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#26 2012-10-10 00:13:25

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,390

Re: Symbol for Infinitesimal?

Hi;

Zero is absolutely no quantity, no space, no area, no length (on a number line), etc. It does not 'exist' because it takes up none of those things. That's how I think of zero.

Where is the √2 on the number line? Is it here 1.414.? Or here 1.414213562373095?
Or here 1.414213562373095048801689? Or here 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573?

You can not say that the hypotenuse of a triangle with sides1 and 1 does not exist because you can not find √2 on your number line. What happens between -1  and 1 on your number line? Just a gap?

Zero has a unique characteristic that sets it apart from numbers we normally think of as numbers. One or two apples, ten apples, half an apple, we're still talking apples...zero of something...what are we talking about?

3 is not 3 apples or 3 canaries or 3 battleships. It is just defined as the integer following 2.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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#27 2012-10-10 07:56:16

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: Symbol for Infinitesimal?

OK, I better start off simple and see if we can agree on five basic ideas.

Okay, so I'm curious just how much we agree on this...

Zero is absolutely no quantity, no space, no area, no length (on a number line), etc. It does not 'exist' because it takes up none of those things. That's how I think of zero. Mathematically, zero over infinity.

I might have been inclined to agree with that myself, except for the last part, "zero over infinity."  I do not know quite what you mean by that...but yet again, you seem to be talking about something else.  I will give a further example later on in my explanation, but for now will just say, that does not make any sense mathematically.

Infinitesimal is an entity that exists between nothing and something. Something more than zero, but not enough to be finite or measurable. Zero plus something immeasurably small. The least quantity needed for quantity to exist. Scaled with any finite number (no matter how small) and zero, it would virtually appear to be in the same spot as zero (but of course, is not). Mathematically, one over infinity.

Again, I have a problem with the last thing you wrote, "Mathematically, one over infinity."  I don't know how to emphasize that idea is no longer valid.  If you need proof on that, I can lead you to a number of websites my brother had to take me to to drill that in my head.  This is an OLDER idea of what was thought to be an infinitesimal back then, and is NO longer accepted.  Infinity can NOT be used as such and I feel like if you understood what infinity was, you'd understand WHY this does not make sense.

Finite numbers take up quantity, space, area, length (on a number line), what have you and are limited. It's what we can deal with, see, touch, measure, conceive of, etc. Mathematically, any finite number over any finite number.

I am not 100% sure what you mean by all of this, mostly the part thats says, "It's what we can deal with, see, touch, measure, conceive of, etc."  Besides for maybe the conceiving part, I don't think I can fully agree with all that.  On top of that, just like in my last example, you can not use infinity like that with normal numbers...I do not know how I can stress this anymore.  This is similar to 1/∞, but instead, now your replacing 1 with x.  This is wrong, I already explained that infinity is a different concept, just like in my earlier example...

can you multiply 1 by +?

You just can not do it.  Granted infinity might not be the same as +, but by the WAY you are using it, it still comes to the same idea.

Infinite is an entity that exists between something and everything. Something less than infinity, but too much to be finite or measurable. Infinity minus something. A growing, uncountable quantity that doesn't encompass all quantity. If scaled with zero and infinity on a number line, it would appear virtually in the same spot as infinity. Mathematically, infinity over one (expresses the idea of approaching because the one in the denomnator is countable - you can start counting).

Well again, this is YOUR idea that you have been trying to argue the whole time.  Yet you seem to mostly be repeating the same thing more or less.  I don't really even feel I need to put yet another argument against this, but for arguments sake, I will.  Now so I'm not repeating the same old stuff, let me first talk about this.  Yet again, another fraction that makes no sense with the use of infinity.  Now, just assuming that infinity is a number (breaking the rules here), wouldn't infinity over 1 = infinity, how is that any different then infinity, you fail to prove the difference between infinite and infinity.  So now go back to some of the more ... repeated stuff.  You at least do manage to say infinity itself is not a number, but seem to have an idea that infinite is a number in the first place.  Pretty much using infinite to take the place of infinity.  Though I can see potential uses for all this, why is this so important that you argue this?  What you should do rather is find a way to prove it.  It's interesting because I had just recently made a topic about discovering something new in math.  If you are so heart set as to continue arguing this, why not at least find a way to prove it first, rather then continue this conflict which will likely get you nowhere?

Infinity absolutely encompasses all quantity (that can ever exist), the unlimited universe of quantity that cannot begin to be measured or even approached. Scaled with any finite number (no matter how large) and zero, that number virtually looks like zero (but of course, is not). Mathematically, infinity over zero (expresses the idea of unapproachable, because the zero in the denominator is not countable - you can't even start counting).

There is quite a lot I disagree with here.  First off, the very first sentence you say, unless I'm misunderstanding it, this yet again sounds like your own definition of it.  The second sentence you say, not only do I not agree with that, but you also have a contradiction in there.  Even based on things of what you've been saying, you used 0 in there, since you keep looking at it as fractions, I'm going to do the same: 0/infinity in YOUR definition does NOT equal 0?  I don't feel like reexplaining the first part, because again, you seem to be going by your own definition there.  Lastly, mathematically, based on current rules, that does not even make any sense.  Throughout this whole post, whether or not you think infinity is a number, you continue to use it as one, demonstrating the difference as you say, "Mathematically."  Yet doesn't this seem contradictory to your other idea of infinite?  Either way, I have mostly made my argument; there are many ways to use infinity, wikipedia shows two such examples under real analysis...http://en.wikipedia.org/wiki/Infinity.  I just hope you eventually begin to understand what infinity really is, so as to avoid all this confusion.

I know, breaking some established ideas on the last two. But seems to me there has to be a practical use for "infinity over one" and "infinity over zero", and a way to appreciate a distinction between the two ideas.

In these fractions, one becomes merely a symbol for a finite quantity, because it's being placed next to infinity, which is a symbol...the importance of what finite number is used is not that important, but one is used because it is a compact, perfect representation of finite.

Remember when I said this...

You can continue to argue this further because you are using a definition separate then the normal one,

You are proving my point exactly; you continue to argue this based on the way you see it and try to enforce it upon us without much proof besides for your own reasoning behind it.  Yet you argue an already established idea and try to change the way things are, which I do not think you realize just how big of an impact that would also have.  You take out the current idea and replace it with your own, what happens to everything that is dependent on the previous idea?  Your new idea isn't a replacement for it, and if tried to, would present so many complications I don't even know where to begin.  I just hope this helps a little...

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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#28 2012-10-10 14:08:39

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Symbol for Infinitesimal?

Hi again!

You might check out "infinitesimal calculus" via Google.  Lots of interesting sites.  Robinson introduced
his views in 1960's.  Kiesler has a text about it.  If I recall correctly infinitesimal was introduced as an
infinite sequence that converges to zero.

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#29 2012-10-10 14:15:40

ssybesma
Member
Registered: 2012-05-10
Posts: 8

Re: Symbol for Infinitesimal?

I don't mind shaking things up a bit here.

I don't think that the basic concept of infinity require a masters degee to understand.

And I am certain that there is honest debate and not half the established consensus on some of the things I put forth as you make out to be.

Some things are not 'surface intuitive' and require an imagination. Some things I think are flatly intuitive. Grasping infinity can be incredibly difficult, and is only made easy when you accept it's not a number, but then it can become very easy. But I do know that the idea of infinity is to express unlimited and all-encompassing, so when I see it in relation to numbers, to me it means one simple thing (an impossible end to numbers and the concept of labeling all potential quantity with a name). It can be thought of as a package outside of which can exist nothing, and inside of which exists everything that can ever exist.

The fractions I put forth are nonsensical. They don't pretend to be numbers, and I'm not meaning them to be or claiming them to be numbers. I'm sorry I gave that impression. A number is a finite, measurable quantity (adding further...located between the infinitesimal and the infinite both of which are not numbers either).

Any fraction that has infinity involved with it, by definition cannot be a number. I have no disagreement and I believe I understand why that is for the same reason you do.

I'm trying to adapt a concept to a shorthand way of expressing it because some things seem unnecessarily muddled, possibly by overthinking them.

I do believe I have a very good understanding of infinity, which is not a number.

I disagree that 1/infinity is not infinitesimal...there are probably as many people defending me on that who are much further along in math than I am, than people who are defending the idea it is not. And why doesn't that make sense to you? Even the number 1 when used with infinity is symbolic and not so much a number at that point, because any finite number will do when placed next to infinity (they all appear to converge on zero). Perhaps we can focus on that piece first, since it's a very interesting subject.

I don't believe zero is a number...but I have a slightly different idea of a number in that case.

I think of a number as a existent quantity, so as to distinguish it from a 'quantity of zero'.

Last edited by ssybesma (2012-10-10 15:01:21)

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#30 2012-10-10 15:12:14

ssybesma
Member
Registered: 2012-05-10
Posts: 8

Re: Symbol for Infinitesimal?

noelevans wrote:

Hi again!

You might check out "infinitesimal calculus" via Google.  Lots of interesting sites.  Robinson introduced
his views in 1960's.  Kiesler has a text about it.  If I recall correctly infinitesimal was introduced as an
infinite sequence that converges to zero.

I will.

Someone back then felt it was a useful concept...and I think it has its place and should not have been dismissed.

I don't think you can do away with it. Because we can think of it, maybe it does exist.

It's not zero. It's not a number either (it's not a static value). It's a concept. The idea of a potential '1' to be reached at the end of an endless bunch of decimal zeroes (0.000... )

What's wrong with it? It gives a 'skin' if you will, to the beginning of quantity and distinguishes it from zero, which has perfectly, no potential for quantity.

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#31 2012-10-10 17:30:08

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: Symbol for Infinitesimal?

I don't think that the basic concept of infinity require a masters degee to understand.

And I am certain that there is honest debate and not half the established consensus on some of the things I put forth as you make out to be.

Some things are not 'surface intuitive' and require an imagination. Some things I think are flatly intuitive. Grasping infinity can be incredibly difficult, and is only made easy when you accept it's not a number, but then it can become very easy. But I do know that the idea of infinity is to express unlimited and all-encompassing, so when I see it in relation to numbers, to me it means one simple thing (an impossible end to numbers and the concept of labeling all potential quantity with a name). It can be thought of as a package outside of which can exist nothing, and inside of which exists everything that can ever exist.

http://www.mathsisfun.com/numbers/infinity.html
http://en.wikipedia.org/wiki/Infinity
http://whatis.techtarget.com/definition/infinity

I could continue to send more, but its pointless, just used google, top 3 results (and yes, I did look at them as well).  First off, infinity does not require a masters degree to understand, I don't know where you got that from.  Second off, sure, there IS debate that goes on about infinity, but it is definitely NOT some half established idea with the other half needing to be figured out.  Grasping infinity can be difficult, but it really isn't that hard of a concept to understand once you know it.

I like the mathisfun one so far (which happens to be from this same site), basically describing it as endless, then going more in-depth with it (better then I could explain myself).  Now let me try to see if this is where you have it wrong...

so when I see it in relation to numbers, to me it means one simple thing (an impossible end to numbers and the concept of labeling all potential quantity with a name). It can be thought of as a package outside of which can exist nothing, and inside of which exists everything that can ever exist.

That isn't quite right.  Infinity is NOT an end to anything, regardless of how "impossible" you make it.  It is just the opposite, endless.  Infinity is not meant to be the final number to the number system, a number that every number equals, or a number being a fraction of infinity.  Infinity just represents that it is endless.  Let me give an example (incase you don't want to just look it up): If you are looking at 0.999... the nines go on forever, or infinitely.  This is a correct way of using infinite.

Look at one of the earlier posts by noelevans, he also tries to show this idea with a different example.  I also showed an earlier example when I gave you the wikipedia link and said to look under real analysis (showing an example of proper use w/ an integral and sigma).  Yet, I am pretty sure the majority of people on this topic realize all this too.  Now I'm not going to try to explain it to you in depth, because I honestly don't think I'd do the best job at doing so, while it is already much better explained at other places like here on this website making it more pointless to do so.

I disagree that 1/infinity is not infinitesimal...there are probably as many people defending me on that who are much further along in math than I am, than people who are defending the idea it is not. And why doesn't that make sense to you? Even the number 1 when used with infinity is symbolic and not so much a number at that point, because any finite number will do when placed next to infinity (they all appear to converge on zero). Perhaps we can focus on that piece first, since it's a very interesting subject.

Actually, you might be right about this.  Back when I was working on it with my brother, he had me looking up a ton of stuff about it, and upon it I discovered that this was no longer an accepted usage of an infinitesimal, yet, trying to look it up now, I can see no evidence to this anymore.  Now I didn't blindly look it up either back then, 1 over infinity I'd still argue from a mathematical standpoint still doesn't make sense.  1 IS a real number, yes, but you can NOT divide it by infinity (regardless of whether or not you are looking for an answer), because as I said earlier, infinity itself is not a number.  The reason for this is because of what infinity is.  Now, I actually can not find evidence that this no longer exists anymore, contrary to multiple posts I said earlier.  Therefore, I can not argue you are actually wrong about that and I must apologize about that.  I will continue to look this up farther to see if I can find evidence of this again...

I don't believe zero is a number...but I have a slightly different idea of a number in that case.

Fair enough.  I have no arguments for that in that case.

Last edited by Calligar (2012-10-10 17:37:03)

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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#32 2012-10-12 19:28:17

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: Symbol for Infinitesimal?

Okay, so I would like to clear up something I did in fact have wrong.  I've said numerous times stuff like this...

Again, I have a problem with the last thing you wrote, "Mathematically, one over infinity."  I don't know how to emphasize that idea is no longer valid.  If you need proof on that, I can lead you to a number of websites my brother had to take me to to drill that in my head.  This is an OLDER idea of what was thought to be an infinitesimal back then, and is NO longer accepted.

I would like to note that I was actually wrong about that.  The problem and reason why I thought that was wrong was confusion when looking at definition of infinity.  I am not misunderstanding infinity but rather what exactly others mean by 1/∞.  See, I was looking at this number more mathematically, looking at what infinity is and one is, and thinking that you can not divide 1 by infinity.  However, this more has to be seen from a different view.  It does seem that it is acceptable to use this for an infinitesimal, because you break 1 up endlessly, thus giving you arguably the smallest possible number.  Though, there is debate about this, it is not considered an officially wrong idea either.  So, ssybesma, and to all others whom this might apply to, I would like to say I'm sorry for presenting any confusion about that...

Last edited by Calligar (2012-10-12 19:30:01)

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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#33 2012-10-13 02:20:44

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

Re: Symbol for Infinitesimal?

Well, if you want to have infinity and -infinity in your number system it wouldn't be the integers. You would have to get a set Z*, change the axiom that says that there exists no biggest number and include infinity with the definition that it is greater than any number from the set Z and -infinity with the definition that it is the number less than any number in Z.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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