A fundamental problem is to determine how closely we can
approximate an irrational number by rational numbers with
bounded denominators. A number p/q is a good approximation
to
an irrational number 
if 
for any
p'/q' with 
. This
means that p/q is closer to than any other number
with smaller denominator. For example, the approximations to
are 3, 22/7, 179/57, 333/100, 355/113 and so on. (The
last approximation 355/113 has been known in China since the
fifth century.)
The chapter proves the basic properties of the good approximations and their relation to the convergents of the continued fractions. The chapter concludes with a discussion of algebraic and transcendental numbers, and the use of Diophantine approximations to prove the existence of transcendental numbers.