0/0- The Definitive Answer

After a quite heated, albeit illuminating class discussion on what 0/0 should be, we can finally put this question (to the best of our ability) to rest. To this aim, we should first discuss each position and see the evidence for each individually before coming to a definite conclusion (and oh boy is the conclusion a doozy.) So the choices supported by class discussion are:

0/0= 0, 1, ∞, or undefined/indeterminate

1. 0/0=0

Examine the function y(x)=0/x. For all x ≠ 0, y(x)=0. Graphically this looks like


So a logical conclusion would to allow the discontinuity to simply equal the limit as x approaches 0

So we would have when x=0, y(0)=0/0=0.

2. 0/0=1

Let us now look the function g(x)=x/x. For all x ≠ 0, g(x)=1. We can see this visually as


Again we should be tempted to fill this discontinuity by the limit as x approaches 0

When x=0, we would have g(0)=0/0=1


3. 0/0= ∞

For any a > 0, 

When a=1, this looks like

We know this is true for any a > 0, so we could think to simply extend this property to when a=0 as well (there is nothing particularly wrong with this assumption.) Therefore we would have 0/0= ∞

There is similar argument when a < 0, except there the limit goes to -∞, 
another logical answer.

4. 0/0 is undefined/indeterminate

This is nothing inherently wrong with refusing to answer the question. In fact why should we answer a question that seems to have multiple correct answers? All three approaches above seem somewhat logical, so we should be worrying about now about how to pin down a single answer.  After all, when we solve something like 2b=5 we don't get several logical non-equivalent answers. So where is the inconsistency? Where was our mistake? The reality is that all of the above answers are correct. 

Consider the equation bx=a, where we can see that x is solved for by x=a/b. Now, consider when a=0, or bx=0. From the properties of the 0, we see that either b or x must be zero. Since we don't want to let x be trivially zero, let us say b=0 instead. Then we have

0x=0

Here is the key part: because ANY number times 0 is 0, x can be any number. Again we can see that

x=0/0

follows trivially from this result. Therefore we concluded that 0/0 can be any number x that has the property 0x=0.

Another way to see this is with the following question: how many times can I add up zeros to get zero? Any number of times!

5. Conclusion

If 0/0 can be any number, what is the point in defining it? The concept of having a "solution" to 0/0 seems rather meaningless, since any number we could possible wish to be the answer is.  So, we shouldn't feel the need to provide a single answer. Our conclusion is then this: we in fact will refuse to answer the question. In other words, since any 0/0 can be any number, 0/0 is undefined/indeterminate.