 Continued Fractions - Maple Help

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