## 1.4. Natural Numbers, Integers, and Rational Numbers

### Theorem 1.4.4: The Rationals

Let

**A**be the set**N**x**N - {0}**and define a relation*r*on**N**x**N - {0}**by saying that*(a, b)*is related to*(a’, b’)*if*a * b’ = a’ * b*. Then this relation is an equivalence relation.
If *[(a, b)]* and *[(a’, b’)]* denotes the equivalence classes
containing *(a, b)* and *(a’, b’)*, respectively, and if we define the
operations

*[(a, b)] + [(a', b')] = [(a * b' + a’ * b, b * b')]**[(a, b)] * [(a’, b’)] = [(a * a’, b * b’)]*

### Proof:

The proof is much similar to proving the similar statement regarding the integers. The details are left as an exercise.
Notice the requirement that
*(a,b) N x N - {0}*
i.e. the integer

*b can not be zero. If we did allow both*

*a*and*b*to become zero, the relation would not be an equivalence relation any longer. As a hint: what pairs*(a, b)*would be related to (0,0) ?