How do we know this is true? What is the proof?

Someone emailed me just recently with this question:

"How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?"
(The person who emailed me this question is from Japan, based on the email address.)

I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi.

You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever.

I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational.

Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number.

It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi and it's irrational. And what does that mean, irrational? they ask. And they're told it means the decimal expansion continues forever without repeating.

School mathematics does a lot of announcing facts. I hope it doesn't kill in the students that small wondering, questioning, interested voice that wants to know more:
But why? How do we know that? Is there a proof?

Do you know?

Now that I have you interested, I'll probably disappoint you by telling that in the case of pi, the proofs about its irrationality require understanding of calculus and "advanced math" - math advanced beyond basic algebra. But here's one anyway:
Proof that Pi is irrational
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