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Homerational number

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# 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

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

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

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

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

2. the inequality $a\cdot d\geq b\cdot c$ 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.

## Mathematics Subject Classification

13B30*no label found*11A99

*no label found*03E99

*no label found*

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## Comments

## alternative definition

Back in my day, we called the rational numebers {p/q : p, q are integers}

-apk

## Re: alternative definition

hopefully q was a non-zero integer in your day :)

## Re: alternative definition

It took you five years to prepare that response?

:)

Cam

## (a,b) definition

In general it has been more usefull, more extensible, to define ”rational numbers

^{}” as a pair (a,b) with a multiplication^{}and addition rule. And then take Integers as an example.Addition rule (a,b)+(c,d) (a*d+b*c,b*d)

Multiplication rule: (a,b)*(c,d) (a*c,b*d)

a,b,c,d ∈Z and b,d != 0

Definitions like this allow one to manipulate items from a more general ring and also use the usuall nomeclature a/b=c/d . This includes certain exotic cases in Algebraic Geometry

^{}.