The field of constructible lengths
October 6, 2011 2 Comments
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 , since that already means complex numbers. So how about . You won’t find this notation elsewhere, but I think its good notation.
Yesterday, we showed that if , then so is , , , and (provided that . About a month ago, we learned the definition of a field. Notice anything?
Woah! is a field! How cool is that? Well, we actually showed one more thing. We showed that whenever , and is positive.
This has some immediate, and amazing consequences. You have probably memorized the fact that
This means that is constructible, and since , we also know that is constructible. Together, this tells us that the angle is constructible, so we can construct a regular heptadecagon (17-gon).
Yeah, that’s pretty awesome.