## The rational numbers

The rational numbers is the set $ℚ = { ab | a,b∈ℤ, b≠0 }$ with $ab = cd if ad=bc,$ and $ab + cd = ad+bc bd and ab ⋅ cd = ac bd .$

Define a relation $\le$ on $ℚ$ by $x≤y if there exists y-x∈ ℚ≥0,$ where $ℚ≥0 = { ab | a,b∈ ℤ≥0, b≠0 }.$

1. The set $ℤ$ with the operations of addition and multiplication is an integral domain.
2. The set $ℚ$ with the operations of addition and multiplication is a field.
3. The set $ℚ$ 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 $ℤ$ is the free group on one generator, and the rationals $ℚ$ is the field of fractions of $ℤ$. Thus, both number systems are universal objects (in the sense of category theory).

The definition of the rational numbers $ℚ$ 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?????