Tag: imaginary numbers

Having the boys try out some fun and surprising ideas about 360 and 720 degree rotations from Eric Weinstein’s interview with Roger Penrose

Yesterday I heard a terrific interview with Roger Penrose on Eric Weinstein’s podcast:

The podcast episode is here:

‘Today the boys and I talked through three fun to see, but maybe tricky to understand, ways that something need to rotate 720 degrees to get back to where it started.

We started by looking at a circle rotating around a second circle of the same size:

Next we looked at the famous “wine glass” problem. I originally wanted to color the water in the glass with food coloring, but chickened out!

Before going on to the Dirac Belt Trick, I showed the boys this really nice video showing the trick in a pretty unusual – and super fun – way:

After the video demonstration, I had the boys try the trick with a belt. At the end my old son made a connection between the belt trick and the complex numbers which was a nice and totally out of the blue surprise to me:

Anyone interested in physics should listen to Weinstein’s interview of Penrose – it is amazing. I was really happy to be able to pull out a few ideas from the interview to share with the kids today!

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!

Ed Frenkel, the square root of 2 and i

A few weeks ago, some thoughts on twitter from Michael Person inspired this talk with my kids:

https://mikesmathpage.wordpress.com/2014/04/19/imaginary-numbers/

Last weekend I picked up the audio book version of Ed Frenkel’s “Love and Math” and Frenkel’s discussion of \sqrt{2} and i made me want to revisit this conversation about properties of numbers.

We started with \sqrt{2}.   Their reaction to hearing that we were talking about \sqrt{2} was to talk about why it was irrational, and since they nearly remembered the proof from last time, this proof made for an instructive start to the conversation today.  It is always nice to review some of the ideas behind these simple proofs with them and watch their ability to make mathematical arguments develop.

Next we moved on to talking about i.  They remembered a few basic properties about i, though my older son still thinks that it is something that math people just made up.  I’m not terribly bothered by that for now, but the ideas in Frenkel’s book are giving me some new perspective on how to present some of these more advanced concepts to the boys.  Hopefully this new perspective is going to lead to a much better approach to teaching them math.  In any case, here’s what we said about i:

The next two videos are the main point of the talk today – in what ways are \sqrt{2} and i similar?  This question is a specific example of the broad question of symmetries in math that Frenkel discusses in his book.  I felt like the book walked up a couple of stairs and then hopped into an elevator to the top floor, though.  The ideas were inspiring, but I was left (i) wanting more and (ii) wanting to fill in a few more details.  One focus of these math conversations with my kids over the next few years will be spent on (ii).  I’ll work on (i) by finishing the audio book on a drive to and from Boston this weekend!

For today, though, let’s just stick with some similarities between \sqrt{2} and i that Frenkel highlights:

So, without digging too deep into the details, it looks like the set of numbers that we get by adding \sqrt{2} to the rational numbers has some nice, simple properties.  If we add or multiply, we seem to never leave the system.  Pretty neat.  i seems to have the same property.  Frenkel make the point that is we aren’t too bothered by \sqrt{2}, we shouldn’t be that bothered by i.  This is a nice point, obviously, and a fun idea to share with kids.  I really loved that my older son made the connection between i and x from algebra.  Only one step away from polynomial rings . . .  ha ha!

So, definitely on the theoretical side, but a definitely a fun morning.  Looking forward to plucking a few more ideas  out of “Love and Math”  to share with the boys.

Imaginary Numbers

A few weeks ago, when I was in London, Michael Pershan wrote an interesting piece about complex numbers on his blog:

In my jet-lagged state I came up with what seemed like a great response, but not too surprisingly, it didn’t seem so great once I wasn’t so jet-lagged!   However, Pershan’s post stayed with me for a couple of weeks.  I’m not sure why – there wasn’t anything specific that bothered me, or at least nothing that I was able to articulate well.  I still couldn’t shake it though, and because I’ve spent the week with my older son talking about logs and imaginary numbers I really wanted to talk to the boys about i.

By lucky coincidence just this morning Steven Strogatz linked  to another blog post on twitter that helped me collect my thoughts on the subject of i (even though the blog post has nothing to do with complex numbers):

The part of this post that got my attention was this line:  “Well, I’ve finally got my answer, and it only takes eleven words: Math is big ideas, approached from as many angles as possible.”

So, a few weeks of thinking things over in my head combined with a little jolt from the Math With Bad Drawing’s post led to this morning’s Family Math conversation:

(1) The first thing I wanted to talk about was the hardest – structure.  I dug up my old copies of Mike Artin’s and van der Waerden’s “Algebra” books and talked about rings.   I don’t think I did a good job here, but all I wanted to do was point out some of the things that mathematicians think are important about the number system.  A lot of this was sort of review – they know the words “associative” and “commutative” though they may not have ever seen all of this structure on the board all at the same time.  They certainly had never heard the word “ring” in this context before:

(2)  With the background structure out of the way, and in particular with the mention of 0 and 1, I wanted to start talking about some specific numbers.  We started with an “easy” number – 0.  What are the important properties of 0?  Why doesn’t 0 seem strange to anyone any more?

 

(3) Next up was -1.   Same sort of idea behind this talk as the talk about 0. -1 is a little hard to understand, but we seem to be pretty comfortable with it and I don’t think many people would think -1 is a number that mathematicians just made up:

 

(4) Next up was \sqrt{2}.  I wanted to talk through some of the ideas behind irrational numbers and hint at some of the confusion that they had caused over the years.  We talk about a simple right triangle and show how \sqrt{2} comes up pretty naturally.  We also then walk through the proof of why it is irrational:

 

(5) Now for two famous irrational numbers \pi and e.   We just touch on a few simple properties of these two numbers and talk about they are in some sense even more strange than \sqrt{2}.

(6) Finally we get to i.  Following Artin and van der Waerden, I introduce the complex numbers by looking at a specific quotient ring in a polynomial ring . . . . Ha!

Actually, we first quickly review a few of the interesting numbers that we’ve talked about already and then point out that we still do not know about any number that satisfies a pretty simple equation – x^2 + 1 = 0.  I call that solution “i” and then we talk about a few interesting properties ranging  from the famous e^{\pi * i} + 1 = 0 to Gauss’s Fundamental Theorem of Algebra.    I also mention a few applications that come up in physics.

So, thanks to Michael Pershan and to Ben Orlin for both causing and helping me to think through talking about i.  Definitely a fun morning.