rational number
The rational numbers $\mathbb{Q}$ are the fraction field of the ring $\mathbb{Z}$ 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:
$$\frac{a}{b}=\frac{c}{d}\iff ad=bc$$ 
Addition and multiplication of fractions are given by the formulae
$\frac{a}{b}}+{\displaystyle \frac{c}{d}$  $=$  $\frac{ad+bc}{bd}$  
$\frac{a}{b}}\cdot {\displaystyle \frac{c}{d}$  $=$  $\frac{ac}{bd}$ 
The field of rational numbers is an ordered field, under the ordering relation $\le $ defined as follows: $a/b\le c/d$ if

1.
the inequality $a\cdot d\le b\cdot c$ holds in the integers, and $b$ has the same sign as $d$, or

2.
the inequality $a\cdot d\ge b\cdot c$ 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  20130322 11:50:30 
Last modified on  20130322 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  $\mathbb{Q}$ 
Related topic  Fraction 
Related topic  ProofThatTheRationalsAreCountable 
Defines  rational 