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?
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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?
How would anything equal multiplied by 0 manage to be equal to 1?Quote:
I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
The way you've set that up sounds wrong.
Soma
I seriously doubt that infinitesimal is considered anything resembling 0. While math is not my area, I think there is a huge difference between something infinitely small and nothing at all and this difference somehow has been expressed in maths. What I remember learning is that an infinitesimal is a quantity approaching zero but not reaching it. It's distance to zero is infinitesimally small.
Infinite quantities may be applied to maths, but you already realize that rules are different for them when for instance we learn that any operation on infinity returns infinity. So, you shouldn't just assume good old algebra is the way to solve your question. 0 * 1 is not what you should be looking for. Infinite quantities aren't numbers at all.
Instead here's three possibilities:
- If we were to represent infinitesimals differently than infinitely large quantities, it would probably make sense to establish a rule based on the position of them in the equation. That is to say an infinitesimally small number multiplied infinite times is still an infinitesimal number, while an infinitely large quantity multiplied by an infinitesimal amount is still an infinitely large quantity. This holds true for all operations.
- We can simply establish that infinitesimal quantities are infinite quantities in fact. Which they are. There's no direction for infinity. So an infinitesimally small quantity is an infinite quantity. Infinite quantities aren't big or small. They are simply infinite.
- Or we can remember that infinitesimal is not the same as 0, as such -- and because any operation on infinity results in infinity -- the answer to your question is infinity.
If you look closely, all the three possibilities are the same. :)
Infinitesimals don't equal 0. If they did, all of calculus would be nonexistent.
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! :D
Oh, it wouldn't, I'm trying to imply that maybe infinitesimal != 0.
Yes this is as far as I know too, but I found some compelling arguments showing that because infinitesimal is so small, that is, if one were to search for it's 'end', they could never find it, thus making it nothing.
Well, I can't find the pages with the proofs anymore. But here's a good reference nonetheless: 0.999... - Wikipedia, the free encyclopedia...
Now, I hope my assuming that saying that 0.(999) = 1 is the same as saying that 0.(000)1 = 0, isn't wrong :)
True, and in real life calculations, they're not used. To me, this seems to lead to holes in the mathimatical system, though, for example the infamous divide by 0 problem. The idea is too see how infinity and numbers would co-exist in the same mathimatical system. Formal math seems to say they don't/can't.
We know that infinity is bigger than 1, and that infinitesimal is smaller than 1, so, when a difference in relationship to tangible real numbers can be shown, doesn't that prove that they aren't the same thing? On other words: Surely something can't be smaller and bigger than something else at the same time ;)
The problem with this is the assumption that any operation on infinity results in infinity, for example, 0 * infinity, is of course, 0.
Just as we have 1000, and 0.001, we have infinity, that is, 1(00)0, and infinitesimal, that is, 0.(00)1, I purposed 1 as an answer because if 1000 * 0.001 = 1, then why not 1(00)0 * 0.(00)1 = 1?
This breaks the rules, though.
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.
They exist, you've just not been introduced to the formal math for dealing with infinities and real numbers. Extended real number line - Wikipedia, the free encyclopedia :p
Do you mean 0 or 1/∞? If the later, then you need to reread my last post. :)
We don't. Infinity isn't measurable. There may be a need to express infinity in two directions in mathematics. But it's probably because we shouldn't, that theorists have so much trouble defining what mathematical rules they will obey.Quote:
We know that infinity is bigger than 1, and that infinitesimal is smaller than 1
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.Quote:
so, when a difference in relationship to tangible real numbers can be shown, doesn't that prove that they aren't the same thing? On other words: Surely something can't be smaller and bigger than something else at the same time
Oh, no it's not :)Quote:
The problem with this is the assumption that any operation on infinity results in infinity, for example, 0 * infinity, is of course, 0.
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. 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.
Indeterminate. See above. :p
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.
:)
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.
This is all very interesting, but please bear with me while I don't get anything. :)
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)
Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x?
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/∞?
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)
Look at Well-definition - Wikipedia, the free encyclopedia
There's It shows that it kind of is.
Yeah probably, but I like to try anyway ;-)
42. That's good :D
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. :confused:
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 haven't even studied limits to a satisfactory degree, but they do have properties like other things, such as logs. lim x to a f(x) * lim x to a g(x) is the same as lim x to a f(x)g(x). Of course, you're asking people to explain calculus to you at this point which is a bit annoying I imagine.Quote:
Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x
My interpretation of what's been said is this: the road is really a number line, with signs posted. Start from an arbitrary number N not forty-two and try to go to 42. I'm willing to bet that there are infinite real numbers between N and N+1 that you won't even make it to the next sign. Now, is infinity bigger or smaller than forty-two? Well, it's damn meaningless to measure!Quote:
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?
Of course, you can deal with infinity by using limits. For example, one of the first things you do, studying limits is find definition of a function. Is that information useful? I dunno. For me math is more about application than anything else, but I can't tell you what limits are applied to yet. I'm still learning calculus myself....
So these algebraic rules are suppose to be transparent of limits? I guess that makes sense. I'll look closer at it again.
Yep, it's basically a number line. Sure, there's an infinite real numbers between N and N+1, but that holds true to even finite number lines, and that doesn't stop one from traversing them by 42.
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)
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.
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.
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.
The derivative of f(x) is defined as f'(x)=limh->0 (f(x+h)-f(x))/h, notice the fact that at x=0, it is 0/0, an indeterminate form. The derivative, intuitively, is how much the function rises, as it the amount it runs approaches zero. Derivatives, an application of limits, have innumerable applications. And integration, the most important application of derivatives, have even more.
How long have you been studying? I did for a few months before I actually started this semester at my local community college. It will definitely help you later. Many have issues when they first encounter a new paradigm of math, and being pre-exposed is like a vaccine against failing. Anyway, good luck, and if you ever get stuck on something, I have my PMs turned on, *unlike some people* cough cough...