# Complex infinity

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?

## Answers and Replies

matt grime
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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.

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

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
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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".

ok dude what you just said had nothing to do with what i was talking about... you dont 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
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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.

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
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ok dude what you just said had nothing to do with what i was talking about... you dont 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
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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.

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
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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
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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
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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.

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matt grime
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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
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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.

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 dont really exist, although they are defined.

Hurkyl
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numbers that dont 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
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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 dont really exist, although they are defined.

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

HallsofIvy
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When using complex math it can have applications because you are using non real numbers... which is exactly what they are... numbers that dont really exist, although they are defined.
Absolute nonsense. Complex number exist as much as, and in the same sense as, real numbers do.

mathwonk
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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.)

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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.

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?

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CRGreathouse
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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
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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.