A rational number is a number that can be expressed as a fraction 
where 
and 
are integers and 
. A rational number 
is said to have numerator 
and denominator 
. Numbers that are not rational are called
irrational numbers. The real
line consists of the union of the rational and irrational numbers. The set of
rational numbers is of measure zero on the real
line, so it is "small" compared to the irrationals
and the continuum.
The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted 
. Here, the symbol 
derives from the German word Quotient, which can be
translated as "ratio," and first appeared in Bourbaki's Algèbre
(reprinted as Bourbaki 1998, p. 671).
Any rational number is trivially also an algebraic number.
Examples of rational numbers include 
, 0, 1, 1/2, 22/7, 12345/67, and so on. Farey
sequences provide a way of systematically enumerating all rational numbers.
The set of rational numbers is denoted Rationals in the Wolfram Language, and a number
can be tested to see if it is rational
using the command Element[x, Rationals].
The elementary algebraic operations for combining rational numbers are exactly the same as for combining fractions.
It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.
For 
,
, and 
any different rational numbers, then
 |
is the square of the rational number
 |
(Honsberger 1991).
The probability that a random rational number has an even denominator is 1/3 (Salamin and Gosper 1972).
It is conjectured that if there exists a real number 
for which both 
and 
are integers, then 
is rational. This result would follow from the four
exponentials conjecture (Finch 2003).
