Is there a sufficiently reachable plausibility argument that $\pi$ is irrational? I was teaching someone earlier today (precisely, a twelve-year-old) and we came upon a problem on circles. Little did I know in what direction it would lead. I was able to give a quick plausibility argument that all circles are similar and that they are related by some constant (most conventionally $π$). We came to the actual computation of $π$ and there was no way I could escape mentioning its irrationality. Indeed, being a somewhat bright student, this person asked, after I had written out the partial expansion $3.141592...$ whether she could not continue computing the digits, and how she could do this. Although I mentioned approximation by polygons as one possible approximation algorithm, she assured me that she would continue to find more digits, to which I asked, To what end? Indeed, I went on, it is impossible to finish computing the digits. But it must start to repeat after some digit, no matter how large, she retorted. No, I said. It is an irrational number (we had earlier talked about irrationality and it was easy enough showing a variant of the classic proof of the irrationality of $\sqrt 2,$ being a simple algebraic number). She then asked for reasons why this is the case. Suddenly, I found myself short of explanation as I had myself never bothered to study any of the known proofs in detail. I tried telling her that the known proofs were impossible for her to understand now, but she persisted nevertheless. Then I promised that I would come with it when next we met.

However, I am sure that this would not benefit her in any way. Therefore, I sought for a simple explanation (not necessarily a proof in the usual sense) that was sufficiently convincing, but so far I have found none. All I have found are variants on Lambert's or Hermite's proofs, and those are far from what I'm looking for.

In consequence, I thought to ask here. Perhaps someone has come across a similar situation and had found an argument sufficiently enlightening at that level (that is, a sketch of ideas or plausibility argument that can lead to a proof -- ideas can always be grasped by anyone, after all). In short, do you know any argument that I could present to this person that could at least slake their curiosity for now until they are ready for the classic proofs (if they continue to be interested)? If so please present them.

Thank you.
But it must start to repeat after some digit, no matter how large, she retorted.


This sounds like a misunderstanding that you could do something about by showing her a concretely defined different irrational where it is clear to see that the digits cannot repeat, such as
$$ \sum_{n=1}^\infty 10^{-n^2} $$
If you manage to convince her that is is possible for a number to have a decimal expansion that will never start (and continue!) repeating, it might become easier to accept that $\pi$ could be one of those numbers too.
Consider regular polygons with $2^n$ sides inscribed in a circle. To determine the ratio of the area of a $2^n$-gon to that of a square, one truncates Viète’s formula at an appropriate number of factors:

$$ \frac{A_4}{A_{2^n}} =\underbrace{{\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots}_\text{$n-2$ factors} $$

The limit of the ratio is of course $\dfrac2\pi$. To the student, the factors should look obviously irrational. And while it takes a rigorous proof (Mathologer’s take on Lambert’s proof is probably the most accessible) to exclude the possibility that the product might somehow end up rational, it should be apparent that it would be exceptional for $\pi$ to be rational if its natural approximations are all irrational. At least the student should see that other quantities, clearly related to $\pi$, are indisputably irrational and it can’t be claimed that digits will eventually repeat, and the same can very likely be said for $\pi$ as well.