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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. –  Brian M. Scott May 1 '12 at 0:24
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We'd all be unemployed =) –  matgaio May 1 '12 at 0:25
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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. –  Brett Frankel May 1 '12 at 0:52
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The 7-o-clock has two solutions {7,-6}. It should rather have {7,-5} –  leonbloy May 1 '12 at 1:30
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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?" –  Qiaochu Yuan May 1 '12 at 1:57
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2 Answers 2

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. –  Ross Millikan May 1 '12 at 1:12
    
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. –  Adrián Barquero May 1 '12 at 1:13
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@AdriánBarquero: no worries, it was fun to write! –  Bruno Joyal May 1 '12 at 1:14
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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... –  Adrián Barquero May 1 '12 at 0:32
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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. –  George Lowther May 1 '12 at 2:16
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