
Symbol for Infinitesimal?
 MathsIsFun
 Administrator
Re: Symbol for Infinitesimal?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 jU
 Real Member
Re: Symbol for Infinitesimal?
 kylekatarn
 Power Member
Re: Symbol for Infinitesimal?
IMHO an infinitesimal is something that converges to 0, for example 1/n>0 if n>+oo
 jU
 Real Member
Re: Symbol for Infinitesimal?
 Jims
 Member
Re: Symbol for Infinitesimal?
8
or
[s]∞[/s]
it works in html ... 'strikeout' ... a horizontal line through the middle of the the text.
Last edited by Jims (20051207 04:21:24)
 Zach
 Super Member

Re: Symbol for Infinitesimal?
Boy let me tell you what: I bet you didn't know it, but I'm a fiddle player too. And if you'd care to take a dare, I'll make a bet with you.
Re: Symbol for Infinitesimal?
 bobbym
 Administrator
Re: Symbol for Infinitesimal?
I think I would go for itty bitty, teeny weeny. That is best described by epsilon.
See my paper entitled "Itty bitty, teeny weeny and epsilon, how small is tiny?"
In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Symbol for Infinitesimal?
Hi bobbym
Where can we find the paper?
The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
 bobbym
 Administrator
Re: Symbol for Infinitesimal?
I have not written it yet. But the title is a killer, a real grabber!
In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Symbol for Infinitesimal?
Question:
Does the mathematics community look at the number line the way I do, thus (I'm only using the positive half, but it also applies to the negative half):
5 basic representations (for lack of a better word) of numbers in a traditional number line (aside from positive/negative):
1. zero (itself) 2. infinitesimal (forever approaching, but never reaching zero) 3. all finite numbers between the infinitesimal and the infinite (including all fractional, irrational and trancendental numbers) 4. infinite (forever approaching, but never reaching infinity) 5. infinity (itself)
I have seen some discussion where people insist that zero and infinitesimal, and infinity and infinite are virtually the same and allow approximations to zero and infinity, but I think that leads to errors in understanding them.
Literal zero and literal infinity should be allowed to stand alone without approximations. They both represent a pure idea.
I understand there is a proposed symbol for the infinitesimal (epsilon  ε). I like it.
Is there also a proposed symbol for the infinite (which, as far as I know is best represented by ∞  ε, not ∞  1)?
==========
Last, this part is interesting about the number line (I wrote this last night):
(Let me know if you think this is off base, or I'm not quite describing the idea I'm trying to convey in the best way.)
Say you have a number line as such (note that there are no arrows on the ends of the line, the ends of the line are in fact  and + infinity):
∞  +∞
Q: where would you put the zero?
A: anywhere you want to in between; infinity has no scale so you don't have to worry about exactly centering it; every possible number in between would crowd adjacent to the zero anyway, making it impossible to plot any other number, no matter how large it was; the other numbers would neither be on top of the zero (equal to it) or away from the zero (visually indicating what portion of infinity they are)
Q: after putting in the zero, can you put in any other numbers?
A: no; it is impossible to put any other number on the line, if you want to avoid imposing a scale; if scale is not important, you can do anything you want; the reason why you cannot put any other number on the line, is that you cannot get close enough to the zero to avoid the problem of dedicating a portion of the line to that number that visually indicates its proportion of infinity  that would not represent the nature of infinity accurately; and of course, you cannot put any number in the same spot as the zero, because you are saying that zero is equal to that number, which would not represent the nature of zero accurately
Q: instead of zero, can I put in a different finite number?
A: yes, but that is the only other number you can put on the line, (again, without imposing a scale)
Last edited by ssybesma (20121010 06:34:55)
 Calligar
 Full Member
Re: Symbol for Infinitesimal?
Why not just make up your own symbol for it. There is no way to represent infinitesimal...1/∞ (contrary to my older belief), doesn't actually make any sense, and this idea has been dropped anyway (can look at things like wikipedia).
Besides for that, to the best of my knowledge, we don't have a character that can truly represent infinitesimals by themselves, we can only really understand the idea of it as the smallest possible number. Making up a character for it could solve that problem, as then we have something to represent it. Just an idea.
Life isn’t a simple Math: 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
Re: Symbol for Infinitesimal?
Calligar wrote:Why not just make up your own symbol for it. There is no way to represent infinitesimal...1/∞ (contrary to my older belief), doesn't actually make any sense, and this idea has been dropped anyway (can look at things like wikipedia).
Besides for that, to the best of my knowledge, we don't have a character that can truly represent infinitesimals by themselves, we can only really understand the idea of it as the smallest possible number. Making up a character for it could solve that problem, as then we have something to represent it. Just an idea.
Epsilon is the proposed symbol for the infinitesmal...along with that, there should be a separate symbol for the infinite, because the infiite is to infinity what the infinitesimal is to zero. Same idea in reverse.
 Calligar
 Full Member
Re: Symbol for Infinitesimal?
Epsilon is the proposed symbol for the infinitesmal
Are you certain? The idea seems arguable, though I wouldn't personally know how to argue it. From what I gathered from it, it seemed more like a different concept, maybe has similarities, but if that's true, I'm not going to even try to argue it. Though I'm not questioning how close it is to an infinitesimal.
along with that, there should be a separate symbol for the infinite, because the infiite is to infinity what the infinitesimal is to zero. Same idea in reverse.
I'm sorry, forgive me if I'm being critical; correct me if I'm wrong. That doesn't even sound like it makes sense, for a number or reasons. Infinity is an arguable opposite to 0 in a different sense, but infinity itself is not a number. Saying that there is a number an infinitesimal away from infinity sounds wrong. I don't know quite what terms are used for it, but that reminds me more of the idea of absolute (by definition in ENGLISH, not math). In otherwords, I don't know quite what that would be called, but what you seem to be talking about sounds like a totally different idea unless I am mistaken somewhere. However, that doesn't quite dismiss the validity of your argument earlier either, which I seemed to have missed....
3. all finite numbers between the infinitesimal and the infinite (including all fractional, irrational and trancendental numbers) 4. infinite (forever approaching, but never reaching infinity)
You keep mentioning infinite as being like....this max number, as I had said earlier, unless I'm mistaken, infinity itself is NOT a number, it is rather an idea of of something limitless or unlimited. However, that doesn't mean that what you are saying either is wrong, the idea itself is still logical, I just think your misunderstanding infinity.
(Please, if I'm wrong about this, tell me! I have spent a LOT of time trying to understand infinity in the past, and I neither want to be misinformed nor misleading someone.)
Life isn’t a simple Math: 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
 MathsIsFun
 Administrator
Re: Symbol for Infinitesimal?
There is also Alephnull, Alephone, etc for size of infinite sets.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
Re: Symbol for Infinitesimal?
True, infinity is not a number...neither is zero. That would be my argument. In order to be a number, it must exist and also be in the finite realm.
Zero has a problem...it doesn't exist...Infinity has a problem...it's not finite.
These ideas work together well in pairs and are opposites of each other:
Zero and infinity  no quantity whatsoever and all quantity
Infinitesimal and infinite  approaching zero and approaching infinity
I roll my eyes everytime I see someone saying ".000 ... 1" equals zero. It makes as little sense as someone saying "999 ..." equals infinity.
The idea of marking the difference between being the extreme and approaching the extreme by allowing those identities are nothing to be ignored, and I think you cannot understand zero and infinity correctly without accepting them.
Re: Symbol for Infinitesimal?
ssybesma wrote:True, infinity is not a number...neither is zero. That would be my argument. In order to be a number, it must exist and also be in the finite realm.
Zero has a problem...it doesn't exist...Infinity has a problem...it's not finite.
These ideas work together well in pairs and are opposites of each other:
Zero and infinity  no quantity whatsoever and all quantity
Infinitesimal and infinite  approaching zero and approaching infinity
I roll my eyes everytime I see someone saying ".000 ... 1" equals zero. It makes as little sense as someone saying "999 ..." equals infinity.
The idea of marking the difference between being the extreme and approaching the extreme by allowing those identities are nothing to be ignored, and I think you cannot understand zero and infinity correctly without accepting them.
There are so many things wrong with this I don't know where to start.
First of all, 0 is a number.
Second, if you include infinity on the number line, including something that approaches infinity makes no sense whatsoever.
Third, I would like to see your definition of this term "infinite".
Fourth, 0.000...01 is not a number. It is not a valid notation. The 1 never comes.
Fifth, 999... is also NaN, but only in the number systems as we know them. Look up padic and 10adic systems.
The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Symbol for Infinitesimal?
If zero is a number then infinity can be a number too.
Ever count to zero?
That's why I say it's not.
====
If zero or any other number are on the line with infinity, you are done. You cannot put any other numbers on the line in their 'proper' location.
I never said anything about putting infinity and the infinitesimal on the number line together...you basically cannot do it.
=====
Definition of infinite: quantity that is dynamic and is constantly approaching infinity (what most people mistakenly think of when they say 'infinity' is really what I define as infinite...infinity is unapproachable, infinite is the futile effort to do that)
=====
I never said that infinitesimal or infinite are numbers...if I did, my pardon, but they are not zero or infinity either...that's exactly why they have their own identity.
=====
When you complicate things unnecessarily, you put off understanding them...so don't ask yourself why things are such a mystery when all you have to do is accept the obvious.
My opinion is that the whole subject has been unnecessarily complicated and possibly some distinctions (such as the idea of infinitesimal and infinite having their own identities separate from zero and infinity) haven't been explored enough.
This idea that zero is a number is misguided. No more or less defined as a number than infinity is but for the opposite reason. Yes, it is routinely placed on the number line because it is the origin (does not mean it's a number as you can easily put infinity on an admittedly lessuseful number line as well). You can also add infinity to the number line with the zero as long as no other numbers are added.
Conversely, you can have a number line that has any finite number on it, plus infinity on it...at that point you cannot place zero on it because it doesn't have a proper location. It will appear to be in the same place as the finite number you put on it if infinity is present on it.
Zero and infinity will sqeeze the other off a proper location on the number line if either of them is included on the number line along with any finite number. Infinity because of the impossible scale, and zero because it starts to appear in the same position as any finite number, which of course it's not.
You should be able to explain accurately why anything I said is wrong if it's not making sense to you. If you can't, maybe you should just accept the simplicity of it.
 bobbym
 Administrator
Re: Symbol for Infinitesimal?
Hi;
Ever count to zero?
That's why I say it's not.
You can count to zero, 10,9,8,7,6,5,4,3,2,1, zero. Blast off! Ever had 5 dollars in your pocket and your bill was 5 dollars. How much is in your pocket now? You just had the store count to zero.
In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Symbol for Infinitesimal?
OK, you are not counting to a finite quantity. You are counting to no quantity. Show me zero dollars and zero cents and then explain why I should accept you have SOME (a finite quantity) of money.
Well, you can't because if you did, it would be at least a penny in this case. Zero dollars is equal to zero apples is equal to zero planet Jupiters. If those things are absolute equals (they are) then zero is not a true number.
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? Battleships? Could be. Does it really matter at that point? No. It matters only if there's a number of something involved, meaning a nonzero quantity.
In order for any of those three things I mentioned NOT to be the same, they have to be a finite quantity, something other than zero. Zero does strange things when you insist it's a number. Why not have infinity be called a number? You can put it on a number line also. I don't think that's the sole justification for zero to be a number.
Like I said, it does have a place on the number line...I'll never disagree why it's there...I just disagee on what it is. It's a very underappreciated entity in mathematics.
Its basic identity is 'diminished' when people insist the infinitesimal is the same.
Re: Symbol for Infinitesimal?
Hi!
The subject of infinity and infinitesimal are definitely troublesome. Here are some ideas.
...................................................................................................................................... 0 1 2 3 4 5...
Suppose the distance twixt 0 and 1 is 1. The dist twixt 1 and 2 is 1/2. The dist twixt 2 and 3 is 1/4. The dist twixt 3 and 4 is 1/8. The distance twixt 4 and 5 is 1/16 and so forth for ever. Then the total distance cannot be 2, but becomes "infinitesimally" close to 2. Call 2 the infinite point; that is, infinity. Then if we wish to include negative numbers in a similiar fashion we have a total length on the "number line" of 4 with minus infinity on one end and plus infinity on the other.
Of course the concept of distance (check out metric spaces) get massacred! Makes for an interesting space.
Suppose + goes with credit and  goes with debt. Then if we have 3 credits and no debts we might picture this as
...........:.........:..........*......> for credits and *...........:............:............:.......... +0 +1 +2 +3 0 1 2 3
Then +0 and 0 mean different things and +0 is not equal to 0
Why do we draw a number line going both ways with zero in the middle and say that +0 equals 0?
Food for thought?
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.
 Calligar
 Full Member
Re: Symbol for Infinitesimal?
What you are saying is most certainly interesting, but I fear you misunderstand certain things. First off, let me clarify this...
Like I said, it does have a place on the number line...I'll never disagree why it's there...I just disagee on what it is.
You pretty much admit there, unless I'm mistaken, that you are not going by the same definition as everyone else. I am just curious, did you really expect people to not argue this if you are going by a different idea in the first place? Now more for the specifics....
I don't disagree with what you are arguing about how both zero and infinity are different then normal numbers, you have proven that thus far..., but zero is already the idea of nothingness, you are already proving something I'm sure most of us know. See, I feel bobbym already answered your question about counting to 0, even in real life, you can keep taking away until you have nothing left, thus you have 0.
Why not have infinity be called a number? You can put it on a number line also.
Infinity ALREADY has its own definition, though it might not be clear. I have already said what it is more....simply put. Infinity itself is not a number because its a completely different concept. For example, can you multiply 1 by +? It doesn't seem to make any sense because they are 2 different concepts. Then you talk about infinite, which is basically the opposite of an infinitesimal, what an infinitesimal is to 0, an infinite is to infinity. So now then, back to the beginning...
You can continue to argue this further because you are using a definition separate then the normal one, but I ask this: are you really using it correctly then? You might have a possible idea that can be used, but whatever it is, it is not infinity, and that is where I think mostly your argument falls apart.
Last edited by Calligar (20121010 14:50:17)
Life isn’t a simple Math: 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
Re: Symbol for Infinitesimal?
ssybesma wrote:If zero is a number then infinity can be a number too.
Why so?
ssybesma wrote:Ever count to zero?
That's why I say it's not.
You can count only natural numbers and you have just proven, though very, very unrigorously, that 0 is not a natural number. Is 3/5 not a number just because you cannot count to it?
ssybesma wrote:If zero or any other number are on the line with infinity, you are done. You cannot put any other numbers on the line in their 'proper' location.
Why not? Putting a number on the line actually doesn't impose a measure.
ssybesma wrote:Definition of infinite: quantity that is dynamic and is constantly approaching infinity (what most people mistakenly think of when they say 'infinity' is really what I define as infinite...infinity is unapproachable, infinite is the futile effort to do that)
That is not a rigorous definition. There is no way of putting that definition in mathematical terms.
ssybesma wrote:When you complicate things unnecessarily, you put off understanding them...so don't ask yourself why things are such a mystery when all you have to do is accept the obvious.
Let me get this straightyou introduced a new idea to a community of mathematicians and yet you are wondering why they aren't accepting it without thinking about it...
ssybesma wrote:This idea that zero is a number is misguided. No more or less defined as a number than infinity is but for the opposite reason. Yes, it is routinely placed on the number line because it is the origin (does not mean it's a number as you can easily put infinity on an admittedly lessuseful number line as well). You can also add infinity to the number line with the zero as long as no other numbers are added.
My first argument still applies. Why would you think that 0 is a number?
ssybesma wrote:Conversely, you can have a number line that has any finite number on it, plus infinity on it...at that point you cannot place zero on it because it doesn't have a proper location. It will appear to be in the same place as the finite number you put on it if infinity is present on it.
This makes no sense whatsoever.
ssybesma wrote:Zero and infinity will sqeeze the other off a proper location on the number line if either of them is included on the number line along with any finite number. Infinity because of the impossible scale, and zero because it starts to appear in the same position as any finite number, which of course it's not.
Again, you seem to be in denial considering 0's realness as a finite number.
ssybesma wrote:You should be able to explain accurately why anything I said is wrong if it's not making sense to you. If you can't, maybe you should just accept the simplicity of it.
Ok, so you are proposing me to, if I don't understand what you are saying, I should just accept it emptymindedly? That makes no sense to me.
The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Symbol for Infinitesimal?
OK, I better start off simple and see if we can agree on five basic ideas.
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.
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.
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.
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).
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).
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.
=====
Just like the rest of you, I don't think I have it all figured out, but I have been thinking about this subject for many years and I tweak things a bit as I find that my definitions aren't good enough to express what I'm trying to say.
Last edited by ssybesma (20121010 23:50:00)
