
Re: What is infinity minus one?
Posted:
Oct 12, 2004 2:28 PM


Tracy Poff <pofft@gmx.net> wrote in news://2t0o2cF1qqiliU1@uniberlin.de:
> Is this because a + (a) = 0 is defined on the set of reals, and > infinity is nonreal? > > This question was posed to my calc teacher in high school when > discussing indeterminate forms, and she replied (in essence) that it > was a magical thing that didn't have a reason, while I disagreed. > > Similarly, if my reasoning is correct, this would mean that 1*infinity > is undefined, since that is defined as the multiplicative identity > property of the reals. Is this correct? > > Tracy Poff >
I think what you have is a pretty good reason. I never thought about it like that, but it is a nice, simple explanation. The way I learned to look at it was to draw a contradiction of some sorts thus showing it is not welldefined (which is essentially the same as undefined).
Here is the way I get a contradiction for oo  oo. Define:
A1 := 1 + 2 + 3 + 4 + 5 + 6 +... > oo
A2 := 2 + 4 + 6 + 8 + 10 + 12 +... > oo
A3 := 3 + 4 + 5 + 6 + 7 + 8 + ... > oo
Look at:
A1  A2 = 1 + 3 + 5 + 7 + ... > oo
Thus oo  oo = oo
A1  A3 = 1 + 2 = 3.
Thus 3 = oo  oo = oo or 3 = oo #
Therefore oo  oo is not defined.
You can actually make oo  oo be any real number you want by starting your series A3 at n+1.
The reason 1*infinity is undefined, I would agree, is that infinity is not really a number (not in the reals). You simply cannot multiply a number by something that is not a number in any mathematical fashion that will make sense all of the time.
3*lemon does not make much sense mathematically.
There are times when you can multiply a number times a nonnumber and have significant results, like a scalar times a matrix or a number times a set. But in these cases, the multiplication has been very clearly defined.
[1, 2] [2*1, 2*2] 2 * [3, 4] = [2*3. 2*4] [5, 6] [2*5, 2*6]
2 * {1, 3, 5} = {2*1, 2*3, 2*5}
But overall, I would say your reasoning is pretty good and catches the essential parts of why arithmetic with infinity is not a good idea.
 Tim
 Timothy M. Brauch NSF Fellow Department of Mathematics University of Louisville
email is: news (dot) post (at) tbrauch (dot) com

