Thread: Concept of Quantity

Originally Posted by Mario F.

But a search on google reveals I would be wrong; It's in fact undetermined.

Indeterminate. See above.

Originally Posted by Mario F.

Which in my own view means we don't have a proper mathematical model to deal with infinity.

Yeah, we do, but it only deals with the sure parts of it, and leaves you to fend for yourself when you get to the indeterminate ones. It does give you a few weapons to help you defend yourself though. I've slaughter many a indeterminate with l'Hôpital's Rule. For more detail, see above.

Originally Posted by Mario F.

The fact we try to give it mathematical properties is probably something someone one day will give a condescending smile at, for our brave but futile attempts.

This is all very interesting, but please bear with me while I don't get anything.

Originally Posted by User Name:

Infinitesimals don't equal 0. If they did, all of calculus would be nonexistent.

I don't understand why that would be so. Does this have to do with calculus equations often have the double mirror effect? Kind of like calculating PI? (Not sure what the real term is)

Originally Posted by User Name:

The problem is that infinity doesn't exist in the real numbers, so it doesn't apply to the same rules as real numbers. The only way of dealing with infinities, without using the extended reals, is to use infinite limits. You're conjecture that 1/∞ * ∞ = ∞/∞ = 1 can be disproven by showing that both limx->∞ 1/x = 0 and limx->∞ 1/x^2 = 0 and that limx->∞ x = ∞, then considering both limx->∞ 1/x * limx->∞ x = limx->∞ x/x = 1 and limx->∞ 1/x^2 * limx->∞ x = limx->∞ x/x^2 = limx->∞ 1/x = 0, to see that there are different possible answers depending on the infinity in question. So, in other words, depending on the "size" of the infinity, you get different results. Ratios of infinities, and a few other special cases like these are called indeterminate forms. There are a few good calculus resources out there, if you want to learn this type of stuff. I particularly like MIT's OCW Calculus, with Wikipedia to fill in the gaps. Unfortunately, they spend absolutely no time learning limits.

I suggest you learn limits. They can be a fun for people who like algebra. You finally get to *almost* divide by 0!

Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x?

Originally Posted by User Name:

No, 1 IS .999... and .999... IS 1, just as 1/3 IS 2/6 and 2/6 IS 1/3. They are two representations of the exact same number. That's why they are the same, not because the nines never end. In terms of math: 1-.999... = 0 not .(000)1.

...

Do you mean 0 or 1/∞? If the later, then you need to reread my last post.

Yep, when I said .(000)1 I meant 1/∞, I guess I should use that form instead, from now on.

So, if .999~ is 1 (which I'm not necessarily disagreeing with), what is 1 - 1/∞?

Originally Posted by Mario F.

We can surely prove, by logic alone, that an infinitesimal is infinite. What you should probably be rid of is the notion that infinity is only something that is bigger than something else.

Why? Because we live such that we can't 'capture' infinity, everytime we try, it just acts like a number so big that it can't be counted - even when not restrained by time. I can't think of an exception to this.
(Yes I do realize that infinity isn't a real number)

Originally Posted by Mario F.

Oh, no it's not
As I said before, you cannot think in algebra terms. I'd say it is infinite. But a search on google reveals I would be wrong; It's in fact undetermined. Which in my own view means we don't have a proper mathematical model to deal with infinity. In any case, no. It's not zero. Again infinity is not a number, it's no longer even a measurable concept.

Look at Well-definition - Wikipedia, the free encyclopedia
There's It shows that it kind of is.

Originally Posted by Mario F.

The fact we try to give it mathematical properties is probably something someone one day will give a condescending smile at, for our brave but futile attempts.

Yeah probably, but I like to try anyway ;-)

Originally Posted by Mario F.

Let me extend on this by giving you a concrete example why infinite is both "bigger" than 42 and "smaller" than 42. Or, in other words why infinity is both infinitely large and infinitely small. Or, to be more precise, why infinity isn't either.

The example is a derivation from Zeno's arrow paradox. If you fire an arrow at a target, that arrow has to travel half its way before it can start doing the other half. But then, it needs to travel half of that before it can travel the other half of the first half. And so on, ad aeternum.

You can suddenly reach the conclusion that the arrow path can be divided infinite times into infinitesimal sections. Note already the need to express "infinity" and "infinitesimal" in the same sentence to mean the same thing. One can attribute that to a trick of the tongue. But if you can divide space into infinitely small portions, the arrow need to take an infinite amount of travelling before it reaches its target. And that's the paradox. It shouldn't ever reach it.

Physics will have something to say about all this. But here I'm just demonstrating to you that infinity is both small and large. And it is both things at the same time. And exactly because it doesn't have any measurable properties, it should in fact not even be determined in terms of being small or big. It's just infinite.

42. That's good

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 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.

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?

Originally Posted by Yarin

I don't understand why that would be so. Does this have to do with calculus equations often have the double mirror effect? Kind of like calculating PI? (Not sure what the real term is)

Differentials(denoted with the d in front of them) are not zero, if they were, the derivative would be meaningless, the integral also, by inclusion. df/dx=limh->0 (f(x+h)-f(x))/h This says that the ratio of two differentials(infinitesimals), is finite. If they were zero, then the ratio would 0/0 , and therefore undefined. I'm not going into detail on what they really are, it takes more explaining than I really want to give, since there are many resources already available. (MIT OCW + Wikipedia = everything I know)

Originally Posted by Yarin

Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x?

A law of limits is that, iff limx->c f(x)*g(x) = L, then (limx->c f(x))*(limx->c g(x)) = L I used the inverse.

Originally Posted by Yarin

Yep, when I said .(000)1 I meant 1/∞, I guess I should use that form instead, from now on.

1/∞ is indeterminate, you can't give it a single representation. Neither can it be represented with a real number(.(000)1 is not a number). 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.

Originally Posted by Yarin

So, if .999~ is 1 (which I'm not necessarily disagreeing with), what is 1 - 1/∞?

Remember that ∞ can't be used without limits(within standard analysis, in nonstandard, they use rules that give the same results as if you were to take a limit, which is why nonstandard analysis is largely considered extraneous). So, you take the limit limx->∞ 1-1/x = 1. In general, any number minus an infinitesimal is that same number. Counterintuitive, maybe, but true, definitely.

To help Mario, what he means by infinities and infinitesimals being the same, is that the are the same in their immeasurableness. On is always, unconditionally larger, and the other is always, unconditionally, smaller.

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.

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)