arXiv Open Access 2022

Parity and Partition of the Rational Numbers

Peter Lynch Michael Mackey

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Abstrak

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.

Topik & Kata Kunci

math.NT

Penulis (2)

Peter Lynch

Michael Mackey

Format Sitasi

APA MLA BibTeX

Lynch, P., Mackey, M. (2022). Parity and Partition of the Rational Numbers. https://arxiv.org/abs/2205.00565

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Informasi Jurnal

Tahun Terbit: 2022
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓