Continued-fraction

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Topic: Continued-fraction

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	Continued fraction - Wikipedia, the free encyclopedia
		Continued fractions also play a role in the study of chaos, where they tie together the Farey fractions which are seen in the Mandelbrot set with the Minkowski question mark function and the modular group Gamma.
		Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers.
		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.
	en.wikipedia.org /wiki/Continued_fraction (2538 words)

	Generalized continued fraction - Wikipedia, the free encyclopedia
		In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity.
		For example, there is a close relationship between the continued fraction for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx.
		If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than x but uniformly decrease; and all convergents of even order are less than x but uniformly increase.
	en.wikipedia.org /wiki/Generalized_continued_fraction (378 words)

	PlanetMath: continued fraction
		This is version 23 of continued fraction, born on 2002-06-12, modified 2006-02-24.
		Any rational number is the value of two and only two finite continued fractions; in one of them, the last denominator is 1.
		For one more example, the distribution of leap years in the 4800-month cycle of the Gregorian calendar can be interpreted (loosely speaking) in terms of the continued fraction expansion of the number of days in a solar year.
	planetmath.org /encyclopedia/ContinuedFraction.html (447 words)

	Continued Fractions
		Continued fraction are extremely important in the theory of rational approximation.
		With this convention, the correspondence between rational numbers and finite continued fractions becomes 1-1.
		For example, for a continued fraction (either finite or infinite) one defines a family of finite segments
	www.cut-the-knot.org /do_you_know/fraction.shtml (609 words)

	Numbers and Functions as Continued Fractions - Numericana
		Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals [also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral coefficients].
		An Introduction to Continued Fractions by Ron Knott of the University of Surrey.
		The rational value whose [finite] continued fraction expansion is a truncation of the continued fraction expansion of a given number is called a convergent of that number.
	home.att.net /~numericana/answer/fractions.htm (3600 words)

	Chaos in Numberland: The secret life of continued fractions
		Continued fractions are a forgotten part of our mathematical education but their properties are vital guides to approximation and important probes of the complexities of dynamical chaos.
		Continued fractions first appeared in the works of the Indian mathematician Aryabhata in the 6th century.
		Continued fractions also provide us with a way of constructing rational approximations to irrational numbers and discovering the most irrational numbers.
	plus.maths.org /issue11/features/cfractions (3357 words)

	Continued Fractions
		Continued fractions were developed (or discovered) in part as a response to a need to approximate irrational numbers.
		Because continued fractions are be developed using Euclid's algorithm it is tempting to believe that the great geometrician used them, but this is probably not true.
		An important fact about continued fractions is that they can be used to represent any real number; a fact which should be stated as a formal theorem.
	www.petrospec-technologies.com /Herkommer/contfrac.htm (1773 words)

	Math Forum - Ask Dr. Math
		This continued fraction may also be written; 1 1+_________ 1 2+_________ 1 3+_________ 1 1+________ 1+.......
		The answer cannot be negative (since all terms in the continued fraction are positive) so you know to choose the positive root.
		Date: 05/15/2002 at 09:11:16 From: Doctor Paul Subject: Re: infinite continued fraction Hi, Let x = [1, 2, 3, 1, 2, 3,...] Then 1 x = ----------- x1 where x1 = [2, 3, 1, 2, 3,...].
	mathforum.org /library/drmath/view/60729.html (236 words)

	HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
		On the theory that continued fractions are underused, probably because of their unfamiliarity, I offer the following propaganda session on the relative merits of continued fractions versus other numerical representations.
		The familiar transcendental functions of rational arguments also have simple continued fractions, but these are generally not regular and cannot be reconstructed from numerical values by a simple algorithm, since nonregular representations aren't unique.
		Thus, to "round off" a continued fraction after a certain term, add in the next term iff it is +-1.
	www.inwap.com /pdp10/hbaker/hakmem/cf.html (2805 words)

	Continued Fractions - An introduction
		The square root as a continued fraction is the initial whole number from Step 1 and the period is all the numbers but adding the final integer of Step 4 to the initial integer to form the period.
		Of all continued fractions, this is the simplest.
		Then express this decimal fraction as an ordinary fraction: use a large enough power of 10 as the denominator so that the numerator and denominator are integers, e.g.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/cfINTRO.html (8242 words)

	cfrac.html
		The input to this routine is the secondary sequence of the continued fraction.
		The number of terms in the Egyptian fraction representation of x/y is the sum of the odd terms after the first in the continued fraction list, which is at most x.
		Termination of the algorithm follows from the termination of the continued fraction representation algorithm, which is essentially the same as Euclid's algorithm for integer GCD's.
	www.ics.uci.edu /~eppstein/numth/egypt/cfrac.html (2028 words)

	This note's for you
		The continued fraction expansion is [2, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18] The convergents are: 7/3, 65/28, 137/59, 339/146, 1493/643,...
		The continued fraction expansion for log[2](3) is [1,1,1,2,2,3,1,5,2,23,2,2,1,...] (see sequence A028507 of the On-Line Encyclopedia of Integer Sequences for more terms).
		Remarkably, this corresponds to the third convergent of the continued fraction expansion.
	www.research.att.com /~njas/sequences/DUNNE/TEMPERAMENT.HTML (3574 words)

	c1
		Including a_0 among the partial quotients being restricted to {1,2} gives a set whose smallest element has continued fraction [_1,2_] and largest element [_2,1_] (where the _ are used to mark a periodic block).
		I'd be interested in hearing more!) For almost all x, and for all a, the frequency with which a appears as a partial quotient in the continued fraction expansion of x is log base 2 of ((a + 1)^2 / (a + 1)^2 - 1).
		What he means is that the set of >a[i] in the continued fraction decomposition are unbounded.
	www.math.niu.edu /~rusin/known-math/98/c1 (1248 words)

	cf
		The continued fraction of e has a nice pattern, but it is not particularly rapidly convergent.
		The top part of the Markov spectrum has numbers whose >continued fraction expansions, in order to converge so very slowly, >are forced to have only partial quotients of just 1 or 2, and also more >delicately to be periodic.
		Those continued fraction expansions >converge more slowly than some quadratic irrationalities, despite >the number being transcendental, and not a lot better than the >golden ratio.
	www.math.niu.edu /~rusin/known-math/99/cf (535 words)

	Generalized continued fraction - Wikipedia, the free encyclopedia
		In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity.
		For example, there is a close relationship between the continued fraction for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx.
		If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than x but uniformly decrease; and all convergents of even order are less than x but uniformly increase.
	en.wikipedia.org /wiki/Generalized_continued_fraction (368 words)

	Continued Fractions
		] This is a representation of a finite Continued Fraction.
		The Finite Continued Fraction represents a rational number and can be calculated using the Euclidean Algorithm.
		A decimal number must be able to have a Continued Fraction representation or it is not a legitimate decimal number.
	home.earthlink.net /~usondermann/contfrac.html (239 words)

	Simple Continued Fraction Expansion of Pi
		In "simple" continued fractions, all the b's are 1 and the number can be re-written as [a0; a1, a2, a3,...].
		C.D. Olds actually gave 23 terms of the simple continued fraction expansion of pi and, in late 1984, I complained about this (paucity of data) to Martin Gardner, then mathematical-games columnist for Scientific American.
		Let [0; 1, 2, 3, 4, 5,...] represent the indices of the continued fraction terms [3; 7, 15, 1, 292, 1,...], or (better, perhaps) think of the terms as having dropped the initial '3'.
	chesswanks.com /pxp/cfpi.html (1110 words)

	Citebase - Multidimensional continued fraction and rational approximation
		The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction.
		Cheng, On the continued fraction and Berlykamp's algorithm, Ieee Trans.Infor.
		E.V.Podsypanin, A generalization of continued fraction algorithm that is related to ViggoBorun algorithm (Russian), Studies in Number Theory (LOMI), Vol.4, Zap.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0401141 (570 words)

	SanAntonio.html
		We hope that bringing continued fraction folks from both number theory and analytic theory together will introduce techniques and methods used in one area to people in the other area, to make each group more aware of the ongoing research in each field and, finally, to stimulate research possibilities.
		We are interested in having talks that will discuss some topic in either number theory or analytic theory of continued fractions.
		At the next Joint Annual Mathematics Meeting, which will be held in San Antonio from January 12-15, 2006, there will be an American Mathematical Society special session on Continued Fractions.
	www.trincoll.edu /~jmclaugh/SanAntonio.html (280 words)

	Continued Education
		Generalized continued fraction 1: action''' is a generalization of the concept of continued fraction in which the numerators are allowed to 3: A generalized continued fraction is an expression such as: 14: nvergents are formed in a similar way to those of continued fractions.
		Continued fraction 1: In mathematics, a '''continued fraction''' is an expression such as 7: nity, the resulting expression is a generalized continued fraction.
		If all $\pm$ signs are pos 27: he positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are great 29: The condition for the limits to be equal (and the continued fraction to have a definite value) is
	www.witchware.com /File/15001-Continued.Education.Html (518 words)

	Numbers and Functions as Continued Fractions - Numericana
		An Introduction to Continued Fractions by Ron Knott of the University of Surrey.
		Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals [also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral coefficients].
		The rational value whose [finite] continued fraction expansion is a truncation of the continued fraction expansion of a given number is called a convergent of that number.
	home.att.net /~numericana/answer/fractions.htm (3614 words)

	Citebase - Real Numbers With Polynomial Continued Fraction Expansions
		This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like π and e, ζ(k) (for each positive integer k≥ 2), various special functions evaluated at integral arguments and various algebraic numbers.
		We also pose several questions about the nature of the set of real numbers which have a polynomial continued fraction expansion.
		[7] Lisa Lorentzen and Haakon Waadeland, Continued Fractions with Applica tions, North-Holland, Amsterdam-London-New York-Tokyo, 1992.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0402462 (413 words)

	Continued Fraction Method
		A way to find integer square roots is to use continued fractions.
		This was the fastest algorithm before the Quadratic Sieve method took over.
	www.frenchfries.net /paul/factoring/theory/cont.frac.html (110 words)

	Continued fraction objects
		Suppose x is a continued fraction, possibly infinite
		But in 1972, Bill Gosper solved the problem of general arithmetic on continued fractions
		The work to calculate the remaining terms will not be done until we need it
	perl.plover.com /yak/cftalk/TALK/slide024.html (116 words)

	Search Encyclopedia.com
		fraction -> Arithmetic Operations Involving Fractions When fractions having the same denominator, as 3/10 and 4/10, are added, only the numerators are added, and their sum is then written over the common denominator: 3/10+4/10=7/10.
		Fractions having unlike denominators, e.g., 1/4 and 1/6, must first be converted into fractions having a common denominator, a denominator in...
		fraction fraction [Lat.,=breaking], in arithmetic, an expression representing a part, or several equal parts, of a unit.
	www.encyclopedia.com /searchpool.asp?target=Continued+fraction+factorization (116 words)

	Integer factorization - Wikipedia, the free encyclopedia
		In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number ; for example, given a number like 91, the challenge is to find a number such as 7 which divides it.
		As of 2005, the largest semiprime factored using general-purpose methods as part of public research is RSA-200, which is 663 bits long.
		A fast integer factorization algorithm would mean that the RSA public-key algorithm was insecure.
	en.wikipedia.org /wiki/Integer_factorization (116 words)

	This note's for you
		The continued fraction expansion is [2, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18] The convergents are: 7/3, 65/28, 137/59, 339/146, 1493/643,...
		The continued fraction expansion for log[2](3) is [1,1,1,2,2,3,1,5,2,23,2,2,1,...] (see sequence A028507 of the On-Line Encyclopedia of Integer Sequences for more terms).
		That is to say, this particular continued fraction does indeed converge to the irrational number it is supposed to represent.
	www.research.att.com /~njas/sequences/DUNNE/TEMPERAMENT.HTML (3574 words)

	Continued Fractions and Modular Forms
		However for continued fractions, two distinct cases have to be considered: the continuous and the discrete case.
		The main object of this lecture is the alternating sum of coefficients of a continued fraction.
		The alternating sum of coefficients of a continued fraction seems to be the first example where one needs not only upper bounds for sums of Kloosterman sums, but also their precise asymptotics.
	algo.inria.fr /seminars/sem99-00/vardi.html (1676 words)

	Continued Fractions - History
		The origin of continued fractions is traditionally placed at the time of the creation of Euclid's Algorithm.
		This brief sketch into the past of continued fractions is intended to provide an overview of the development of this field.
		Continued fractions have also been utilized within computer algorithms for computing rational approximations to real numbers, as well as solving indeterminate equations.
	archives.math.utk.edu /articles/atuyl/confrac/history.html (933 words)

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