Repeating Continued Fractions

The continued fraction expansion of a quadratic irrational is eventually periodic; the converse is also true.
The repeating sequence, the initial part of the continued fraction form, the value, and the first few convergents to that value are shown.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The continued fraction expansions of the square roots of the integers that are not perfect squares possess the so-called "palindromic property". That means that their periodic element sequences without the last element are palindromic (i. e., can be reversed), and the last element is twice the initial term. For example, the continued fraction expansion of is {5, {1, 1, 3, 5, 3, 1, 1, 10}}. The sequence 1, 1, 3, 5, 3, 1, 1 is palindromic, and the 10 is two times the initial 5.
The following shows this for the first 200 integers:
DeleteCases[ContinuedFraction@Sqrt@Range@200, {_}]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+