Defining 0/0Date: 01/29/2001 at 15:02:10 From: Jon Howard Subject: Zero/zero, finally defined? Hey, I'm in the tenth grade and I recently posed a question to my Algebra teacher on defining 0/0. Based on our own rules of math, I argued my teacher into agreeing that 0/0 must be defined as 1 simply because, even though zero is undefined, 0 = 0. And our math laws say that anything divided by itself equals 1. So my question is, based on these simple laws, shouldn't 0/0 = 1? Thanks a lot. Jon Howard Date: 01/29/2001 at 16:35:07 From: Doctor Shawn Subject: Re: Zero/zero, finally defined? Jon, That argument sounds convincing at first, but it'll get you into trouble. Zero is, in fact, defined - it's zero. It might seem strange that that's a defined value, but you can tell the difference between one apple and zero apples, can't you? Division by zero, on the other hand, IS undefined. The reason for that is that it leads to contradictions, and from that you can prove anything you like. Also, no definition of division by zero can be entirely consistent. Check out the Dr. Math FAQ: Dividing by 0 http://mathforum.org/dr.math/faq/faq.divideby0.html I bet you'll be able to see the flaw in your argument after you read through those articles. Good luck, and feel free to write back if you have more questions. - Doctor Shawn, The Math Forum http://mathforum.org/dr.math/ Date: 01/30/2001 at 20:51:31 From: Jon Howard Subject: RE: When 0/0=1, then all of its combinations are real. Hi again, Sorry to get back on the same subject, but I don't think I expressed my question thoroughly. When 0/0 = 1, that means that 0/1 = 0, 0 = 0, 0 = 1*0, and 1 = 1. This doesn't include 1/0 = 0, which isn't true. And it is only undefined after 1. See, when you do 0/0 = 2, then you're saying that 1 = 2, which isn't true. Also, if 0/0 = 1 is true, then it has an effect on prime numbers as well, because 0 would have two factors, 0 and 1. So, to further my question, when we say that 2*0 = 0, then we are saying that 0/0 = 2, or 1 = 2, which isn't true. With this in mind, isn't it true that beyond the number 1, anything times 0 is undefined, and that 0 would be considered a prime number? Thanks, Jon Date: 01/30/2001 at 22:38:02 From: Doctor Peterson Subject: RE: When 0/0=1, then all of its combinations are real. Hi, Jon. It seems to me that you are trying to resolve an impossible problem by destroying one of the fundamental facts of math, that zero times anything is zero. Why would you want to do away with that, just so you can define 0/0? The fact is, we KNOW that 0*1 = 0 AND 0*2 = 0, so your reasoning shows equally well that 0/0 = 1 and that 0/0 = 2. When we find that an operation can produce two different results (and in fact, it could be anything at all), then we just accept that it is not defined. Any definition would cause inconsistencies in the rest of math, such as implying that 1 = 2, as you said. Rather than deny that 0*2 = 0 (which would mess everything else up anyway), we just call 0/0 indeterminate. As for zero being prime, the definition requires exactly two POSITIVE divisors. See our FAQ on prime numbers: http://mathforum.org/dr.math/faq/faq.prime.num.html And if zero were prime, it would mess up everything we do with primes, without contributing anything useful to number theory. The fact is, zero is divisible by everything, not just by itself and 1. But you might be interested in some deeper facts about that indeterminate 0/0. When you get into calculus, you will learn that there are many problems that seem to lead to this "value," but you can often approach the problem differently and find which value "0/0" has in the specific case. You'll find that, in fact, it can have any value, not only 1. If we defined 0/0 to have one particular value, then all that work would become wrong! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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