Philosoraptor - Episode 10

Were the first statement true, then yes. But it isn't - precisely because you can't divide by zero. The concept of division is predicated upon the concept of multiplication. Saying that 0/0 = 1 would be saying that the equation 0*x = 0 implies the equation x = 1. This is not the case, since here x could be any other number. So... when your teacher told you division by zero was undefined, it wasn't a cop-out. It's undefined because you can't define it consistently. 

More succinctly, just consider the following contradiction:

3*0 = 0
3*0/0 = 0/0
3*(0/0) = (0/0)
3*1 = 1
3 = 1,
which is clearly false. 

Obviously, any number would work in place of the 3.

Why not define division by zero to be infinity, then? Because that too is inconsistent. In the equation 0 * x = 0, x doesn't have to be a positive number. It could be -1 or even an imaginary number, which is different from a nonexistent number and a perfectly valid solution to the equation. While infinity is itself not a number, it IS associated with positive real numbers, and negative infinity IS associated with the negative reals. Since neither of these seems a 'better' candidate for the result of division by zero, we can't arbitrarily anoint one as being 'correct'.

Since infinity is not a number but a concept associated with the concept of a limit, providing a symbolic contradiction as above would probably require calculus. You guys don't want to see that, and I don't want to spend any more time on this post.

The key take-away is that defining division by zero is inherently inconsistent. Similarly 0^0 is undefined...