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I found this article on pi: http://blog.plover.com/math/pi.html and while I found it very interesting, it seemed unfinished. The basic point of the article is that pi is complex (for example e has a simple continued fraction representation: [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, ...], but pi does not), and the author claims that this complexity is due to the nonlinear nature of the euclidean distance metric. However, the author doesn't really have a conclusion, and I felt that I still had some questions that weren't satisfied:

1) How does such a complex constant (pi) arise from such a simple definition (a circle)?

a) Is it because the base-10 decimal representation is flawed, and there is another representation of numbers where pi is simple? If so, what is this representation?

b) Or is it because of some property of euclidean space, like the nonlinear nature of the distance metric. If so, where exactly does this property come into play in the definition of pi, and how does it create such complexity? It seems like a the simple square root of sum of squares metric shouldn't create such a bizarre constant (or if it did, that the constant would have something to do with the number 2).

Furthermore, if the answer is b, then are there any geometries or spaces that don't have this property, such that pi would be a simple constant?

I hope my questions aren't too vague! Thanks!

edit: By complex (I probably should have said complicated) I mean that, as pointed out in the article, whereas other irrational numbers like sqrt(2) or e have nice representations (in those two cases, they have nice continued fraction forms), pi does not have a nice continued fraction form. That's why I was wondering if there are any real number representations where pi does have a nice form, akin to e's representation in continued fraction form.

My main line of inquiry (which is the same line of inquiry of the linked incomplete article), is: how does such a simple definition of a circle: all points that are distance r away from a center, create such an incredibly complicated number?

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    On a tangent, the continued fraction of $e$ should really be written $[1, 0, 1, 1, 2, 1, 1, 4, 1, \ldots]$, because then it's made of these nice repeating $[1, 2n, 1]$ blocks with no broken symmetry in the beginning. (Douglas Hofstadter credits Bill Gosper for this "discovery" in his book *Fluid Concepts and Creative Analogies*.)2012-01-06
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    What do you mean by complex? For the question of irrationality, there is a related mathoverflow question: http://mathoverflow.net/questions/35341/are-there-any-interesting-consequences-of-the-irrationality-of. Nothing you've mentioned has anything to do with base $10$. $\pi$ is $\pi$, whether or not you expand it with respect to some base, and for example the continued fraction has nothing to do with base $10$.2012-01-06
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    @JonasMeyer I think by "complex" he means "cannot be written as a finite sequence of digits".2012-01-06
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    @AlexBecker: I don't see that in the question. How is that related to continued fractions? Finite decimal expansions are only a subset of rational numbers, and I do not see even rational numbers mentioned outside of the title.2012-01-06
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    I suspect he means there aren't any nice representations of $\pi$, among various common representations of real numbers.2012-01-06
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    Of course, that doesn't quite cut it either, because algebraic numbers don't necessarily have a nice representation.2012-01-06
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    @JonasMeyer This looks a lot like many irrational-number-related questions I've been asked over the years, and that's what I based my judgement off of. Often people don't really have any handle at all to grasp irrational numbers, and so their questions look very different than the idea they intended to convey because that idea does not have solid form in their mind.2012-01-06
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    @AlexBecker: I agree.2012-01-06
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    By complex (I probably should have said complicated) I mean that, as pointed out in the article, whereas other irrational numbers like sqrt(2) or e have nice representations (in those two cases, they have nice continued fraction forms), pi does not have a nice continued fraction form. That's why I was wondering if there are any real number representations where pi does have a nice form, akin to e's representation in continued fraction form.2012-01-06
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    Maybe $\pi$ is complicated with respect to continued fraction representations, but why is that the standard for being "complicated"? $\pi$ arises as the value of several "simple" infinite series, notably $\sum_{n=0}^{\infty}(-1)^n\frac{1}{2n+1}$.2012-01-06
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    [Brouncker's formula](http://en.wikipedia.org/wiki/William_Brouncker,_2nd_Viscount_Brouncker) gives a "nice" continued fraction for $\pi$ at the expense of allowing improper fractions into the construction.2012-01-06

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