rational number
The rational numbers ℚ are the fraction field of the ring ℤ 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 equivalent if the product of the cross terms is equal:
ab=cd⇔ad=bc |
Addition and multiplication of fractions are given by the formulae
ab+cd | = | ad+bcbd | ||
ab⋅cd | = | acbd |
The field of rational numbers is an ordered field, under the ordering relation ≤ defined as follows: a/b≤c/d if
-
1.
the inequality a⋅d≤b⋅c holds in the integers, and b has the same sign as d, or
-
2.
the inequality a⋅d≥b⋅c holds in the integers, and b has the opposite sign as d.
Under this ordering relation, the rational numbers form a topological space 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 |
Synonym | ℚ |
Related topic | Fraction |
Related topic | ProofThatTheRationalsAreCountable |
Defines | rational |