From: bumby@no.rutgers.edu (Richard Bumby)
Newsgroups: sci.math
Subject: Re: Continued Fractions
Keywords: Help, Continued Fractions
Date: 20 Aug 92 21:18:52 GMT
Distribution: na
Organization: Rutgers Univ., New Brunswick, N.J.

andyc@eliot.uucp (Andy Collins) writes:

>I am looking for information on continued fractions. I would
>hope that there are books on the subject out there, but have
>not had the time to do a full-out search. I would appreciate
>any information.

>Thanks

>andy

>PS (not a trademark)
>Where are continued fractions used?
>Who has done the most work on these things?

>If you can could you post replies, my mail will be straigtened out
>by next week.

Continued fractions are the most exciting objects in all of
mathematics.  I would not be surprised to find love poetry inspired by
continued fractions!  As for references ...

Perron's "Kettenbrueche" (1929) remains the standard reference.  It
has been reprinted by Chelsea, but appears to be available only in the
original German.  This book covers both the arithmetic and analytic
theory. 

The chief use of continued fractions arises from the fact that they
describe a procedure for efficient generation of good rational
approximations.  A good treatment of the arithmetic theory from this
point of view can be found in Cassels, "An Introduction to Diophantine
Approximation", Cambridge, 1957.

Any reasonable thorough textbook on elementary number theory would
have an introduction to the arithmetic theory of continued fractions.
The analytic theory still seems to be treated somewhat as an advanced
subject. 

The continued fraction representation of a real number seems to be a
very rigid object.  Whatever property you choose as the basis of your
study, you seem to be led to the same construction.  The question of
approximation by rational numbers have analogs concerning simultaneous
approximation of several real numbers by rationals with the same
denominator.  It is tempting to think that there should be something
like a continued fraction that could be used to organize work on this
subject.  No comparable theory has been found, although some
constructions have been proposed that have a large number of desirable
properties.  A very thorough discussion of this can be found in
Brentjes, "Multidimensional Continued Fraction Algorithms",
Mathematical Centre Tract 145 (Amsterdam, 1981).

The subject is important enough to have its own section heading in the
Encyclopedic Dictionary of Mathematics, though the 3 pages allocated
do not give enough space to describe the subject as it relates to
current research.  The fact that the continued fraction expansion of
every quadratic irrational is eventually periodic (published by Lagrange
in 1770) continues to be the star of the show.  Combining this with
the result that says that the convergents of the continued fraction
give _all_ good rational approximations (in a sense which I will not
make precise here), shows that these periods allow the fundamental
units of real quadratic fields to be computed.

The relation between the partial quotients in the continued fraction
of a number and the approximation properties of the number allow
examples to be constructed of numbers having prescribed approximation
properties.  I find this to be extremely important, even though it is
obtained fairly early in the development of the theory.  Its
importance is somewhat subtle since it depends on an interpretation of
theorems which is not easily included in their statement.  In a sense,
the pattern of the sequence of partial quotients includes more
significant information about the number than, for example, its string
of decimal digits.
-- 
R. T. Bumby **  Rutgers Math ||   Amer. Math. Monthly Problems Editor
bumby@math.rutgers.edu       || P.O. Box 10971 New Brunswick, NJ08906-0971
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