Parity and Partition of the Rational Numbers

Abstract

Summary 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.