0/0

An undefined quantity which has had several proposed solutions:

Solution One: 0/0 = 0
Reason: Zero divided by anything is still zero.

Solution Two: 0/0 = 1
Reason: Anything divided by itself equals one.

Solution Three: 0/0 = infinity
Reason: Division by zero can result in infinity.

Solution Four: 0/0 = undefined
Reason: There is no answer!

Sometimes, zero divided by zero can equal some other number, when the limit theorem is applied. For instance, in the function Y=3X(X-2)/(X-2), what is Y when X=2? Taking the limit as X approaches 2 will give the answer. This is really just a way of applying Solution Two above, but some mathematicians seem to care.

With my half-semester's worth of basic university level maths, I will tentatively make the argument that Solutions One, Two and Three are invalid. I base this on the fact that the mathematical smallprint has not been taken into account. That's right, in today's dodgey world even mathematics has smallprint.

The fundamental laws regarding variables being operated on by zero that are used in the the first to third solution have the condition that the variable itself can't be zero:

0/x = 0 For all x where x != 0

x/x = 1 For all x where x != 0

x/0 = infinity for all x where x != 0

So the only viable solution is Four, undefined. The damn thing's meaningless. There is probably more smallprint involving complex numbers but it's irrelevant (I hope) and I'll be damned if i know it. I'm also probably failing my course (Intro to Calculus and Linear Algebra) so by no means take this as the be all and end all.

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Bob9000 I believe you're idea follows the same idea of the proof that 1=2, which is obviously flawed in some way (Look if it isn't then well... I'll be more than happy to concede to you as the universe collapses on our heads.

In the field of calculus, 0/0 is sometimes referred to as indeterminate, meaning it could potentially equal any number it wants to. Examples:

If x = 0, divide both sides by x. Result: 1 = 0/x.

If 2x = 0, divide both sides by x. Result: 2 = 0/x.
OR, divide both sides by 2, and find that x = 0.

Therefore, 0/x = 1 and 0/x = 2. The same is true of any other real number. 0/0 = all real numbers. From this, you could easily conclude that all real numbers are equal. But that's another story...

0/0 is referred to as an indeterminate form, along with 0*infinity, infinity/infinity, infinity^0, 0^infinity, 1^infinity, infinity - infinity, and 0^0.
(More precisely, the limits of these quantities are indeterminate forms; the quantities themselves are always undefined.)

Indeterminate forms can have various meanings depending on how you got the values in them. 0/0 as such is meaningless -- it's undefined. But in a limit, X*5/X = 5, _even if X is 0_! That is, X*5/X, when X is 0, becomes 0/0, which is undefined; but the limit of X*5/X as X approaches 0 is 5, because you are allowed to cancel the X's. Thus, 0/0 can have different values depending on where the 0's came from, so it is called indeterminate.

The other types of indeterminate forms are harder to deal with, but the same logic applies.

Solution 5: 0/0 = 0/0
Reason: Humans divide by zero all the time.

KNaN AM-150 has a contest where one of the afternoon DJs ask a question, and then get answers from the first 5 callers. The prize is split between the 5 callers with the correct answer. If one of the callers gets the correct answer, he wins $150. If two of those callers get the correct answer, both win $75. If three callers get the answer, they each win $50, and so on. If none of the 5 callers can come up with the correct answer, no division of the prize money takes place (and no, the station hosting the contest is not defined as "a winner").

If you try to simplify it, it will not produce a number. That doesn't make it meaningless or undefined. Even in the case of 0/0, it communicates something. In this case it means I've got nothing to give and no one to give it to.