Consider the definition of division. For arbitrary real numbers x, y, and z, if
xy=zxy=z
then
x=y×zx=y×z
for nonzero y. But if we let x and y = 0, then:
0=0×z0=0×z
This holds for any z: 0, 42, −1−−−√−1, and so on. This means that 0000 is undefined.
Strictly speaking, no algebraic expression can be ∞∞ or −∞−∞, which are values that are defined to be the absolute upper and absolute lower bounds of all real numbers. When an expression with a free variable, like 1/x1/x, is said to be infinity, it usually means that the expression grows larger and larger as its free variable is brought closer to some focal point. For example, 1/x1/x gets larger and larger when the variable xx is positive and is swept closer and closer to zero. In this case, the "limit" of the expression as x approaches zero from the right is said to be infinity (∞∞) because 1/x1/x grows without bounds. Similarly, if xx is assumed to be negative and is brought closer and closer to zero, the expression 1/x1/x will grow in the negative direction. So that limit is said to be −∞−∞. The expression 1/01/0 is still said to be "undefined" because it makes no sense (i.e., nothing multiplied by zero is equal to 1) and moreover could be interpreted to be positive infinity or negative infinity depending on how the zero got in the denominator.
A critical difference between limits that evaluate to 1/01/0 and limits that evaluate to 0/00/0 is that we have absolutely no intuition about what the latter case does. In the former case, we know that the expression that led to 1/01/0 either exploded to positive or negative infinity. However, 0/00/0 might have resulted from an expression whose limit is either finite or infinite. For example, as xx approaches 0, the expression:
x/xx/x approaches 1
4x/x4x/x approaches 4
x/(4x)x/(4x) approaches 1/4
x2/xx2/x approaches 0
x/x2x/x2 approaches ∞∞ or −∞−∞
Consequently, 0/00/0 is not only undefined, but it is a so-called indeterminate form. So it is especially troubling. Moreover, when 0/00/0 shows up in your problem, it means you've made a mistake in the formulation and you need to reconsider exactly what the question means.
It's actually a frequent problem on the hairier edges of physics, and there's a whole set of techniques called "renormalization" to rephrase the problems in terms that don't involve undefined quantities. These techniques are very sensitive to the precise formulation of the problem, which is why some recent scientific results (like faster-than-light neutrinos or variable values of alpha) are so puzzling: they make the renormalization techniques impossible, and you end up having to throw out all of physics!
