Difference between revisions of "Rational number"

Revision as of 11:44, 24 June 2006

A number is called rational if it can be represented as a ratio of two integers.

All integers are rational because every integer $a$ can be represented as $a=\frac a1$ (or $\frac{2a}2$ , or...)
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 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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 Actually, the last property characterizes rationals among all [[real number]]s.
 ==Properties==
-# Rational numbers form a [[commutative]] [[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.
 # 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.
 ==See also==
 [[rational approximation]]