# Part 2 of talking about infinity with my younger son

Yesterday my younger son and I talked about why the set of rational numbers has the same size as the set of positive integers. That project is here:

Today we followed up on that project by taking a look at why the set of real numbers is larger. First, though, I wanted to tie up one loose end from yesterday. Unfortunately, though, I left things a little too open ended and we didn’t tie up that loose end in one video:

Now that we understood that we’d over counted (in some sense) yesterday, I wanted to show him why that over counting didn’t really change the proof. I also wanted to show him one real curiosity that comes up with infinite sets:

Now we moved on to talking about real numbers. I suspected that he’d seen Cantor’s diagonal argument before (and he had), so I asked him to sketch the proof. He got most of the way there:

Finally, we tied up the loose ends from his summary of Cantor’s diagonal argument and talked about one other surprise with infinite sets.

We had a great time talking about infinity this weekend. It amazes me that kids can get their heads around ideas in math that were absolutely cutting edge just over 100 years ago!

# Talking about infinity with my younger son

My younger son is reading Bridges to Infinity right now as part of the math / reading project that we are doing in 2021.

The chapter discussing various types of infinities caught his eye this week, so today we talked about infinity. The specific goal of the project (for me, anyway) was to help him understand why the size of the set of positive integers and the size of the set of positive rational numbers was the same infinity.

We started with a quick introduction about infinity and what it meant for two infinite sets to be the same size:

Next we moved on to talking about the positive rational numbers and how to make a 1 to 1 map between them and the positive integers. This is a longer than usual video, but it turned out my son had an idea that I wanted to pursue to the end.

In the last video we figured out that if we could get a map from the integers to the rational numbers between 0 and 1 we’d be done. Here my son shows his idea for how to create that map:

This project is going to be one I remember for a long time! It is really fun to see a kid get some ideas on a really challenging math problem, and then follow them all the way through to the end!

# Sharing Tamás Görbe’s circle packing problem with my younger son

Yesterday I saw a really neat tweet from Tamás Görbe:

It turns out that we looked at this problem a few years ago, but for reasons I don’t remember my younger son wasn’t part of that project:

https://mikesmathpage.wordpress.com/2018/04/03/a-terrific-problem-to-share-with-calculus-and-geometry-students-from-the-iowa-state-problem-collection/

Today I thought it would be really fun to tackle the problem with my younger son. We started with a quick introduction to the problem and then a discussion of how to approach the solution:

The first thing my son tried was finding the radius of a single inscribed circle inside of an equilateral triangle:

Next he tried to find the radius of the circles when there were 3 inscribed circles. This part was pretty challenging for him, but his work really shows what a kid struggling through a math problem can look like:

Now we got to the heart of the problem – what is the radius when there are “n” inscribed circles:

Finally, we looked what the area covered by the circles would be in the limit as n goes to infinity. We also talked a bit about the surprise – why isn’t all of the area of the triangle covered?

I really think this problem is a great one to share with kids who have see geometry. It is great to see how they approach the problem, and also really nice to see how they thinking about the area in the limit.

# Revisiting The Cat in Numberland

Yesterday this tweet happened to appear in my feed:

It reminded me of an old project on infinity that I’d done with the boys:

Talking about The Cat in Numberland”

Last night I had the boys read the book again and write down three things that they found interesting. This morning we talked through some of those ideas. I’d forgotten that my older son had an appointment this morning, so we were unexpectedly pressed for time. So, sorry if parts of this project feel a bit rushed – I think I panicked a bit more about the time than I should have.

Here’s what my younger son had to say about the book – he was interested in a few of the twists and turns that happened in Hilbert’s hotel:

My older son interpreted my instructions in a different way and came up with a few conjectures instead. In this video we talk about whether or not the complex numbers (with only integer coefficients) could fit inside of Hilbert’s hotel.

This was a pretty lucky break as my younger son had wondered about the rationals, which is essentially the same problem:

Finally, we discussed the real numbers and the boys both guessed that they wouldn’t fit in. At the end I showed them Cantor’s diagonal argument . . . just in time for my older son to head out!

# *Revisiting Richard Evan Schwartz’s Gallery of the Infinite

At the beginning of this year I bought a copy of Richard Evan Schwartz’s Gallery of the Infinite:

We ended up doing a fun project using the book:

Talking through Ricahrd Evan Schwartz’s Gallery of the Infinite with kids

Yesterday I saw the book on our shelf and asked the boys to find something in it that they thought was interesting. The section on bijections caught the eye of my younger son, and we used that section for a project today.

First we talked about the basic idea of bijections and how you could use bijections to tell if two sets were the same size:

Next we talked about a bijection that is pretty challenging for kids to find -> a bijection between the positive integers and the set of all integers:

Finally, we talked through Cantor’s “diagonal argument” which shows that there is no bijection between the integers and the real numbers (and, thus, that the “infinity” of real numbers is somehow larger than the “infinity” of integers!):

Tomorrow we’ll talk through the section of the book that my older son thought was interesting -> the Cantor set.

# A simplified version of the Banach-Tarski paradox for kids

Yesterday we were listening to Patrick Honner’s appearance on the My Favorite Theorem podcast. Honner was discussing Varignon’s Theorem. We actually have discussed this appearance before, but the kids hadn’t listened to the podcast, yet:

After listening to the podcast I asked my older son what his favorite theorem was:

However, after giving up on the idea initially (!) I looked at the Wikipedia page for the Banach-Tarski paradox and found an idea that I thought might work. Here’s the page:

The idea was to share the first step in the proof – exploring the Cayley graph of $F_2$ – with kids. Here’s the picture from Wikipedia:

So, here’s what I did.

First I introduced the boys to some basic ideas about a free group on two generators. I used a Rubik’s cube to both demonstrate the ideas and to show why a Rubik’s cube didn’t quite work for a perfect demonstration (I know that part of the video drags on a bit, but stay with it – there is a nice surprise):

Next we talked about the free group with two generators in more detail. My younger son accidentally came up with a fantastic example that helped clarify how this free group worked.

Then there was a bit of a surprise misconception that I only uncovered by accident. That led to another important clarification.

So, completely by accident, we had a great conversation here.

In the last video they boys thought you could use the “letters” $x, x^{-1}, y, y^{-1}$ only once. In the beginning of this video I clarified the rules.

Next we began to talk about the representation of our free group by the Cayley graph from Wikipedia pictured above. I was really fun to hear how the boys described what they saw in this graph.

Finally, we looked at two different ways to break the Cayley graph into pieces. This video is a little long, but it has a simplified version of the main idea in the Banach-Tarski paradox.

The first decomposition of the Cayley graph is into 5 pieces -> the identity element, words that start with $a$, words that start with $a^{-1}$, words that start with $b$, and words that start with $b^{-1}$. This decomposition is pretty easy to see in the picture.

The second – and very surprising decomposition is as follows:

The combination of (i) the words that start with $a$ and (ii) $a$ multiplying (on the left) all the words that start with $a^{-1}$ gives the entire set. The same is true for the combination of (i) the words that start with $b$ and (ii) $b$ multiplying (on the left) all the words that start with $b^{-1}$

Although the words describing this decomposition might not make sense right away, you’ll see that the boys had a few questions about what was going on and eventually were able to see how this second decomposition worked.

And this second decomposition gives a huge surprise -> we’ve taken 4 subsets, combined them in pairs and created two exact copies of the original set. Ta da 🙂

This project is an incredibly fun one to share with kids. I’m pretty surprised that *any* ideas related to the Banach-Tarski paradox are accessible to kids, but the simple ideas about the Cayley graph of $F_2$ really are. Using those ideas you can show the main idea behind the sphere paradox without having to dive all the way into rotation groups which I think are a little more abstract and harder to understand.

Anyway, this one was a blast!

# Talking though Richard Evan Schwartz’s Gallery of the Infinite with kids

We received Richard Evan Schwartz’s Gallery of the Infinite in the mail this week:

I thought that the boys would love reading the book and asked them to each read it twice prior to today’s project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :

The first thing the boys wanted to talk about was the “smallest” infinity -> $\aleph_0$. Here we talked about that infinity and other sets of integers that were the same size.

Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the “size” of the rational numbers was the same as the positive integers.

This argument is represented in the book by a painting of a shark!

Now my older son wanted to talk about Cantor’s diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.

Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.

Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we’ve only had the book for a few days, it is falling to pieces. Boo 😦

# Sharing Kelsey Houston-Edwards’s Axiom of Choice video with kids

Kelsey Houston-Edwards has a new video out about the Axiom of Choice:

The video is amazing (as usual) and I wanted to be able to share it with the boys. This one is a bit hard than usual – the topic is pretty advanced to begin with and is also pretty far outside of my own knowledge – but we gave it a shot.

Here’s what the boys thought after seeing the video:

Next we reviewed how Houston-Edwards divided the numbers from 0 to 1 into buckets. The boys didn’t quite have the details right, but that actually made talking through the idea pretty easy – I learned from their explanation what points needed to be re-emphasized.

Now we talked through the really challenging part of the video -> creating the set with no size. Given the challenge of explaining this idea to kids, I’m pretty happy with how the conversation went here. Also, I only finally understood the argument myself while I was explaining it to them!

Now we backed away from the complexity of the Axiom of Choice and reviewed two other slightly easier ideas that came up in our discussion. Here we discuss why $\sqrt{2}$ is irrational:

Finally, we wrapped up by discussing why the rational numbers are countable:

Although kids will have a hard time understanding all of the ideas that Kelsey Houston-Edwards brings up in her Axiom of Choice video, I think it is fun to see which ideas grab their attention. The idea that you can have a set that doesn’t have a size is pretty amazing. I was pretty happy with how things went today – exploring the ultra complex idea first and then backing off to discuss slightly easier ideas involving infinity. Definitely a fun set of ideas to plant in the minds of younger kids 🙂

# Sharing Kelsey Houston-Edwards’s video about Pi and e with kids

Yesterday I a new video from Kelsey Houston-Edwards that just blew me away. At this point I don’t have the words to describe how much I admire her work. What she is doing to make challenging, high level math both accessible and fun for everyone is amazing.

If I exchange Infinitely many digits of Pi and E are the two resulting numbers transendental?

Before showing the boys Houston-Edwards’s video, I wanted to see what they thought about the question. So, we just dove in:

Next, I took a great warm up idea from Houston-Edwards’s video and asked the boys if they could find *any* two irrational numbers that you could use to swap digits and produce a rational number.

Now, with that little bit of prep work, we watched the new video:

After the video we talked about what we learned. I think just tiny bit of prep work we did really helped the boys get a lot more out of the video.

One of the fun little challenge questions from the video was to show that (assuming $\pi$ and $e$ differ in infinitely many digits, then you will produce uncountably many different numbers by swapping different digits. I didn’t expect that the boys would be able to construct this proof, so I gave them a sketch of how I thought about it (and hopefully my idea was right . . . . )

I think that kids will find the ideas in Houston-Edward’s new video to be fascinating. It is so fun (and sadly so rare) to be able to share ideas that are genuinely interesting to professional mathematicians with kids. As always, I can’t wait for next week’s PBS Infinite series video!

# Talking about “The Cat in Numberland”

Last we did a couple of projects based on Kelsey Houston-Edwards’s video about infinity:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Extending our project on Kelsey Houston-Edwards’s infinity video

I got a comment from Allen Knutson on the 2nd project recommending using “The Cat in Numberland” to talk about infinity with kids. I ordered the book immediately and had the boys read it a few times this week. We got around to talking about it this afternoon.

Here’s their initial reaction to the book:

In the last video we I asked the boys for 3 ideas from the book that they wanted to talk about. They chose:

(1) When “Hilbert’s Hotel” is full, how do you fit one more person in?

(2) How about fitting in 26 more people?

(3) When you take away half the people how can the hotel still be full?

Here’s the explanation for part 1 – the idea here shows one strange thing about infinity!

Here’s part 2:

My older son got a little confused by the numbering of the hotel rooms in this video. The numbering of the rooms is hardly the main point, but it is nice to be able to review / revisit some counting ideas in this unusual context:

For part 3 we had a nice conversation about how you can form a bijection between the counting numbers and the non-negative even integers. That conversation went pretty fast so I asked the boys to each find another bijection and got really lucky when they picked two pretty cool ideas – powers of 2 and prime numbers.

The last movie ended with a question about whether or not the primes were infinite. This was also hardly the main point of the project, but turned out to be a fun way to end the conversation today.

So, thanks to Allen Knutson for pointing me to the book and to Kelsey Houston-Edwards for the Infinity video which has now led to three fun projects with the boys!