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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
by xamuel 12y ago
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.
- impendia 12y agoOn 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.