continued fraction

Given a sequence of positive real numbers $(a_{n})_{n\geq 1}$ , with $a_{0}$ any real number. Consider the sequence

	$\displaystyle c_{1}$	$\displaystyle=a_{0}+\cfrac{1}{a_{1}}$
	$\displaystyle c_{2}$	$\displaystyle=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}}}$
	$\displaystyle c_{3}$	$\displaystyle=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}}}}$
	$\displaystyle c_{4}$	$\displaystyle=\ldots$

The limit $c$ of this sequence, if it exists, is called the value or limit of the infinite continued fraction with convergents $(c_{n})$ , and is denoted by

a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\ldots}}}

or by

a_{0}+\frac{1}{a_{1}+}\frac{1}{a_{2}+}\frac{1}{a_{3}+}\ldots

In the same way, a finite sequence

(a_{n})_{1\leq n\leq k}

defines a finite sequence

(c_{n})_{1\leq n\leq k}\;.

We then speak of a finite continued fraction with value $c_{k}$ .

An archaic word for a continued fraction is anthyphairetic ratio.

If the denominators $a_{n}$ are all (positive) integers, we speak of a simple continued fraction. We then use the notation $q=\langle a_{0};a_{1},a_{2},a_{3},\ldots\rangle$ or, in the finite case, $q=\langle a_{0};a_{1},a_{2},a_{3},\ldots,a_{n}\rangle\;.$

It is not hard to prove that any irrational number $c$ is the value of a unique infinite simple continued fraction. Moreover, if $c_{n}$ denotes its $n$ th convergent, then $c-c_{n}$ is an alternating sequence and $|c-c_{n}|$ is decreasing (as well as convergent to zero). Also, the value of an infinite simple continued fraction is perforce irrational.

Any rational number is the value of two and only two finite continued fractions; in one of them, the last denominator is 1. E.g.

\frac{43}{30}=\langle 1;2,3,4\rangle=\langle 1;2,3,3,1\rangle\;.

These two conditions on a real number $c$ are equivalent:

1. $c$ is a root of an irreducible quadratic polynomial with integer coefficients.

2. $c$ is irrational and its simple continued fraction is “eventually periodic”; i.e.

c=\langle a_{0};a_{1},a_{2},\ldots\rangle

and, for some integer $m$ and some integer $k>0$ , we have $a_{n}=a_{n+k}$ for all $n\geq m$ .

For example, consider the quadratic equation for the golden ratio:

x^{2}=x+1

or equivalently

x=1+\frac{1}{x}\;.

We get

	$\displaystyle x$	$\displaystyle=$	$\displaystyle 1+\cfrac{1}{1+\cfrac{1}{x}}$
		$\displaystyle=$	$\displaystyle 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}$

and so on. If $x>0$ , we therefore expect

x=\langle 1;1,1,1,\ldots\rangle

which indeed can be proved. As an exercise, you might like to look for a continued fraction expansion of the other solution of $x^{2}=x+1$ .

Although $e$ is transcendental, there is a surprising pattern in its simple continued fraction expansion.

e=\langle 2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,\ldots\rangle

No pattern is apparent in the expansions some other well-known transcendental constants, such as $\pi$ and Apéry’s constant $\zeta(3)$ .

Owing to a kinship with the Euclidean division algorithm, continued fractions arise naturally in number theory. An interesting example is the Pell diophantine equation

x^{2}-Dy^{2}=1

where $D$ is a nonsquare integer $>0$ . It turns out that if $(x,y)$ is any solution of the Pell equation other than $(\pm 1,0)$ , then $|\frac{x}{y}|$ is a convergent to $\sqrt{D}$ .

$\frac{22}{7}$ and $\frac{355}{113}$ are well-known rational approximations to $\pi$ , and indeed both are convergents to $\pi$ :

$\displaystyle 3.14159265\ldots$	$\displaystyle=$	$\displaystyle\pi=\langle 3;7,15,1,292,...\rangle$
$\displaystyle 3.14285714\ldots$	$\displaystyle=$	$\displaystyle\frac{22}{7}=\langle 3;7\rangle$
$\displaystyle 3.14159292\ldots$	$\displaystyle=$	$\displaystyle\frac{355}{113}=\langle 3;7,15,1\rangle=\langle 3;7,16\rangle$

For one more example, the distribution of leap years in the 4800-month cycle of the Gregorian calendar can be interpreted (loosely speaking) in terms of the continued fraction expansion of the number of days in a solar year.

Title	continued fraction
Canonical name	ContinuedFraction
Date of creation	2013-03-22 12:47:12
Last modified on	2013-03-22 12:47:12
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	29
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11Y65
Classification	msc 11J70
Classification	msc 11A55
Synonym	chain fraction
Related topic	FareySequence
Related topic	AdjacentFraction
Related topic	SternBrocotTree
Related topic	FareyPair
Defines	anthyphairetic ratio
Defines	simple continued fraction