{1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}} In mathematics, a continued fraction is an expression obtained through an iterative process of representing...

a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial...

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a...

Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions...

In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning...

a common fraction. But every number, including π, can be represented by an infinite series of nested fractions, called a continued fraction: π = 3 + 1...

continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction...

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English...

The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related...

truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal...

analytical theory of continued fractions. Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with...

{\displaystyle \varphi =1+1/\varphi } can be expanded recursively to obtain a continued fraction for the golden ratio: φ = [ 1 ; 1 , 1 , 1 , … ] = 1 + 1 1 + 1 1 +...

irreducible quadratic). A simple continued fraction is a continued fraction where the denominator is 1. The simple continued fraction expansion of Champernowne's...

a rational number is a number that can be expressed as the quotient or fraction ⁠ p q {\displaystyle {\tfrac {p}{q}}} ⁠ of two integers, a numerator p...

continued fractions algorithm to find integers b {\textstyle b} and c {\textstyle c} , where b c {\textstyle {\frac {b}{c}}} gives the best fraction approximation...

based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of 2 {\displaystyle {\sqrt {2}}} . Despite having a smaller...

continued fraction of a given real number. A consequence of this criterion, often called Legendre's theorem within the study of continued fractions,...

"two twos"), or equivalently "2 - 2", is the only fixed point. A continued fraction for e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 8 , . . . ] {\displaystyle...

[citation needed] In the following year Galois's first paper, on continued fractions, was published. It was at around the same time that he began making...

integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense...

b)&=(-1)^{a}\mathrm {B} \left({\frac {x}{x-1}};a,1-a-b\right)\end{aligned}}} The continued fraction expansion B ( x ; a , b ) = x a ( 1 − x ) b a ( 1 + d 1 1 + d 2 1...

between numbers in the Stern–Brocot tree may be defined in terms of continued fractions or mediants, and a path in the tree from the root to any other number...

transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator...

represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series,...

convergents of a continued fraction share the same property: If pk−1/qk−1 and pk/qk are two successive convergents of a continued fraction, then the matrix...

Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction...

} Every real number can be essentially uniquely represented as a continued fraction, namely as the sum of its integer part and the reciprocal of its fractional...

Rational approximations of square roots may be calculated using continued fraction expansions. The method employed depends on the needed accuracy, and...

tables: The following continued fraction R ( q ) {\displaystyle R(q)} is called Rogers-Ramanujan continued fraction, Continuing fraction S ( q ) {\displaystyle...

"Exponential integral E: Continued fraction representations (Formula 06.34.10.0005)". "Exponential integral E: Continued fraction representations (Formula...