Parity and Partition of the Rational Numbers

TL;DR

This paper extends the concept of parity from integers to rational numbers, revealing a rich algebraic structure and equal natural density among three parity classes, with applications to the Calkin-Wilf and Stern-Brocot trees.

Contribution

It introduces a novel parity classification for rationals and analyzes their distribution using 2-adic valuation and natural density, connecting to well-known trees.

Findings

Three parity classes (even, odd, none) partition the rationals.

The parity pattern in the Calkin-Wilf tree repeats as odd/none/even.

All three classes have equal natural density in the rationals.

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.