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.
Best Answer
The original Liouville's number is probably the easiest, but most of the proofs tend to invoke calculus (because why not?), so let me try to show it in a more 7th-grade friendly way. I'll call this the swaths-of-zero approach.
So we know that Liouville's number $L$ looks like this: .1100010000000000000000010... with a 1 in the $n!$ places.
When we square it, we get this: .012100220001000000000000220002...
What happens is that in the $2n!$ places we get a 1, and in the $p!+q!$ places we get a 2. (The great thing about this is that it can be explained using the elementary-school algorithm, the one they are all familiar with, for multiplication.)
If we multiply $L$ by an integer and write down the answer, the value of that integer will be "laid bare" as we go deeply enough into $L$'s decimal expansion, as eventually the 1s are far enough away to become that integer without stepping on each other.
Similarly, if we multiply $L^2$ by an integer, we will see that integer in some places, and 2 times that integer in others. For large enough $n,$ if we look between the $n!$ place and the $(n+1)!$ place, the last thing we'll see is that integer written at the $2n!$ place.
Thus the swaths of zero in the multiple of $L$ are, $n!-(n-1)!=(n-1)(n-1)!$ long (minus a constant), whereas the widest swaths of zero in the multiple of $L^2$ are $n!-2(n-1)!=(n-2)(n-1)!$ (minus a constant) long, which is shorter, so there is no way to add positive multiples of $L$ and $L^2$ together to clear everything after the decimal point, or find positive multiples of each so that everything after the decimal point is equal.
More generally:
Suppose $a_jL^j+...$ and $a_kL^k+...$ are integer polynomials in $L,$ where $j>k.$ We show that their values cannot match up fully past the decimal point. The swaths of zero in the first polynomial, moving back from the $n!$ spot, are a constant away from $(n-j)(n-1)!$ long (the constant being the length of the sum of the coefficients), whereas in the second they are a constant away from $(n-k)(n-1)!$ long, in the same place (moving back from the $n!$ spot).
I don't know if this explanation holds up to the standards of rigor you like to maintain when teaching them, but I think they will find it fascinating.