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Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?

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    Sounds hard; its being transcendental seems to preclude the existence of a proof that won't appeal to the concept of limits.2010-08-24
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    @Chris, the problem is with defining the length of a circle without appealing to a limit!2010-08-24
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    well, intuitively you can define the length of the circumference by rolling the circle along a line, but that probably doesn't help much2010-08-24
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    Chris: that can be shown to be equivalent to "slicing up" the circle to form a "parallelogram" of appropriate dimensions; unfortunately for you this too involves limits.2010-08-24
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    If you are going to work 'intuitively', then it is pretty obvious that zooming in or out does not change proportions of lengths, so in particular it does change the proportion between the circumference and the diameter! Now, if you want to actually prove something, you need to define things precisely, and you are more or less stuck with limits.2010-08-24
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    that was my question and it seems the answer is no.2010-08-24
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    @Chris, Your question was «how can we define the length the circumference without using limits?»?2010-08-24
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    Actually it started "Is there a proof ... ?" :-) I'm happy to accept "no" as an answer, if backed up by a convincing argument, e.g. "any such proof would involve defining the length of the circumference and that requires using limits."2010-08-24
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    @Chris, what I am asking is: since the length of the circumference is defined in terms of limits, there is no possible way to prove anything about it without invoking limits. If what you want to know is if one can define the length of the circumference without using limits, then your question should ask that :)2010-08-24
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    it's not quite obvious to me, since we're actually interested in a ratio of lengths rather than the length of the circumference itself. Plus, is it obvious that the only way to define the length of the circumference is by using limits?2010-08-24
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    @Chris: how else would you define the length of something that isn't straight? You should think very hard about what you think a length is. (Is it something you can measure with a ruler? How do you measure the length of something curved with a straight ruler?)2010-08-24
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    you could use a bendy ruler! (not totally serious comment BTW)2010-08-24
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    @Mariano and Qiaochu: underlying the circle-independent limit is a circle-independent sequence of approximations. It is enough to show independence for the sequence and this does not require limits.2010-08-24
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    @T: but the very definition of "length of the circle" uses a limit. The only reason why your finite polygonal objects can be thought of a "set of aproximations", to use the language in your answer, is because the length of the circle is a limit; and you have to know, for example, that the limit exist for it to even make sense to aproximate it &c.2010-08-24
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    My answer does not depend on any notion of approximation (except to help connect it to other better-known explanations that do involve approximation, that is, convergence of the sequences). Pi as conventionally defined using limits is a limit of *something* and my answer was that the *something*, which you can call a "sequence of approximants" or by any other name, is the same for any two circles. Accordingly, whatever "limits" are and whenever they exist, they would be the same for the two circles. Equality of two limits is easier to prove than existence or evaluation of either one alone.2010-08-24
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    This is an instance of the common pattern that proving a mathematical object is well-defined (e.g., a sequence associated to a circle does not depend on the choice of circle), is easier than proving more specific properties of the object (e.g., the sequence has a limit, that limit is calculated by a particular integral, the integral has some invariance properties with respect to rotation, the numerical value of the integral is a transcendental number).2010-08-24
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    I don't usually add a new answer when five answers are already there, but I posted one, and up-voted several.2013-08-18
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    You can avoid introducing any new notion of limit altogether if you just define reals using Dedekind cuts. Define the space of cuts D as all nonempty collections of rationals bounded below. Then define R as the image of D under upward closure (of sets) in Q. Then T.'s proof that \pi_k(C)=\pi_k(C') (say by circumscribed polygons) for all k and all pairs of circles C, C' yields that the map \pi:\mbox{Circles}\rightarrow D defined by \pi(C)=\{\pi_k(C) : k\in \mathbb{N}\} is constant, so its projection onto R consists of a single point.2013-11-13

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