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:

 $\mathrm{\pi }\=3\+\frac{1}{7\+\frac{1}{15\+\frac{1}{1\+\frac{1}{..\.}}}}$

Continued fractions can be used to find rational approximations to real numbers, by simply truncating the resulting fraction at a certain point. For example, $\mathrm{\π} \approx 3\+\frac{1}{7}$.

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 $\mathit{x}$, the geometric mean of the coefficients of the continued fraction expansion of $\mathit{x}$ is the following number:   2.6854520010653064453097148...  which is known as Khinchin's constant.

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MathApps/RealAndComplexNumbers