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

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David W. Cantrell

Posts: 3,395
Registered: 12/3/04
Re: What is infinity minus one?
Posted: Oct 12, 2004 5:14 PM
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Tim Brauch <RnEeMwOs.pVoEst@tbrauch.cNOoSPAMm> wrote:
> 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?


That's certainly not quite the reason. By that very reasoning
Sqrt(-1) + (-Sqrt(-1)) should be undefined too. But after all,
we don't want to deal strictly with reals all the time!

> > 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.


Again, by reasoning such as that, 1*Sqrt(-1) should also be undefined. Of
course, it _is_ undefined in the reals. But 1 is also the multiplicative
identity both in the system of complex numbers and in the system of
extended real numbers, thereby allowing us to claim 1*Sqrt(-1) = Sqrt(-1)
and 1*infinity = infinity in appropriate systems.

> I think what you have is a pretty good reason. I never thought about it
> like that, but it is a nice, simple explanation.


And of course I disagree!

> 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.


Your idea is not bad. But using series of _integers_, it's hard to see
how you'd "make oo - oo be any real number". It would be hard to get pi,
for example.

> The reason 1*infinity is undefined, I would agree, is that infinity is
> not really a number (not in the reals).


Well, _of course_, since the reals don't include infinity, it's not defined
_there_. (But so what? The same argument would allow us to say that
Sqrt(-1) is undefined.) Infinity _is_ an element of the extended reals (and
whether we choose to call it a "number" or not is essentially irrelevant),
and so 1*infinity is defined there.

> 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.


The only products which are normally undefined in the extended reals are
those of 0 and +-infinity. For nonzero x, you can indeed simply multiply:

x * infinity = infinity if x > 0,
-infinity if x < 0.

[snip]
> 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.


If you think it's a bad idea, then perhaps you should contemplate why
the most used (in terms of number of computations performed per day)
number system in the world, floating-point arithmetic, incorporates
arithmetic with infinity (and does so according to an internationally
accepted standard). You might also ask yourself why, if it's a bad idea,
all computer algebra systems (or at least all known to me) implement
arithmetic with infinity.

David Cantrell





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