Difference between revisions of "Rational number"
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* All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac a1</math> (or <math>\frac{2a}2</math>, or...) | * All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac a1</math> (or <math>\frac{2a}2</math>, or...) | ||
* All numbers whose [[decimal expansion]] or expansion in some other number [[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]] is finite are rational (say, <math>12.345=\frac{12345}{1000}</math>) | ||
| − | * All numbers whose decimal expansion is [[periodic]] are rationals. | + | * All numbers whose decimal expansion is [[periodic]] in some base are rationals. |
Actually, the last property characterizes rationals among all [[real number]]s. | Actually, the last property characterizes rationals among all [[real number]]s. | ||
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==Properties== | ==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 <math>0</math>) and the result of each such operation is again a rational number. | # 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 <math>0</math>) and the result of each such operation is again a rational number. | ||
Revision as of 13:47, 29 June 2006
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 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.
