Another example of this kind of thing is demonstrating that the set of rational numbers are countable. Each rational number has the form a / b for a pair of in

Another example of this kind of thing is demonstrating that the set of rational numbers are countable. Each rational number has the form a / b for a pair of integers (a, b). This representation could be non-unique, e.g. 1/2 = -3/-6 are two different representations of the same rational number as fractions of integers. For this representation to be unique let's require the denominator b to be positive and a, b to have no common divisor. Let's consider (a, b) a point with integer coordinates on the lattice Z x Z, where Z is the set of integers (Z = { ..., -2, -1, 0, 1, 2, ... }). We can count each of these points if we begin by labeling the origin (0, 0) as "0" then (1, 0) as "1" then (0, 1) as "2",(-1, 0) as "3", etc. We can order these integer lattice points first by their increasing magnitude |a| + |b| from the origin and then by the angle they make relative to the origin and count them all one by one. Informally this shows that the set of rational numbers is not larger (in the sense of cardinality) as N_{>=0}, the set of non-negative integers. It's not smaller either as the set of non-negative integers N_{>=0} are a subset of rational numbers.