Is it easier to prove that pi is transcendental or not constructible?

[eddybob123]

hi. I've been working on a project lately about pi. and its unconstructiveness doesn't make sense. can you think of a way to possibly do this?

 

[HallsofIvy]

The fact that something doesn't make sense to you does not mean it isn't true! (It never made sense to me that George W. Bush was elected president of the United States, but...)

Do you understand what "constructible numbers" are? The only constructible numbers, in the sense of geometry (given a line segment of length "1" we can construct a line segment of this length using only straightedge and compasses), are those that are "algebraic of order a power of 2" (algebraic of order 1, 2, 4, 8, ...). $\pi$ is a transcendental number, not algebraic of any order, and so is not constructible.

 

[micromass]

hi. I've been working on a project lately about pi. and its unconstructiveness doesn't make sense. can you think of a way to possibly do this?

Well, you're in good company. The unconstructiveness of pi didn't make sense to a lot of people. In fact, for longer than 2000 year, people tried to prove it in one way or another. Starting from the ancient greeks with their question of squaring the circle. In attempts to show that pi was constructive, people invented variants of integral calculus before integrals were around! So the unconstructiveness of pi was really a very popular problem, and people couldn't imagine that it wasn't true.

Therefore, I consider it a real triomph of mathematics that pi was shown to be transcendental. For over 2000 years people have struggled to find a solution for the problem, and it was only with the new developed methods that they could find an answer. And from that moment on it appeared to people that they really could find answers to these problems, using these methods. The solution to this problem (and other related problems: the solvability of the quintic, the parallel postulate,...) is the start of modern mathematics (in my opinion).

 

[eddybob123]

but can you prove that it is transcendental? I have already come up with a small method.

 

[micromass]

but can you prove that it is transcendental? I have already come up with a small method.

Yes, you can prove that it's transcendental. A proof is given in the following link: http://myyn.org/m/article/proof-of-lindemann-weierstrass-theorem-and-that-e-and-pi-are-transcendental2/
Beware however, since the proof is quite involved...

 

[lavinia]

The construcable numbers are of a special type and do not include all algebraic numbers. Is it easier to show that pi is not algebraic? Showing that it is transcendental seems to be overkill.

 

[disregardthat]

The construcable numbers are of a special type and do not include all algebraic numbers. Is it easier to show that pi is not algebraic? Showing that it is transcendental seems to be overkill.

To show that an algebraic number is not constructible one usually shows that the degree of its algebraic extension of Q over Q is not a power of two. My point is that to show non-constructibility usually requires its irreducible polynomial over Q. So I wouldn't be surprised that its anymore difficult to show that a number is not algebraic than not constructible.