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Defining 0/0

Date: 01/29/2001 at 15:02:10
From: Jon Howard
Subject: Zero/zero, finally defined?


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?


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   

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   

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?


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:   

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   
Associated Topics:
High School Number Theory

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