What Math Topics Are Next for Welch Labs?

I’ve really enjoyed making the Imaginary Numbers Are Real series thus far. And I promise, I’ll be wrapping it up as soon as possible. What I really love about this series was presenting what I felt to be a complete picture - all the way from the origin and very basics of imaginary numbers, to the modern mathematics they enable. I’m excited to make more series like this, and am working through possible topics. Three of my favorite are:

1. Factoring

2. Euler’s Formula

3. Introductory Statistics

What do you think?

What mathematics material would you like to see from Welch Labs in the future? If you’re a teacher, what topics/formats/approaches would really help you? Please let me know what you think in the comments section below. Thank you!

 

Imaginary Numbers Are Real [Part 8: Math Wizardry]

This week we use the complex plane to solve algebra problems. I think the idea of solving tough algebra problems visually is pretty fantastic. The problems I present in this episode, \(X^3-1=0\) and \(X^8-1=0\) give solutions that are roots of unity, numbers that equal one when raised to integer powers.  These numbers have a special place in number theory, and show up in one of my favorite pieces of mathematics: The Discrete Fourier Transform (DFT):

$$X[k] = \frac{1}{N_F}\sum^{N_F-1}_{n=0}h_w[n]x[n]e^{-j2\pi k/N_F}$$

The DFT is a huge part of signal processing, allowing us to convert time series, such as audio signals, into frequency representations in the complex domain. Frequency representations of signals are crucial for all kinds of applications, such as audio filtering, image processing (e.g. Instagram filters), audio and image compression (e.g. mp3s, jpeg). The Fourier Transform also has lots of interesting overlap with our hearing systems work – which we’ll talk about a little in part 11.

 

 

 

 

Imaginary Numbers Are Real [Part 7: Complex Multiplication]

This week we uncover the connection between complex multiplications and the complex plane. Our result is another approach to complex multiplication. As shown in figure 1, we know have two completely valid, but completely separate ways to multiply complex numbers. There are certainly other math problems that are solvable by various methods - but I really like this one because it reminds me that there's more to math than what we see on the page. Since these two methods look so different, but do the same exact thing, this suggests that we are only glimpsing a deeper process from different perspectives. 

Figure 1. Two ways to multiply complex numbers and the Milky Way hangin out. 

This must raise the question, what is the deeper process? What is the connection between math and our universe? Why is math unreasonably good at predicting reality? Questions like these land is firmly in the realm of metaphysics - and are questions that people have asked for thousands of years. In fact, as we saw earlier in the series, questions like these historically have slowed down the development of mathematics. 

There's no simple answer to questions like these, but they should serve to remind us that at their core, math and science are ways that we make meaning out of the world around us.

Which is pretty cool.  

Imaginary Numbers are Real [Part 6: The Complex Plane]

“Imagine you had an art class in which they taught you how to paint a fence, but never showed you the great masters. Of course, you would say; ‘I hate art.’ You were bad at painting the fence but you wouldn’t know what else there is to art. Unfortunately, that is exactly what happens with mathematics. What we study at school is a tiny little part of mathematics. I want people to discover the magic world of mathematics, almost like a parallel universe, that most of us aren’t aware even exists.”
— Edward Frenkel

Let's Do Some Real Math

This week I'm leaving you with a challenge. The quote above is from Edward Frenkel, a Berkeley mathematics professor, author of Love and Math, and generally cool guy. Frenkel draws a really interesting analogy about how we teach math. People who do math for a living exist in an uncertain world of creativity and discovery, while math classes are typically quite the opposite. Math is, by nature, a highly technical subject - meaning that just wrapping your head around things can take quite a bit of time. This leaves many students too tired for "creative discovery". The unfortunate side effect here is that, just as Mr. Frenkel says, many students do end up feeling like fence painters, and not explorers. 

With this in mind, between this episode and the next, I'm leaving you with some actual math. A real problem. No fence painting.

Forgive me in advance for the vague nature of the problem statement - but when you come across a real problem in STEM, this is what it feels like. This is the nature of mathematical discovery - we don't know what we're searching for. The upside is that when we do find something, this makes it all the more exciting. And after all, it wouldn't really be searching or discovery if we knew what we are going to find beforehand. When you hop on a plane to fly to Albuquerque, you aren't "discovering it".

Your Assignment

So this is your job. I would like you to discover for yourself what it means to multiply complex numbers on the complex plane. 

There's a very specific, and very useful interpretation of complex multiplication using the complex plane, and I want you to find it.

Figure 1. Tools you need.

Figure 2. Examples to try.

The tools you need our colorfully summarized in figure 1. Our approach next time will make use of four examples, these are (less colorfully) shown in figure 2. I recommend for each example plotting the two numbers we're multiplying and the result on the complex plane. From there, look for patterns, make theories, test your assumptions, and do try to have some fun. Good luck!

Imaginary Numbers Are Real [Part 5: Numbers are Two Dimensional]

In part 5 we begin to see what makes imaginary numbers so unique, by investigating their special relationship with real numbers. It's interesting that mathematicians regularly made use of \(\sqrt{-1}\) for the 200 years after Bombelli's death without recognizing its deeper connection to the real numbers. I think this says a lot about how difficult it is to wrap your head around imaginary numbers. If math geniuses like Euler missed this, it certainly isn't obvious. 

This episode concludes with a concept we'll dig into more next time: the complex plane. This is the plane formed by adding the imaginary axis to the real axis. The complex plane is more useful than ordinary planes because of the special relationship between its axes - this is what we'll explore next time. 

Enjoy!