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单词 ContinuedFraction
释义

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

a0+1a1+1a2+1a3+

or by

a0+1a1+1a2+1a3+

In the same way, a finite sequence

(an)1nk

defines a finite sequence

(cn)1nk.

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 perforceirrational.

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.

4330=1;2,3,4=1;2,3,3,1.

These two conditions on a real number c are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

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

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

c=a0;a1,a2,

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

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

x2=x+1

or equivalently

x=1+1x.

We get

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

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

x=1;1,1,1,

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.

e=2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,

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. Aninteresting example is the Pell diophantine equationMathworldPlanetmath

x2-Dy2=1

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 daysin a solar year.

Titlecontinued fraction
Canonical nameContinuedFraction
Date of creation2013-03-22 12:47:12
Last modified on2013-03-22 12:47:12
OwnerPrimeFan (13766)
Last modified byPrimeFan (13766)
Numerical id29
AuthorPrimeFan (13766)
Entry typeDefinition
Classificationmsc 11Y65
Classificationmsc 11J70
Classificationmsc 11A55
Synonymchain fraction
Related topicFareySequence
Related topicAdjacentFraction
Related topicSternBrocotTree
Related topicFareyPair
Definesanthyphairetic ratio
Definessimple continued fraction
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更新时间:2025/5/5 0:18:45