Continued Fractions

A continued fraction is an expression of the form

a_{0} + 1a_{1} + 1a_{2} + 1a_{3} + ⋯

which may or may not go on indefinitely. We denoteWarning: This notation clashes with the notation used in Davenport. Our notation is standard.he value of this continued fraction by

[a_{0}, a_{1}, a_{2}, …].

The a_{n} are called the partial quotients of the continued fraction (we will see why at the end of this lecture). Thus, e.g.,

[1, 2] = 1 + 12 = 32,

and

17251 = [3, 2, 1, 2, 6] = 3 + 12 + 11 + 12 + 16.

Continued fractions have many applications, from the abstract to the concrete. They give good rational approximations to irrational numbers, and the have been used to understand why you can’t tune a

piano perfectly.See http://www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTMLontinued

fractions also suggest a sense in which e appears to be “less transcendental” than π.

There are many places to read about continued fractions, including Chapter X of Hardy and Wright’s Intro. to the Theory of Numbers, §13.3 of Burton’s Elementary Number Theory, Chapter IV of Davenport, and Khintchine’s Continued Fractions. The notes you’re reading right now draw primarily on Hardy and Wright, since their exposition is very clear and to the point. I found Davenport’s chapter IV uneccessarily tedious; I felt marched through a thick jungle to see a beautiful river.

Introduction Finite Continued Fractions definition A finite continued fraction is an expression

a_{0} + 1a_{1} + 1a_{2} + ⋯ + 1a_{m},

where each a_{n} is a rational number and a_{n} > 0 for all n ≥ 1. If the a_{n} are integers, we say that the continued fraction is integral.

Warning: This notation clashes with the notation used in Davenport. Our notation is standard. t

See http://www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML C