February 23, 2010
In other news, I’ve been thinking about writing another essay, similar to Convolutions and the Weierstrass Approximation Theorem. This one will be about the Fundamental Theorem of Algebra. It won’t be a totally rigorous paper – the intent will be to present a very simple idea that a bright high schooler should be able to understand: Euler’s formula for e^(x+iy) tells us that polynomials in a complex variable z = x+iy grow in all directions as |z| increases. It’s such a simple idea that that quickly leads to at least a convincing heuristic argument that it’s a shame that texts often opt to emphasize how weird it is that e^(i*pi) + 1 = 0, instead, while saying nothing more on the FToA than that Gauss proved it seven different ways because he was smart, and don’t you wish you could be smart like Gauss too?
It will be a heuristic proof, though maybe I will formally reduce it to several assertions, leaving a very specific hole in a very plausible-looking theorem unfilled by a fundamental group argument.
This is just one of those things I could’ve understood much sooner than I actually did. This stuff is supposed to make sense, after all.