11.6 Extending \(\mathbb Z\) to \(\mathbb Q\).

We have extended the ‘definition’ of the natural numbers (in terms of Peano’s axioms and Von Neumann’s model) to \(\mathbb Z\) by using equivalence relations. There is a very natural way to extend the integers \(\mathbb Z\) to the rationals \(\mathbb Q\) in a similar way. Define a relation \(\approx\) on \(\mathbb Z\times\mathbb Z\setminus\{0\}\) by \[ (a,b)\approx(c,d)\text{ if and only if }a\ast d = b\ast c. \] You should check that this is an equivalence relation and that the following ‘arithmetic’ operations on the set of equivalence classes, \(Q=\{[a,b]|a,b\in\mathbb Z, b\not=0\}\), are all well-defined \[ [a,b]+[c,d] = [a\ast d + b\ast c,b\ast d] \] \[ [a,b]-[c,d]= [a\ast d - b\ast c,b\ast d] \] \[ [a,b]\ast[c,d] = [a\ast c,b\ast d] \] and if \(c\not=0\) \[ [a,b]\doteqdot[c,d] = [a\ast d,b\ast c]. \] In a similar way to above we can deduce

Theorem 11.16

The function \(g:Q\to\mathbb Q\) given by \(g([a,b]) = a/b\) is a well-defined bijection that preserved the four operations of addition, subtraction, multiplication and division.

It should be clear that the ‘number’ \([a,b]\) in \(Q\) is meant to represent the rational number \(a/b\).