Continued Fractions - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Continued Fractions

Main Concept

A continued fraction is a unique representation of a number, obtained by recursively subtracting the integer part of that number and then computing the continued fraction of the reciprocal of the remainder, if it is non-zero. If the number is rational, this process terminates with a finite continued fraction:

$\frac{123}{45}=2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{3}}}}$

Otherwise, the result is called an infinite continued fraction:

Continued fractions can be used to find rational approximations to real numbers, by simply truncating the resulting fraction at a certain point. For example, .

The numbers appearing on the left of the expansion (the integer parts) are called coefficients.

Coefficient facts

 • The continued fraction coefficients of quadratic numbers (solutions of a quadratic equation with integer coefficients) eventually repeat.
 • For some non-quadratic numbers such as Euler's number $e=2.718...$, the coefficients have an obvious pattern: 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...
 • However, for almost all real numbers $\mathbit{x}$, the geometric mean of the coefficients of the continued fraction expansion of $\mathbit{x}$ is the following number:

2.6854520010653064453097148...

which is known as Khinchin's constant.

Input a Maple expression in the box below that evaluates to a real number and click Enter, or choose one from the drop-down box. Adjust the number of approximating coefficients using the slider, and see how the coefficient frequency is affected in the graph.

 Pi22/7sqrt(2)exp(1)ln(3)(1+sqrt(5))/2Other    # of coefficients =



 More MathApps
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 10, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}