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 đŸ™‚

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.

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