Parity and Partition of the Rational Numbers

Abstract

We define an extension of parity from the integers to the rational numbers. Three parity classes are found -- even, odd and `none'. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural density provides a means of distinguishing the sizes of countably infinite sets. The Calkin-Wilf tree has a remarkably simple parity pattern, with the sequence `odd/none/even' repeating indefinitely. This pattern means that the three parity classes have equal natural density in the rationals. A similar result holds for the Stern-Brocot tree.