Talking about infinity with my younger son

My younger son is reading Bridges to Infinity right now as part of the math / reading project that we are doing in 2021.

The chapter discussing various types of infinities caught his eye this week, so today we talked about infinity. The specific goal of the project (for me, anyway) was to help him understand why the size of the set of positive integers and the size of the set of positive rational numbers was the same infinity.

We started with a quick introduction about infinity and what it meant for two infinite sets to be the same size:

Next we moved on to talking about the positive rational numbers and how to make a 1 to 1 map between them and the positive integers. This is a longer than usual video, but it turned out my son had an idea that I wanted to pursue to the end.

In the last video we figured out that if we could get a map from the integers to the rational numbers between 0 and 1 we’d be done. Here my son shows his idea for how to create that map:

This project is going to be one I remember for a long time! It is really fun to see a kid get some ideas on a really challenging math problem, and then follow them all the way through to the end!