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.