continued fraction

Given a sequenceMathworldPlanetmath of positive real numbers (an)n1, with a0 any real number. Consider the sequence

c1 =a0+1a1
c2 =a0+1a1+1a2
c3 =a0+1a1+1a2+1a3
c4 =

The limit c of this sequence, if it exists, is called the value or limit of the infinite continued fraction with convergentsMathworldPlanetmath (cn), and is denoted by


or by


In the same way, a finite sequence


defines a finite sequence


We then speak of a finite continued fraction with value ck.

An archaic word for a continued fractionMathworldPlanetmath is anthyphairetic ratio.

If the denominators an are all (positive) integers, we speak of a simple continued fraction. We then use the notation q=a0;a1,a2,a3, or, in the finite case, q=a0;a1,a2,a3,,an.

It is not hard to prove that any irrational number c is the value of a unique infiniteMathworldPlanetmathPlanetmath simple continued fraction. Moreover, if cn denotes its nth convergent, then c-cn is an alternating sequence and |c-cn| 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.


These two conditions on a real number c are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

1. c is a root of an irreduciblePlanetmathPlanetmath quadratic polynomial with integer coefficientsMathworldPlanetmath.

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


and, for some integer m and some integer k>0, we have an=an+k for all nm.

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


or equivalently


We get

x = 1+11+1x
= 1+11+11+1x

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


which indeed can be proved. As an exercise, you might like to look for a continued fraction expansion of the other solution of x2=x+1.

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


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

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


where D is a nonsquare integer >0. It turns out that if (x,y) is any solution of the Pell equationMathworldPlanetmath other than (±1,0), then |xy| is a convergent to D.

227 and 355113 are well-known rational approximations to π, and indeed both are convergents to π:

3.14159265 = π=3;7,15,1,292,
3.14285714 = 227=3;7
3.14159292 = 355113=3;7,15,1=3;7,16

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