Division by zero is done on the Riemann sphere in the complex plane and its result tends to infinity whereas zero divided by any other number tends to one. It is also important to note that in the real number system (and on better graphing calculators), division by zero is not impossible but remains undefined. http://en.wikipedia.org/wiki/Riemann_sphere

Georg Friedrich Bernhard Riemann, German mathematician, student of Carl Friedrich Gauss, and mathematical genius developed numerous theories in mathematics such as Riemann partitions, The Generalized Riemann Hypothesis, Non-Euclidean Geometry, and extensive work in the realm of imaginary (complex) numbers, and was the mathematician Einstein selected for use in the General Theory of Relativity. Riemann's work in Non-Euclidean geometry just happens to fit exactly with what is actually experienced when taking measurements on the surface of the Earth.

That is, triangles laid upon the surface of the Earth add to 182 degrees instead of the Euclidean 180 degrees. This is due to the failure of the Euclidean fifth axiom, the Parallel Postulate. Its failure is immediately evident to anyone looking at a spherical model of planet Earth where longitude lines at the equator cross at the North Pole proving failure of the parallel postulate (parallel at the equator and crossing at the North Pole). Carl Friedrich Gauss suspected this as well as the existence of non-Euclidean geometry, and the number of primes less than a given magnitude yet did not disclose choosing to let his students discover and receive credit for the items. His most significantly accomplished student was Georg Friedrich Bernhard Riemann.

It is fascinating to note that real numbers generally tend to give imaginary results and imaginary numbers tend to give real results. Meaning that Euclidean results tend to give ideal answers verses real answers and non-Euclidean geometry gives answers consistent with what is really observed in our natural world. This is not to detract from Euclidean mathematics merely that a perfect triangle is merely an idealization and not typically encountered in the real world. Certainly, man can create one but trying to find one in the real world would be as challenging as trying to find a straight line. A straight line laid out on the surface of the Earth would tend to describe a geodesic. Nevertheless, this remains for another topic.

Since the consequences of Riemann's interpretation of non-Euclidean geometry tend to reflect what is actually observed here on Earth, it is tantalizingly imminent that Einstein would indeed use this mathematical model for the description of gravitation. How else could one conceivably proceed? This also provides compelling evidence of Riemann's stature as a founding father of modern physics, a circumstance that will inevitably become more prevalent upon a conclusive solution to the Riemann hypothesis.

In order to provide some degree of logical reasoning for the necessity of division by zero in some (but not all) circumstances, examine the incompleteness theorem presented by Kurt Godel.

"Godel showed that truth cannot be contained within the limits of strict logic. Only if we allow paradox can truth completely reveal itself in form. These two sides of Godel's proof represent the apophatic (via negativa) and cataphatic (via positiva) approaches to truth, respectively. In the apophatic approach, one adheres to strict logic to show that any attempt to represent or speak truth necessarily failsthe truth is beyond all rational comprehension. In the cataphatic approach, on the other hand, one embraces paradox and the coincidence of opposites to demonstrate the tangible presence of truth in all its limitless expressions. Like the ancient mathematics of Pythagoras, Godel's mathematical proof can be seen as a symbol of profound truths about the relationship between the limited and the unlimited, form and formlessness, transcendence and immanence, Godel's postmodern mathematics undermines any attempt to fixate on any totalizing axiomatic system for mathematical discourse, and reveals the essential ambiguity, openness, and emptiness of mathematical activity." Thomas J. McFarlane Spring 2000 Revised and Edited for the Web March 2004.

Inherent to Godel's proof is the conclusion that if a system is consistent, then it is incomplete. Conversely, if a system is complete, then it must be inconsistent. Considering this fact when examining such a precept as the impossibility of division by zero, we would be relegated to an inconsistent mathematics. Alternatively, entertaining the possibility of division by zero generates consistency, yet leaves us with an incomplete understanding of mathematics, which is precisely the circumstance with which we are dealing. Therefore, in some circumstances, one can in fact divide by zero.

http://en.wikipedia.org/wiki/Bernhard_Riemannnn

http://en.wikipedia.org/wiki/Riemann_sphere

Mathematical Poetics of Enlightenment Thomas J. McFarlane Spring 2000, Revised and edited for the web March 2004. http://www.integralscience.org