Dividing by Zero

Here's a question that my students never ask, but really should: Why can't we divide by zero?

That's a good question. We learn in school that dividing by zero is 'undefined'. We even often have to write restrictions on rational expressions to make sure that the denominator of a fraction is not zero, because when we have a fraction we are actually dividing by the denominator. So we can't have the denominator being zero because that would mean we were dividing by zero. Now why is it that we can't divide by zero? It's simple...

Suppose, in some world, we could divide by zero. Now we know that:

0 = 0 (obviously)

We can rewrite this as:

0x1 = 0x2

If we could divide by zero, then in this expression we could divide both sides of the equation by zero and we'd get:

1=2

Which obviously makes no sense (1 does not, will never, ever equal 2).

So there you have it. We can't divide by zero simply because, if you could, 1 would equal 2 and that would just be crazy. The whole universe would collapse or something.

If you're a math teacher, I suggest you share this information with your students. It's surprising: I've never had a student who knew why we can't divide by zero. They would know how to write restrictions, they would know that dividing by zero is not allowed, but they had no idea why. They just simply accept it as fact, for some reason.


Note: The choice of using 1 and 2 were arbitrary. I could have picked any other two numbers, say x and y, and I would have been able to show that x=y (if dividing by zero were allowed).