## The Rational Numbers

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

Last updates: 12 June 2012

## The rational numbers

The **rational numbers** is the set
$$\mathbb{Q}=\left\{\frac{a}{b}\phantom{\rule{0.5em}{0ex}}\right|\phantom{\rule{0.5em}{0ex}}a,b\in \mathbb{Z},\phantom{\rule{0.5em}{0ex}}b\ne 0\}$$
with
$$\frac{a}{b}=\frac{c}{d}\phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{2em}{0ex}}ad=bc,$$
and
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.$$

Define a relation

$\le $ on

$\mathbb{Q}$ by

$$x\le y\phantom{\rule{2em}{0ex}}\text{if there exists}\phantom{\rule{1em}{0ex}}y-x\in {\mathbb{Q}}_{\ge 0},$$
where

$${\mathbb{Q}}_{\ge 0}=\left\{\frac{a}{b}\phantom{\rule{0.5em}{0ex}}\right|\phantom{\rule{0.5em}{0ex}}a,b\in {\mathbb{Z}}_{\ge 0},\phantom{\rule{0.5em}{0ex}}b\ne 0\}.$$

- The set $\mathbb{Z}$ with the operations of addition and multiplication is an integral domain.
- The set $\mathbb{Q}$ with the operations of addition and multiplication is a field.
- The set $\mathbb{Q}$ with the operations of addition and multiplication and
the total order $\le $ is an ordered field.

## Notes and References

A child learns about fractions (half a cookie) in the first couple of years of life. The point of this
section is the mechanics of computation with fractions,
something often included in a primary school curriculum.
The integers $\mathbb{Z}$ is the free group on one generator,
and the rationals $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$. Thus, both number
systems are universal objects (in the sense of category theory).

The definition of the rational numbers $\mathbb{Q}$ is treated
in [Bou, Alg. Ch. I § 9 no. 4].

## References

[BouAlg]
N. Bourbaki, *Algebra I*,
Chapter I, Section 9 No. 4, Springer-Verlag, Berlin 1989.
MR?????

[BouTop]
N. Bourbaki,
*General Topology*, Chapter IV, Section 1, Springer-Verlag, Berlin 1989.
MR?????

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