Tool to compute continued fractions. A continued fraction is the representation of a number N in a form of a series of integers (a0, a1, ..., an) such as N = (a0+1/(a1+1/(a2+1/(...1/(an))).
Continued Fractions - dCode
Tag(s) : Series
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Continued Fractions
Continued Fraction Calculator
Continued Fraction to Number Converter
Answers to Questions (FAQ)
How to calculate a continued fraction?
Continued fraction expansion is close to algorithm of euclidean division, as for PGCD.
Example: If the fraction approximating pi is $ 355/113 = 3.14159292035... $
$$ 355 = 3 \times 113 + 16 \\ 113 = 7 \times 16 + 1 \\ 16 = 16 \times 1 + 0 $$
The continued fraction is [3,7,16]
Some developments of continuous fractions are infinite
To find the corresponding fraction, use the irreducible fraction tool.
How to calculate the continued fraction of a root?
Calculate an approximate value of the root (approximation as accurate as possible) and dCode will provide the corresponding continuous fraction.
How to write a continued fraction in LaTex?
The easiest way is to use cfrac: $$ e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{ 1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cdots}}}}}}}} $$
But the shortest way is to write $$ e = [2 ; 1, 2, 1, 1, 4, 1, 1, 6, \cdots] $$
Which are the most remarquable continued fractions?
Most known continued fractions are:
— Square Root of 2: $ \sqrt{2} = [1;2,2,2,2,2,\cdots] $
— Golden Ratio: $ \Phi = [1;1,1,1,1,1,\cdots] $
Source code
dCode retains ownership of the "Continued Fractions" source code. Except explicit open source licence (indicated Creative Commons / free), the "Continued Fractions" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Continued Fractions" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Continued Fractions" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Continued Fractions" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Continued Fractions on dCode.fr [online website], retrieved on 2023-12-05,
