Talk:Pi: Difference between revisions
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:: <math>e^{i\pi} + 1 = 0</math>
: That is an amazingly significant formula, “simply” linking 5 of the key constants into one piece. There is no way that <math>\tau</math> would work nearly so well in it; it's not coupled to the transcendental functions in such a direct fashion. (Also, you use <math>\pi</math> directly when working with circular areas, and neither <math>\pi</math> nor <math>\tau</math> is great for spherical volumes or their equivalents in higher dimensions.) –[[User:Dkf|Donal Fellows]] 21:28, 11 February 2012 (UTC)
:: Well, of course -- if you replace the terms in the equation with something different, you need to change the equation if you want the result to be valid. That's just basic math. In this case, a valid pair of changes are: "replace pi with tau (or vice versa)" and "swap the positions of the + and the =". In other words: '''<math>e^{i\tau} = 1 + 0</math>'''. You might also want to swap <math>i</math> and <math>\tau</math> depending on the font you are using (<math>e^{\tau i} = 1 + 0</math>), but multiplication is commutative with complex numbers, so that will not change the equation's validity.
:: As for the general case of n dimensions... I was going to look that up, but currently wikipedia says that the proportionality constant for volume of a [[wp:N-sphere|hypersphere]] is <math>V_n (R) = C_n R^n</math> where <math>C_n = \frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} + 1)} \,</math> but that gives the same result for <math>C_n</math> for both n=2 and n=3 so it can't be right. So, for now, I will take your word that it gets messy. It's been ages since I thought about this issue. Still, as motivation for a task? I do not think that we exclude tasks based on this kind of mathematical popularity issue... --[[User:Rdm|Rdm]] 16:34, 12 February 2012 (UTC)
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