The cube root of 1

After a week of doing a little bit of practice for the AMC 8, my older son has returned to Art of Problem Solving’s Precalculus book. The chapter he’s on know is about trig identities.

Unrelated to his work in that book, the cube root of 1 came up tonight and he said “that’s just 1, right?” So, we chatted . . .

First we talked about the equation $x^3 - 1 = 0$:

For the second part of the talk, we discussed the numbers $\frac{1}{2} \pm \frac{\sqrt{3}}{2}$ and their relation to the equation $e^{i\theta} = \cos(\theta) + i\sin(\theta)$

Finally, I connected the discussion with the double angle (and then the triple angle) formulas that he was learning today. You can use the same idea in this video with $\cos(5\theta)$ to find that $\cos(75^{o}) = (\sqrt{5} - 1)/4$:

So, a lucky comment from my son led to a fun discussion about some ideas from trig that he happened to be studying today 🙂

Working through some introductory trig ideas with my son

My older son is working through Art of Problem Solving’s Precalculus book this year. I love this book and am happy to help him work through it. For now the topic is trig functions. Here are three nice conversations we’ve had in the last week:

(1) Exploring some parametric plots:

We had been looking at graphs of sin(x) and cos(x), but what if I made the x-coordinate a function cos(t) and the y-coordinate function of sin(t)? What would that look like?

(2) A clever homework problem that gave him some trouble

The question to asked him to find the product:

$\tan(\pi/12)*\tan(2\pi/12)*\tan(3\pi/12)*\tan(4\pi/12)*\tan(5\pi/12)$

(3) Talking about the sum and difference formulas

This morning he was working through the geometric proofs in the book that give you the sum and difference formulas for cos(a + b) and sin(a + b). After that I showed him how the formulas come from the formula:

$e^{i \theta} = \cos{\theta} + i\sin{\theta}$

I’ve never taught a precalculus class before, so I’m explaining most of these ideas for the first time.  Hopefully the book will give him a good foundation and my fumbling around will show him one or two fun ideas.   I think this will be a fun year.

Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map $Z \rightarrow Z^2.$ The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map $Z \rightarrow Z^2.$

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at $(1 + i)^2$. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

Using Jacob Lurie’s Breakthrough Prize lecture to inspire kids

I was a little sad that the Breakthrough Prize switched the idea behind the lectures from prize winners from (what I understood to be) a public lecture to one pitched at first year graduate students.

Granted, it can be difficult in math to deliver public lectures about your work, but Jacob Lurie’s Breakthrough Prize lecture after winning the 2014 Breakthrough Prize is a public lecture masterpiece:

To show how lectures like Lurie’s can inspire kids, this morning I watched the first 10 to 15 minutes of his lecture with my kids and then asked them about what they thought. It led to a really fun conversation.

To start – my younger son (who is in 4th grade) was interested in Lurie’s discussion of modular arithmetic, and we used some clock arithmetic to work through a few examples:

After that my older son said that the ideas about the ever-growing sets of different kinds of numbers was something he thought was interesting:

After hearing about the ideas that they found interesting, I asked each of the boys to come up with a question about the integers that they thought would be interesting. My younger son asked a question about $i$ instead – his question led to a super fun talk about properties of $i.$ You never know what is going to grab a kid’s attention!

Finally, I asked my older son to come up with a question about integers, and he wondered if we’d be able to prove that $\sqrt{3}$ was irrational!

Oh yes!! Another great conversation!

So, I really hope the Breakthrough Prize folks go back to more of a “public lecture” format than a “first year graduate student” format. The public lectures can be such a powerful tool for inspiring kids.

A neat complex number program from Dan Anderson

I drove back to Boston from NYC today and was pretty tired when I got home. BUT, upon arrival I saw this incredible tweet from Dan Anderson:

I had to ask Dan (and click through a few of the related tweets) to figure out what was going on, but the final shape in the Gif is the image of the unit circle under the map:

$z -> z + z^4 / 4 + z^9 / 9 + \ldots + z^{n^2} / n^2 + \ldots$

Amazing!

It was clear that the boys would find Dan’s tweet interesting, so I thought up a short project over dinner. Before diving in to that project, though, I just showed them the gif that Dan made and asked for their thoughts. The pings that you hear from my phone during the four videos are a barrage of more neat gifs from Dan!

After hearing what the kids thought about the shape, we moved to the white board to talk a little bit about complex numbers. The kids have heard a bit about complex numbers in the past – just not recently. Once we finished a quick review of some of the basics of complex numbers, we talked about what the image of the unit circle looks like when you take powers.

The talk here was obviously not intended to be a comprehensive talk about complex numbers. The ideas here are just what came to mind when I saw Dan’s picture plus my attempt to explain those ideas to kids on the fly.

Now I explained the picture Dan was making. When you first write down the map (as above) it looks really complicated. We tried to simplify as much as possible by walking through the images of some easy numbers. I was happy that we were actually able to make some good progress here. (Also happy that my older son thinks every infinite series adds up to -1/12. ha ha – thanks Numberphile!)

After we finished this part, Dan actually published a gif showing the first 20 steps – that picture helps you see the images that we talked about in this video:

Not having Dan’s latest gif handy during our project, we went back to his original picture to see if we had understood how 1, -1, i, and -i behave. It is really neat that you can explain the behavior of these special points to young kids 🙂

We end this video by looking at another new gif that Dan made while we were working out our project – it shows the images of circles with radius ranging from 0 to 1 under the map. Great work by Dan!

So, a really fun project for kids thanks to Dan’s incredible programming work. Such a great way to introduce kids to the cool behavior of complex numbers.

Stuart Price and Joshua Bowman’s PIth roots of unity exercise

Saw this amazing post about the $\pi^{th}$ roots of unity yesterday and wanted to use it for a quick extension of our $\pi$ day activity:

Unfortunately, kids up at 11:00 pm, dog up at 1:00 am, cat up at 4:00 am and then everyone up at 6:00 am let to “one of those nights” . . . . So, instead of making use of a really great exercise, I sort of totally butchered it – but it is the idea that counts, right 🙂 Despite stumbling through our project this morning, I can’t recommend Price’s post and Bowman’s Desmos program enough.

For the kids to understand the project a little better, I wanted to do a quick introduction to the complex plan and how the roots of unity show up on the unit circle. We’ve talked a little bit about $i$ before, so the ideas here aren’t totally new to the kids, but a quick re-introduction seemed appropriate:

Next up was a reminder of some of the rational approximate to $\pi$ that we found yesterday in our activity inspired by Evelyn Lamb:

The fractions that we found that approximate $\pi$ are 22/7, 333/106, and 355/113. We reviewed these fractions and also what the similar approximations to $2\pi$ would be.

With the background out of the way, we moved on to Bowman’s Desmos activity. First I just like the kids play around with it and see if they could find a situation in which we nearly had a regular polygon using powers of the $\pi^{th}$ roots of unity. This was a fun “what do you notice” exercise.

Also, sorry for the extra blue screen – don’t know what happened to the camera here. Double also, ignore all of my talking for the first minute, please . . . . I was tired, confused, and incoherent.

Finally, having found the number 44 as a case where the dots where nearly equally spaced and having seen that this approximation was the same number we saw in the numerator of our 44/7 approximate for $2\pi$, we looked to see if we’d see something interesting at 666 and 710. Right around 2:00 is the “wow” moment.

So, a fun project showing a geometric representation of some continued fraction approximations for $\pi$. Definitely one I’d like to have a 2nd, non-exhausted chance at, but oh well. Hopefully the awesome work of Stuart Price and Joshua Bowman shines through over my several stumbles in this project.

I asked a kid what his favorite number was, you won’t believe what happened next :)

A friend from graduate school and her son stopped by yesterday on their way down to NYC.  Her son and my kids have a lot of common interests, including math.  Chit chatting over dinner I asked him what his favorite number was and got quite a surprise:

$\pi^{e^i}$.

Now that’s a favorite number!

Although his particular favorite number is a little difficult to talk about without getting into things like cos(1) and sin(1), I though that it would be fun to show the kids a little bit about complex numbers.   My end goal was to show the kids the value of $i^i$ rather than $\pi^{e^i}$, but first we had to have a little discussion about powers:

With some of the simple properties of exponents out of the way, the one last thing we needed to touch on before taking about $i^i$ was square roots. Despite intending to be informal here, I was unfortunately a little too informal and I think that caused a little bit of confusion. It took a few extra examples to get us back on track.

Next up was the fun identity $e^{\pi i} + 1 = 0.$ I didn’t have any interest in deriving this equation, rather I just wanted to use it as a starting point. With just a few manipulations of this equation we can come to a value for $i^i.$ It was super fun to see all three kids react to the surprising value of $i^i.$ At the end I mention the approximate value of $\pi^{e^i}$ as well.

So, a nice little math talk this morning sparked by this surprise favorite number!