Revisiting writing 1/3 in binary

A few years ago we did a project about writing 1/3 in binary:

Writing 1/3 in Binary

Earlier this week my older son was working on a probability problem about flipping coins and that problem reminded me of that old project. So, today we revisited that old project. We also got a really fun surprise at the end.

Here’s the introduction – what to the boys remember about binary and, in particular, about writing 1/3 in binary?

Next I asked the boys to come up with a similar problem. They suggested trying to write 1/6 in base 5:

For what I thought would be the last part of the project I suggested that we take a look at the number 0.01010101… in base 5. They boys solved this problem pretty quickly based on what they learned in the last video and then my older son suggested a new problem -> what is 0.1001001001…. in base 2?

After I turned off the camera, just for fun I showed the boys what $\pi$ in base 2 was (using Wolfram Alpha). I noticed a fun connection with the number we just saw, so we added one final note to this project:

So, a fun project with a neat little surprise at the end. I think project like this are a great way for kids to expand their own ideas about numbers.