My favorite proof that the square root of 2 is irrational -> continued fractions

Last week my younger son and I did two fun projects studying the proof in this tweet from Lior Patcher:

This weekend I thought it would be fun to explore my favorite proof – the approach using continued fractions.

We’ve talked about continued fractions before, but probably not for a few years, so I started the project today by asking my son what he remembered about them:

Before moving on to the square root of 2, we talked about why rational numbers would always have finite continued fractions:

Now we calculated the continued fraction for the square root of 2 – it has a pretty fun surprise:

Finally, and this part was just for fun, I showed him the neat little mathematical trick for quickly calculating the convergents. We looked at the first few fractions that were good approximations to the square root of 2.