Listing the Positive Rationals

Abstract

The rational numbers are countable. The usual proof demonstrates that there exists a one-to-one function from the natural numbers onto the positive rational numbers or simply a list of all positive rationals (without repeats). But the list is not explicitly given. That is, there is no easy way to say which rational is 150th in the list or where 21 13 appears in the list. In this paper, we discuss a different listing of the positive rationals for which we can easily answer such questions. Start with a quick review of the “traditional” list. Consider the infinite array with the rows and columns indexed by the positive integers. The entry in the jth row and ith column is i j . The list is then formed by moving along the diagonals from lower left to upper right including only the rational numbers that are reduced (numerator and denominator have no common factor other than 1). The first thirty entries in this list are included. 1 2 3 4 5 6 7 8 9 . . . 1 11 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 . . . 2 12 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 . . . 3 13 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 . . . 4 14 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4 . . . 5 15 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 5 . . . 6 16 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 . . . 7 17 2 7 3 7 4 7 5 7 6 7 7 7 8 7 9 7 . . . 8 18 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 . . . 9 19 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 . . . .. .. .. .. .. .. .. .. .. .. 1 11 11 5 1 21 7 1