Difference between revisions of "Rational number"
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| − | A '''rational number''' is a [[number]] that can be | + | A '''rational number''' is a [[number]] that can be expressed as the [[ratio]] of two [[integer]]s. |
| + | == Examples == | ||
| − | + | * All integers are rational because every integer <math>a</math> can be represented as <math>\frac{a}{1}</math> | |
| − | * All integers are rational because every integer <math>a</math> can be represented as <math> | + | * Every number with a finite [[decimal expansion]] is rational (for example, <math>12.345=\frac{12345}{1000}</math>) |
| − | * Every number with a finite [[decimal expansion]] is rational ( | + | * Every number with a periodic decimal expansion (for example, 0.314314314...) is also rational. |
| − | * Every number with a | ||
| − | Moreover, any rational number satisfies exactly one of the last two conditions | + | Moreover, any rational number in any [[base numbers|base]] satisfies exactly one of 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 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 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. | ||
| + | * Despite this, the set of rational numbers is [[countable]], i.e. the same size as the set of integers. | ||
| + | |||
| + | == See also == | ||
| − | |||
* [[Fraction]] | * [[Fraction]] | ||
* [[Rational approximation]] | * [[Rational approximation]] | ||
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[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
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| + | {{stub}} | ||
Latest revision as of 22:00, 3 February 2025
A rational number is a number that can be expressed as the ratio of two integers.
Examples
- All integers are rational because every integer
can be represented as 
- Every number with a finite decimal expansion is rational (for example,
) - Every number with a periodic decimal expansion (for example, 0.314314314...) is also rational.
can be represented as 
)Moreover, any rational number in any base satisfies exactly one of 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.
- Despite this, the set of rational numbers is countable, i.e. the same size as the set of integers.
) and the result of each such operation is again a rational number.See also
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