 We can reorder the integers, alternating between positive and negative: 0, -1, 1, -2, 2, -3, 3, …. This ordering has the same order type as the natural numbers, and thus enables a one-to-one correspondence between the natural numbers and the integers. One of Cantor’s most striking early observations is that the same is possible with the positive rational numbers. Every positive rational number can be written uniquely in lowest terms as some fraction \(p/q\), where \(p\) and \(q\) are positive integers with no common factor. We can then order these fractions by first comparing the sum \(p+q\) of numerator and denominator, and then if the sum is equal for two fractions, put the one with lower numerator \(p\) first. This ordering begins 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, …. (Note that fractions like 2/2, 2/4, 3/3, 4/2, are missing from this list, because they are not written in lowest terms.) Every positive rational number must appear in this list (because it can be written in lowest terms with some particular finite sum of numerator and denominator), and only has finitely many predecessors (because there are at most \(n+1\) fractions whose sum of numerator and denominator is \(n\)).