Rational number
Revision as of 11:44, 24 June 2006 by ComplexZeta (talk | contribs) (→Properties: modern definition is that fields are automatically commutative)
Contents
Definition
A number is called rational if it can be represented as a ratio of two integers.
Examples
- All integers are rational because every integer
can be represented as 
(or 
, or...) - All numbers whose decimal expansion or expansion in some other number base is finite are rational (say,
) - All numbers whose decimal expansion is periodic are rationals.
can be represented as 
(or 
, or...)
)Actually, the last property characterizes rationals among all real numbers.
Properties
- Rational numbers form a field. In plain English it means that you can add, subtract, multiply, and divide them (with the obvious exception of division by
) and the result of each such operation is again a rational number. - Rational numbers are dense in the set of reals. This means that every non-empty open interval on the real line contains at least one (actually, infinitely many) rationals. Alternatively, it means that every real number can be represented as a limit of a sequence of rational numbers.
) and the result of each such operation is again a rational number.
