Continued fractions are expressions of the form
where the 
's are usually integers. This expression is denoted by
.
The simplest continued fraction expansions are those of rational numbers a/b and the entries are just the quotients occurring in the Euclidean algorithm computation of the gcd of a and b. For example,
This expansion follows from the following computation of the greatest common divisor.
We can rewrite these as fractions, and by combining them, write
in terms of the successive quotients:
Combining these equations yields the formula
where in (1) we have replaced 
with
from (2), replace 
by
\;, and so on.
Continued fractions occur naturally in approximation of real
numbers by rational numbers with bounded denominators. These approximation
properties lead to efficient formulas for the computation of many classical
transcendental functions. Additional
applications of continued fractions are in the solution of
Diophantine equations such as 
, and more generally
.
This chapter studies the basic properties of finite and infinite continued fractions, periodic continued fractions, and the continued fraction expansion of e. The continued fraction expansion is
