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Possible Duplicates:
Why is Euclidean geometry scale-invariant?

Proof that Pi is constant (the same for all circles), without using limits

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    Did you mean why is the ratio of circumference of a circle to diamter independent of the circle?2011-02-26
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    What definition of $\pi$ are you using that might make it not a constant? What would cause it to vary? Vote to close as not a real question.2011-02-26
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    +1 @Moron: I would like to see the answer for your question. I had some doubts too but not a clear picture about it. Thanks2011-02-26
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    @Moron: Yes, that's correct -- updated the question, thanks!2011-02-26
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    @Sunil: Updated my question to Moron's wording, since he's meaning was the intent of my question.2011-02-26
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    @Ross Millikan: Updated the core question per Moron's suggestion, and removed the secondary question deal with any reasons why Pi might not be a constant. Have any additional feedback? Thanks!2011-02-26
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    I think the new version of the question is a good one.2011-02-26
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    @Mike Spivey: +1 Thanks for the edits to the question.2011-02-26
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    Duplicate: http://math.stackexchange.com/questions/23129/why-is-euclidean-geometry-scale-invariant2011-02-26
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    @Qiaochu Yuan: +1 thanks, did search for a duplicate first, and figured there might be one - I'll take a look at the question you linked to. Cheers!2011-02-26
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    @blunders: Take a look at this also: http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits2011-02-26
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    @blunders: Here's another link: http://www.oocities.org/cf/ilanpi/pi-exists.html2011-02-26

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Because when you change the scale, both the diameter and the circumference change by multiplying by the same scale factor, since they are both of dimension 1. Thus, their ratio is independent of the scale factor. Since all circles are similar, the ratio is the same for all circles.

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    +1 Funny thing is, your answer makes sense to me -- the answer and question linked to as a duplicate... does not - oh, well.2011-02-26
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    @blunders: You may appreciate these answers (added again here as the identical link is a bit lost in the comments to the question): http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits2011-02-26
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    @Derek Jennings: +1 Thanks!2011-02-26