Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex infinity

  1. May 23, 2008 #1
    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?
     
  2. jcsd
  3. May 23, 2008 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
    Last edited: May 23, 2008
  4. May 23, 2008 #3
    Thanks for the reply, something tells me if I encountered the Riemann sphere before complex infinity, I would have been less confused.
     
  5. May 29, 2008 #4
    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.
     
  6. May 29, 2008 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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 [itex]a\le x \le b[/itex] 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".
     
  7. May 29, 2008 #6
    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
     
  8. May 30, 2008 #7

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  9. May 30, 2008 #8
    Well then you can understand my confusion about complex infinity.
     
  10. May 30, 2008 #9

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  11. May 31, 2008 #10

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Where?

    Again, where?

    Ditto.


    Nope, 1/0 has plenty of meaning in many places in mathematics.
     
  12. Jun 2, 2008 #11
    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.
     
  13. Jun 2, 2008 #12

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  14. Jun 2, 2008 #13

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  15. Jun 2, 2008 #14

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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.
     
    Last edited: Jun 2, 2008
  16. Jun 2, 2008 #15

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  17. Jun 2, 2008 #16

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  18. Jun 2, 2008 #17
    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.
     
  19. Jun 2, 2008 #18

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  20. Jun 2, 2008 #19

    matt grime

    User Avatar
    Science Advisor
    Homework Helper


    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'.

    I can add no more than Hurkyl did: you usage if 'real' and 'imaginary' is completely wrong.
     
  21. Jun 3, 2008 #20

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Absolute nonsense. Complex number exist as much as, and in the same sense as, real numbers do.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?