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I'm tutoring a girl in 8th grade (so she is 14 years old) and she recently had a mathematics chapter about numbers. In the last paragraph they introduced the difference between rational and irrational numbers. After that they gave two examples of an irrational number, namely $\pi$ and $\sqrt 2$. In the book it wasn't proved these numbers really were irrational.

The exercises started with a few easy questions, but then the following was asked:

Is $\pi^2$ rational or irrational?

She immediately thought it had to be irrational because $\pi$ is. I explained to her this argument is false since $\sqrt{2}^2=2\in\mathbb{Q}$. I remembered that $\pi$ is transcedental so $\pi^2$ cannot be rational. However, since she is only in 8th grade and the notion of irrational was just introduced I couldn't talk about fields, minimal polynomials and such.

Does anyone know an elementary proof of the fact that $\pi^2$ is not rational?

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    $\begingroup$ Unless the book said that $\pi$ was transcendental, I don't think she's given enough information to do this. (Assuming she's not expected to use calculus and the like.) $\endgroup$ – Akiva Weinberger May 22 '15 at 20:06
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    $\begingroup$ I already suspected that ;-) $\endgroup$ – Jolien May 22 '15 at 20:06
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    $\begingroup$ As alex says, the author probably meant to ask if $\sqrt{\pi}$ is irrational. $\endgroup$ – anon May 22 '15 at 20:10
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    $\begingroup$ The author may be testing the students' intuition with this question. A typical 8th grader wouldn't have nearly enough methods to prove or understand a proof. Rather, I say it's up to them to discover that this number $\pi$ has a different $flavor$ than the others, and it's up to you to then refine their suspicions as a story-teller would, with grandeur and mystique. They ought to walk away realizing that exponents cannot force rationality and that non-rational numbers can leave us mere humans a bit dumbstruck with how to handle them. $\endgroup$ – zahbaz May 22 '15 at 20:34
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Maybe the author made a mistake, and meant to ask something like "is $\sqrt{\pi}$ irrational?"? Or maybe the author just intends to spark open-ended curiosity.

If there was an elementary standalone proof that $\pi^2$ was irrational, then it would imply $\pi$ was too. But I don't think there is a straightforward proof for $\pi$.

Making use of the given that $\pi$ is irrational doesn't help either of course.

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  • $\begingroup$ Seeing the proofs that $\pi^2$ is irrational I believe the only option left is a mistake made by the author. $\endgroup$ – Jolien May 22 '15 at 20:27
  • $\begingroup$ Maybe. But my other suggestion is that it is intentionally too hard for that level. It could just be trying to get students to think hard for the logical exercise of it all. There is value in that even if the answer to the question ends up being inaccessible to the readers. $\endgroup$ – alex.jordan May 22 '15 at 20:35
  • $\begingroup$ That is certainly true. $\endgroup$ – Jolien May 22 '15 at 20:36
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    $\begingroup$ There is an elementary proof of the irrationality of $\pi^2$ in Michael Spivak's calculus text, but it's not something to expect a students to come up with as an exercise, and even for a student to understand it is a lot of work. I suspect the author is essentially a dogmatist who says "Here are the facts of mathematics. Believe and obey!!". Lots of math teachers do that. Its disgraceful. The author was probably an idiot who didn't notice that the irrationality of $\pi^2$ does not follow from that of $\pi$. ${}\qquad{}$ $\endgroup$ – Michael Hardy May 22 '15 at 20:58
  • $\begingroup$ Perhaps it's easier to prove the only class of irrationals that become rational when squared are the roots. $\endgroup$ – Joshua Aug 27 '18 at 15:09
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Hermite's proof that $\pi^2$ is irrational is here (found via a Google search for proof that pi squared is irrational ): http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

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  • $\begingroup$ Thanks for the proof but she also doesn't know how to integrate. $\endgroup$ – Jolien May 22 '15 at 20:09
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    $\begingroup$ Then she will have to wait until she can. $\endgroup$ – marty cohen May 22 '15 at 20:12
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The identity: $$ \pi^2 = 18\sum_{n\geq 1}\frac{1}{n^2\binom{2n}{n}} \tag{1}$$ comes from the Euler series acceleration method and it can be used to prove the irrationality of $\pi^2$ and even more, for instance providing a (rather crude) upper bound for the irrationality measure of $\pi^2$. In fact, the existence of an identity similar to $(1)$ is the key of Apery's proof about the irrationality of $\zeta(3)$. However, I wouldn't try to prove $(1)$ or to explain a good portion of the technicalities of diophantine approximation to an $8$th-grader, no matter how brilliant he/she is.

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  • $\begingroup$ Thank you for your answer! Could you maybe give me a link with the details of this proof? (I never heard of irrationality measure before.) $\endgroup$ – Jolien May 23 '15 at 6:20
  • $\begingroup$ @Jolien: a pretty good sketch is given here: paramanands.blogspot.it/2013/10/… $\endgroup$ – Jack D'Aurizio May 23 '15 at 8:26
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Ivan Niven's proof is the simplest one (IMHO) with math level of medium-to-high-graders. Here it is.

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  • $\begingroup$ Isn't this a proof of the irrationality of $\pi$ itself? $\endgroup$ – Jolien May 22 '15 at 20:17
  • $\begingroup$ Right, my misread. $\endgroup$ – hOff May 22 '15 at 20:20
  • $\begingroup$ It also uses integrals. $\endgroup$ – David K May 22 '15 at 20:24
  • $\begingroup$ The pdf is no longer available... $\endgroup$ – Klangen Sep 24 '18 at 19:46

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