Continued fraction expansion of a real number

Definition

Let $x$ be a real number. Put $x_0 = x$ and $a_0 = \lfloor x_0 \rfloor$ . For $n \ge 0$ , if $x_n \neq a_n$ , put $x_{n+1} = \dfrac{1}{x_n - a_n}$ and $a_{n+1} = \lfloor x_{n+1} \rfloor$ . If $x_n = a_n$ , the process stops. The continued fraction expansion of $x$ is the simple continued fraction $[a_0; a_1, a_2, \dots]$ , finite or infinite.

Remarks

Here $\lfloor y \rfloor$ is the largest integer that is at most $y$ . Since $0 < x_n - a_n < 1$ , each $a_{n+1}$ with $n \ge 0$ is a positive integer. For example, $\frac{43}{19} = [2; 3, 1, 4]$ , and $\sqrt{2} = [1; 2, 2, 2, \dots]$ . The process stops if and only if $x$ is rational, by the termination theorem. For a rational $x$ , the steps are the steps of the Euclidean algorithm on its numerator and denominator.

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Continued fraction definition
Simple continued fraction definition
Continued fraction expansion of a real number definition

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Tag	`0SJ7Z6VKWK`
URI	`m2://0SJ7Z6VKWK`
Kind	definition
Writing	W0 stub
MSC	11A55
Defines	Continued fraction expansion of a real number
Uses	Simple continued fraction

Continued fraction expansion of a real number

Definition

Remarks

Reading path

Where this leads

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