe initial \"convergents\" (required for the first two terms) are 0⁄1 and 1⁄0.\nFor example, here are the convergents for [0;1,5,2,2].\n! n !\nan\n!\nhn\n!\nkn\nWhen using the [Babylonian method](https://bliptext.com/articles/methods-of-computing-square-roots) 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.\nSpecifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ...\n,\n2k−1\n, ... For example, the continued fraction expansion for [\\sqrt3](https://bliptext.com/articles/square-root-of-3) is\n[1; 1, 2, 1, 2, 1, 2, 1, 2, ...]\n. Comparing the convergents with the approximants derived from the Babylonian method:\n! n !\nan\n!\nhn\n!\nkn\n# [Convergents approaching the golden ratio | upload.wikimedia.org/wikipedia/commons/b/b4/Golden///ration///convergents.svg]\n\n## Properties\nThe [Baire space](https://bliptext.com/articles/baire-space-set-theory) is a topological space on infinite sequences of natural numbers.\nThe infinite continued fraction provides a [homeomorphism](https://bliptext.com/articles/homeomorphism) from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the [usual topology](https://bliptext.com/articles/euclidean-topology) on the reals).\nThe infinite continued fraction also provides a map between the [quadratic irrationals](https://bliptext.com/articles/quadratic-irrational) and the [dyadic rationals](https://bliptext.com/articles/dyadic-rational), and from other irrationals to the set of infinite strings of binary numbers (i.e. the [Cantor set](https://bliptext.com/articles/cantor-set)); this map is called the [Minkowski question-mark function](https://bliptext.com/articles/minkowski-question-mark-function).\nThe mapping has interesting self-similar [fractal](https://bliptext.com/articles/fractal) properties; these are given by the [modular group](https://bliptext.com/articles/modular-group), which is the subgroup of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) having integer values in the transform.\nRoughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) [upper half-plane](https://bliptext.com/articles/upper-half-plane); this is what leads to the fractal self-symmetry.\nThe 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](https://bliptext.com/articles/gauss-kuzmin-distribution).\n\n## Some useful theorems\nIf \\ a_0\\, a_1\\ , a_2\\ , \\ \\ldots\\ is an infinite sequence of positive integers, define the sequences \\ h_n\\ and \\ k_n\\ recursively:\n**Theorem 1.** For any positive real number \\ x\\\n**Theorem 2.** The convergents of \\ [\\ a_0\\ ; a_1\\, a_2\\ , \\ldots\\ ]\\ are given by\nor in matrix form,\n**Theorem 3.** If the \\ nth convergent to a continued fraction is then\nor equivalently\n**Corollary 1:** Each convergent is in its lowest terms (for if \\ h_n\\ and \\ k_n\\ had a nontrivial common divisor it would divide which is impossible).\n**Corollary 2:** The difference between successive convergents is a fraction whose numerator is unity:\n**Corollary 3:** The continued fraction is equivalent to a series of alternating terms:\n**Corollary 4:** The matrix\nhas [determinant](https://bliptext.com/articles/determinant) (-1)^{n+1}, and thus belongs to the group of [unimodular matrices](https://bliptext.com/articles/unimodular-matrix)\n**Corollary 5:** The matrix\nhas determinant (-1)^na_n, or equivalently,meaning that the odd terms monotonically decrease, while the even terms monotonically increase.\n**Corollary 6:** The denominator sequence satisfies the recurrence relation, and grows at least as fast as the [Fibonacci sequence](https://bliptext.com/articles/fibonacci-sequence), which itself grows like O(\\phi^n) where is the [golden ratio](https://bliptext.com/articles/golden-ratio).\n**Theorem 4.** Each (\\ sth) convergent is nearer to a subsequent (\\ nth) convergent than any preceding (\\ rth) convergent is.\nIn symbols, if the \\ nth convergent is taken to be then\nfor all\n**Corollary 1:** The even convergents (before the \\ nth) continually increase, but are always less than \\ x_n ~.\n**Corollary 2:** The odd convergents (before the \\ nth) continually decrease, but are always greater than \\ x_n ~.\n**Theorem 5.**\n**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.\n**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.\"Theorem 6: Consider the set of all open intervals with end-points . Denote it as \\mathcal C. Any open subset of is a disjoint union of sets from \\mathcal C.\"**Corollary:** The infinite continued fraction provides a homeomorphism from the Baire space to.\n\n# Semiconvergents\nIf\nare consecutive convergents, then any fractions of the form\nwhere m is an integer such that, are called semiconvergents, secondary convergents, or intermediate fractions.\nThe (m+1)-st semiconvergent equals the [mediant](https://bliptext.com/articles/mediant-mathematics) of the m-th one and the convergent.\nSometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., 0 0\n, that is closer to x than any approximation with a smaller or equal denominator.\nThe simple continued fraction for x can be used to generate all of the best rational approximations for x by applying these three rules:\nFor example, 0.84375 has continued fraction [0;1,5,2,2].\nHere are all of its best rational approximations.\n! Continued fraction\n! Rational approximation\n! Decimal equivalent\n! Error\nThe 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.\nThe \"half rule\" mentioned above requires that when ak is even, the halved term ak/2 is admissible if and only if\nThis is equivalent to:.\n[ak; ak − 1, ..., a1] > [ak; ak + 1, ...]\n.\nThe convergents to x are \"best approximations\" in a much stronger sense than the one defined above.\nNamely, n/d is a convergent for x if and only if\nhas the smallest value among the analogous expressions for all rational approximations m/c with\nc ≤ d\nso long as\nc < d\n. (Note also that\nas\nk → ∞\n.)\n\n## Best rational within an interval\nA rational that falls within the interval, for\n0 < x < y\n, can be found with the continued fractions for x and y. When both x and y are irrational and\nwhere x and y have identical continued fraction expansions up through\nak−1\n, a rational that falls within the interval is given by the finite continued fraction,\nz(x,y) = [a0; a1, a2, ..., ak − 1, min(ak, bk) + 1]\nThis rational will be best in the sense that no other rational in will have a smaller numerator or a smaller denominator.\nIf x is rational, it will have two continued fraction representations that are finite,\nx1\nand\nx2\n, and similarly a rational y will have two representations,\ny1\nand\ny2\n. The coefficients beyond the last in any of these representations should be interpreted as\n+∞\nz(x1, y1)\n,\nz(x1, y2)\n,\nz(x2, y1)\n, or\nz(x2, y2)\n.\nFor example, the decimal representation 3.1416 could be rounded from any number in the interval.\nThe continued fraction representations of 3.14155 and 3.14165 are\nand the best rational between these two is\n[3; 7, 16] = 355⁄113 = 3.1415929....\nThus, 355⁄113 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.\n\n## Interval for a convergent\nA rational number, which can be expressed as finite continued fraction in two ways,\nwill 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](https://bliptext.com/articles/undefined))\nand\nThe numbers x and y are formed by incrementing the last coefficient in the two representations for z. It is the case that\nx < y\nwhen k is even, and\nx > y\nwhen k is odd.\nFor example, the number 355⁄113 has the continued fraction representations\nand thus 355⁄113 is a convergent of any number strictly between\n[3; 7, 15, 2]\n||=||\n688⁄219 ≈ 3.1415525\n[3; 7, 17]\n||=||\n377⁄120 ≈ 3.1416667\n\n## Legendre's theorem on continued fractions\nIn his Essai sur la théorie des nombres (1798), [Adrien-Marie Legendre](https://bliptext.com/articles/adrien-marie-legendre) derives a necessary and sufficient condition for a rational number to be a convergent of the continued fraction of a given real number.\nA consequence of this criterion, often called **Legendre's theorem** within the study of continued fractions, is as follows:\n**Theorem**.\nIf α is a real number and p, q are positive integers such that, then p/q is a convergent of the continued fraction of α.\n**Proof**.\nWe follow the proof given in [An Introduction to the Theory of Numbers](https://bliptext.com/articles/an-introduction-to-the-theory-of-numbers) by [G. H. Hardy](https://bliptext.com/articles/g-h-hardy) and [E. M. Wright](https://bliptext.com/articles/e-m-wright).\nSuppose α, p, q are such that, and assume that α > p/q.\nThen we may write, where 0 < θ < 1/2.\nWe write p/q as a finite continued fraction [a0; a1, ..., an], where due to the fact that each rational number has two distinct representations as finite continued fractions differing in length by one (namely, one where an = 1 and one where an ≠ 1), we may choose n to be even.\n(In the case where α < p/q, we would choose n to be odd.)\nLet p0/q0, ..., pn/qn = p/q be the convergents of this continued fraction expansion.\nSet, so that and thus,where we have used the fact that pn-1 qn - pn qn-1 = (-1)n and that n is even.\nNow, this equation implies that α = [a0; a1, ..., an, ω].\nSince the fact that 0 < θ < 1/2 implies that ω > 1, we conclude that the continued fraction expansion of α must be [a0; a1, ..., an, b0, b1, ...], where [b0; b1, ...] is the continued fraction expansion of ω, and therefore that pn/qn = p/q is a convergent of the continued fraction of α.\nThis theorem forms the basis for [Wiener's attack](https://bliptext.com/articles/wiener-s-attack), a polynomial-time exploit of the [RSA cryptographic protocol](https://bliptext.com/articles/rsa-cryptosystem) that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key n = pq satisfy p < q < 2p and the private key d is less than (1/3)n1/4).\n\n# Comparison\nConsider\nand\n. If k is the smallest index for which\nak\nis unequal to\nbk\nthen\nx < y\nif\n(−1)k(ak − bk) < 0\nand\ny < x\notherwise.\nIf there is no such k, but one expansion is shorter than the other, say\nand\nwith\nfor\n0 ≤ i ≤ n\n, then\nx < y\nif n is even and\ny < x\nif n is odd.\n\n# Continued fraction expansion of π and its convergents\nTo calculate the convergents of [π](https://bliptext.com/articles/pi) we may set\n, define\nand\n,\nand\n,\n. Continuing like this, one can determine the infinite continued fraction of π as The fourth convergent of π is [3;7,15,1] = 355⁄113 = 3.14159292035..., sometimes called [Milü](https://bliptext.com/articles/mil), which is fairly close to the true value of π.\nLet us suppose that the quotients found are, as above, [3;7,15,1].\nThe 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.\nThe first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3⁄1.\nThen, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22⁄7, which will be too large.\nMultiplying 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.\nThus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106.\nThe third convergent, therefore, is 333⁄106.\nWe proceed in the same manner for the fourth convergent.\nThe 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.\nIn this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:\nTo sum up, the pattern is\nThese convergents are alternately smaller and larger than the true value of π, and approach nearer and nearer to π.\nThe difference between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent.\nFor example, the fraction 22⁄7 is greater than π, but 22⁄7 − π is less than 1⁄7 × 106 = 1⁄742 (in fact, 22⁄7 − π is just more than 1⁄791 = 1⁄7 × 113).\nThe 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.\nThus the difference between 22⁄7 and 3⁄1 is 1⁄7, in excess; between 333⁄106 and 22⁄7, 1⁄742, in deficit; between 355⁄113 and 333⁄106, 1⁄11978, in excess; and so on.\nThe 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.\nInstead of the fractions written above, we have thus the series:\nThe first term, as we see, is the first fraction; the first and second together give the second fraction, 22⁄7; the first, the second and the third give the third fraction 333⁄106, and so on with the rest; the result being that the series entire is equivalent to the original value.\n# [The following Maple code will generate continued fraction expansions of pi | upload.wikimedia.org/wikipedia/commons/9/91/Continued///fraction///expansion///of///Pi///.png]\n\n# Non-simple continued fraction\nA non-simple continued fraction is an expression of the form\nwhere the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.\nTo illustrate the use of non-simple continued fractions, consider the following example.\nThe sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:\nor\nHowever, several non-simple continued fractions for π have a perfectly regular structure, such as:\nThe first two of these are special cases of the [arctangent](https://bliptext.com/articles/inverse-trigonometric-functions) function with π = 4 arctan (1) and the fourth and fifth one can be derived using the [Wallis product](https://bliptext.com/articles/wallis-product).\nThe continued fraction of \\pi above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.\n\n# Other continued fraction expansions\n\n## Periodic continued fractions\nThe numbers with periodic continued fraction expansion are precisely the [irrational solutions](https://bliptext.com/articles/quadratic-irrational) of [quadratic equations](https://bliptext.com/articles/quadratic-equation) with rational coefficients; rational solutions have finite continued fraction expansions as previously stated.\nThe simplest examples are the [golden ratio](https://bliptext.com/articles/golden-ratio) φ = [1;1,1,1,1,1,...] and √2 = [1;2,2,2,2,...], while √14 = [3;1,2,1,6,1,2,1,6...] and √42 = [6;2,12,2,12,2,12...].\nAll irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for √2) or 1,2,1 (for √14), followed by the double of the leading integer.\n\n## A property of the golden ratio φ\nBecause the continued fraction expansion for [φ](https://bliptext.com/articles/golden-ratio) doesn't use any integers greater than 1, [φ](https://bliptext.com/articles/golden-ratio) is one of the most \"difficult\" real numbers to approximate with rational numbers.\n[Hurwitz's theorem](https://bliptext.com/articles/hurwitz-s-theorem-number-theory) states that any irrational number k can be approximated by infinitely many rational m⁄n with\nWhile virtually all real numbers k will eventually have infinitely many convergents m⁄n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5⁄3, 8⁄5, 13⁄8, 21⁄13, 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, [355⁄113](https://bliptext.com/articles/mil) for [π](https://bliptext.com/articles/pi).\nIt can also be shown that every real number of the form a + bφ⁄c + dφ, where a, b, c, and d are integers such that\na d − b c = ±1\n, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.\n\n## Regular patterns in continued fractions\nWhile there is no discernible pattern in the simple continued fraction expansion of π, there is one for\ne\n, the [base of the natural logarithm](https://bliptext.com/articles/e-mathematical-constant):\nwhich is a special case of this general expression for positive integer n:\nAnother, more complex pattern appears in this continued fraction expansion for positive odd n:\nwith a special case for\nn = 1\nOther continued fractions of this sort are\nwhere n is a positive integer; also, for integer n:\nwith a special case for\nn = 1\nIf\nIn(x)\nis the modified, or hyperbolic, [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind, we may define a function on the rationals p⁄q by\nwhich is defined for all rational numbers, with p and q in lowest terms.\nThen for all nonnegative rationals, we have\nwith similar formulas for negative rationals; in particular we have\nMany of the formulas can be proved using [Gauss's continued fraction](https://bliptext.com/articles/gauss-s-continued-fraction).\n\n## Typical continued fractions\nMost irrational numbers do not have any periodic or regular behavior in their continued fraction expansion.\nNevertheless, for [almost all](https://bliptext.com/articles/almost-all) numbers on the unit interval, they have the same limit behavior.\nThe arithmetic average diverges:, and so the coefficients grow arbitrarily large:.\nIn particular, this implies that almost all numbers are well-approximable, in the sense that[Khinchin](https://bliptext.com/articles/aleksandr-khinchin) proved that the [geometric mean](https://bliptext.com/articles/geometric-mean) of\nai\ntends to a constant (known as [Khinchin's constant](https://bliptext.com/articles/khinchin-s-constant)):[Paul Lévy](https://bliptext.com/articles/paul-l-vy-mathematician) proved that the nth root of the denominator of the nth convergent converges to [Lévy's constant](https://bliptext.com/articles/l-vy-s-constant)\n[Lochs' theorem](https://bliptext.com/articles/lochs-theorem) states that the convergents converge exponentially at the rate of\n\n# Applications\n\n## Pell's equation\nContinued fractions play an essential role in the solution of [Pell's equation](https://bliptext.com/articles/pell-s-equation).\nFor example, for positive integers p and q, and non-square n, it is true that if\np2 − nq2 = ±1\n, then\np⁄q\nis a convergent of the regular continued fraction for √n.\nThe converse holds if the period of the regular continued fraction for √n is 1, and in general the period describes which convergents give solutions to Pell's equation.\n\n## Dynamical systems\nContinued fractions also play a role in the study of [dynamical systems](https://bliptext.com/articles/dynamical-system), where they tie together the [Farey fractions](https://bliptext.com/articles/farey-sequence) which are seen in the [Mandelbrot set](https://bliptext.com/articles/mandelbrot-set) with [Minkowski's question-mark function](https://bliptext.com/articles/minkowski-s-question-mark-function) and the [modular group](https://bliptext.com/articles/modular-group) Gamma.\nThe backwards [shift operator](https://bliptext.com/articles/shift-operator) for continued fractions is the map\ncalled the **[Gauss map](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator)**, which lops off digits of a continued fraction expansion:\nh([0; a1, a2, a3, ...]) = [0; a2, a3, ...]\n. The [transfer operator](https://bliptext.com/articles/transfer-operator) of this map is called the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator).\nThe distribution of the digits in continued fractions is given by the zero'th [eigenvector](https://bliptext.com/articles/eigenvector) of this operator, and is called the [Gauss–Kuzmin distribution](https://bliptext.com/articles/gauss-kuzmin-distribution).\n\n# History","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:36:56.368Z","updated_at":"2025-01-25T17:36:56.368Z","search_vector":"'continu':2 'fraction':3 'simpl':1","search_content":" \n \n\n# \nA **simple** or **regular [continued fraction](https://bliptext.com/articles/continued-fraction)** is a [continued fraction](https://bliptext.com/articles/continued-fraction) with numerators all equal one, and denominators built from a sequence \\{a_i\\} of integer numbers.\nThe sequence can be **finite** or in**finite**, resulting in a **finite** (or **terminated**) continued fraction like\nor an **infinite** continued fraction like\nTypically, such a continued fraction is obtained through an [iterative](https://bliptext.com/articles/iterative) process of representing a number as the sum of its [integer part](https://bliptext.com/articles/integer-part) and the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of another number, then writing this other number as the sum of its [integer part](https://bliptext.com/articles/integer-part) and another [reciprocal](https://bliptext.com/articles/multiplicative-inverse), and so on.\nIn the finite case, the iteration/[recursion](https://bliptext.com/articles/recursion) is stopped after finitely many steps by using an integer in lieu of another continued fraction.\nIn contrast, an infinite continued fraction is an [infinite expression](https://bliptext.com/articles/infinite-expression-mathematics).\nIn either case, all integers in the sequence, other than the first, must be [positive](https://bliptext.com/articles/positive-number).\nThe integers a_i are called the [coefficients](https://bliptext.com/articles/coefficient) or terms of the continued fraction.\nSimple continued fractions have a number of remarkable properties related to the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) for integers or [real numbers](https://bliptext.com/articles/real-number).\nEvery [rational number](https://bliptext.com/articles/rational-number) p⁄q has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to (p,q).\nThe numerical value of an infinite continued fraction is [irrational](https://bliptext.com/articles/irrational-number); it is defined from its infinite sequence of integers as the [limit](https://bliptext.com/articles/limit-mathematics) of a sequence of values for finite continued fractions.\nEach finite continued fraction of the sequence is obtained by using a finite [prefix](https://bliptext.com/articles/prefix-computer-science) of the infinite continued fraction's defining sequence of integers.\nMoreover, every irrational number \\alpha 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](https://bliptext.com/articles/commensurability-mathematics) values \\alpha and 1.\nThis way of expressing real numbers (rational and irrational) is called their continued fraction representation.\n\n# Motivation and notation\nConsider, for example, the [rational number](https://bliptext.com/articles/rational-number) 415⁄93, which is around 4.4624.\nAs a first [approximation](https://bliptext.com/articles/approximation), start with 4, which is the [integer part](https://bliptext.com/articles/floor-and-ceiling-functions); 415⁄93 = 4 + 43⁄93.\nThe fractional part is the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of 93⁄43 which is about 2.1628.\nUse the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of 4 + 1⁄2 = 4.5.\nNow, 93⁄43 = 2 + 7⁄43;\nthe remaining fractional part, 7⁄43, is the reciprocal of 43⁄7, and 43⁄7 is around 6.1429.\nUse 6 as an approximation for this to obtain 2 + 1⁄6 as an approximation for 93⁄43 and 4 + 1⁄2 + 1⁄6, about 4.4615, as the third approximation.\nFurther, 43⁄7 = 6 + 1⁄7.\nFinally, the fractional part, 1⁄7, is the reciprocal of 7, so its approximation in this scheme, 7, is exact (7⁄1 = 7 + 0⁄1) and produces the exact expression for 415⁄93.\nThat expression is called the continued fraction representation of 415⁄93.\nThis can be represented by the abbreviated notation 415⁄93 = [4; 2, 6, 7].\n(It is customary to replace only the first comma by a semicolon to indicate that the preceding number is the whole part.) Some older textbooks use all commas in the\n(n + 1)\n-tuple, for example, [4, 2, 6, 7].\nIf the starting number is rational, then this process exactly parallels the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) applied to the numerator and denominator of the number.\nIn particular, it must terminate and produce a finite continued fraction representation of the number.\nThe sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm.\nIf the starting number is [irrational](https://bliptext.com/articles/irrational-number), then the process continues indefinitely.\nThis produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit.\nThis is the (infinite) continued fraction representation of the number.\nExamples of continued fraction representations of irrational numbers are:\n√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...]\n. The pattern repeats indefinitely with a period of 6.\ne = [2;1,2,1,1,4,1,1,6,1,1,8,...]\n. The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.\nπ = [3;7,15,1,292,1,1,1,2,1,3,1,...]\n. No pattern has ever been found in this representation.\nΦ = [1;1,1,1,1,1,1,1,1,1,1,1,...]\n. The [golden ratio](https://bliptext.com/articles/golden-ratio), the irrational number that is the \"most difficult\" to approximate rationally.\nγ = [0;1,1,2,1,2,1,4,3,13,5,1,...]\n. The [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant), which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.\nContinued fractions are, in some ways, more \"mathematically natural\" representations of a [real number](https://bliptext.com/articles/real-number) than other representations such as [decimal representations](https://bliptext.com/articles/decimal-representation), and they have several desirable properties:\n[a0;a1,... an−1,an] = [a0;a1,...\nan−1,(an−1),1]\n. Usually the first, shorter one is chosen as the [canonical representation](https://bliptext.com/articles/canonical-form).\n\n# Formulation\nA continued fraction in canonical form is an expression of the form\nwhere ai are integer numbers, called the coefficients or terms of the continued fraction.\nWhen the expression contains finitely many terms, it is called a finite continued fraction.\nWhen the expression contains infinitely many terms, it is called an infinite continued fraction.\nWhen the terms eventually repeat from some point onwards, the continued fraction is called [periodic](https://bliptext.com/articles/periodic-continued-fraction).\nThus, all of the following illustrate valid finite simple continued fractions:\nFor simple continued fractions of the form\nthe a_n term can be calculated using the following recursive formula:\nwhere and\nfrom which it can be understood that the a_n sequence stops if N_{n+1} = 0.\n\n## Notations\nConsider a continued fraction expressed as\nBecause such a continued fraction expression may take a significant amount of vertical space, a number of methods have been tried to shrink it.\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz) sometimes used the notation\nand later the same idea was taken even further with the nested fraction bars drawn aligned, for example by [Alfred Pringsheim](https://bliptext.com/articles/alfred-pringsheim) as\nor in more common related notations as\nor\n[Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) used a notation reminiscent of [summation notation](https://bliptext.com/articles/summation),\nor in cases where the numerator is always 1, eliminated the fraction bars altogether, writing a list-style\nSometimes list-style notation uses angle brackets instead,\nThe semicolon in the square and angle bracket notations is sometimes replaced by a comma.\nOne may also define infinite simple continued fractions as [limits](https://bliptext.com/articles/limit-of-a-sequence):\nThis limit exists for any choice of a_0 and positive integers.\n\n# Calculating continued fraction representations\nConsider a real number r. Let and let f = r - i. When f\\neq 0, the continued fraction representation of r is [i;a_1,a_2,\\ldots], where is the continued fraction representation of 1/f.\nWhen r\\geq 0, then i is the [integer part](https://bliptext.com/articles/integer-part) of r, and f is the [fractional part](https://bliptext.com/articles/fractional-part) of r.\nIn order to calculate a continued fraction representation of a number r, write down the [floor](https://bliptext.com/articles/floor-function) of r. Subtract this value from r. If the difference is 0, stop; otherwise find the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the difference and repeat.\nThe procedure will halt if and only if r is rational.\nThis process can be efficiently implemented using the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) when the number is rational.\nThe table below shows an implementation of this procedure for the number 3.245 = 649/200:\n!Step\n!Real Number\n!Integer part\n!Fractional part\n!Simplified\n!Reciprocal of f !1\n!2 !3 !4\nThe continued fraction for 3.245 is thus or, expanded:\n\n# Reciprocals\nThe continued fraction representations of a positive rational number and its [reciprocal](https://bliptext.com/articles/multiplicative-inverse) are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively.\nIn other words, the numbers represented by and are reciprocals.\nFor instance if a is an integer and x 1 then\nThe last number that generates the remainder of the continued fraction is the same for both x and its reciprocal.\nFor example,\n\n# Finite continued fractions\nEvery finite continued fraction represents a [rational number](https://bliptext.com/articles/rational-number), and every [rational number](https://bliptext.com/articles/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.\nThese two representations agree except in their final terms.\nIn 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.\nThe final element in the short representation is therefore always greater than 1, if present.\nIn symbols:\n[a0; a1, a2, ..., an − 1, an, 1] = [a0; a1, a2, ..., an − 1, an + 1]\n.\n[a0; 1] = [a0 + 1]\n.\n\n# Infinite continued fractions and convergents\nEvery infinite continued fraction is [irrational](https://bliptext.com/articles/irrational-number), and every [irrational](https://bliptext.com/articles/irrational-number) number can be represented in precisely one way as an infinite continued fraction.\nAn infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number.\nThese rational numbers are called the **convergents** of the continued fraction.\nThe larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated.\nNumbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers.\nOther numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally.\nThe [golden ratio](https://bliptext.com/articles/golden-ratio) Φ has terms equal to 1 everywhere—the smallest values possible—which makes Φ the most difficult number to approximate rationally.\nIn this sense, therefore, it is the \"most irrational\" of all irrational numbers.\nEven-numbered convergents are smaller than the original number, while odd-numbered ones are larger.\nFor a continued fraction\n[a0; a1, a2, ...]\n, the first four convergents (numbered 0 through 3) are\nThe 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.\nThe denominators are formed similarly.\nTherefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain [multivariate polynomials](https://bliptext.com/articles/multivariate-polynomial) called [continuants](https://bliptext.com/articles/continuant-mathematics).\nIf successive convergents are found, with numerators h1, h2, ... and denominators k1, k2, ...\nthen the relevant recursive relation is that of [Gaussian brackets](https://bliptext.com/articles/gaussian-brackets):\nThe successive convergents are given by the formula\nThus to incorporate a new term into a rational approximation, only the two previous convergents are necessary.\nThe initial \"convergents\" (required for the first two terms) are 0⁄1 and 1⁄0.\nFor example, here are the convergents for [0;1,5,2,2].\n! n !\nan\n!\nhn\n!\nkn\nWhen using the [Babylonian method](https://bliptext.com/articles/methods-of-computing-square-roots) 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.\nSpecifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ...\n,\n2k−1\n, ... For example, the continued fraction expansion for [\\sqrt3](https://bliptext.com/articles/square-root-of-3) is\n[1; 1, 2, 1, 2, 1, 2, 1, 2, ...]\n. Comparing the convergents with the approximants derived from the Babylonian method:\n! n !\nan\n!\nhn\n!\nkn\n# [Convergents approaching the golden ratio | upload.wikimedia.org/wikipedia/commons/b/b4/Golden///ration///convergents.svg]\n\n## Properties\nThe [Baire space](https://bliptext.com/articles/baire-space-set-theory) is a topological space on infinite sequences of natural numbers.\nThe infinite continued fraction provides a [homeomorphism](https://bliptext.com/articles/homeomorphism) from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the [usual topology](https://bliptext.com/articles/euclidean-topology) on the reals).\nThe infinite continued fraction also provides a map between the [quadratic irrationals](https://bliptext.com/articles/quadratic-irrational) and the [dyadic rationals](https://bliptext.com/articles/dyadic-rational), and from other irrationals to the set of infinite strings of binary numbers (i.e. the [Cantor set](https://bliptext.com/articles/cantor-set)); this map is called the [Minkowski question-mark function](https://bliptext.com/articles/minkowski-question-mark-function).\nThe mapping has interesting self-similar [fractal](https://bliptext.com/articles/fractal) properties; these are given by the [modular group](https://bliptext.com/articles/modular-group), which is the subgroup of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) having integer values in the transform.\nRoughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) [upper half-plane](https://bliptext.com/articles/upper-half-plane); this is what leads to the fractal self-symmetry.\nThe 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](https://bliptext.com/articles/gauss-kuzmin-distribution).\n\n## Some useful theorems\nIf \\ a_0\\, a_1\\ , a_2\\ , \\ \\ldots\\ is an infinite sequence of positive integers, define the sequences \\ h_n\\ and \\ k_n\\ recursively:\n**Theorem 1.** For any positive real number \\ x\\\n**Theorem 2.** The convergents of \\ [\\ a_0\\ ; a_1\\, a_2\\ , \\ldots\\ ]\\ are given by\nor in matrix form,\n**Theorem 3.** If the \\ nth convergent to a continued fraction is then\nor equivalently\n**Corollary 1:** Each convergent is in its lowest terms (for if \\ h_n\\ and \\ k_n\\ had a nontrivial common divisor it would divide which is impossible).\n**Corollary 2:** The difference between successive convergents is a fraction whose numerator is unity:\n**Corollary 3:** The continued fraction is equivalent to a series of alternating terms:\n**Corollary 4:** The matrix\nhas [determinant](https://bliptext.com/articles/determinant) (-1)^{n+1}, and thus belongs to the group of [unimodular matrices](https://bliptext.com/articles/unimodular-matrix)\n**Corollary 5:** The matrix\nhas determinant (-1)^na_n, or equivalently,meaning that the odd terms monotonically decrease, while the even terms monotonically increase.\n**Corollary 6:** The denominator sequence satisfies the recurrence relation, and grows at least as fast as the [Fibonacci sequence](https://bliptext.com/articles/fibonacci-sequence), which itself grows like O(\\phi^n) where is the [golden ratio](https://bliptext.com/articles/golden-ratio).\n**Theorem 4.** Each (\\ sth) convergent is nearer to a subsequent (\\ nth) convergent than any preceding (\\ rth) convergent is.\nIn symbols, if the \\ nth convergent is taken to be then\nfor all\n**Corollary 1:** The even convergents (before the \\ nth) continually increase, but are always less than \\ x_n ~.\n**Corollary 2:** The odd convergents (before the \\ nth) continually decrease, but are always greater than \\ x_n ~.\n**Theorem 5.**\n**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.\n**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.\"Theorem 6: Consider the set of all open intervals with end-points . Denote it as \\mathcal C. Any open subset of is a disjoint union of sets from \\mathcal C.\"**Corollary:** The infinite continued fraction provides a homeomorphism from the Baire space to.\n\n# Semiconvergents\nIf\nare consecutive convergents, then any fractions of the form\nwhere m is an integer such that, are called semiconvergents, secondary convergents, or intermediate fractions.\nThe (m+1)-st semiconvergent equals the [mediant](https://bliptext.com/articles/mediant-mathematics) of the m-th one and the convergent.\nSometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., 0 0\n, that is closer to x than any approximation with a smaller or equal denominator.\nThe simple continued fraction for x can be used to generate all of the best rational approximations for x by applying these three rules:\nFor example, 0.84375 has continued fraction [0;1,5,2,2].\nHere are all of its best rational approximations.\n! Continued fraction\n! Rational approximation\n! Decimal equivalent\n! Error\nThe 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.\nThe \"half rule\" mentioned above requires that when ak is even, the halved term ak/2 is admissible if and only if\nThis is equivalent to:.\n[ak; ak − 1, ..., a1] > [ak; ak + 1, ...]\n.\nThe convergents to x are \"best approximations\" in a much stronger sense than the one defined above.\nNamely, n/d is a convergent for x if and only if\nhas the smallest value among the analogous expressions for all rational approximations m/c with\nc ≤ d\nso long as\nc y\nwhen k is odd.\nFor example, the number 355⁄113 has the continued fraction representations\nand thus 355⁄113 is a convergent of any number strictly between\n[3; 7, 15, 2]\n||=||\n688⁄219 ≈ 3.1415525\n[3; 7, 17]\n||=||\n377⁄120 ≈ 3.1416667\n\n## Legendre's theorem on continued fractions\nIn his Essai sur la théorie des nombres (1798), [Adrien-Marie Legendre](https://bliptext.com/articles/adrien-marie-legendre) derives a necessary and sufficient condition for a rational number to be a convergent of the continued fraction of a given real number.\nA consequence of this criterion, often called **Legendre's theorem** within the study of continued fractions, is as follows:\n**Theorem**.\nIf α is a real number and p, q are positive integers such that, then p/q is a convergent of the continued fraction of α.\n**Proof**.\nWe follow the proof given in [An Introduction to the Theory of Numbers](https://bliptext.com/articles/an-introduction-to-the-theory-of-numbers) by [G. H. Hardy](https://bliptext.com/articles/g-h-hardy) and [E. M. Wright](https://bliptext.com/articles/e-m-wright).\nSuppose α, p, q are such that, and assume that α > p/q.\nThen we may write, where 0 1, we conclude that the continued fraction expansion of α must be [a0; a1, ..., an, b0, b1, ...], where [b0; b1, ...] is the continued fraction expansion of ω, and therefore that pn/qn = p/q is a convergent of the continued fraction of α.\nThis theorem forms the basis for [Wiener's attack](https://bliptext.com/articles/wiener-s-attack), a polynomial-time exploit of the [RSA cryptographic protocol](https://bliptext.com/articles/rsa-cryptosystem) that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key n = pq satisfy p < q < 2p and the private key d is less than (1/3)n1/4).\n\n# Comparison\nConsider\nand\n. If k is the smallest index for which\nak\nis unequal to\nbk\nthen\nx < y\nif\n(−1)k(ak − bk) < 0\nand\ny < x\notherwise.\nIf there is no such k, but one expansion is shorter than the other, say\nand\nwith\nfor\n0 ≤ i ≤ n\n, then\nx < y\nif n is even and\ny < x\nif n is odd.\n\n# Continued fraction expansion of π and its convergents\nTo calculate the convergents of [π](https://bliptext.com/articles/pi) we may set\n, define\nand\n,\nand\n,\n. Continuing like this, one can determine the infinite continued fraction of π as The fourth convergent of π is [3;7,15,1] = 355⁄113 = 3.14159292035..., sometimes called [Milü](https://bliptext.com/articles/mil), which is fairly close to the true value of π.\nLet us suppose that the quotients found are, as above, [3;7,15,1].\nThe 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.\nThe first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3⁄1.\nThen, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22⁄7, which will be too large.\nMultiplying 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.\nThus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106.\nThe third convergent, therefore, is 333⁄106.\nWe proceed in the same manner for the fourth convergent.\nThe 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.\nIn this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:\nTo sum up, the pattern is\nThese convergents are alternately smaller and larger than the true value of π, and approach nearer and nearer to π.\nThe difference between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent.\nFor example, the fraction 22⁄7 is greater than π, but 22⁄7 − π is less than 1⁄7 × 106 = 1⁄742 (in fact, 22⁄7 − π is just more than 1⁄791 = 1⁄7 × 113).\nThe 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.\nThus the difference between 22⁄7 and 3⁄1 is 1⁄7, in excess; between 333⁄106 and 22⁄7, 1⁄742, in deficit; between 355⁄113 and 333⁄106, 1⁄11978, in excess; and so on.\nThe 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.\nInstead of the fractions written above, we have thus the series:\nThe first term, as we see, is the first fraction; the first and second together give the second fraction, 22⁄7; the first, the second and the third give the third fraction 333⁄106, and so on with the rest; the result being that the series entire is equivalent to the original value.\n# [The following Maple code will generate continued fraction expansions of pi | upload.wikimedia.org/wikipedia/commons/9/91/Continued///fraction///expansion///of///Pi///.png]\n\n# Non-simple continued fraction\nA non-simple continued fraction is an expression of the form\nwhere the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.\nTo illustrate the use of non-simple continued fractions, consider the following example.\nThe sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:\nor\nHowever, several non-simple continued fractions for π have a perfectly regular structure, such as:\nThe first two of these are special cases of the [arctangent](https://bliptext.com/articles/inverse-trigonometric-functions) function with π = 4 arctan (1) and the fourth and fifth one can be derived using the [Wallis product](https://bliptext.com/articles/wallis-product).\nThe continued fraction of \\pi above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.\n\n# Other continued fraction expansions\n\n## Periodic continued fractions\nThe numbers with periodic continued fraction expansion are precisely the [irrational solutions](https://bliptext.com/articles/quadratic-irrational) of [quadratic equations](https://bliptext.com/articles/quadratic-equation) with rational coefficients; rational solutions have finite continued fraction expansions as previously stated.\nThe simplest examples are the [golden ratio](https://bliptext.com/articles/golden-ratio) φ = [1;1,1,1,1,1,...] and √2 = [1;2,2,2,2,...], while √14 = [3;1,2,1,6,1,2,1,6...] and √42 = [6;2,12,2,12,2,12...].\nAll irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for √2) or 1,2,1 (for √14), followed by the double of the leading integer.\n\n## A property of the golden ratio φ\nBecause the continued fraction expansion for [φ](https://bliptext.com/articles/golden-ratio) doesn't use any integers greater than 1, [φ](https://bliptext.com/articles/golden-ratio) is one of the most \"difficult\" real numbers to approximate with rational numbers.\n[Hurwitz's theorem](https://bliptext.com/articles/hurwitz-s-theorem-number-theory) states that any irrational number k can be approximated by infinitely many rational m⁄n with\nWhile virtually all real numbers k will eventually have infinitely many convergents m⁄n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5⁄3, 8⁄5, 13⁄8, 21⁄13, 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, [355⁄113](https://bliptext.com/articles/mil) for [π](https://bliptext.com/articles/pi).\nIt can also be shown that every real number of the form a + bφ⁄c + dφ, where a, b, c, and d are integers such that\na d − b c = ±1\n, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.\n\n## Regular patterns in continued fractions\nWhile there is no discernible pattern in the simple continued fraction expansion of π, there is one for\ne\n, the [base of the natural logarithm](https://bliptext.com/articles/e-mathematical-constant):\nwhich is a special case of this general expression for positive integer n:\nAnother, more complex pattern appears in this continued fraction expansion for positive odd n:\nwith a special case for\nn = 1\nOther continued fractions of this sort are\nwhere n is a positive integer; also, for integer n:\nwith a special case for\nn = 1\nIf\nIn(x)\nis the modified, or hyperbolic, [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind, we may define a function on the rationals p⁄q by\nwhich is defined for all rational numbers, with p and q in lowest terms.\nThen for all nonnegative rationals, we have\nwith similar formulas for negative rationals; in particular we have\nMany of the formulas can be proved using [Gauss's continued fraction](https://bliptext.com/articles/gauss-s-continued-fraction).\n\n## Typical continued fractions\nMost irrational numbers do not have any periodic or regular behavior in their continued fraction expansion.\nNevertheless, for [almost all](https://bliptext.com/articles/almost-all) numbers on the unit interval, they have the same limit behavior.\nThe arithmetic average diverges:, and so the coefficients grow arbitrarily large:.\nIn particular, this implies that almost all numbers are well-approximable, in the sense that[Khinchin](https://bliptext.com/articles/aleksandr-khinchin) proved that the [geometric mean](https://bliptext.com/articles/geometric-mean) of\nai\ntends to a constant (known as [Khinchin's constant](https://bliptext.com/articles/khinchin-s-constant)):[Paul Lévy](https://bliptext.com/articles/paul-l-vy-mathematician) proved that the nth root of the denominator of the nth convergent converges to [Lévy's constant](https://bliptext.com/articles/l-vy-s-constant)\n[Lochs' theorem](https://bliptext.com/articles/lochs-theorem) states that the convergents converge exponentially at the rate of\n\n# Applications\n\n## Pell's equation\nContinued fractions play an essential role in the solution of [Pell's equation](https://bliptext.com/articles/pell-s-equation).\nFor example, for positive integers p and q, and non-square n, it is true that if\np2 − nq2 = ±1\n, then\np⁄q\nis a convergent of the regular continued fraction for √n.\nThe converse holds if the period of the regular continued fraction for √n is 1, and in general the period describes which convergents give solutions to Pell's equation.\n\n## Dynamical systems\nContinued fractions also play a role in the study of [dynamical systems](https://bliptext.com/articles/dynamical-system), where they tie together the [Farey fractions](https://bliptext.com/articles/farey-sequence) which are seen in the [Mandelbrot set](https://bliptext.com/articles/mandelbrot-set) with [Minkowski's question-mark function](https://bliptext.com/articles/minkowski-s-question-mark-function) and the [modular group](https://bliptext.com/articles/modular-group) Gamma.\nThe backwards [shift operator](https://bliptext.com/articles/shift-operator) for continued fractions is the map\ncalled the **[Gauss map](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator)**, which lops off digits of a continued fraction expansion:\nh([0; a1, a2, a3, ...]) = [0; a2, a3, ...]\n. The [transfer operator](https://bliptext.com/articles/transfer-operator) of this map is called the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator).\nThe distribution of the digits in continued fractions is given by the zero'th [eigenvector](https://bliptext.com/articles/eigenvector) of this operator, and is called the [Gauss–Kuzmin distribution](https://bliptext.com/articles/gauss-kuzmin-distribution).\n\n# History","revisionCount":0},"hasMore":false,"nextCursor":null}