Difference between revisions of "Rational number"
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==Examples== | ==Examples== | ||
| − | * All integers are rational because every integer <math>a</math> can be represented as <math>a= | + | * All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac{a}{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. | ||
| − | + | ||
| + | Moreover, any rational number satisfies the last two conditions. | ||
==Properties== | ==Properties== | ||
| − | # Rational numbers form a [[field]]. In plain English it means that you can add, subtract, multiply, and divide them (with the | + | # Rational numbers form a [[field]]. In plain English it means that you can add, subtract, multiply, and divide them (with the exception of division by <math>0</math>) 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. | + | # Rational numbers are [[dense]] in the set of reals. This means that every non-[[empty set | 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. |
Revision as of 10:47, 22 July 2009
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 
- 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 
)Moreover, any rational number satisfies the last two conditions.
Properties
- Rational numbers form a field. In plain English it means that you can add, subtract, multiply, and divide them (with the 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.
