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Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it funny too (I just hope she never gets to see this question).

enter image description here

Of course the first thing I noticed was the gross mistake at 9 o-clock. But then this mistake got me thinking about what would happen if $\pi$ were a rational number. By this I mean,

What sort of important results depend crucially on the irrationality of $\pi$?

The only example I was able to come up with is the good old greek problem of the squaring of the circle, which basically asks for the constructibility of $\sqrt{\pi}$ and thus if $\pi$ were rational, its square root would be constructible and thus the problem would not be impossible.

NOTE

I edited the question title and a part of my question because as was pointed out in some of the comments, that part didn't make much sense. Although I didn't know that at the moment, so it wasn't such a bad thing that I included that "nonsense" in my question at first. But as Bruno's answer explains, some part of my original misunderstanding can be given some sense after all.

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    Mathematics would be inconsistent.2012-05-01
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    We'd all be unemployed =)2012-05-01
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    @BrianM.Scott Can you please explain why? Maybe you can add that as an answer if you don't mind?2012-05-01
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    @Adrián: http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational2012-05-01
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    @AdriánBarquero: Basically, there's no "if" or "when" about the irrationality of $\pi$. So it makes no sense to ask "what if $\pi$ is rational".2012-05-01
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    @sdcvvc Thanks for the link, but I already know that $\pi$ is an irrational number. I was trying to think about what results would not hold if this were not the case.2012-05-01
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    @ZhenLin Of course it's a bit of a nonsense question, since math would thus be inconsistent and you one can derive anything from a contradiction. But the OP asks "What sort of mathematical results depend crucially on the irrationality of $\pi$?" which is by no means a vapid question.2012-05-01
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    @Adrian: But there is a **a proof** $\pi$ is irrational. It would not somehow vanish if you assumed that $\pi$ is rational in first place. Your first question - what proofs use the fact that $\pi$ is irrational - is OK; the second question - "what if $\pi$ was rational" is like asking "what if $2+2=5$"?2012-05-01
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    @ZhenLin Yes I see your point. But as Brett Frankel said, what I am really interested in, is in knowing which results make use of the fact that $\pi$ is irrational. Maybe the problem is with the title of my question. But the way in which I wrote in the title was precisely the way in which the thought first came to my mind.2012-05-01
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    (if you are really interested in a model of arithmetic where something that might be named $\pi$ is rational, check http://mathoverflow.net/questions/21367/. But this is Forbidden Knowledge and please be very very careful with drawing any conclusions.)2012-05-01
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    @sdcvvc I see. Then maybe I can delete that part of my question and I'll try to edit the question title. Thanks for pointing that out.2012-05-01
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    If $\pi$ were rational, then the sines and cosines corresponding to "nice angles" couldn't be constructible lengths, since by Niven's theorem, sines and cosines of rational numbers are necessarily transcendental.2012-05-01
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    The 7-o-clock has two solutions {7,-6}. It should rather have {7,-5}2012-05-01
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    If $\pi$ were rational, it would also exist in the $p$-adic setting, and there would be no theory of Fontaine's rings (http://www.math.jussieu.fr/~colmez/tsinghua.pdf p.131 for more details).2012-05-01
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    @JoelCohen So then, that's a whole portion of a theory that depends on the fact that $\pi$ is irrational? Why don't you add that as an answer?2012-05-01
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    @J.M. Niven's theorem? There is a "Niven's theorem" that asserts that $r$ and $\cos(r \pi)$ are both rational only in the cases where $2 \cos(r \pi)$ is an integer. Or do you mean Lindemann's theorem that $e^x$ is transcendental whenever $x$ is nonzero and algebraic (which has the corollary that $\pi$ is transcendental)?2012-05-01
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    @Adrián: in what sense is an answer to this question not also an answer to the question "what is a proof by contradiction that $\pi$ is irrational?"2012-05-01
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    @Robert: oops, I misremembered the precise statement. I must have mixed them up...2012-05-01
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    @Qiaochu Well now that you put it in that way, I don't really think there's any difference between the two. Although in that case, if you just ask "what is a proof by contradiction that $\pi$ is irrational?", then maybe interesting things like what Joel Cohen mentioned about the theory of Fontaine rings wouldn't have been mentioned. But in any case, why do you ask that?2012-05-01
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    @Adrián: I think there is a matter of emphasis which should be made explicit. The question it seems to me you really want to ask is more like "where are we naturally led to use the irrationality of $\pi$ as a hypothesis when developing some other mathematics?"2012-05-01
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    @Qiaochu That sounds great. But do you think that this question should be edited to include that or that a different question should be asked? If you feel like you can make this question better and more focused by editing it please by all means do so.2012-05-01
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    I received that clock as a gift last year :) It's ticking in my room now.2012-05-16

2 Answers 2

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Edit: the question was changed by the time I had finished writing this post, but I'll leave it up.

I'm going to say that this question, as I interpret it, does not really make much sense. At the very least, it is not a mathematical question. From my understanding, it can be interpreted in two different ways (which I am wording very loosely):

  1. Could $\pi$ have a different value?
  2. In a different Universe, could $\pi$ have a different value?

Question 1, which I think is the one you mean to ask, has a simple answer: mathematics would be inconsistent, as pointed out by Brian in the comments. The simple reason is that $\pi$ can be proven to be irrational (and in fact transcendental), hence if we could also prove it to be rational, we'd be in big trouble.

Question 2 is more interesting but requires a little interpretation. We can define $\pi$ to be the circumference of a circle of diameter $1$ in the Euclidean plane. We can mimick this definition by changing the rules of the game a little bit. For instance, if we change the metric on the plane, then a "circle" takes on a whole new meaning. For instance, in the taxicab metric, a circle of diameter $1$ is simply a square with side length $1$, and its circumference is $4$. Thus in a taxicab Universe, $\pi$ would have a different (and rational!) value, but that's simply because it would have a different definition...

There are also spaces in which the circumference of a circle of fixed radius varies with the position of the circle in the space.

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    I saw a line from Douglas Hofstadter "If $\pi$ were 3, this sentence would look like this". All the o's were hexagons.2012-05-01
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    Sorry for changing the question title and a part of the question. Please leave this answer. In fact you're also right. I was in some way also thinking about what would happen if $\pi$ had a different value (although I hadn't realized that at first), a rational value actually. Of course at first I didn't even realize that that part of my original question didn't make much sense. But actually asking it has made me realize that, because of the comments and this answer.2012-05-01
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    @AdriánBarquero: no worries, it was fun to write!2012-05-01
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If that happens, the circle would not be a differentiable manifold ... And so most manifolds would not be smooth ... The world would be very painful to live.

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    Eduardo, that's really interesting. Can you please add some details please? I don't see why it wouldn't be a differentiable manifold...2012-05-01
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    Well, the circle clearly *is* a differentiable manifold. So, if this answer is true, then this gives a proof of the irrationality of pi.2012-05-01