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Comment by ahelwer | Hacker News Reader

I think we can all relate to an unclear sense of awe upon first encountering something like e^(i * pi) = -1 (which might have occurred around the first time reading Contact). It takes exposure to a certain amount of math to switch paradigms: it would be weirder if that identity weren't the case.

Dazzling inexperienced students with magical identities and coincidences isn't teaching mathematics, it's teaching numerology. I really appreciated this article.

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In an ideal world, anyone trying to dazzle students with e^(i*pi) should immediately be rebuffed with the demand: "define complex exponentiation". While I was teaching calculus, I found that students have a tendency to assume that any function sufficiently familiar to them, must be defined for whatever values you can think to stick in it; if you were teaching the mathematics of chairs and tables, they wouldn't bat an eye if you mentioned cos(chair). This phenomenon also explains why there's such a ridiculously big deal made by laymen about the undefinedness of division-by-zero.

On the one hand, I agree with you.

On the other hand, to answer your demand: I define complex exponentiation by the formula e^(it) = -1 for all real, nonzero t.

Clearly stupid. Why? There are a lot of properties that one might naturally ask e^(it) to satisfy (for example, that e^(it) * e^(iu) = e^(i(t + u)), or that its derivative is ie^(it)), and the usual complex exponential satisfies all the right ones. So writing e^(i * pi) with no context has, I submit, at least some content.

Yeah, once you work through the explanation of that kind of thing the magic kind of falls out of it when you realize how the definitions fit together to produce that result.

However, I think there are certainly some areas of Math that are still pretty mysterious to the human intellect, no matter how advanced. The distribution of prime numbers and chaos come to mind.

The Monster group is my favorite example. It may very well have the same nature of "just falling out of definitions in an unsurprising manner" but it's certainly too large for any human to ever feel that way. Monstrous Moonshine was this project and is mentioned in the article.

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