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# Division by Zero AuthorMessage

Euclid  Posts : 95
Join date : 2009-09-15
Age : 57
Location : Alexandria  Subject: Division by Zero Mon Sep 28, 2009 10:16 am A nice example of division by zero may be found at chapter 141. in Burnside's "Theory of Finite Groups".Let i be a primitive root of the congruence i^(p^m -1) = 1 (modulo p).Let a, b, c, d be powers of i with ad-bc <> 0 (modulo p).Consider the operations (ax+b)/(cx+d) (modulo p).Quote :Moreover, if we represent i^x/0 by infinity for all values of x, any operation of this group, when carried out on the set of quantities. infinity, 0, i, i^2,......, i^(p^m-1),will change each of them into another of the set; [...]Hence the permutation-group is triply transitive, since it contains an operation transforming any three of the p^m + 1 symbols into any other three.   Euclid  Posts : 95
Join date : 2009-09-15
Age : 57
Location : Alexandria  Subject: Re: Division by Zero Wed Nov 10, 2010 1:42 pm Another OneLast Sunday, one of our colleagues presented a nice division by zero in the very applied probabilities domain.Biasing the very clear presentation, that definitively deserves an accessible to students published note, I will say that was about the :dF/dGdivision where, par Newton!, dF and dG are infinitesimally small quantities.Par l'Hôspital! dF/dG makes sense and dF/dG = F'/G', even when F and G are probabilities.In practice, to apply l'Hôspital's rule to zero-probability events requires - A missing definition (that is equivalent to introducing infinitesimal dP probabilities) - A postulate that the definition fits to the real insurance world as well as the Topology and the Theory of Measure, and - A double column paper comparing the dF calculus to the delta-epsilon calculus.By the way, here is a less known construction of the real numbers,Definition: A real number is an equivalence class of slopes.where a slope is a map L : Z -> Z, with the property that the set {L(m + n) − L(m) − L(n) | m, n in Z} is finite.   Division by Zero Page 1 of 1

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