Thursday, March 7, 2013

Dividing by Zero

Usually, dividing by zero is a problem.

Like most problems, it can be solved if you put your mind to it:  Start at the beginning and take it one step at a time.  Such is the beauty of mathematics:  If each step is correct, and each prior true, then the result is also true.

The definition of division is that it is the inverse of multiplication, so, to understand division, we must first study multiplication.  Multiplication may be represented as a function:

mult(x, y) ≡ x × y = z

This maps a two-dimensional domain onto a one-dimensional co-domain, causing degeneracy.  If we invert multiplication, this mapping is reversed, and each point of the inverse's one-dimensional domain maps to an at-least-one-dimensional subspace of its co-domain.

mult-1(z) = {x, y} with x × y = z

For mult-1 to be usable as a function, we must add a constraint, reducing the co-domain's dimensions by one.  This is why division has a second input:

div(z, y) ≡ ÷ y ≡ mult-1(z) with y constrained to the given value.
i.e. div(z, y) ≡ mult-1(z) | y

However, if y = 0, then z = 0, independent of x, and the constraint fails to reduce the co-domain's dimensions.

In essence, the problem is that multiplication by zero causes information to be lost.  The salient solution is then to keep the information for as long as possible.

Because multiplication is associative, there is a simple methodology; simply remove the following axiom:

x × 0 = 0, for all x

Further investigation will need to be done to determine whether this solves all the problems (e.g. tan(π/2) = ?), whether this arithmetic is a useful abstraction of reality, and how it fits in with Aleph-null, Peano arithmetic, and other such mathematical concepts.

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