There is one situation in which you can divide, not by zero, but by "infinitely small numbers"--the field of surreal numbers (it's on Wikipedia) of John H. Conway (the Game of Life guy). It's pretty...surreal in that there is no set of all surreal numbers, but it includes all real numbers, every cardinal number including the infinite ones, something much like the ordinals (but using different rules of arithmetic), and lots more different kinds of infinity, and their reciprocals which are infinitesimals. Still, division by zero in the surreal field is forbidden.

A modification of this (some guy name Robinson I think) leads to Freshman calculus, with all the tricks of dividing by infintessimals to get derivatives--tricks invented by Newton--for which he was criticized because, though the methods just happened to produce the right answers, were not very rigorous. Weierstrass invented the modern definition of the limit to make it rigorous, and Robinson figured out how to make Newton's manipulation of infintessimals rigorous, showing why it worked (i.e. it didn't just accidentally work, there was something to it).

__________________
-----
Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info