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Message:: [QUOTE="Shoewrecky, post: 941130, member: 22350"]ok my definition0 / 0 = 0+0 = 0 x 0 = 0 - 0 = ?5/0 = 5.0 or piggies / smokies / bears / popo's /cops Real Answer from Mathforum.org (I take no credit unless it's right <img src="styles/default/xenforo/clear.png" class="mceSmilieSprite mceSmilie1" alt=":)" unselectable="on" unselectable="on" /> ) 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." 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?  5/0 = ?The reason is related to the associated multiplication question. If you divide 6 by 3 the answer is 2 because 2 times 3 IS 6. If you divide 6 by zero, then you are asking the question, "What number times zero gives 6?" The answer to that one, of course, is no number, for we know that zero times any real number is zero not 6. So we say that division by zero is undefined, for it is not consistent with division by other numbers. For example, we could say that 1/0 = 5. But there's a rule in arithmetic that a(b/a) = b, and if 1/0 = 5, 0(1/0) = 0*5 = 0 doesn't work, so you could never use the rule. If you changed every rule to specifically say that it doesn't work for zero in the denominator, what's the point of making 1/0 = 5 in the first place? You can't use any rules on it. But maybe you're thinking of saying that 1/0 = infinity. Well then, what's "infinity"? How does it work in all the otherequations?  Does infinity - infinity = 0? Does 1 + infinity = infinity? If so, the associative rule doesn't work, since (a+b)+c = a+(b+c) will not always work: 1 + (infinity - infinity) = 1 + 0 = 1, but (1 + infinity) - infinity = infinity - infinity = 0. You can try to make up a good set of rules, but it always leads to nonsense, so to avoid all the trouble we just say that it doesn't make sense to divide by zero. What happens if you add apples to oranges? It just doesn't make sense, so the easiest thing is just to say that it doesn't make sense, or, as a mathematician would say, "it is undefined". Maybe that's the best way to look at it. When, in mathematics, you see a statement like "operation XYZ is undefined", you should translate it in your head to "operation XYZ doesn't make sense".[/QUOTE]

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