PlanetMath: rational number

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rational number

(Definition)

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 of two integers and , where is nonzero. Two fractions and are equivalent if the product of the cross terms is equal:

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

Addition and multiplication of fractions are given by the formulae

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

The field of rational numbers is an ordered field, under the ordering relation $\leq$ defined as follows: $a/b \leq c/d$ if

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

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.

"rational number" is owned by djao.

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Other names:

$\mathbb{Q}$

Also defines:

rational

Attachments:: decimal expansion (Result) by pahio
rings of rational numbers (Theorem) by pahio

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AMS MSC:	03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
	11A99 (Number theory :: Elementary number theory :: Miscellaneous)
	13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

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