Definition
Rational Number
A rational number is a number that can be written as the quotient of two integers with a non-zero denominator.
The set of all rational numbers is denoted by
. Any rational number has a decimal expansion that either terminates or repeats periodically.
Countability
Countability via reduced forms (implicit composition)
Every rational number can be written uniquely in reduced form as
Here,
denotes the greatest common divisor. Consider the set The map
is a bijection, because each rational number has exactly one reduced representation. Since
and is countable, it follows that is countable. Therefore, is countable.
Countability via grouping
First write each rational number in unique reduced form,
For such a reduced pair
, define Then
belongs to the single group This is unique because the reduced form of
is unique, so the value of is unique as well. Each is finite. Indeed, if , then and , so only finitely many pairs can occur. In fact, . Since
and each
is finite, is a countable union of finite sets. Therefore, is countable.