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 $$\mathbb{Q}=\left\{\frac{a}{b}\right|a,b\in \mathbb{Z},b\ne 0\}$$ with $$\frac{a}{b}=\frac{c}{d}\text{if}ad=bc,$$ and $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\text{and}\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.$$

Define a relation $\le$ on $\mathbb{Q}$ by $$x\le y\text{if there exists}y-x\in {\mathbb{Q}}_{\ge 0},$$ where $${\mathbb{Q}}_{\ge 0}=\left\{\frac{a}{b}\right|a,b\in {\mathbb{Z}}_{\ge 0},b\ne 0\}.$$

1. The set $\mathbb{Z}$ with the operations of addition and multiplication is an integral domain.
2. The set $\mathbb{Q}$ with the operations of addition and multiplication is a field.
3. 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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