Rational number
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.
Actually, the last property characterizes rationals among all real numbers.
Properties
- Rational numbers form a commutative 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.