The field of constructible lengths

Let’s say that a real number is constructible if it can be the length of a line segment which can be constructed via the methods described yesterday. But in reality, we’ll be slightly more general than that. We’ll allow negative numbers as well, (sort of like drawing the line in the opposite direction). This will make everything very nice.

We should have a set of all constructible numbers. We of course can’t use \mathbb C, since that already means complex numbers. So how about \mathbb K. You won’t find this notation elsewhere, but I think its good notation.

Yesterday, we showed that if a,b\in K, then so is a+b, a-b, a\cdot b, and \frac ab (provided that b\neq0. About a month ago, we learned the definition of a field. Notice anything?

Woah! \mathbb K is a field! How cool is that? Well, we actually showed one more thing. We showed that \sqrt a\in \mathbb K whenever a\in\mathbb K, and a is positive.

This has some immediate, and amazing consequences. You have probably memorized the fact that

\cos\frac{2\pi}{17}=\frac1{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}\right)

This means that \cos\frac{2\pi}{17} is constructible, and since \sin\frac{2\pi}{17}=\sqrt{1-\left(\cos\frac{2\pi}{17}\right)^2}, we also know that \sin\frac{2\pi}{17} is constructible. Together, this tells us that the angle \frac{2\pi}{17} is constructible, so we can construct a regular heptadecagon (17-gon).

Yeah, that’s pretty awesome.

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2 Responses to The field of constructible lengths

  1. Pingback: The field of constructible lengths, part 2 « Andy Soffer

  2. Pingback: More awesomeness of the tower rule « Andy Soffer

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