Difference between revisions of "Rational number"
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| − | A '''rational number''' is a [[ | + | A '''rational number''' is a [[number]] that can be represented as a ratio of two [[integer]]s. |
Revision as of 19:46, 23 July 2006
A rational number is a number that 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 (repeating, i.e. 0.314314314...) in some base 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.
