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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.) –  Rahul Jan 6 '12 at 4:19
    
What do you mean by complex? For the question of irrationality, there is a related mathoverflow question: mathoverflow.net/questions/35341/…. 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 $10$. –  Jonas Meyer Jan 6 '12 at 4:19
    
@JonasMeyer I think by "complex" he means "cannot be written as a finite sequence of digits". –  Alex Becker Jan 6 '12 at 4:24
    
@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. –  Jonas Meyer Jan 6 '12 at 4:27
    
I suspect he means there aren't any nice representations of $\pi$, among various common representations of real numbers. –  Dustan Levenstein Jan 6 '12 at 4:29

2 Answers 2

For your point 1(a), you could always write numbers using a non-rational base. Trivially $\pi$ can be written as $10_{\pi}$ in base $\pi$,

but you have now lost the ability to write four simply in the same base: it becomes about $10.220122\ldots_{\pi}$ and even has other representations such as $3.301102\ldots_{\pi}$. Addition is not easy using this base.

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First of all, for modern mathematicians $\pi$ is a particular number that can be defined in many ways, many of which have little to do with geometry. In a different geometry the ratio of circumference to diameter of a circle might be different, but it wouldn't be $\pi$.

Another bit of terminology: don't say "complex" to a mathematician when you mean "complicated". Complex numbers are something completely different.

For example, if you use the "$1$-norm" so the distance from $(x_1,y_1)$ to $(x_2,y_2)$ is $|x_1 - x_2| + |y_1 - y_2|$, then a unit "circle" has circumference $8$ and diameter $2$, so the ratio of circumference to diameter is $4$. That's not $\pi$.

You could represent numbers in lots of ways. For example, why not represent a number $x$ using the decimal representation of $x/\pi$? Thus $\pi$ is represented by $1.0$ in this system. It does not have many other virtues, though.

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OP, thanks for the link to the Dominus talk -- fascinating! Robert Israel, the OP could object that you're just correcting his terminology without addressing the substance of his/her question. Re whether there are other spaces where pi might be nicer, this may be relevant: math.stackexchange.com/questions/53023/… Note that in "most" spaces (i.e., most Riemannian geometries), the ratio of the circumference of a circle to its diameter is not a constant, so one can't weasel out by saying that one wanted "the" geometrical value of pi. –  Ben Crowell Jan 6 '12 at 4:40
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Yes, I was a bit loose with notation in my original post. Your example of a "1-norm" circle is interesting, because if I'm not mistaken that norm is linear, and "pi" in that space is nice, which seems to support Dominus's claim that "pi" is complex because of the nonlinearity of the Euclidean metric. If this is the case, where does the nonlinearity come in to make things complicated? Of course if you use x/pi then pi will be 1.0, but I was looking for a more sensible representation where pi would have a nice form, like how e has a nice form in continued fraction represenation. –  stackedAE Jan 6 '12 at 5:22
    
Also, in response to your first sentence, the author of the article wrote blog.plover.com/math/pi-2.html : "A couple of people have written to me to point out that π also appears in a number of constants and laws from physics, such as Coulomb's law. I believe that these appearances are invariably derived from the appearance of π as the circumference, and, in many cases, that this is quite obvious. Is it true that all definitions of Pi are implicitly linked to Euclidean space? If not, then where do they come from and how are they relevant to the real world? –  stackedAE Jan 6 '12 at 5:34
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I think it's accurate to say that there must some link, however convoluted, i.e. there must be some way to prove that the number you're talking about is the same as $\pi$, and this will involve some equation that your number satisfies which $\pi$ also satisfies; the proof that $\pi$ satisfies this will involve the circle in some way. For example, $\pi$ could be defined in terms of the asymptotics of the probability that $n$ independent flips of a fair coin will produce between $n/2 - c \sqrt{n}$ and $n/2 + c \sqrt{n}$ heads. –  Robert Israel Jan 6 '12 at 5:54
    
... and this comes from the fact that $\int_{-\infty}^\infty e^{-x^2}\ dx = \sqrt{\pi}$. One proof of that identity involves a change to polar coordinates in a double integral -- clearly circles coming in there. –  Robert Israel Jan 6 '12 at 5:58

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