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. Any algorithm for the "Continued Fractions" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "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.) or any database download or API access for "Continued Fractions" or any other element are not public (except explicit open source licence). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Continued Fractions" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source (Creative Commons CC-BY free distribution license).
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Continued Fractions on dCode.fr [online website], retrieved on 2025-07-15,
