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### Defining 0/0

```
Date: 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/
```
Associated Topics:
High School Number Theory

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