{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...

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

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

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...

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,...

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...

"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...

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...

[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...

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,...

} 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...

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

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...

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

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

rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals...