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Topic: Continued fraction

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	math lessons - Continued fraction
		Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers.
		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.
		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.
	www.mathdaily.com /lessons/Continued_fraction (2024 words)

	continued fraction - Article and Reference from OnPedia.com
		Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers.
		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.
		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.
	www.onpedia.com /encyclopedia/continued-fraction (2106 words)

	Continued fractions - LoveToKnow 1911
		In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the I last partial quotient, a n as a n - I +1.
		Thus a continued fraction equivalent to 7r (the ratio of the circumference to the diameter of a circle) is I I I I II 3+ 7+15+7+292+i-1-i+..
		Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. Serret's Cours d'Algbbre Superieure.
	www.1911encyclopedia.org /Continued_fractions (3103 words)

	Continued fraction - Wikipedia, the free encyclopedia
		Continued fraction notation is a representation for the real numbers that avoids both these problems.
		The continued fraction representations of a 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.
		The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with rational coefficients.
	en.wikipedia.org /wiki/Continued_fraction (2810 words)

	Numbers and Functions as Continued Fractions - Numericana
		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.
		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].
	home.att.net /~numericana/answer/fractions.htm (3600 words)

	Elementary Introduction to Continued Fractions
		It is a consequence of the Euclidean Algorithm that the continued fraction representation of all rational fractions must terminate.
		When this happens, we call the continued fraction "periodic." Joseph Louis Lagrange discovered in 1770 A.D. that the continued fraction expansion of any quadratic irrational is periodic after a while.
		The convergents in Continued Fractions can also be employed to find successively closer approximations of the roots - that is, values of x that satisfy f(x)=0) - of a function f of x, provided f has continuous derivatives of several orders[1].
	www.mathpath.org /concepts/cont.frac.htm (785 words)

	A little about Continued Fractions (Site not responding. Last check: 2007-11-07)
		Continued fractions have been used for an amazing variety of things, including solving quadratic equations, proving that Pi is irrational, calculating the date of Easter, predicting eclipses, and investigating chaos.
		General continued fractions are sometimes used in analysis to find the value of special functions in regions where power series converge poorly.
		General continued fractions also crop up when one is inverting a tridiagonal matrix, and as a substitute for the transfer matrix method.
	ourworld.cs.com /christopherereed/confracs.htm (913 words)

	On Shanks' algorithm for computing the continued fraction of log_b a (Site not responding. Last check: 2007-11-07)
		On Shanks' algorithm for computing the continued fraction of log_b a
		We give a more practical variant of Shanks' 1954 algorithm for computing the continued fraction of log_b a, for integers a > b > 1, using the floor and ceiling functions and an integer parameter c > 1.
		The variant, when repeated for a few values of c = 10^r, enables one to guess if log_b a is rational and to find approximately r partial quotients.
	www.cs.uwaterloo.ca /journals/JIS/VOL5/Jackson/matthews3.html (115 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)

	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.
		All of the above fractions are periodic in the sense we apply to decimal fractions.
	www.cut-the-knot.com /do_you_know/fraction.html (609 words)

	Continued Fractions
		The whole continued fraction itself is an element of the field that is a superset of the set containing the partial numerators and denominators.
		These two theorems are the only ones from the theory of finite continued fractions, although they apply to infinite continued fractions as well, that will be needed for the study of families of continued fractions.
		Groups of continued fractions whose periods are predictable and static abound, but the more interesting groups whose periods grow are rare and harder to detect.
	www.u.arizona.edu /~miller/webreport/node2.html (583 words)

	Stern-Brocot Tree and Continued Fractions
		As we saw, the left-side encoding is useful in determining location of a fraction on the Stern-Brocot tree.
		In the Stern-Brocot tree, every fraction (except for 1/0 and 0/1) has two children: the left offspring and the right offspring that are located just below the fraction, one a little to the left, the other a little to the right from their parent.
		These are the fractions whose mediant equals the given one.
	www.cut-the-knot.org /blue/ContinuedFractions.shtml (814 words)

	Chaos in Numberland: The secret life of continued fractions
		Continued fractions also provide us with a way of constructing rational approximations to irrational numbers and discovering the most irrational numbers.
		Continued fractions first appeared in the works of the Indian mathematician Aryabhata in the 6th century.
		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.
	pass.maths.org.uk /issue11/features/cfractions/index.html (3357 words)

	In today’s world of e-commerce and mass digital communication, there is a clear need for the ability to securely ... (Site not responding. Last check: 2007-11-07)
		As the remainder is a proper fraction with a as the denominator, r
		Irrational numbers can be rewritten as continued fractions by repetitively extracting the largest integer and leaving a fractional remainder in terms of the original number.
		While the algorithm is longer than that for brute force factoring, the continued fraction factoring method has shown itself in examples to be a more efficient process for determining factors of a large number.
	www.mit.edu /~rkabir/research/Fractions.htm (3138 words)

	[No title]
		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.
		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.
		The input to this routine is the secondary sequence of the continued fraction.
	www.ics.uci.edu /~eppstein/numth/egypt/cfrac.html (2028 words)

	Continued Fractions
		It isn't only roots of quadratic equations that have simple patterns in their continued fraction expansions, though a periodic expansion implies that the number is the root of a quadratic equation.
		For instance, the continued fraction for e goes [2,1,2,1,1,4,1,1,6,1,1,...], which is better written as [1,0,1, 1,2,1, 1,4,1, 1,6,1,...].
		The sizes of coefficients in the continued fraction expansion are closely related to how closely a number can be approximated by rationals.
	c2.com /cgi/wiki?ContinuedFractions (970 words)

	Continued Fractions - An introduction
		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.
		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.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/cfINTRO.html (8265 words)

	Continued Fraction
		Continued fraction is a very useful tool of analysis.
		Continued fraction has nice property that it converge very fast (some are better than power series) and it can be useful to approximate the real number as ratio of two integers.
		This tutorial is an introduction to regular continued fraction (finite, infinite and periodic) and its application to convert decimal to fraction, compute Pi and Euler number.
	people.revoledu.com /kardi/tutorial/ContinuedFraction/index.html (95 words)

	CONTINUED FRACTIONS WITH A CALCULATOR (Site not responding. Last check: 2007-11-07)
		Should even this become boring, give the expression for tan z as a Continued Fraction and try it out for several values of z in radian measure.
		Barnes and Noble's Dictionary of Mathematics--1972, Millington and Millinton-- defines a continued fraction as "an integer and a fraction, the denominator of which is also an integer and a fraction, etc." This definition is the springboard for my lesson.
		Conversely every infinite continued fraction in an irrational number." I shall not get to these ideas in this introductory mini-teach, but they could be dealt with in a future class, should sufficient interest arise.
	www.iit.edu /~smile/ma8621.html (398 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.
		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.
		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.
	en.wikipedia.org /wiki/Generalized_continued_fraction (422 words)

	Continued fractions
		The history of continued fractions can be traced back to an algorithm of Euclid.
		The speed of convergence of continued fractions to a number they represent varies from number to number (but it is always very very fast).
		Now that you know that continued fractions are very good at approximating numbers rational and irrational, it is not surprising to find them in many unusual (at first glance) places.
	www.math.temple.edu /~yury/calendar/node2.html (586 words)

	Continued Fractions, Math 413 (Number Theory)
		This is the case for most continued fraction expansions of irrational numbers.
		Periodic continued fractions are expressed by using the form {tail, {cycle}}, and are correctly converted back to quadratic surds.
		Maple Commands for continued fraction expansions are implemented in the default library with convert[confrac] and in the numtheory package with the cfrac command.
	www.math.umbc.edu /~campbell/Math413Spr05/Notes/ContinuedFractions.html (1168 words)

	Section 13.1: Approximation with Continued Fractions
		As we saw in the Prelab, continued fractions provide a good means for approximating real numbers by rational numbers.
		The applet takes two arguments as input: the first is the real number we want to approximate, and the second is the number of terms to use from that real number's continued fraction expansion.
		A good approach is to look at the convergents from the continued fraction expansion of our number, because they will be rational numbers which give good approximations to the value.
	www.math.mtu.edu /mathlab/COURSES/holt/dnt/pell1.html (598 words)

	CONTINUED FRACTIONS
		The continued fraction of the square root of 2 is
		The continued fraction of R is studied largely through the behavior of a sequence of rational numbers called the convergents to R. To define convergents, begin with
		One of the main applications of continued fractions is to best approximates of irrational numbers by rational numbers.
	faculty.evansville.edu /ck6/integer/contfr.html (693 words)

	[No title]
		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.
		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).
	www.math.niu.edu /~rusin/known-math/98/c1 (1248 words)

	Continued Fractions - Introduction
		For many, this may be a first introduction to continued fractions since this subject, if it is taught at all, is restricted to a single chapter in a number theory text.
		Let X be a continued fraction that is represented by n+1 terms.
		But B is a continued fraction with n terms, and by our induction hypothesis, it can be written as a rational p/q.
	archives.math.utk.edu /articles/atuyl/confrac/intro.html (1227 words)

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