Infinity...

[WillJ]

The difference might as well be zero because there is only one way you could tell the difference between 5 and 5+1/infinity, and that is to multiply by infinity, which is impossible, since the answer is always infinity. As such, no matter what you do with the numbers, the answer is the same. They therefore might as well be the same. If someone offered you 1/infinity of a pint of beer, you'd probably not feel too grateful. That's not even a molecule - it's not even a quark. It's an infinitely small fraction of the smallest thing there could possibly be, which is already pretty small.

So, 1/infinity is zero. Any integer divided by infinity might as well be zero. You can't divide by infinity. How many ones make one infinity? Might as well ask how many ones make the Second Boer War.

Where do you get that the only way to tell the difference is to multiply by infinity? That might clear the denominator, but isn't there already a difference between the two numbers (1/infinity)? And there are plenty of things you can do to them, and they'll still be different. Multiply them by 6, and you get 30 versus 30+30/infinity. The only reason to say that these aren't different is if you don't consider 30/infinity a valid difference, or rather anything divided by infinity as a valid difference, and I don't see why that would be the case. It looks like you're using circular logic in saying that 1/infinity equals zero because +/- 1/infinity doesn't give you a difference because 1/infinity equals zero (of course, you never said the third part, but there's no other explanation that I can think of for the second part).

 

[Perfection]

Well, a lot of applications of infinity involve expressions that are known as indeterminate forms, these would be things like infinity/infinity or 0/0 or 0*infinity

 

[Halcyon]

Where do you get that the only way to tell the difference is to multiply by infinity? That might clear the denominator, but isn't there already a difference between the two numbers (1/infinity)? And there are plenty of things you can do to them, and they'll still be different. Multiply them by 6, and you get 30 versus 30+30/infinity. The only reason to say that these aren't different is if you don't consider 30/infinity a valid difference, or rather anything divided by infinity as a valid difference, and I don't see why that would be the case. It looks like you're using circular logic in saying that 1/infinity equals zero because +/- 1/infinity doesn't give you a difference because 1/infinity equals zero (of course, you never said the third part, but there's no other explanation that I can think of for the second part).

30/infinity is pretty much equal to 1/infinity - cancel both sides by thirty. An infinite number of thirties divide into infinity. You can't really do simple calculations involving infinity. The answer is always 0 or infinite. Infinity is more of a concept than a number, really. That doesn't stop some infinities being bigger than others, of course. Omega infinity versus beta infinity, for instance. They're not numbers, but the former is still larger. It doesn't have to make sense, because they're mathematical constructs, and useful ones on occasion.

Regardless - 1/infinity is 0 to an infinite number of significant figures, and it is therefore infinitely accurate to say that 1/infinity equals 0. No?

 

[newfangle]

I'd like to think of infinity as a number arbitrarily large enough to make things nice.

 

[Perfection]

That's usually how I like to think of it as well.

 

[The Last Conformist]

In what sense can the human mind comprehend mathematical concepts at all?

If you can use infinity (and appaling numbers of supposedly educated people cannot), I don't think there is anything more you can sensibly ask for.

 

[Erik Mesoy]

Huh? First of all, I always thought "->" means "yields." What does it mean here? "Approaches," perhaps? And does N represent a number?

Oh, and let me ask you this. If 1/infinity equals zero, since cross-products are always equal, does that mean zero times infinity equals anything you want it to?

Yep, "Approaches", a key point of many such functions. Let me spell it out:
As a variable N approaches Infinity, its inverse, the fraction 1 over N, approaches Zero.
Approaches is the keyword. N never actually reaches infinity! If it did, our mathematics would collapse.
That's why your cross-product goes wrong too. We never reach 1 or infinity.

*pulls out calculus book* Ah. Here's a good function, showing how you use limits to work out what you cannot calculate directly, much like these infinities and infinitesimals.
Evaluate lim (N->0) of (((sqrt(N+25))-5)/N.
Go on, work it out. If you try to put in N=0, you're dividing by zero. Bad boy.
But look at these values:
For N=1, it is 0,09902.
For N=0,01, it is 0,09999.
For N=0,0001, it rounds to 0,1.
So as N approaches 0, the limit of "(((sqrt(N+25))-5)/N" is one-tenth.

Now, WillJ, see how (1-(1/Infinity))=1, or 0,99999...=1 ? Approach it ever more closely, you'll get an ever better match.
Dividing by infinity is nonsense, as a rule. So we use limit functions and other forms to appraoch it without having to calculate anything actually involving that horrid number.

 

[The Last Conformist]

There's maths and maths. There's this funny thing called arithmetic with extended real numbers, in which dividing and multiplying with infinity is all fine.

 

[Erik Mesoy]

Yes yes yes, that's why I said

Dividing by infinity is nonsense, as a rule.

Most rules have exceptions. I'm trying to keep this within the area of standard maths... the four basic operations and square roots.

 

[TubaGuy]

Once you hit Calculus you have to just accept the fact that n/infinity, where n is any real number, is going to be zero for the purposes of that problem. You have to use that alot to find either Horizontal asymptotes or tanget lines, I can't remember which one at this point. Crazy what a month and a half of not using it can do... No wonder my parents can't remember any of it.

 

[pboily]

Infinity's not all that difficult to comprehend. It's just hard to visualise.

If anyone is looking for an easy way to visualize infinity, there is the stereographic projection of the extended complex plane on the sphere, in which infinity actuallly maps to ... the North Pole! So, depending on your "view point", infinity is no harder to visualize then good ole Santa.

(Of course, there is the little matter of multiplication being hard to visualize on the sphere, but...)

 

[pboily]

I guess I was looking for a philosophical answer, not a mathematical one...

I'm not sure there is a difference.

 

[Perfection]

I'm trying to keep this within the area of standard maths... the four basic operations and square roots.

Your idea standard mathematics doesn't meet my mathematical standards!

 

[Azadre]

Your idea standard mathematics doesn't meet my mathematical standards!

If he includes integrals, derivatives, and logerithms, I'd agree with him.

 

[Perfection]

If he includes integrals, derivatives, and logerithms, I'd agree with him.

Nah, there's a whole lot more that needs to be added.

 

[MSTK]

Infinity is not a number. It's just a word, like "go", "house", etc. The definition is "going on without ending".

But anyways, try and think of infinity in terms of time and space.

If you keep on going in one direction forever and ever and ever, where will you get? Will there be a big giant "wall" to the universe that you can't walk out of? Or will the universe go on forever? If so, then how can something fill all that space? Because there will be no way to fill up the universe. Say that the Big Bang happened. It's just one spec. There could have been a big bang at the same time far away, and the results from both would have never touched. That is...unless the universe is not infinite. For all we know, there is an infinite amount of Big Bangs to fill up this infinite universe. Or maybe you can say that Big Bangs only happen at the center of the universe. But there is no such thing as a center! Maybe you can say that the conditions of a Big Bang are so specific that the odds are so slim of it occurring ever again that only one could happen. Well, you have an infinite amount of space here. Even if the odds were 1/10^10000000000000, there could be that many big bangs (10^10000000000000) happening at once, each creating a little section of the universe that would never touch. This could happen forever, because you can never fill up the universe to the point where there is no room at all. Also consider that there are an infinite amount of big bangs happening every nanosecond, and by the time your brain processes any information, an infinite amount of big bangs and little tiny universes of matter had sprung up.

Now couple that up with the infinite reaches of time. Even if time were not a "dimension", time is just a room for everything to happen in, just like the universe. But expotentially. You're squaring infinity, essentially, by factoring in time as well.

So, no matter how old a "big bang" is, there is always one that is infinitely older. And big bangs will pop up an infinite times a second, because there are an infinite amount of space for them to pop up, and atleast a tiny fraction of infinity will have the right factors. And this will go on for ever and ever. For an "eternity", new universes (stars and stuff) will emerge from big bangs in this giant universe that we all live in, and it will never end. Ever. It will just keep on popping up, an infinite times every second. It has been doing it, forever, as well. No matter how far back in time you go, it will have been popping up these big bangs. Time stretches both ways, too, ya know.

And through an eternity of an infinite amount of big bangs and universes occuring at any giving moment, the universe isn't close to being filled up yet. And it never will be.

Kind of makes you feel kind of insignificant, huh?
I guess you can take my liver now.

 

[WillJ]

30/infinity is pretty much equal to 1/infinity - cancel both sides by thirty. An infinite number of thirties divide into infinity.

Hmm, you're right about that. Then again, couldn't you also say that 30/infinity is 30 times 1/infinity, for obvious reasons?

You can't really do simple calculations involving infinity.

Looking at what I just said, I'd tend to agree.

Regardless - 1/infinity is 0 to an infinite number of significant figures, and it is therefore infinitely accurate to say that 1/infinity equals 0. No?

Yes, I realized before I even brought it up that 1/infinity would have to be .0 repeating and THEN a 1 (which is of course pretty ********), but I was thinking that even though it can't be expressed in numericals, the concept should still exist. I mean, why is it illogical to imagine the concept of a number closer to a certain number than any other number but not quite that number? I suppose that no matter how many mathematical arguments you throw at me, it'll always go against my common sense.

Yep, "Approaches", a key point of many such functions. Let me spell it out:
As a variable N approaches Infinity, its inverse, the fraction 1 over N, approaches Zero.
Approaches is the keyword. N never actually reaches infinity! If it did, our mathematics would collapse.
That's why your cross-product goes wrong too. We never reach 1 or infinity.

I don't see how that can nullify the existance of "infinitely close" but not infinity itself.

*pulls out calculus book* Ah. Here's a good function, showing how you use limits to work out what you cannot calculate directly, much like these infinities and infinitesimals.
Evaluate lim (N->0) of (((sqrt(N+25))-5)/N.
Go on, work it out. If you try to put in N=0, you're dividing by zero. Bad boy.
But look at these values:
For N=1, it is 0,09902.
For N=0,01, it is 0,09999.
For N=0,0001, it rounds to 0,1.
So as N approaches 0, the limit of "(((sqrt(N+25))-5)/N" is one-tenth.

Now, WillJ, see how (1-(1/Infinity))=1, or 0,99999...=1 ? Approach it ever more closely, you'll get an ever better match.
Dividing by infinity is nonsense, as a rule. So we use limit functions and other forms to appraoch it without having to calculate anything actually involving that horrid number.

I'm convinced you just made that up, and are waiting for me to avoid looking stupid and say, "Ah yes, that's interesting," so you can then laugh your ass off at me. Well I'm on to your plan!

 

[Erik Mesoy]

I didn't make it up! Page 41 of the book "Calculus and Analytic Geometry", Edwards and Penney.
Nor do you have to look stupid. This is a common phobia among my classmates- fear of any form of maths that looks complicated.
All that's new is the "limit" which means to calculate it for values of N closer and closer to the limit. There's addition, subtraction, division and a square root in there. Set N=1 and work it out yourself, it's not that hard.

Bad ASCII Art:
     ______
    [U]√(N+25)-5[/U]
        5

It's not really the sort of thing I can make up.

If you really want to get technical on infinity, I'll bring out the aleph-(cardinal) infinities.

I don't see how that can nullify the existance of "infinitely close" but not infinity itself.

We don't have an infinity, really. We have "bigger than anything else, big enough to make the equation work out" for most situations.
For infinitesimals, this becomes "Smaller than anything else" where it's not zero, but not anything above zero either.
Infinitesimals are... zero. It's just not polite to mention the fact, since it's technically not so, but you can't ever get accurate enough for a small enough number.
Infinity is... infinite. We can't ever get an accurately high number.

Ahh, you'll learn about it when you get into advanced maths classes.

 

[Mise]

I mean, why is it illogical to imagine the concept of a number closer to a certain number than any other number but not quite that number?

You hit the nail on the head that time. If infinity was a number, you could add 1 to it and that would be bigger than infinity. In the same way, if a number is infinitely close to another number, but not quite that number, then the difference of those two numbers yields another real number, which could be halved (say) and added to the "infinitely close" number, making THAT number closer. E.g. say A was "infinitely close" to B. A - B = C. D = B + (C/2). D is closer to A than B is, but B is supposed to be infinitely close!

 

[Plotinus]

Descartes distinguished between "conceiving" of something and "imagining" it by saying that we can "conceive" of a chiliagon (a shape with a thousand sides) but we can't imagine it. It's one thing to know what a word means and another to form a mental image of it (indeed, some people, such as the philosopher Gilbert Ryle, are unable to form mental images of anything).