Continued Fraction

Continued Fraction

A finite continued fraction, where n is a non-negative integer, a0 is an integer, and ai is a positive integer, for i=1,…,n.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every 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). The numerical value of an infinite continued fraction will be irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.

The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term see Padé approximation and Chebyshev rational functions.

Read more about Continued Fraction:  Motivation and Notation, Basic Formulae, Calculating Continued Fraction Representations, Notations For Continued Fractions, Finite Continued Fractions, Continued Fractions of Reciprocals, Infinite Continued Fractions, Some Useful Theorems, Semiconvergents, Best Rational Approximations, Comparison of Continued Fractions, Continued Fraction Expansions of π, Generalized Continued Fraction, Generalized Continued Fraction For Square Roots, Pell's Equation, Continued Fractions and Chaos, Eigenvalues and Eigenvectors, History of Continued Fractions

Other articles related to "continued fraction, continued fractions":

Convergence Problem - Elementary Results - Worpitzky's Theorem
... The continued fraction x converges to a finite value, and converges uniformly if the partial numerators ai are complex variables ... Ω is the smallest image space that contains all possible values of the continued fraction x ... is apparently the oldest published proof that a continued fraction with complex elements actually converges ...
History of Continued Fractions
... the greatest common divisor which generates a continued fraction as a by-product 499 The Aryabhatiya contains the solution of indeterminate equations using continued fractions 1579 Rafael Bombelli, L'Alg. 1695 John Wallis, Opera Mathematica – introduction of the term "continued fraction" 1737 Leonhard Euler, De fractionibus continuis dissertatio – Provided the first then-comprehensive account of ... proved the equivalence of a certain form of continued fraction and a generalized infinite series, proved that every rational number can be written as a finite continued fraction, and proved that the ...
Pell's Equation - Solutions - Fundamental Solution Via Continued Fractions
... Let denote the sequence of convergents to the continued fraction for ... Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is ... As Lenstra (2002) describes, the time for finding the fundamental solution using the continued fraction method, with the aid of the Schönhage–Strassen ...
List Of Representations Of E - As A Continued Fraction
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Lochs' Theorem
... theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number ... numbers in the interval (0,1), the number of terms m of the number's continued fraction expansion that are required to determine the first n places of the ... than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the ...

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