Today I decided to revisit an absolutely terrific number theory book – Martin Weissman’s *An Illustrated Theory of Numbers*:

I started first by talking about regular old integers and then introduced the idea of Gaussian integers. The boys have seen complex numbers before, so although the concept of of Gaussian integers might be new, they’ve done computations with imaginary numbers before.

They had lots of interesting thoughts and ideas – the ones I chose to explore in the future sections were addition, multiplication, and what “primes” might look like:

First up – we talked a little bit about the geometry of Gaussian integers, and what adding two Gaussian integers would look like:

Next we looked at the geometry of multiplication. It is a little harder to see what is going on here, but luckily in the previous video my older son had thought to compare the length of the points (or their distance from the origin). So, that approach at least gave us a nice place to start:

Finally, we took a look at the concept of prime numbers, and found a few regular integer primes