A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence { a i } {\displaystyle...

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or...

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

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

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

For the continued fraction expansion of a number, see simple continued fraction, of a function, see continued fraction. This disambiguation page lists...

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

continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such...

a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction...

irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (with numerators all 1, shown above) also...

seven years later). He computed the representation of e as a simple continued fraction, which is e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8...

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

digits seem to indicate that it could be a normal number. The simple continued fraction expansion of Euler's constant is given by: γ = 0 + 1 1 + 1 1 +...

a_{n}=34n^{3}+51n^{2}+27n+5} and b n = − n 6 {\displaystyle b_{n}=-n^{6}} . Its simple continued fraction is given by: ζ ( 3 ) = 1 + 1 4 + 1 1 + 1 18 + 1 1 + 1 1 + 1 …...

} .: 368  For a real number x {\displaystyle x} given by its simple continued fraction expansion x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1}...

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

Friedrich Gauss to represent the convergents of a simple continued fraction in the form of a simple fraction. Gauss used this notation in the context of finding...

of b ( n ) {\displaystyle b(n)} for all n {\displaystyle n} . Simple continued fractions for the lemniscate constant and related constants include ϖ =...

Euler proved that the number e is represented as the infinite simple continued fraction (sequence A003417 in the OEIS): e = [ 2 ; 1 , 2 , 1 , 1 , 4 ,...

{x*(1+8x+x^{2})}{(1-x)^{3}}}} 5 C D n {\displaystyle {\sqrt {5CD_{n}}}} has the simple continued fraction [5n-3;{2,2n-2,2,10n-6}]. [ordinary] decagonal number Deza, Elena;...

represented with a simple repeating pattern of integers. The square bracket notation used above is a short form for a continued fraction. Written in the...

factor of 99999. the smallest integer whose square root has a simple continued fraction with period 3. a prime index prime, as 13 is prime. In Mexico...

a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the simple continued fraction e =...

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

5 {\textstyle 1\times 10^{-5}} ). It can be expressed as the simple continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS). So...

This problem was solved during the 18th century by means of simple continued fractions. Knowing the "best" approximations of a given number, the main...

Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular Aleksandr Yakovlevich...

not known whether this constant is rational or irrational. Its simple continued fraction is given by [ 0 ; 1 , 1 , 1 , 1 , 1 , 22 , 1 , 2 , 3 , 1 , . ...

{5^{4}}{24+{\cfrac {7^{4}}{32+{\cfrac {9^{4}}{40+\ddots }}}}}}}}}}}}} The simple continued fraction is given by: G = 1 1 + 1 10 + 1 1 + 1 8 + 1 1 + 1 88 + ⋱ {\displaystyle...

sympathetic to him. In the following year Galois's first paper, on simple continued fractions, was published. It was at around the same time that he began making...