# A fun conversation about complex numbers from a problem that didn’t look so interesting at first

Last night my younger son was working in Art of Problem Solving’s Precalculus book and came across this problem:

Following the fun conversation we had last night, I thought it would be good to revisit the problem tonight for a project. We started by talking through the problem and reviewing the first solution that came to his mind:

Next we moved to a second solution – this one involving geometric series. Here we also talked through a solution that didn’t come up last night.

Finally, we talked through a solution involving trig. As I mentioned in the twitter thread, this was a fun one for me because on a stand alone basis, calculating Cos(36) – Cos(72) is a pretty difficult problem:

When I first saw the problem here, I have to admit that I thought it looked pretty dull. Turned out to be a terrific problem to illustrate a variety of different ways to talk about complex numbers! Definitely a fun surprise.

# What a kid learning about the basics of complex numbers can look like

My younger son s studying complex numbers in Art of Problem Solving’s Precalculus book. The book has a nice problem asking about cube roots:

Find the values of $z$ satisfying $z^3 = -4 \sqrt{2}$ + $4 \sqrt{2}*i$.

Although his first approach the the problem is not correct, it is interesting to see that he is close to grasping the geometric ideas about complex numbers:

In the last video he’s found 3 numbers that he thinks will satisfy the original equation. Here we check them and find that they don’t:

Seeing why his original solutions didn’t work allowed him to find his mistake from the first video and here he finds the correct solutions:

I really liked how this seemingly straightforward problem helped my son learn about complex numbers. It is always fun to see a kid starting to understand a subject right from the beginning!

# Revisiting the angle sum arctan(1/2) + arctan(1/3)

Today we did a 3d printing project revisiting an angle sum that we’d looked at last week -> arctan(1/2) + arctan(1/3).

We started by reviewing how to approach the sum using complex numbers:

Next my older son explained a geometric way to approach the problem:

Now we went to Mathematica to create the 4 triangles using the RegionPlot3D function. It is a nice geometry exercise to have kids describe the boundary of a simple 2d object:

At the end of the day I had my younger son use the shapes to assemble the 3×2 rectangle and describe how this arrangement showed that the original angles added up to 45 degrees:

I like using 3d printing to help kids see math in a different way. The problem today was originally inspired from a section on complex numbers in Art of Problem Solving’s Precalculus book. It was nice to be able to use it to explore a little bit of 2d geometry, too.

# Using Po-Shen Loh’s quadratic formula idea to calculate the Cosine of 72 degrees

We did a project using the ideas of sums and products of roots at the time, and I wanted to revisit the idea tonight now that my son is studying complex numbers. An idea I thought would be fun was to explore was how to use the sum and product of roots ideas for quadratic functions to calculate the cosine of 72 degrees.

I started with a quick review of the main ideas we’d be using in this project as it has been several months since we went through Po-Shen Loh’s idea:

Now we dove into the problem of finding the roots of the equation $x^5 - 1 = 0$.

Now we moved to the main idea in the project – how can we factor the polynomial we found in the last video – $x^4 + x^3 + x^2 + x + 1$ – into two quadratic polynomials?

The work here is a little tricky, but my son got through it really well. The ideas here are definitely accessible to students who have learned a little bit about polyomials and sums and products of roots.

Finally, we solved for the roots of the quadratic equation $x^2 + x - 1 = 0$ (I accidentally wrote this equation wrong, so we get off to a bad start. Luckily we caught the error after about a minute.)

Solving this equation gives us the value of cos(72)!

It was really fun to see that the combination of introductory ideas from complex numbers, polynomials, and sums / products of roots of quadratics could help us calculate the value of cos(72). I’m excited to play around with Po-Shen Loh’s idea a bit more and see where else we can find some fun applications!

# A neat problem from AoPS’s Precalculus book showing the connection between complex numbers and trig functions

Last week my younger son was working through a really neat problem from Art of Problem Solving Precalculus book:

We actually did a project on it at the end of last week, but the video for the last part of that project got messed up in the camera memory, unfortunately. I’ll put the first two videos of the original project at the bottom of this post – the calculations that start this project are what was in the video that didn’t record properly.

So, to start today’s project by showing how to solve the original problem by doing calculations with complex numbers:

Next I had my older son look at the same approach on a pretty famous problem that also boils down to calculating the sum of arctan values. I had planned to look at the geometric interpretation of this problem in the next video, but by happy coincidence my older son saw the geometric idea right away.

Since my older son saw the geometric connection in the last problem much more quickly than I thought he would, I had my younger son talk through the geometry here. At the end I asked the boys if they thought the geometric solution or the complex number solution was more illuminating:

Finally, off camera I asked the kids to come up with two similar problems that they thought would be fun to try to solve. They came up with two interesting ones that we played around with for about 20 min and then discussed what we found in the video below:

Definitely a fun project. The connection between trig functions and complex numbers is something that I think many kids would find fascinating. I love that that Art of Problem Solving took the time to illustrate this amazing connection!

Below are the two videos from the project with my younger son working through the original problem. I’m sad that them video with the final calculations didn’t record properly, but at least that part was mostly mechanical and easy enough to repeat.

Part 1:

Part 2:

# Using Dillon Berger’s complex exponential sum tweet to review calculus ideas with my son

I saw an amazing tweet from Dillon Berger last week:

Today I had my older son study the animation for a few minutes and then we talked about about some of the ideas it is illustrating:

Now we pulled up the tweet and had him talk about what he was seeing. It is always fun to see advanced math ideas through the eyes of kids:

Finally, we played around with the ideas in Mathematica. Although we didn’t replicate the incredible animation that Berger did, we found some fun mathematical ideas to talk through.

Berger’s tweet is great for sharing with students learning (or reviewing) ideas in calculus. It is also really fun to use to hint at some of the amazing ideas that’ll come later when students study complex numbers in more detail.

# What a kid learning math can look like -> talking through two interesting homework problems my younger son had this week

This week my son had two pretty neat homework problems – one from his math class at school and one from Art of Problem Solving’s Precalculus book. I thought it would be a nice and easy project today to go back and review these two problems.

The first one was a geometry problem from his school math homework:

The second is a problem about complex numbers from Art of Problem Solving’s Precalculus book:

# Using Juan Carlos’s Complex Function Plotter with my younger son.

At the end of January I saw a really neat tweet from Juan Carlos:

My younger son has been learning about complex numbers in Art of Problem Solving’s Precalculus book lately, and today I thought it would be fun for him to see a little bit about what complex functions “look” like.

We started with a quick review of what he knows about complex numbers and then we talked a bit about why graphing a function of a complex variable is difficult:

Next we talked about graphing a specific function -> $f(z) = z^2$

Now we moved to Juan Carlos’s plotter. Here we discussed $f(z) = z^2$ and then $f(z) = z^2 + t$ where $t$ varies between -1 and 1.

We had a couple of slips of the hand trying to type in between the tripod in this and the next video – that’s why you’ll see a few jumps in the vids.

Finally, we wrapped up by having my son graph a few functions that he created. It was fun to see what he thought about those functions:

# Working through some introductory problems on matrix powers

I saw a really neat problem in Strang’s Linear Algebra book earlier this week:

Tonight I had my son work through them on camera. These problems bring together ideas not just from linear algebra, but also from a high school algebra class.

Here’s his work on the first problem:

Here’s his work on the 2nd problem – this one is a fun surprise. The numbers don’t get big at all. In fact, this matrix has powers that are the identity matrix:

Here’s the third problem – a lot of the work in this problem is him remembering how to multiply complex numbers. I really like this problem because it brings in quite a bit of math from outside of linear algebra:

Here’s the last problem which is another fun surprise. We change one entry of the matrix in the previous problem by a tiny amount, and the powers of the matrix behave in a completely different way:

# When Cos(x) is larger than 1

My son stumbled on an amazing graph completely by accident the other day. He’s doing some work reviewing trig functions this week and I asked him to just play around with some graphs in Mathematica to get a feel for how Sin[x] and Cos[x] behave. One of the graphs he drew was:

$y = \cos( \sqrt{x})$ from $x = -100$ to $100$:

I certainly wasn’t expecting him to make a graph like this one, but was happy that he did. Yesterday we talked through what was going on.

We started by discussing why the graph seemed so strange:

Now we dove into some of the details – which involve complex numbers and the definition:

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

as well as the definition of even and odd functions. So, there’s a lot of math to that we need to bring to the table to understand what’s going on in our graph.

Finally, we calculated the exact value of $\cos(7i)$. Again, there’s a lot of advanced math that comes in to the calculation here – but even if some of the math ideas took a bit to sink in, I’d say that all in all it was a good conversation: