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Topic: What is infinity minus one?
Replies: 25   Last Post: Jun 7, 2013 10:01 PM

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Tim Brauch

Posts: 201
Registered: 12/6/04
Re: What is infinity minus one?
Posted: Oct 12, 2004 2:28 PM
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Tracy Poff <pofft@gmx.net> wrote in news://2t0o2cF1qqiliU1@uni-berlin.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 well-defined (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 non-number 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





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