Convergents of a continued fraction

Definition

Let $[a_0; a_1, a_2, \dots]$ be a simple continued fraction. For each $n$ such that $a_n$ exists, the $n$ -th convergent is the rational number $\dfrac{p_n}{q_n} = [a_0; a_1, \dots, a_n]$ . Here $p_n$ and $q_n$ are given by $p_{-2} = 0$ , $p_{-1} = 1$ , $q_{-2} = 1$ , $q_{-1} = 0$ and $p_n = a_n p_{n-1} + p_{n-2}, \qquad q_n = a_n q_{n-1} + q_{n-2} \quad (n \ge 0).$

Remarks

The recurrences give $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$ . So $\gcd(p_n, q_n) = 1$ , and the fraction $p_n / q_n$ is in lowest terms. For $\sqrt{2} = [1; 2, 2, \dots]$ , the convergents are $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \dots$ . For a general continued fraction, the same recurrences hold with $p_n = b_n p_{n-1} + a_n p_{n-2}$ and $q_n = b_n q_{n-1} + a_n q_{n-2}$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Continued fraction definition
Simple continued fraction definition
Convergents of a continued fraction definition

Where this leads

6 pages build on this one.

Tag	`0SRWNR42BQ`
URI	`m2://0SRWNR42BQ`
Kind	definition
Writing	W0 stub
MSC	11A55
Defines	Convergents of a continued fraction
Uses	Simple continued fraction

Convergents of a continued fraction

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by