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

${\displaystyle \frac{a}{b}}+{\displaystyle \frac{c}{d}}$ $=$ ${\displaystyle \frac{ad+bc}{bd}}$

${\displaystyle \frac{a}{b}}\cdot {\displaystyle \frac{c}{d}}$ $=$ ${\displaystyle \frac{ac}{bd}}$

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	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	$\mathbb{Q}$
Related topic	Fraction
Related topic	ProofThatTheRationalsAreCountable
Defines	rational