In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory. First, the natural numbers (including zero) are established on an axiomatic basis such as Peano's axiom system and then this is expanded to the ring of integers. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of ordered pairs of integers, } with , define a binary relation on this set by if and only if . This relation is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying transitivity).