1. Assume a rectangle with a width of 1 and a height of pi. The area of the rectangle is width x height = 1 x pi = pi.

Divide the width of the rectangle by two, and multiply the height of the rectangle by two. The area of the rectangle is width x height = 1/2 x 2pi = pi.

Repeat ad infinitum and one arrives at the case of inifinitesimally small x infinitely large. The result is ... pi.

Then again, were the original rectangle of width 1 and height e, then infinitesimally small x infinitely large would equal e.

However useful, infinity and infinitesimality are concepts, not numbers. You cannot just multiply them and expect to get a defined answer in return.

2. Originally Posted by User Name:
To prove .(000)1 is not a number, AFAIK, requires a real analysis concept called Cauchy Sequences. Basically, it means that any real number can be expressed as a convergent sequence of differences. There is not sequence that can converge to .(000)1.
Hm.... I think I may have to change my mind on this. Using nested intervals theorem, you could say .(000)1 = 0, because both 0 and .(000)1 is in the infinitely nested intervals [0, 1], [0, .1], [0, .01], ad infinitum.

3. Originally Posted by Yarin
Why do you say "infinity and infinitesimal in the same sentence to mean the same thing"? It seemed to me, that you meant different things. It seems like you said "infinite" in reference to "so many that the amount has no end, and "infinitesimal" in reference "so small their smallness has no end".
I think you pretty much answered yourself in the last sentence. What's an infinitesimal quantity if nothing else than an infinite quantity? Can you bound it? More on that below, where I revisit 42.

I don't see the paradox your talking about. If the length is infinite of course the arrow would take an infinite amount of time to reach the target.
The paradox is achieved the moment you realize that yet still the arrow always reaches the target. Strengthening the idea that infinity boundless quality is only possible due to it's dimensionless attribute. And "dimensionless" doesn't fit with "big" or "small".

My thinking is, your right, infinity can't be measured. Let me give an illustration of what I'm thinking. Say, in space (by space I mean an imaginary void, not real space), there's a road, who's size is infinite (NOT infitesimal), remember, infinity means "boundless", so the road's size wouldn't be bound, and without bound, things keep going. So while on the road, we take a measuring tape and measure part of the road we're standing on, which we measure to be... 42 inches. Now, if we want to, we can walk a few yards and measure the same length out again. How does that not prove that 42 is smaller than infinity? Now, let's say we can pick up the road. Now matter which way we place it, we won't be able to put it within the 42 inches we have measured out. How does that not prove that infinity is bigger than 42?
I didn't say it wasn't. I said it is also smaller than 42. It is both things at the same time in fact. Or more probably, neither one of them. It does not respect our puny attempts at giving it a dimensional property. If I tell you that I measured the distance between the arrow and the target as being 42 paces, won't you immediately reach the realization that infinity is after all also smaller than 42 paces?

Infinity is boundless, dimensionless, nor greater or smaller, unless you get a specific need to bring it down to our level of comprehension, in which case you will use math to try and define a more or less formalized set of rules to try to explain/comprehend/use it. Problem is you are probably always going to find these paradoxes and you are going always to have to resort to mathematical fallguys like indeterminates... and accept the practical consequences of your audacity (NaNs). I don't think you can represent infinity within a dimensional field without that removing its properties. The moment you explain it in terms of size, you know you missed the point.

4. Originally Posted by Mario F.
I didn't say it wasn't. I said it is also smaller than 42. It is both things at the same time in fact. Or more probably, neither one of them.
ah, so infinity is probably 42..., thank you, this information is sure be useful to figuring out what the Ultimate Question is

5. Here's a BBC documentary on infinity, with some of it's apparent paradoxes and peculiarities, if you have an hour to kill.

YouTube - BBC Horizon (2010) - To Infinity and Beyond (complete, uncut)

6. Fascinating! You stole one hour from my workday. Well done

Interesting prospect the idea of an infinite number of universes, each of them infinite. I was thinking this same issue yesterday. How could one describe this concept, given that an universe being infinite, because its boundless, would certainly have to preclude the existence of any other universe.

Then the thought occurred to me as I was rereading about the arrow paradox. And it starts with the premise, "Is infinity boundless? Think again". Let's look at the following bounded interval of real numbers; [0, 42]. It's perfectly bounded. Yet, there's an infinite number of elements inside. The universe of elements inside this closed interval is infinite. It's an infinite bounded universe -- another delicious paradox on what possibility the biggest mass producer of those in cosmology; Infinity.

ed: well, by definition infinity is bound to produce an infinite number of paradoxes... big deal

7. Originally Posted by Mario F.
And it starts with the premise, "Is infinity boundless? Think again". Let's look at the following bounded interval of real numbers; [0, 42]. It's perfectly bounded. Yet, there's an infinite number of elements inside. The universe of elements inside this closed interval is infinite. It's an infinite bounded universe -- another delicious paradox on what possibility the biggest mass producer of those in cosmology; Infinity.
That's a different kind of infinity. The measure of members in a set is called cardinality. There are infinitely many infinities(cardinal numbers) used for denoting which infinite sets are bigger than others. (That was purposefully overcomplicated, it's actually pretty simple.) Like, for example, that the set of real is bigger than that of naturals.

Cardinality - Wikipedia, the free encyclopedia

And, yes, it is boundless, by definition. Wikipedia says that the word itself is from the Latin for "unboundedness."

8. Originally Posted by Yarin
So I know infinitesimility is suppose to = 0, but I was thinking, what does infinitesimility * infinity equal? I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
What do you guys think?
You don't argue in math. You need to have a clear definition of the concepts you are using.

As for your example, if f(x) -> oo and g(x) -> oo , we still cannot say much about f(x)/g(x). The limit can be 0, it can be oo, or something in between.

EDIT:
And no, 0.(000)1 is not a number. What would it mean? 0.(9) means lim(N->oo) sum(1...N) 9/10^N.

9. Originally Posted by Sang-drax
And no, 0.(000)1 is not a number. What would it mean?
Why wouldn't it be number?

Originally Posted by Sang-drax
0.(9) means lim(N->oo) sum(1...N) 9/10^N.
That's a good way to evaluate .999... but, .999... = lim(N->oo) sum(1...N) 9/10^N doesn't imply .999... := lim(N->oo) sum(1...N) 9/10^N.

10. Originally Posted by User Name:
Why wouldn't it be number?
EDIT: I see that you agree with me in a post above.

Originally Posted by User Name:
That's a good way to evaluate .999... but, .999... = lim(N->oo) sum(1...N) 9/10^N doesn't imply .999... := lim(N->oo) sum(1...N) 9/10^N.
Then what is the definition of 0.(9)?

11. Originally Posted by Sang-drax
Then what is the definition of 0.(9)?
.(9) is the definition of .(9), just like 2 is the definition of 2 and pi is the definition of pi.

12. Suppose that 0.999... != 1.

Okay, what's the average of the two? For that matter give me ANY number that exists between them.

13. If there's one thing I've learned about math, it's that Sang-drax is always right.

Originally Posted by brewbuck
Suppose that 0.999... != 1.

Okay, what's the average of the two? For that matter give me ANY number that exists between them.
What's the average of infinity and 12? You can't plug concepts like that into normal math equations. The idea of 0.999... is different from a specific number.

And certainly 0.999... is not equal to 1. Just because you can't find a number in between them doesn't make them the same. The inequality 0.999... < 1 still holds for any number of nines.

14. Originally Posted by Perspective
And certainly 0.999... is not equal to 1. (...) The inequality 0.999... < 1 still holds for any number of nines.
Prove it. (The qualifier "any number of nines" sounds strange to me, given the context: we're talking about the 9s repeating without end.)

15. Isn't it self-evident that an irrational number composed entirely 9s is smaller than its closest higher integer? Maybe you math folks have some way to formalize this proof. I do wonder though if it is required when we have already been expressing such things in other ways. Like with an interval, for instance ( [0, 1[ ).

Incidentally, Sang-drax expression above has a quality that is, I think, not being properly acknowledged: N->oo meaning isn't really that N grows towards infinity. Although that certainly can be safely implied, that description can hide one important piece of information for this purpose. What N->oo truly means is that N grows unbounded. And given the accountability properties of Real Numbers I think a simple Cantor's diagonal is enough to prove that 0.(9) is not 1.