Sharing a new result about the Cantor set with kids

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

The new result is that any number in the intrrval [0,1] can be written as the product $x^2 * y$ where $x$ and $y$ are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!