Rationals

The rationals are obtained from the integers by insisting that all division problems with nonzero divisor have answers. This can be done by formally adding multiplicative inverses (the reciprocals) and closing under multiplacation and addition or by thinking of rationals as equivalence classes of pairs of natural numbers giving division problems with the same answer. This last approach is what our usual notation for rationals does, so this extension will look familiar.

Definition 57   The number system of rationals (written ) is obtained from using an equivalence relation on pairs $(p,q)$ with $p\in \ensuremath{\mathbb{Z}}$ and $q\in \ensuremath{\mathbb{N}}^+$ as follows:

1. the pair $(a,b)$ is equivalent to $(c,d)$ precisely when $ad=bc$. We write $[\frac{a}{b}]$ for the equivalence class of $(a,b)$ under this relation.
2. addition is defined by
   
   $$[\frac{a}{b}]+[\frac{c}{d}]=[\frac{ad+bc}{bd}]$$
   
   

3. multiplication is defined by
   
   $$[\frac{a}{b}][\frac{c}{d}]=[\frac{ac}{bd}]$$
   
   

4. order is defined by
   
   $$[\frac{a}{b}]<[\frac{c}{d}] \mbox{ if and only if } ad < bc$$
   
   

5. the inclusion of into is given by $\iota: \ensuremath{\mathbb{Z}}\to \ensuremath{\mathbb{Q}}$ with $\iota(a)=[\frac{a}{1}]$

In order for this definition to make good sense it is important that the proposed equivalence relation is in fact an equivalence and that the addition, multiplication, and order do not depend on the choice of representative of the equivalence clases:

Proposition 117   The relation on pairs numbers given by $(a,b)\approx (c,d)$ if and only if $ad=bc$ is an equivalence relation.

(3 Points )

Proposition 118   The addition operation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 119   The multiplication operation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 120   The order relation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 121   The inclusion $\iota: \ensuremath{\mathbb{Z}}\to \ensuremath{\mathbb{Q}}$ preserves addition, multiplication, and order.

(3 Points )

Since our object is to extend the integers to a system in which every non-zero number has a multiplicative inverse we want to be sure that we do indeed get multiplicative inverses, but also that we have not lost any of the nice properties of the operations that we had before we extended:

Proposition 122   The addition operation on is associative, commutative, and has an identity $[\frac{0}{1}]$.

(3 Points )

Proposition 123   The multiplication operation is associative, commutative, has an identity $[\frac{1}{1}]$, and distributes over addition.

(3 Points )

Proposition 124   The order relation on is transitive: if $[\frac{a}{b}]<[\frac{c}{d}]$ and $[c,d]<[u,v]$ then $[a,b]<[u,v]$.

(2 Points )

Proposition 125   Order on satisfies trichotomy: exactly one of $[\frac{a}{b}]=[\frac{c}{d}], [\frac{a}{b}]<[\frac{c}{d}], [\frac{c}{d}]<[\frac{a}{b}]$ holds.

(2 Points )

Definition 58   A rational $[\frac{a}{b}]$ is called negative if $a<0$ and positive if $a>0$.

Note that because of trichotomy any rational number is either positive, negative, or zero.

Proposition 126   Every integer except 0 has a multiplicative inverse.

(2 Points )

These propositions tell us that we have been successful in our attempt to extend the integers to include multiplicative inverses without losing any of the nice algebraic properties of the operations. In addition we have retained all of the nice properties of the order and, in fact, have gained some properties for the order.

Proposition 127   The order on is dense: if $[\frac{a}{b}]<[\frac{c}{d}]$ then there is a rational $[\frac{u}{v}]$ with $[\frac{a}{b}]<[\frac{u}{v}]<[\frac{c}{d}]$.

(2 Points )

Proposition 128   The order on is Archidemdian: Given any rational $[\frac{a}{b}]$ there is a natural number $n$ with $[\frac{a}{b}]<[\frac{n}{1}]$.

(2 Points )