 
Summary: Rational and Irrational Numbers
Putnam Practice
September 7, 2004
A rational number is one that can be expressed in the form a/b, where
a, b are integers and b = 0. To represent a given nonzero rational number,
we can choose a/b such that a is an integer, b is a natural number, and
(a, b) = 1. We shall say then that the representative fraction is in lowest
terms. An easy consequence of the definition is that any rational number
has a periodic decimal expansion. Real numbers with nonrepeating decimal
expansions cannot be expressed in the form a/b and are called irrational.
A number that satisfies an equation of the form
c0xn
+ c1xn1
+ ... + cn1x + cn = 0;
where c0, ..., cn are integers and c0 = 0 is called algebraic. A number that
is not algebraic is called transcendental. It is known that and e are
transcendental numbers.
Theorem 1 (Rational Root Theorem) If c0, c1, ...cn are integers, a/b
is in lowest terms and x = a/b is a root of the equation
c0xn
