Since we’ve talked a little bit about continued fractions in the past, this seemed like a great way to celebrate day. We started with a quick reminder about continued fractions:

After the quick introduction, we used my high school teacher’s fun continued fraction technique – Split, Flip, and Rat – to calculate the continued fraction for . This exercise gives you a great opportunity to talk with kids about fractions and decimals.

Next up was today’s activity – the continued fraction for ! Unfortunately, for this continued fraction split, flip, and rat doesn’t work so well. Nonetheless, we do get to have a good discussion about decimals while calculating the first two pieces of the continued fraction for

To calculate a few more parts of the continued fraction we went to Wolfram Alpha. Turned out to be a pretty neat way (and obviously a much quicker way) to see the next few numbers in the continued fraction. Again, we got to have a great discussion about decimals and reciprocals.

Now, having found a few terms in the continued fraction, we went and looked at what fractions other than 22/7 were good approximations to Happy 333/106 day everyone 🙂

Finaly (and sorry for the camera screw up on this one), I wanted to show a different continued fraction for . In a previous video my younger son thought that we’d find a pattern in the continued fraction for We didn’t in the first one that we looked at, but there are indeed continued fractions for that do have amazingly simple patterns.

So, a fun little project for day. A great opportunity to review lots of arithmetic in the context of learning about continued fractions and

The simple continued fraction can also be written in a linear form, for example the 4th convergent to π can be written as [3; 7, 15, 1, 292].