0 Divided by 0

Date: 11/27/97 at 17:19:53
From: Peter Rimshick
Subject: 0 divided by 0

What is the answer to 0 divided by 0? I think it is undefined because 
of this equation:

   0/0 = x/1

X can equal any number and still satisfy that equation by the cross 
multiplication method. But I am starting to doubt myself because I was 
talking to a couple of friends of mine about it and they said that it 
is due to  L'Hopital's Rule. I have looked it up on the Internet but 
all the explanations are in calculus terms, which are like Greek to 
me. Please help me figure this out.

Thanks.

Date: 12/15/97 at 10:39:53
From: Doctor Mark
Subject: Re: 0 divided by 0

Hi Peter,

Well, your friends are wrong. L'Hopital's rule does *not* tell you 
what 0/0 is, because 0/0 is what is called an "indeterminate" 
quantity, which is to say that its value depends on what the situation 
is. To convince your friends of this, ask them the following question:

"Find the limit of (ax)/x as a approaches 0 by using L'Hopital's 
rule."

They will get "a" (trust me!).  

But if you just put x = 0 in this expression, you get 0/0.  So, 
according to L'Hopital, 0/0 is equal to a.

Did you notice that I didn't say what "a" was?  That's because it 
doesn't matter. You can pick a equal to anything you want. For 
instance, you could pick a = 1.  Then you would get

   0/0 = 1

Or pick a = - 3.14159.  Then:

   0/0 = - 3.14159.

So as you can see, 0/0 can be anything you want it to be. On the other 
hand, in a particular problem, 0/0 might turn out to be something very 
precise (and that's where you really do need calculus to understand 
it!).

I think your argument for why 0/0 is undefined is a really good one.  

However, I have another way of understanding why 0/0 doesn't make 
sense, and it goes like this.

One way of understanding the fraction a/b is to think of it as the 
answer to the following question:

"If I had a dollars, and b friends, and I distributed those a dollars 
equally amongst my b friends, then how much money would each of my 
friends get?"

The answer is that they would each get a/b dollars.

You can see that this works for fractions like 6/3, or 5/10, and so 
on.

But try it for 0/3. If you have 0 dollars, and 3 friends, and you 
distribute those 0 dollars (you're feeling generous...) equally 
amongst each of them, how much would each of your 3 friends get?  
Clearly, they would each get 0 dollars!

Now try it for 3/0.  If you have 3 dollars and 0 friends, and you....
but how can you distribute any amount of money amongst friends who 
don't exist? So the question of what 3/0 means makes no sense!

Now here's the kicker: What if you have 0 dollars and 0 friends? If 
you distribute those 0 dollars equally amongst your 0 friends, how 
much does each of those (nonexistent) friends get? Do you see that 
this question makes no sense either? In particular, if 0/0 = 1, then 
that would mean that each of your nonexistent friends got 1 dollar! 
How could that be?  Where would that dollar have come from?  

Stand your ground, Peter...you're right, and they are wrong, and if 
they don't believe you, tell them to write to me, and I will set them 
straight.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/