In the theory of equations, we have this nifty rule that in an equation in one variable of degree n (highest power of the variable is n), there are exactly n (not necessarily distinct) solutions. If you allow division by 0, this rule is completely trashed-- division of an equation by 0 reduces its information content to zilch-- nothing is solvable. The numbers we are most comfortable with possess a very comforting property-- that of unique factorization. Except for the order of the factors, all integers can be factored in exactly one way, and we use this property to develop general solutions to equations with integer coefficients. To tell it short, if division by 0 is allowed, each integer has an indefinitely large number of factorizations, and no unique solutions are possible to the equations.

Even without division by 0, there exist certain types of number whose factorization is not unique-- the people who study these algebras are very.. strange, and very busy. It makes number theory very difficult to prove anything with, and even makes a self-evident property like equality difficult to prove.

The devil we know is not very frightening. Steve

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Ignoramus et ignorabamus.-- Reymond

Wir mussen wissen. Wir werden wissen.--Hilbert

Pick one.