Difference between revisions of "Rational number"

Revision as of 19:10, 25 November 2007

A rational number is a number that can be represented as the ratio of two integers.

All integers are rational because every integer $a$ can be represented as $a=\frac a1$ (or $\frac{a^{n}}{a^{n-1}}$
All numbers whose decimal expansion or expansion in some other number base is finite are rational (say, $12.345=\frac{12345}{1000}$ )
All numbers whose decimal expansion is periodic (repeating, i.e. 0.314314314...) in some base are rationals.

Actually, the last property characterizes rationals among all real numbers.

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 $0$ ) 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.

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 ==Examples==
-* 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{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 is [[periodic]] (repeating, i.e. 0.314314314...) in some base are rationals.
@@ Line 17: / Line 17: @@
 * [[Fraction]]
 * [[Rational approximation]]
+[[Category:Definition]]
+[[Category:Number theory]]