Rational number: Difference between revisions
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==Examples== | ==Examples== | ||
* All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac a1</math> (or <math>\frac{ | * All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac a1</math> (or <math>\frac{a^{n}}{a^{n-1}}</math> | ||
* All numbers whose [[decimal expansion]] or expansion in some other number [[base numbers |base]] is finite are rational (say, <math>12.345=\frac{12345}{1000}</math>) | * All numbers whose [[decimal expansion]] or expansion in some other number [[base numbers |base]] is finite are rational (say, <math>12.345=\frac{12345}{1000}</math>) | ||
* All numbers whose decimal expansion is [[periodic]] (repeating, i.e. 0.314314314...) in some base are rationals. | * All numbers whose decimal expansion is [[periodic]] (repeating, i.e. 0.314314314...) in some base are rationals. | ||
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* [[Fraction]] | * [[Fraction]] | ||
* [[Rational approximation]] | * [[Rational approximation]] | ||
[[Category:Definition]] | |||
[[Category:Number theory]] | |||
Revision as of 18:10, 25 November 2007
A rational number is a number that can be represented as the ratio of two integers.
Examples
- All integers are rational because every integer
can be represented as 
(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 
)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.
