Arithmetic with infinities

1. Sep 24, 2009

AxiomOfChoice

Am I correct in assuming that you can make sense of $\infty + \infty$ and $\infty + c$ for any $c\in \mathbb{R}$ (both evaluate to $\infty$), but that we can make no sense of $\infty - \infty$?

Are there any other arithmetic operations one can perform with infinities that are undefined?

2. Sep 24, 2009

rasmhop

What do you mean make sense of? I can define $\infty -\infty = 8$ 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 $\infty[/tex], but a sequence can approach [itex]\infty$).
2) Introduce some operations using infinity like you have. We normally call the set of real numbers together with $\infty,-\infty$ the extended real line. In the extended real line we have defined all the usual operations for all values except $\infty - \infty[/tex], [itex]0\times(\pm\infty)$, $(\pm \infty)/(\pm \infty)$, 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 $\infty$, 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 ($-\infty=\infty$). I still believe $\infty - \infty$ is left undefined though, but on the real projective line $\infty+\infty$ is also undefined. You can look it up on for instance Wikipedia if you want more in depth information.

Last edited: Sep 24, 2009