[quote name='Eelco' timestamp='1325243504' post='4898100']
[quote name='cowsarenotevil' timestamp='1325199551' post='4897959']
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.

Right. Defining to the power zero as one is something I think nobody takes exception to.

The analogy with Cantor's theory is slightly off though; the rules of exponentiation dont tell us what to the power zero should be, so we pick the continuous continuation for convenience.[/quote]

I actually meant 0^0 which is not well-defined by limiting behavior, even though it's often convenient to treat it as 1.[/quote]
One could say it is well defined by first taking for the limit with respect to the exponent and then the limit with respect to the base, no?

Cantors diagonal argument relies not on the introduction of another axiom to constrain previously undefined notions; it relies on the rejection of a previously embraced axiom to render an overconstrained argument consistent. That is, it is rightly controversial. Or bullshit, one might say, if one rejects this choice of axioms.[/quote]

I'm not quite sure I understand. I'm not convinced that the notion that infinite sets are equal in size if and only if there is a bijection between them was widely accepted beforehand.[/quote]
It was quite widely accepted that this was the case; think of Galileo's paradox, that highlights these issues, and even though nobody attached his name to it then, these things were already noted in classical antiquity. What was not widely accepted was the 'only if'. That is, the consistency of cantors argument hinges on rejecting the common-sense axiom that two sets can not be identical in size if the one is a subset of the other. Which I think is a manifestly absurd move; id much rather shove the consistency of infinity under the train. But thats a rather long story.

In any case the only point I wanted to make is that there are different (actually useful) contexts that call for different axioms which define things like a/0, 0^0 or infinity in different ways, as well as sometimes not at all, and that in any of these cases it's disingenuous to say "this is obviously this" or "this doesn't make sense so it can only be undefined" to the exclusion of all other sets of axioms.
[/quote]
Agreed.

[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]

Unless you want to write lots of exceptions and if's in your code, or similarly mention lots of special cases in your mathematical work, there is only one useful answer to what a non-zero value divided through zero is: infinity.

As far as I know, division by zero is defined officially as infinity as well. 0/0 is undefined yes, but non-zero / 0 is infinity. The sign or phase of this infinity surely isn't defined (mathematically speaking, not IEEE floating point), but what's for sure is that a value divided through that infinity, gives you zero again.

You're right, but I don't think that's really any different from what I said.

I was agreeing with you, just elaborating on the thought process that led me to my confusion.

Unless you want to write lots of exceptions and if's in your code, or similarly mention lots of special cases in your mathematical work, there is only one useful answer to what a non-zero value divided through zero is: infinity.

As far as I know, division by zero is defined officially as infinity as well. 0/0 is undefined yes, but non-zero / 0 is infinity. The sign or phase of this infinity surely isn't defined (mathematically speaking, not IEEE floating point), but what's for sure is that a value divided through that infinity, gives you zero again.

This all depends. Aftter more research, non-zero/0 can be infinity depending on what system you are using, but for any mathematical system that holds to the axioms of a field it is undefined.

I think also the a/infinity=0 is also undefined for some systems because infinity isn't a number in some systems; however the limit of a/x as x=>infinity is 0 despite infinity not being a number. This I am not positive of, but I feel like I heard this at one point in time.

[quote name='cowsarenotevil' timestamp='1325203962' post='4897983']
You can, for instance, create a perfectly reasonable set of operations that includes division by zero on a real projective line.

I have a question on your link that I'm confused about that's not perfectly related to the OP, but somewhat related.

How do they justify A*infinity=infinity and infinity+infinity=undefined? Why is infinity*infinity=/=2*infinity=infinity? I can understand it for all their other operations except this one.
[/quote]

A very important point that it seems a few people are missing:

Infinity is a concept, not a number. This is critically important in understanding infinities.

Non-mathematicians often use infinity to mean "any number larger than my container", such as a number that won't fit on the line of your answer sheet or larger than your floating point number width, but again, those are NOT mathematicians.

Infinity in mathematics has a much more formal definition; for example there are countable infinities and uncountable infinities. Infinity can be used with some of the same properties as more concrete numbers, but it is still fundamentally different and is not fully interchangeable with them.

In real analysis that uses the projective line discussed above, infinity is again a concept and not a real number. To the mathematician it has a specific meaning as a limit and NOT a real number. Since the whole point of real analysis is to study real numbers, the real operations on non-real infinity don't make sense.

In that subset of math the concept of inf is basically the same as NaN; infinity is a concept and not a number, so it spreads and infects other numbers it touches. An operation that involves a real number and the concept results in a concept. Operations that involve two concepts are undefined since the math is only concerned with real numbers.

[font=arial,helvetica,sans-serif]

A very important point that it seems a few people are missing:

Infinity is a concept, not a number. This is critically important in understanding infinities.

[/font]

[font=arial,helvetica,sans-serif]Numbers are concepts, not numbers.[/font]

[font=arial,helvetica,sans-serif]

Infinity in mathematics has a much more formal definition; for example there are countable infinities and uncountable infinities.[/quote][/font]

[font=arial,helvetica,sans-serif]As has been pointed out countless times, this is only true in a select set of formal systems.

To the mathematician it has a specific meaning as a limit and NOT a real number.[/quote][/font]

[font=arial,helvetica,sans-serif]On a real projective line, ? is ? in the same way that 6 is 6. They are both points on the line and do not require limits to exist. This is not the case in other systems, but it is in this one. Note also that defining infinity in this way or in the usual way in real analysis is completely incompatible with the cardinality of infinite sets, i.e. countable and uncountable infinities; none of them can be used interchangeably.[/font] I point this out because you seem to be saying that "countable" and "uncountable" is a general property of all "infinities."

Since the whole point of real analysis is to study real numbers, the real operations on non-real infinity don't make sense.[/quote]

Unless you develop a formal set of axioms and rules wherein it does make sense.

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In that subset of math the concept of inf is basically the same as NaN; infinity is a concept and not a number, so it spreads and infects other numbers it touches. An operation that involves a real number and the concept results in a concept. Operations that involve two concepts are undefined since the math is only concerned with real numbers.[/quote][/font]

[font=arial,helvetica,sans-serif]You're talking about it as if it's somehow more like poetry than real math. It's a well-defined construct like any other number or any other property of sets or any other whatever it is in the particular system you're working in. Either that, or you're working in a system where it's not well-defined, and then there's not much use worrying about it.[/font]

[font=arial,helvetica,sans-serif]There's a precise reason why, on a real projective line, you can't add infinity to itself, and it has nothing to do with the fact that "operations that involve two concepts are undefined." That doesn't even make sense, because infinity times infinity is well defined, and nothing you're saying gives a formal distinction between (infinity * infinity) and (infinity + infinity)[/font] that would justify why one is defined and the other is not.

(Sorry for formatting issues)

Unless you want to write lots of exceptions and if's in your code, or similarly mention lots of special cases in your mathematical work, there is only one useful answer to what a non-zero value divided through zero is: infinity.

As far as I know, division by zero is defined officially as infinity as well. 0/0 is undefined yes, but non-zero / 0 is infinity.

Who makes this "offical" definition? I've never heard of it.

As for it being a useful answer, consider: b = a/n; c = b * n.

What happens if n = 0, and you allow this with your definition?

b = infinity.
c = infinity * 0.

So what is infinity * 0? Is that undefined? In which case you've just delayed the problem. For any other value of n, the answer is b - but how can infinity * 0 equal b, when b could be any value?

If you are writing a program, isn't it better to throw an error as soon as a problem occurs, rather than pretending it doesn't exist, and running into undefined or incorrect behaviour later on?

As for the debate on whether infinity is a number - "number" is a concept that generally applies to various sets. The integers, the reals, the complex numbers. Infinity is not a real number of a complex number.

It is however worth noting the extended complex plane, or Riemann sphere ( http://en.wikipedia..../Riemann_sphere ), which is the set of complex numbers and infinity. With this set, you can do 1/0 = infinity. There is also the real projective line that cowsarenotevil refers to above. However, this doesn't make it the one single accepted definition - I would argue most of the time when mathematicians do maths, they are not working with this extended set, but instead either the reals of complex numbers. And whilst various operations are well defined, others (such as infinity * 0) are not well defined, and hence you still can run into problems. But it's also wrong to argue that infinity isn't a number, because it can be included as such.

There is no right answer here - arguing whether infinity is a number or not is like arguing whether i or -1 or sqrt(2) are numbers or not. It depends on what set you have chosen to work with.

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?

Simple:
1/10 = 0.1
1/5 = 0.2
1/2 = 0.5
1/1 = 1.0
1/0.5 = 2.0
1/0.25 = 4
1/0.1 = 10
...
1/0 = ? (Infinity)

Fact: Our math is flawed.

[quote name='Lode' timestamp='1325275454' post='4898228']
Unless you want to write lots of exceptions and if's in your code, or similarly mention lots of special cases in your mathematical work, there is only one useful answer to what a non-zero value divided through zero is: infinity.

As far as I know, division by zero is defined officially as infinity as well. 0/0 is undefined yes, but non-zero / 0 is infinity.

Who makes this "offical" definition? I've never heard of it.

As for it being a useful answer, consider: b = a/n; c = b * n.

What happens if n = 0, and you allow this with your definition?

b = infinity.
c = infinity * 0.

So what is infinity * 0? Is that undefined? In which case you've just delayed the problem. For any other value of n, the answer is b - but how can infinity * 0 equal b, when b could be any value?

If you are writing a program, isn't it better to throw an error as soon as a problem occurs, rather than pretending it doesn't exist, and running into undefined or incorrect behaviour later on?

As for the debate on whether infinity is a number - "number" is a concept that generally applies to various sets. The integers, the reals, the complex numbers. Infinity is not a real number of a complex number.

It is however worth noting the extended complex plane, or Riemann sphere ( http://en.wikipedia..../Riemann_sphere ), which is the set of complex numbers and infinity. With this set, you can do 1/0 = infinity. There is also the real projective line that cowsarenotevil refers to above. However, this doesn't make it the one single accepted definition - I would argue most of the time when mathematicians do maths, they are not working with this extended set, but instead either the reals of complex numbers. And whilst various operations are well defined, others (such as infinity * 0) are not well defined, and hence you still can run into problems. But it's also wrong to argue that infinity isn't a number, because it can be included as such.

There is no right answer here - arguing whether infinity is a number or not is like arguing whether i or -1 or sqrt(2) are numbers or not. It depends on what set you have chosen to work with.
[/quote]

Simple:
1* 0 = 0
2*0 = 0
3*0 = 0
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Infinity*0 = 0

[/font]

Simple:
1* 0 = 0
2*0 = 0
3*0 = 0
Infinity*0 = 0

In the real projective line and Reimann sphere Infinity*0 != 0, it is undefined, and in the systems where infinity*0 = 0, x/0 is undefined as far as I know. It seems like it would follow that (1/x)*x is always undefined at x=0 as in the systems that allow division by 0 the second operation is undefined and in systems that do not, the first is undefined.

Fact: Our math is flawed.

What?