rational number

The rational numbers are the fraction field of the ring of integers. In more elementary terms, a rational number is a quotient a/b of two integers a and b, where b is nonzero. Two fractions a/b and c/d are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if the productPlanetmathPlanetmathPlanetmath of the cross terms is equal:


Addition and multiplication of fractions are given by the formulae

ab+cd = ad+bcbd
abcd = acbd

The field of rational numbers is an ordered field, under the ordering relation defined as follows: a/bc/d if

  1. 1.

    the inequality adbc holds in the integers, and b has the same sign as d, or

  2. 2.

    the inequality adbc holds in the integers, and b has the opposite sign as d.

Under this ordering relation, the rational numbers form a topological spaceMathworldPlanetmath under the order topology. The set of rational numbers is dense when considered as a subset of the real numbers.

Title rational number
Canonical name RationalNumber
Date of creation 2013-03-22 11:50:30
Last modified on 2013-03-22 11:50:30
Owner djao (24)
Last modified by djao (24)
Numerical id 15
Author djao (24)
Entry type Definition
Classification msc 13B30
Classification msc 11A99
Classification msc 03E99
Related topic Fraction
Related topic ProofThatTheRationalsAreCountable
Defines rational