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continued fraction

Defines: 
anthyphairetic ratio, simple continued fraction
Keywords: 
number theory, rational number, irrational number, Pell
Synonym: 
chain fraction
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

11Y65 no label found11J70 no label found11A55 no label found

Comments

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

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

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