Divide By 0

I've always thought that the number zero, has a concept, was a remarkable achievement. Here's a detailed and technical explanation as to why we can't divide by 0.

The reason that the result of a division by zero is undefined is the fact that any attempt at a definition leads to a contradiction.

To begin with, how do we define division? The ratio r of two numbers a and b:

r=a/b

is that number r that satisfies

a=r*b.

Well, if b=0, i.e., we are trying to divide by zero, we have to find a number r such that

r*0=a. (1) But r*0=0

for all numbers r, and so unless a=0 there is no solution of equation (1).

Now you could say that r=infinity satisfies (1). That's a common way of putting things, but what's infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then

infinity = infinity+1 = infinity + 2

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.