Why Can't 0 Divided By 0 Be 0?
Date: 11/25/2003 at 17:27:49 From: Steven Subject: Dividing 0 by 0. Why can't you divide 0 by 0? I've thought about it and it seems that dividing 0 objects, into 0 groups will result in 0 groups which makes sense. I've read over your "Why dividing by 0 doesn't work" posts, but I don't see any real reference to this. Here's my take on the classic 'proof' that 2 = 1: a = b a^2 = ab a^2 - b^2 = ab - b^2 (a-b)(a+b) = b(a-b) a+b = b b+b = b 2b = b 2 = 1 In this situation, if we assume a = b = 0, wouldn't it work out? Thanks for your help.
Date: 11/26/2003 at 23:39:03 From: Doctor Ian Subject: Re: Dividing 0 by 0. Hi Steven, >Why can't you divide 0 by 0? Because division of anything by zero is undefined. It's undefined, because any possible number that might be assigned to it would cause most of arithmetic to fall apart, by allowing false statements (like the one you 'prove' below) to be proved. >I've thought about it and it seems that dividing 0 objects, into 0 >groups will result in 0 groups which makes sense. I've read over >your "Why dividing by 0 doesn't work" posts, but I don't see any real >reference to this. Because it's irrelevant. Numbers are used to model various objects and activities in the world, but you can't draw conclusions about numbers based on those objects or activities, because numbers aren't constrained at all by what goes on in the world. For example, if you take an object in the real world and start halving it, you eventually get to the point where you've got something indivisible (e.g., a quark). Should we then conclude that if we keep halving a number, eventually we'll get to a number that can't be halved again? No, we shouldn't. Why not? Because numbers aren't objects. They're just concepts that we sometimes use to represent objects, the way we might use sugar cubes to represent battleships when planning a naval battle. >In the case of (2=1): > >a = b >a^2 = ab >a^2 - b^2 = ab - b^2 >(a-b)(a+b) = b(a-b) If a and b are equal, dividing by (a-b) is dividing by zero, and that's undefined. So you can't draw any valid conclusions from it. >a+b = b >b+b = b >2b = b >2 = 1 > >In this situation, if we assume a = b = 0, wouldn't it work out? No, because you're still dividing by zero. Let's put it this way: You _could_ define 0/0 to be zero, but only at the expense of being able to prove things like 2 = 1. But once you can prove that 2 = 1, you can prove _anything_, and math becomes useless. So what you'd be doing would be sort of like saying, "I'm going to change the rules of chess, so that a queen can move directly to any square, even if there are other pieces in the way, and even if that square isn't in the same rank, file, or diagonal." Now, you might find that game entertaining, but most people would find it completely uninteresting, because both kings begin the game in checkmate. Similarly, a mathematical system in which it's possible to prove that 2 = 1 might be interesting to you, but it would hold no interest for mathematicians. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 11/27/2003 at 17:04:55 From: Steven Subject: Thank you (Dividing 0 by 0.) Ok, I understand perfectly now. Your explanation was much better than the one my teacher gave me which was simply, "You just can't." Thanks a lot. I find your website incredibly helpful.
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