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=\frac | + | * All integers are rational because every integer <math>a</math> can be represented as <math>a=\frac{a}{1}</math> |
| − | * | + | * Every number with a finite [[decimal expansion]] is rational (say, <math>12.345=\frac{12345}{1000}</math>) |
| − | * | + | * Every number with a periodic decimal expansion (e.g. 0.314314314...) is also rational. |
| − | |||
| + | Moreover, any rational number satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other [[number base|base]]. | ||
==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. |
| + | # Despite this, the set of rational numbers is [[countable]], i.e. the same size as the set of integers. | ||
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* [[Fraction]] | * [[Fraction]] | ||
* [[Rational approximation]] | * [[Rational approximation]] | ||
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| + | [[Category:Definition]] | ||
| + | [[Category:Number theory]] | ||
Latest revision as of 14:51, 25 August 2022
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 
- Every number with a finite decimal expansion is rational (say,
) - Every number with a periodic decimal expansion (e.g. 0.314314314...) is also rational.
can be represented as 
)Moreover, any rational number satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other base.
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.
