First, understand that my math skills do not go above the high school level.

I was thinking about division by zero last night for some strange reason. I remember reading somewhere that there is some debate among mathematicians as to what the result actually would be. To me, it seems like a NULL operation. Let's say that I have five dots (.....). As I am typing this, I am currently dividing them by zero. In essence, I am doing nothing. Therefore, the result would be five. Or... ??

That led me to thinking about multiplication by zero. As has always been taught, I know that multiplying anything by zero is zero. However, isn't this also a NULL op? If I again have five dots (.....) and I multiply them by nothing, I am doing nothing, so I am left with five. However, this isn't necessarily commutative. If I have zero dots and I multiply them by five, I am left with nothing since I started out with nothing. However, non- matrix multiplication *IS* commutative. Why would me multiplying my five dots by nothing remove my five dots from existence?

First, understand that my math skills do not go above the high school level.

I was thinking about division by zero last night for some strange reason. I remember reading somewhere that there is some debate among mathematicians as to what the result actually would be. To me, it seems like a NULL operation. Let's say that I have five dots (.....). As I am typing this, I am currently dividing them by zero. In essence, I am doing nothing. Therefore, the result would be five. Or... ??

That led me to thinking about multiplication by zero. As has always been taught, I know that multiplying anything by zero is zero. However, isn't this also a NULL op? If I again have five dots (.....) and I multiply them by nothing, I am doing nothing, so I am left with five. However, this isn't necessarily commutative. If I have zero dots and I multiply them by five, I am left with nothing since I started out with nothing. However, non- matrix multiplication *IS* commutative. Why would me multiplying my five dots by nothing remove my five dots from existence?

Think of it this way:

Each pile of dots contain 5 dots.

If you take 0 such piles of many dots do you have ? (The answer would be 0)

For division you can think of it like this instead:
You have 5 dots and are distributing them evenly among 0 people, how many dots do each person get, (This is harder to define, If you distribute it evenly among 0.5 people then half a person gets 5 dots (Which means that 1 person getting 10 dots, as the divisor gets closer to 0 the result of the division aproaches infinity),

I remember reading somewhere that there is some debate among mathematicians as to what the result actually would be.

I don't know of any mathematician who would say that. Division by zero is undefined; it just doesn't make any sense. It's like dividing by purple or schadenfreude. You can say it, but it doesn't make any sense to actually try to carry it out.

Are you sure you're not thinking of 0/0? This has indeterminate value, which isn't the same as there being 'debate as to what the result would actually be'. (It basically means the result, or whether one even exists, depends on how you reached that point - but there are well-defined ways to determine this for any particular case.)

Are you sure you're not thinking of 0/0? This has indeterminate value, which isn't the same as there being 'debate as to what the result would actually be'. (It basically means the result, or whether one even exists, depends on how you reached that point - but there are well-defined ways to determine this for any particular case.)

I really don't know. I have a memory of reading that somewhere, but I don't remember where or even the context. I believe that I read this somewhere here, but it could have been through a search and who know when it was posted.

I was thinking about division by zero last night for some strange reason. I remember reading somewhere that there is some debate among mathematicians as to what the result actually would be.

The only debate would have been many centuries ago.

For an interesting read on the history of the number, read the book "Zero: The Biography of a Dangerous Idea".

Look back almost 1000 years ago and you'll find that division by zero was one reason some mathematicians rejected the concept of zero. But as zero gained acceptance as an actual number, several definitions needed to be resolved.

Division by zero is undefined. It does not make sense as an operation.


Things get tricky with more advanced math:

In calculus you need to approach (but not hit) the undefined point of x/0; that gives +inf and -inf depending on the direction you approach, or a value of 1 for the result x/x as x approaches zero.

In certain physical sciences like computer hardware engineering you frequently see 0/0 = 1 due to the nature of the physical world. You need a lot of calculus and physics to understand it, but L'Hopital's Rule can help explain it. Again you are looking back 400 years in the past for the controversy. About the only issue left is if the rule should be attributed to L'Hopital or Bernuli.



In the computer's floating point math everything gets an actual result. On the PC you can get any of +inf, -inf and NAN if you attempt to divide by zero. You also can get a floating point exception if those are turned on.

In a computer's integer math system you don't have NAN so it typically generates an exception, but the results vary based on programming language.

That led me to thinking about multiplication by zero. As has always been taught, I know that multiplying anything by zero is zero. However, isn't this also a NULL op? If I again have five dots (.....) and I multiply them by nothing, I am doing nothing, so I am left with five. However, this isn't necessarily commutative. If I have zero dots and I multiply them by five, I am left with nothing since I started out with nothing. However, non- matrix multiplication *IS* commutative. Why would me multiplying my five dots by nothing remove my five dots from existence?
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You have X groups of 5. Zero groups of 5 is zero.

Zero groups of anything is zero. It doesn't matter how many things are in the groups. There can be one thing or ten things or a million things in each group, but since you don't have any groups, your total is zero.

[quote name='MarkS' timestamp='1325194871' post='4897931']
I remember reading somewhere that there is some debate among mathematicians as to what the result actually would be.

I don't know of any mathematician who would say that. Division by zero is undefined; it just doesn't make any sense. It's like dividing by purple or schadenfreude. You can say it, but it doesn't make any sense to actually try to carry it out.
[/quote]

You're right, it doesn't make much sense for there to be debate, but that's different from saying it's a complete non-issue. In different contexts it makes perfectly good sense to give it different meanings. Division by purple doesn't really make sense, even with additional information. Division by zero, though, could be either positive or negative infinity if division by zero is treated as a limit where the denominator approaches zero, or an indeterminate form as TheUnbeliever mentioned. In day-to-day life I think it's pretty common to think of a x/0 as infinite and 0/0 as indeterminate; usually the situations where you would want these operations to exist suggest treating them as limits.

It's just like defining taking numbers to the zero power or saying that the number of real numbers is greater than the number of integers: the concepts are "arbitrarily" defined and so not debatable per se, but somehow Cantor's theory of infinite sets was quite controversial, mainly because the usefulness of defining operations in some particular way is not always so apparent.

You're right, it doesn't make much sense for there to be debate, but that's different from saying it's a complete non-issue.

I didn't say to an amateur there wouldn't be some value to the sloppy mathematics of believing division by zero has some value, I specifically responded to the mathematicians debating it. To mathematicians there is no debate. For them it is a complete non-issue (or at least the living ones as frob pointed out). Limits might be useful, but a trained mathematician understands the difference between the limit and the actual thing.

It's 0/0 which can in theory be treated many possible ways:
Lim X->0 of x/x = 1
Lim X->0 of 0/x = 0
Lim X->0 of x/0 = undefined

Heck really we can justifyably make 0/0 any value we want:
Lim X->0 of 42x/x = 42

In practice the actual value of 0/0 is treated as undefined.

The concept of zero has not been around all that long. There's a recent article on newscientist.com about it.

[quote name='cowsarenotevil' timestamp='1325199551' post='4897959']
You're right, it doesn't make much sense for there to be debate, but that's different from saying it's a complete non-issue.

I didn't say to an amateur there wouldn't be some value to the sloppy mathematics of believing division by zero has some value, I specifically responded to the mathematicians debating it. To mathematicians there is no debate. For them it is a complete non-issue (or at least the living ones as frob pointed out). Limits might be useful, but a trained mathematician understands the difference between the limit and the actual thing.
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I don't understand why assuming (without making explicit) a set of axioms wherein division by zero is any less "sloppy" than treating it as a real operation. I imagine there are plenty of algebraic structures that behave "normally" in most cases but have well-defined notions of division by zero. In any case, different contexts require different assumptions/definitions/axioms/inference rules (or whatever you want to call them) and I don't know any mathematicians who would say that division by zero is a "complete non-issue" unless they had the opportunity to specify context more than, you know, not at all.