On 2010-10-13 14:20:30 +0100, Steven D'Aprano said:
ncorrect -- it's not necessarily so that the ratio of the circumference to the radius of a circle is always the same number. It could have turned out that different circles had different ratios.
But pi is much more basic than that, I think. My background is in physics so I tend to do things from the geometrical point of view - and obviously you are correct that there are non-euclidean geometries. But pi crops up, for instance, when dealing with complex numbers (e^(i pi) = -1 is the poster-child formula for this), and there are all sorts of series expressions for pi which have no really obvious geometrical interpretation.
(Of course, my view of the pi-in-complex-numbers is that this is because complex numbers turn out to essentially //be// two-dimensional euclidean geometry, but that's mostly because I want eerything to be geometry I think. In any case, I think you can get to pi being important in the same sort of way that you can get to e being important.)
(And, it sounds in the above like I think you might not know that pi crops up in complex numbers: that's just clumsy wording, sorry).
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