A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

What an amazing result – there are only finitely many numbers in the Cantor set with terminating decimals – and they can be listed in one line! https://t.co/qeTVWP2ddE pic.twitter.com/jrxEyKUYMo

— Mike Lawler (@mikeandallie) March 19, 2021

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

Idle question that came up in real analysis. 1/4 is in the Cantor set. Is 2^-n in the Cantor set for any n>2?

— Jordan Ellenberg (@JSEllenberg) March 19, 2021

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.