User justin lanier - MathOverflow

Answer by Justin Lanier for Mathematical ideas named after places

2011-05-12T05:58:26Z

What is the simplest, most elementary proof that a particular number is transcendental?

2010-12-14T04:15:33Z

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even quaternions and surreals). One thing that's been hanging in the air is giving a proof that there really do exist transcendental numbers (and in particular, real ones). They're willing to take my word for it, but I'd really like to show them if I can.

I've brainstormed two possible approaches:

1) Use diagonalization on a list of algebraic numbers enumerated by their heights (in the usual way) to construct a transcendental number. This seems doable to me, and would let me share some cool facts about cardinality along the way. The asterisk by it is that, while the argument is constructive, we don't start with a number in hand and then prove that it's transcendental--a feature that I think would be nice.

2) More or less use Liouville's original proof, put as simply as I can manage. The upshots of this route are that we start with a number in hand, it's a nice bit of history, and there are some cool fraction things that we could talk about (we've been discussing repeating decimals and continued fractions). The downside is that I'm not sure if I can actually make it accessible to my students.

So here is where you come in. Is there a simple, elementary proof that some particular number is transcendental? Two kinds of responses that would be helpful would be:

a) to point out some different kind of argument that has a chance of being elementary enough, and

b) to suggest how to recouch or bring to its essence a Liouville-like argument. My model for this is the proof Conway popularized of the fact that $\sqrt{2}$ is irrational. You can find it as proof 8''' on this page.

I realize that transcendence is deep waters, and I certainly don't expect something easy to arise, but I thought I'd tap this community's expertise and ingenuity. Thanks for thinking on it.

English translation of Lambert's Theorie der Parallellinien?

2010-11-08T05:32:54Z

Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the bibliography of Jeremy Gray's Worlds out of Nothing at this link, but I have been unable to find it in English. Any help would be appreciated.

Comment by Justin Lanier

2011-05-12T05:50:22Z

@Michael Probably because of the layout of the DC Metro--to move between two outer locations (say Shady Grove and Vienna/Fairfax), one often has to travel to the center of the District along the way.

Comment by Justin Lanier

2010-12-15T03:21:51Z

Thanks! Will do!

Comment by Justin Lanier

2010-12-15T01:21:29Z

Hi, David. Thanks for your answer. Can you expand upon why "With n large enough, nothing arising out of p_(L) can balance these contributions"? I like the approach of focusing on the term of highest power, but I only see that for large n, the contribution is very small, and so I don't see the contradiction. I also like you remark about Cantor and Liouville being equivalent. But I don't see how exactly it could be that each nonzero digit "kills" a collection of polynomials, since if you removed one 1 and left the rest of L the same, wouldn't it still be transcendental? Thanks again.

Comment by Justin Lanier

2010-12-15T01:08:16Z

Daniel, thanks so much for your thoughtful answer. A few clarifying questions: 1) How do you know that a 4 doesn't pop up somewhere in L^2, or more generally that the sum of some factorials doesn't equal the sum of some others? 2) Do you mean if we look between the 2(n-1)! and the 2n! place for large enough n, we'll see the integer multiplier bare at the end of that stretch? If so, how do we know that there aren't any 2's cropping up along the way that would mess things up? 3) What with the 2's, I feel lost on how you're calculating the length of the swaths of zeros. Could you expand on this?

Comment by Justin Lanier

2010-11-14T19:05:09Z

Thanks, Franz! Maarten Bullynck directed me to "From Kant to Hilbert" by William Ewald. In the first volume (pp. 152-167), the first ten or so pages of Lambert's paper are translated by Ewald himself. Ewald's introductory essay is also really interesting. I'm continuing to research, but I've yet to find a full translation of Lambert's essay on parallels into English. I'll post something here if anything turns up.