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Arithmetic with infinities

  1. Sep 24, 2009 #1
    Am I correct in assuming that you can make sense of [itex]\infty + \infty[/itex] and [itex]\infty + c[/itex] for any [itex]c\in \mathbb{R}[/itex] (both evaluate to [itex]\infty[/itex]), but that we can make no sense of [itex]\infty - \infty[/itex]?

    Are there any other arithmetic operations one can perform with infinities that are undefined?
  2. jcsd
  3. Sep 24, 2009 #2
    What do you mean make sense of? I can define [itex]\infty -\infty = 8[/itex] if I want. We would lose some properties of our numbers, but there would be nothing contradictory about it. Usually we choose one of two approaches:
    1) Never do arithmetic with infinities, but instead speak about infinity using the concept of limits (that is a number can't be [itex]\infty[/tex], but a sequence can approach [itex]\infty[/itex]).
    2) Introduce some operations using infinity like you have. We normally call the set of real numbers together with [itex]\infty,-\infty[/itex] the extended real line. In the extended real line we have defined all the usual operations for all values except [itex]\infty - \infty[/tex], [itex]0\times(\pm\infty)[/itex], [itex](\pm \infty)/(\pm \infty)[/itex], and the real values at which they are normally not defined (such as 0/0). However it comes at a cost. For instance we can usually say that if a+b = a+c, then b=c (this is known as the cancellation law), but it's not true in the extended reals.

    Personally I have only seen one use of the extended real line and that was in measure theory. Apart from that I think people should stick to the real numbers unless the extended real lines provides distinct advantages over the real number system.

    EDIT: I may inadvertently have given the impression that the extended real line is the only way to add [itex]\infty[/itex], but this is wrong. Another common way is known as the real projective line. I'm not really familiar with this except as an example from topology so I haven't looked into its arithmetic properties, but it differs in the way that there is only one extra element ([itex]-\infty=\infty[/itex]). I still believe [itex]\infty - \infty[/itex] is left undefined though, but on the real projective line [itex]\infty+\infty[/itex] is also undefined. You can look it up on for instance Wikipedia if you want more in depth information.
    Last edited: Sep 24, 2009
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