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 Division by Zero

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nick
Euclid
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Join date : 2009-09-15
Age : 56
Location : Alexandria

PostSubject: 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.
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nick
Euclid
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Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

PostSubject: Re: Division by Zero   Wed Nov 10, 2010 1:42 pm

Another One

Last 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/dG
division 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.
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