Examples for

Continued Fractions

The continued fraction representation of a number is a sum of two terms. The first is the number's integer part. The second is recursively defined as the reciprocal of the continued fraction form of the reciprocal of the original number's fractional part. Rational numbers can be represented by finite continued fractions while irrationals require infinitely deep representations. Wolfram|Alpha can be used to convert between and utilize these representations. It also possesses knowledge about symbolic continued fractions as well as related theorems and algorithms.
Numbers

Convert between decimal and continued fraction representations of numbers.

Find the continued fraction representation of a number:

continued fraction 12/67
continued fraction sqrt(2)
continued fraction pi
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Functions

Convert between regular and continued fraction representations of functions.

Find continued fraction representations for a function:

continued fraction sqrt(1+x^2)
continued fraction tan x
continued fractions for BesselJ
hypergeometric cf identities
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Definitions of Terms

Learn about the meanings of terms and phrases relevant to the area of continued fractions.

Find definitions for continued fraction terminology:

what is a regular continued fraction
common notations for continued fractions
Gauss-Kuzmin-Wirsing operator
C-dual convergents
What is a modified Stieltjes fraction?
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Theorems

Learn about theorems related to continued fractions. Apply them and see where and how they are applied.

Get information about a continued fraction theorem:

Worpitzky theorem
Hamburger convergence theorem
Euler-Minding formula
Seidel multiplication theorem
circular convergence theorem
Zaremba conjecture

Find theorems that apply to a specified expression:

apply continued fraction theorems to sqrt(7)

Find theorems by application:

cf theorems that hold for regular continued fractions
continued fraction theorems involving almost all real numbers
continued fraction results involving quadratic irrationals

Find theorems by prover:

Euler's continued fraction theorems
Lagrange's continued fraction theorems
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Algorithms

Learn about and use different algorithms for finding and working with continued fraction representations.

Get information about a continued fraction algorithm:

backward algorithm for a continued fraction
nearest integer continued fraction expansion algorithm
Hurwitz continued fraction algorithm
Schmidt algorithm
Jacobi-Perron algorithm
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Literature

Search for literature relevant to the area of continued fractions.

Find continued fraction papers by author:

continued fraction papers by Stern

Find continued fraction papers by publication:

continued fraction papers published in the Monthly
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