What are some applications of the concept of complex infinity?

[Beer w/Straw]

I am interested in learning more about this concept where 1/0 can in this context be defined as infinity, however, I hear that infinity times 0 still will not reproduce 1 in this context. I have looked up some youtube mobius transformations and have read that "A Möbius transformation is a bijective conformal map of the extended complex plane (i.e. the complex plane augmented by the point at infinity)"

At first guess, I would think that this is somehow related to string theory but am curious as to what other applications, if any, that it arises in.

Please if anyone can shed some light on the subject it would be appreciated.Thanks.
:EDIT: I'm not allowed to posts links to other sites unless I have made more than 15 posts?

 

[matt grime]

Well, yes and no is the answer as always.

String theories (well, lots of them) happen to use 'complex infinity' but that is only because they want to use compact oriented Riemannian manifolds, sometimes with handles, or holes or marked points, to model things. The Riemann sphere (i.e. the complex plane union the point at infinity) is in some sense just the most basic tool of this analysis.

But the point at infinity is no more 'related to' string theory than it is to fluid mechanics or any other area which happens to use complex analysis or the geometry of complex manifolds. I.e. don't attach anything causal to it.

And I don't need to invoke 'the point at infinity' to construct the Riemann sphere - I just glue two copies of C along the open sets

C\{0} --> C\{0}

via

z --> 1/z.

 

[Beer w/Straw]

Thanks for the reply, something tells me if I encountered the Riemann sphere before complex infinity, I would have been less confused.

 

[shamrock5585]

ok well i just wanted to say that 1/0 is not infinity... it is undefined... as you pointed out that infinity times 0 does not give you back 1... it technically would give you zero because if you multiply anything by zero it gives you zero... so that would make the equation wrong in the first place so in a math sense 1/0 has no value it is undefined.

 

[HallsofIvy]

There are a number of different ways to define "infinity"- none of them give algebraic calculations in which you can say "1/0= infinity". However, they can give nicer geometric properties.

For example, the closed interval $a\le x \le b$ is often easier to work with than the open interval a< x< b (the first is "compact" in technical terms)- geometric things, like limits, etc., work better. But another way to make an open interval "compact" is to add a single point that is at both ends at the same time. Geometrically, it now looks like loop.

We can do the same kind of thing to the entire "number line" by adding "positive infinity" at one end and "negative infinity" at the other end of the real number system- converting it from an "open" interval to a "closed" interval.

But another way to do that is to add simply "infinity" and define "infinite distances" so that geometrically the number line "looks like" a loop.

For the complex plane we can add an entire circle of infinities- one at each end of every line through the origin- or we can add single "infinity" so that the complex plane has the geometry of a sphere.

That may be hard to imagine so here's a possible model: Imagine a sphere, of whatever radius you want sitting on the complex plane +so its "south pole" is at z= 0. For every complex number z, draw a line from the "north pole" to z. The point where that line crosses through the circle "corresponds" to z. That maps every point in the complex plane to a point on the sphere. Every point on the sphere except the "north pole" itself corresponds to a complex number. It is the "north pole" that corresponds to the single "infinity" in this case.

Those are, by the way, technically called the "Stone-Cech compactification" and the "one point compactification".

 

[shamrock5585]

ok dude what you just said had nothing to do with what i was talking about... you don't need to explain to me how to use infinity... i understand its application... when applied to 1/0 it seems that this would equal infinity but you would be incorrect in saying this because 1/0 can never be equal to anything when analyzed algebraically... it is undefined

 

[Hurkyl]

1/0 can never be equal to anything when analyzed algebraically... it is undefined

Yes, if '/' is the division operation on real numbers. (Or on complex numbers, integers, et cetera) But if we let '/' denote the division operation for projective numbers 1/0 is projective infinity. The point is, to do this, you have to use an entirely different function; the fact we usually denote those functions by the same symbol is irrelevant: they are still different functions.

More verbosely, RealDivision(1,0) is undefined, while ProjectiveRealDivision(1,0)=ProjectiveInfinity.

 

[Beer w/Straw]

ok well i just wanted to say that 1/0 is not infinity... it is undefined... .

Well then you can understand my confusion about complex infinity.

 

[HallsofIvy]

ok dude what you just said had nothing to do with what i was talking about... you don't need to explain to me how to use infinity... i understand its application... when applied to 1/0 it seems that this would equal infinity but you would be incorrect in saying this because 1/0 can never be equal to anything when analyzed algebraically... it is undefined

Yes, I understand that what I said had nothing to do with what you were talking about. I wasn't responding to you, I was responding to the original post.

 

[matt grime]

ok well i just wanted to say that 1/0 is not infinity...

Where?

it is undefined...

Again, where?

as you pointed out that infinity times 0 does not give you back 1... it technically would give you zero because if you multiply anything by zero it gives you zero...

Ditto.

so that would make the equation wrong in the first place so in a math sense 1/0 has no value it is undefined.

Nope, 1/0 has plenty of meaning in many places in mathematics.

 

[shamrock5585]

I know in computer logic it has applications... In an algebraic sense, if you do all your calculations and you come up with 1/0 you do not have an answer... your answer is not infinity... it is undefined and you must find another way to reach your answer.

 

[HallsofIvy]

No, matt grime's point is that there exist many branches of mathematics in which infinity is defined in such a way that it can be taken equal to 1/0. It just happens that it is NOT defined that way (and not defined at all) in the real number system. Your post should have had "in the real numbers" appended.

 

[matt grime]

Please read up a little on, say, complex analysis and algebraic geometry before asserting that "1/0" means you've gone wrong. The study of poles and zeroes of e.g. holomorphic functions is one of the most important elementary results in mathematics, see the work of Riemann.

 

[mathwonk]

the point is that in the theory matt is describing one backs off a bit and studies the conditions which give rise to an expression like 1/0.

it turns out there is often more information available that does let you decide how to assign a value to it.

namely it often arises from evaluating a quotient of functions like 1/z or 1/z^2,...

in that case one assigns a value to 0 times 1/0, or 0/0 in the same way as when computing derivatives, namely by taking a limit.

so one deals with 0 times 1/0 by asking how it arose,

say as (z^2) times 1/(z^3) and in this case it gives 1/z which then has limit infinity again, although in case of say z^3 times (1/z^3) it does give 1.

so one does not assign a single fixed value of infinity to 1/0, rather one considers the function which gave rise to this, and if it was say 1/z^3, then more accurately one assigns the value "triple infinity", or in matt's language a "pole" of order 3, which needs to be multiplied by a "triple zero", i.e. by z^3, to give 1.

so one introduces the theory of infinite values with multiplicities, mimicking the concept of multiplicity of a root of a polynomial.

 

[matt grime]

Here's a theorem about poles: every non-constant holomorphic map from the Riemann sphere to itself has a pole.

Proof: let f be a holomorphic map from the complex sphere to itself. The image is compact. If the point at infinity is not in the image (i.e. no poles) then it must be bounded, and hence constant by a theorem of Lousiville.

 

[matt grime]

I'm now risking straying into something I don't understand well enough to write about spontaneously, so take it with a pinch of salt; I'm sure mathwonk can correct me.

Take homolomorphic functions on the Riemann sphere again. Not only do non-constant ones have poles, but they have as many poles as zeros (counted with multiplicity). Pass to a Riemann surface of genus g, and by the Riemann-Roch theorem, you can say something, though I can't quite recall what.

 

[shamrock5585]

Has anyone seen this proof before?

http://mathmojo.com/interestinglessons/1equals2/1equals2.html

Basically i was saying, algebraically if you divide by zero you will have an answer that cannot be defined and is wrong.

When using complex math it can have applications because you are using non real numbers... which is exactly what they are... numbers that don't really exist, although they are defined.

 

[Hurkyl]

numbers that don't really exist

Complex numbers are no more or no less 'existant' than the real numbers. The meaning of the technical terms 'real' and 'imaginary', as applied to numbers, have absolutely nothing to do with their colloquial usage.

 

[matt grime]

if you divide by zero you will have an answer that cannot be defined and is wrong.

Yes, we know what you were saying. You are wrong. It is all a matter of qualification.

It is certainly true that 0*1=0*2, and if 0 had mulitiplicative inverse then 1=2, but that is nought to do with anything about 1/0 being inherently undefined and 'wrong'.

When using complex math it can have applications because you are using non real numbers... which is exactly what they are... numbers that don't really exist, although they are defined.

I can add no more than Hurkyl did: you usage if 'real' and 'imaginary' is completely wrong.

 

[HallsofIvy]

When using complex math it can have applications because you are using non real numbers... which is exactly what they are... numbers that don't really exist, although they are defined.

Absolute nonsense. Complex number exist as much as, and in the same sense as, real numbers do.

 

[mathwonk]

i agree with matt that every non constant meromorphic function on any compact riemann surface, i.e. every holomorphic map from the surface to the sphere, has (when counted properly) the same number of zeroes and poles, where counted properly means taking multiplicities into account.

one approach is to remove from the sphere the image of all points where the derivative is zero, and remove their preimages from the surface as well. by the implicit function theorem one then has a proper finite degree covering space, and the number of preimages of every point is then constant.

in fact the same result, that the divisor of zeroes and the divisor of poles have the same degree, holds for every non constant rational function on any complete connected non singular curve over any algebraically closed field, as proved in shafarevich, BAG, using the theory of "finite", i.e. finitely generated , modules over a pid. (the local ring of rational functions on a curve which are regular at a given point, is a pid precisely when the point is non singular.)

 

[shamrock5585]

Yes, we know what you were saying. You are wrong. It is all a matter of qualification.

It is certainly true that 0*1=0*2, and if 0 had mulitiplicative inverse then 1=2, but that is nought to do with anything about 1/0 being inherently undefined and 'wrong'.



Division is defined as the inverse of multiplication, so if you can divide by zero than 1/0 can equal anything based on what you just said.

 

[shamrock5585]

i love ppl just saying that i am wrong, they exist... u have no evidence or anything useful to say other than you are wrong... thanks for the input guys...

hey guys i got a good question for you. Newton said F=m*a. well what if an object is not accelerating. So divide your force by zero acceleration.. is the mass of the object infinite? or could it be 4 or 5... define it for me...


ps think about the word exist... if i have "i" apples how many do i have?... do dragons exist? well someone came up for a word for it so the word dragons exists and it has applications in random stories but do they really exist?

 

[CRGreathouse]

i love ppl just saying that i am wrong, they exist... u have no evidence or anything useful to say other than you are wrong... thanks for the input guys...

I imagine these same posters wonder about your claim that real numbers exist: you haven't given any evidence yourself. Integers existing is one thing, but real numbers? An uncountable infinity of them? Almost all of which can never be constructed?

 

[matt grime]

Tell, you what shamrock, please point out to me anywhere where 1, 2, 3 etc "exist". I can show you real life things that have properties that we label 1, 2, or 3, but for the life of me I can't seem to find a '1' lying around. We're telling you you're wrong to make blanket assertions as the weight of the last century and a half of mathematics demonstrates.

We have repeatedly said that 1/0 is not something one finds in R, but that there are plenty of places where it does come up. Projective geometry is one very good example. Projective geometry is very useful in the real world.

Division by non-zero real numbers is, loosely, defined as the inverse of multiplication. But that's for non-zero real numbers. What makes you think that we can't make a useful and consistent theory by moving outside that idea? I know why I think we can do so: we have.

You need to look beyond what you know about mathematics, to what has been developed over the millennia.

 

[shamrock5585]

I imagine these same posters wonder about your claim that real numbers exist: you haven't given any evidence yourself. Integers existing is one thing, but real numbers? An uncountable infinity of them? Almost all of which can never be constructed?

maybe you can refer to the theorem i posted above and then retract your post

 

[shamrock5585]

ok all I am saying is that 1/0 can have applications just like imaginary numbers but if you are working on dynamics or physics with actual measureable quantities and you end up dividing by zero you will end up with an undefined answer that can equal anything... you must find an alternative route to finding your answer... fair enough?

 

[matt grime]

Division is defined as the inverse of multiplication, so if you can divide by zero than 1/0 can equal anything based on what you just said.

Erm, where did I define 1/0 to be the multiplicative inverse of 0? I didn't. That still doesn't mean that 1/0 is not a useful introduction into complex analysis, or the extended real numbers.

In the extended complex plane x/0 is defined to be infinity for any x not equal to 0. 0/0 is still not defined.

 

[shamrock5585]

key words... extended complex plane...

 

[matt grime]

And the topic of this thread is...?

 

[shamrock5585]

hey man i made a very basic observation that i thought all math nerds knew and you guys picked it apart and told me i was an idiot... just wanted to put a small note that in real numbers you divide by zero you get undefined not infinity and here we are debating it still...

 

[matt grime]

Since the reals do not contain infinity, you cannot divide by zero in them. The purpose of this forum is to enlighten, surely, not to make incorrect and unqualified statements as if they were fact. Mathematicians are pedants, aren't they?

 

[LukeD]

hey man i made a very basic observation that i thought all math nerds knew and you guys picked it apart and told me i was an idiot... just wanted to put a small note that in real numbers you divide by zero you get undefined not infinity and here we are debating it still...

Because you've been arguing about it even though people have told you repeatedly that the topic at hand has nothing at all to do with that discussion.

By the way, when you divide by zero in the real numbers, you don't get "undefined". When you divide by zero, you've written nothing but nonsense; you've written something that's not necessarily even a number. If you're manipulating an equation, you've written something that does not follow from the axioms of the real numbers and therefore, something that you do not know to be true.

You also said that complex numbers aren't "real" because nothing "measurable" can be assigned a complex number. Well that's true, nothing that can be said to be "large" or "small" could possibly be assigned a complex number because complex numbers cannot be ordered (i.e., there is no consistent way to say that for any complex x and y that you can have x < y, x = y, or x > y. This is very easy to prove.), so you should not expect to be able to do such a thing. The complex numbers instead have other uses.

Even in modeling real life things. In fact, even the extended complex plane (that's right, where 1/0 = infinity) is used to model things in physics. The catch is that is not used to "measure" anything; it is instead used to model certain properties of a system.

 

[CRGreathouse]

I imagine these same posters wonder about your claim that real numbers exist: you haven't given any evidence yourself. Integers existing is one thing, but real numbers? An uncountable infinity of them? Almost all of which can never be constructed?

maybe you can refer to the theorem i posted above and then retract your post

You mean when you said "F = ma"? So would

$$C^{mn}_{pq}=g^{ma}\epsilon_{abpq}\sqrt{-g}$$

(related to Hodge duality of Maxwell's equations, apparently) convince you that complex numbers exist? If not, why? This is no less physical than "F = ma".

 

[shamrock5585]

By the way, when you divide by zero in the real numbers, you don't get "undefined". When you divide by zero, you've written nothing but nonsense; you've written something that's not necessarily even a number.

hence the reason we call it undefined haha

ok I am going to put a link here and then ill quote it...

http://en.wikipedia.org/wiki/Division_by_zero

"It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, division by zero is UNDEFINED. Division by zero must be left UNDEFINED in any mathematical system that obeys the axioms of a field. The reason is that division is defined to be the INVERSE OPERATION OF MULTIPLICATION. This means that the value of is the solution x of the equation bx = a whenever such a value exists and is unique. Otherwise the value is left UNDEFINED."