A Conceptual intro to Imaginaries

I feel like every single year I put in my first-draft sketchy lesson plans “conceptual intro to imaginary numbers.” Then I google that phrase and find a bunch of boring-seeming articles and dumbed-down explanations, stare at them really hard, and then end up just doing what I did the year before.

Ugh.

This year I actually came up with something, finally! We had a nice long discussion about the history of number systems, guided by this powerpoint:

Number Systems

This just goes through the different number systems in the order they were devised. It starts with counting numbers, and I asked them why people originally needed this type of number.

The next set of numbers is the natural numbers, which are the same just with 0 added in. Here I just told them that adding zero was a huge deal because humans used to think of it as useless but modern mathematics doesn’t function without it. We do a ton with zero later on for asymptotes, and my [totally realistic] hope is they will be remembering this moment in suspense!

The students correctly predicted that the next set was the integers, with the edition of negative numbers. I asked them why we have negative numbers and several of them independently came up with debt. I told them that one of the first uses was for pyramids, because they had structure above and below ground – this is a spoiler for that Crest of the Peacock chapter on 0. Here we started talking about infinity – the natural numbers are already infinite, because theoretically one can count forever. With negatives, the numbers are now infinite in both directions. Whoa.

They also saw rational numbers coming. I like to tie these into Mitch Hedberg’s joke about the 2-in-one shampoo/conditioner bottles, which he didn’t get since 2 could never fit in 1. Before rational numbers, there was no way to represent 1 divided by 2, and humans’ ancient minds were as boggled as Mitch Hedberg’s. I pointed out that with rational numbers, there are now an infinite amount of numbers BETWEEN all the whole numbers – embedded infinity! My brilliant and autistic student piped up here that of course, there are multiple levels of infinity, he already fully understood that concept. This did nothing to draw in my two math anxiety girls.

For irrational numbers, a term they remembered, I tied in Pythagoras’ cult, and the original discovery that the square root of 2 is irrational. I forget where I read this story but I assume somewhere reliable. Pythagoras, and therefore all of his followers, believed that every number could be expressed as a fraction. This one time, they were all on a boat, and one of the math guys was like “Pythagoras, I reallllllly don’t think that the square root of 2 can be expressed rationally,” demonstrating using a right triangle and Pythagoras’ own Theorem. This was unacceptable, and the cult threw him overboard. But now we all believe in irrational numbers (for the most part), and most of the students remembered pi as being labeled as one.

Oh also, we’d been talking about how the natural numbers are closed under addition, but not subtraction, so we had to add the integers to account for stuff like 4 minus 7. Then the integers are closed under multiplication but not division, so we had to have rational numbers to account for stuff like 1 divided by 2. The first example of an irrational number was the square root of a non-square number. I was like “so is there anything we can’t take the square root of even still?” and they were like “uhhhhhhhhhhhhh” and I was like “OK I’ll just tell you! It’s negative numbers! i is what you get when you take the square root of something negative!” I guess I could have handled that better?

As we were working with imaginary operations like multiplication and addition, I kept finding myself drawing out some ghosts to represent these imaginary numbers:

imaginary numbers

I started doing this last year. This is such an abstract concept that they sometimes feel like all bets are off when it came to things like how addition works. The imaginary number i was so associated with negatives that they would assume all imaginary parts should be written negative. They also sometimes want 2 imaginary numbers multiplied to still have i. I tried to come up with a metaphor – ghosts are pretty transparent, but when 2 of them multiply/ overlap you can see them, they’re just negative. This doesn’t really need a gimmick, though, since they just need to know that i * i = -1, the central concept. Drawing out the ghosts helped them to see that 2i + 3i would just be 5i, since you have 2 imaginary things and then 3 more imaginary things.

For the journal this week I had them write about imaginary/ totally constructed things that serve a purpose. They came up with a huge variety of things, from Santa Claus to the value of money to “basically everything.” These [obviously] got really philosophical, which I appreciated. One student, actually the one who talked about levels of infinity, said “imaginary numbers are extremely real, they just can’t be physically counted,” which I like a lot. I wanted to push back against the idea that these constructed things are purposeless since they aren’t “real.” Like what even IS real?!

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