# Hyperbinary numbers and fraction trees

With Paul Ellis and Japheth Wood. (article)

*Abstract*: Questions about infinity are fascinating, and can lead into deep mathematical topics in set theory. The mathematics of infinite sets wasn’t clearly understood until Cantor defined cardinal numbers in the late 19th century, stating that two sets are the same size if there is a one-to-one correspondence between them. One surprising result from set theory, first proved by Cantor in 1873, is that there are precisely as many rational numbers (fractions) as there are counting numbers. Over one hundred years later, mathematicians Neil Calkin and Herbert S. Wilf published a more elegant proof of this fact.

This article is the result of our work to develop the ideas in the Calkin-Wilf proof, so that they would be accessible to the teachers in our three different Math Teachers’ Circles. We designed an investigation into the hyperbinary numbers (itself a 19th century topic that predates Cantor’s work on cardinality) and developed the Tree of Fractions, much in the style of Calkin and Wilf. We asked teachers to make observations, ask questions, and convince each other of the veracity of their claims.