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# continued fraction

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

$\displaystyle c_{1}$ | $\displaystyle=a_{0}+\frac{1}{a_{1}}$ | ||

$\displaystyle c_{2}$ | $\displaystyle=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}}}$ | ||

$\displaystyle c_{3}$ | $\displaystyle=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}}}}$ | ||

$\displaystyle c_{4}$ | $\displaystyle=\ldots$ |

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

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

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

$\displaystyle x$ | $\displaystyle=$ | $\displaystyle 1+\frac{1}{1+\frac{1}{x}}$ | ||

$\displaystyle=$ | $\displaystyle 1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}$ |

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.

## Mathematics Subject Classification

11Y65*no label found*11J70

*no label found*11A55

*no label found*

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## Comments

## ContinuedFraction: suggestions invited

I plan to enlarge the item ContinuedFraction, and make it a full-fledged topic entry. It'll include proofs, and definitions of additional related jargon, such as "partial denominator". But what else? Maybe:

-- def'n with numerators other than 1. (I don't like this idea.)

-- more on its use in number theory.

-- illustration of a cont'd fraction decomposition of a polynomial, not just a real.

-- an illustrative practical application of the continued fraction algorithm, maybe to some sort of differential equation.

Any other suggestions?

Larry

## Re: ContinuedFraction: suggestions invited

Here is a list of things that you might consider:

1. Explain that the fractions that have only 1's in numerator are called usually ``simple continued fractions''. In general one can have numbers other than 1 in denominator.

2. Treat the convergence of the continued fractions. The simple continued fractions with integer coefficients are always convergent (theorem 166 in Hardy and Wright), but for non-integer coefficients and/or non-simple continued fractions there is a convergence criterion, but I don't remember how it looked like or where I have seen it.

3. Methodically consider approximation of numbers by continued fractions. Mention that convergents of simple continued fractions provide the best approximation to the given number. Also, you might want to mention the related results concerning approximation of irrational by rationals: every irrational is approximable to order 2 (theorem 187 in Hardy and Wright), no algebraic is appoximable to order higher than 2 (Roth's theorem; see PlanetMath entry http://planetmath.org/encyclopedia/LiouvillesTheorem.html)

4. If you are very brave, then you can also consider various multidimensional analogues of continued fractions, but problem is that there are several of them and the material is not so commonly available.

Also, you might want to check out \cfrac command that amsmath package provides for typesetting better-looking continued fractions.

Boris

If you include non-simple continued fractions (i.e., fractions with denominator other than 1), you should treat the convergence.

> I plan to enlarge the item ContinuedFraction, and make it a

> full-fledged topic entry. It'll include proofs, and

> definitions of additional related jargon, such as "partial

> denominator". But what else? Maybe:

> -- def'n with numerators other than 1. (I don't like this

> idea.)

> -- more on its use in number theory.

> -- illustration of a cont'd fraction decomposition of a

> polynomial, not just a real.

> -- an illustrative practical application of the continued

> fraction algorithm, maybe to some sort of differential

> equation.

> Any other suggestions?

> Larry