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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.