Complex Numbers: Nothing Is Real Anymore

Numbers are actually two dimensional. Functions can have multiple outputs. Prime numbers aren’t actually prime.

Andrew M.H. Alexander

I taught complex numbers to high schoolers for the first time in the winter of 2020. I couldn’t find any textbooks that describe complex numbers or lead students through them in a way that is remotely thoughtful or motivated, so I had to make everything up myself. I thought deeply about how to organize, connect, motivate, and justify the mathematics. This is my summary, with links to all my notes and worksheets.

I wanted the kids to discover for themselves the algebra and geometry of complex numbers, and through both a) their own discovery and b) thoughtful practice, gain a lucid intuition for how complex numbers behave.

My primary curricular goal was for us to build to understanding how complex roots work. This seems like the normal amount of stuff that gets done at this level. Plus, it’s a goal that both requires and allows lots of antecedent work and side journeys.

Of course, the roots of any complex number is “just a formula,” which the textbooks happily provide with minimal motivation or intuition. But I wanted the kids to understand how complex roots behave well enough to come up with the formula themselves. The formula is scary and complicated if you don’t know why it works. One thing well-meaning teachers often do is help kids reverse-engineer scary formulas: “See, this variable is in the formula because of such-and-such!” That’s well-intended. But it’s treating the symptoms rather than the root cause. I wanted my kids to forwards-engineer the formula, for themselves. And not to forward-engineer it in a way that required them to either a) perform heroic intellectual feats or b) rely on blinding insights from the gods—but to build it a way that is natural and easy, as a result of being carefully guided, bit by bit, slowly chipping away at the margins. Formulas should be formalized versions of our intuition, not talismans or tablets of received wisdom.

the powers of \(i\) are cyclicbasic arithmetic and algebrathe square roots of \(i\)finding roots of \(i\) in rectangularmultiplying by \(i\) is rotationthe six sixth roots of \(64\)Fermat’s Last Theoremmultiplication is adding anglespolar coordinatesdifferent representations\(re^{i\theta}\) formcomplex roots, revisited\(i^i\) and complex exponentiationImaginary Numbers Are Real

unused ideas: the fundamental theorem of algebrageometry problemsword problemshyperbolic trig functionsprime numbers aren't primequaternionsbaby linear algebraquiz questions

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