Help on infinitesimal calculation

In summary, the conversation discusses the concept of infinitesimal numbers and their representation in non-standard or Robinsonian analysis. It is suggested to search online for more information on the topic and a comparison is made between infinitesimal numbers and the space of Laurent series in one variable with coefficients in R. The conversation also touches on the set theory implications of this concept and the potential for alternative definitions and representations of infinitesimal numbers. The speaker expresses interest in learning more about the topology and norm of this extended set.
  • #1
seratend
318
1
Hi everybody,

I am trying to get addtionnal data on "infinitesimal numbers" dx. I am not sure about the terminology, I have heard it a long ... long time ago during a lecture (my memory may be wrong, so may be I was sleepind and it was during a dream? :rolleyes: ).
I think (memory) that the definition may be something like "whatever e>o, 0<|dx|<e" where dx is the "object" satisfying such definition (i.e. the "infinitesimal number" definition may be the "smallest" open set in a topology?).

Does anyone can provide some information about this?
Thanks a lot in advance.

Seratend.
 
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  • #2
Non-standard or Robinsonian analysis. Google is your friend. (Your topology thing isn't correct - (dx)^2 is even smaller than dx.)
 
  • #3
matt grime said:
Non-standard or Robinsonian analysis. Google is your friend. (Your topology thing isn't correct - (dx)^2 is even smaller than dx.)

Thanks for this initial input. I will try to get some information. Googling almost always displays the classical differential analysis topics and that is not what I am looking for.
I have an initial look, and I am surprised; I didn't think in going into the hyper real numbers space :yuck: !

matt grime said:
Your topology thing isn't correct - (dx)^2 is even smaller than dx.)

Well, I have already underline that my definition may be not consistent or correct at all (It is all what I can remember of, and my memory is poor).
However, assuming the usual topology on |R, dx in this definition may be viewed as a set rather than a number. In that case, It can be rewriten as the intersection of all the open sets containing x for example. In addtion, I do not assume that this "limit" is unique.

Seratend.
 
  • #4
Your topological interpretation really isn't correct. The intersection of all open sets that contain x is the (closed) one point set that contains x. The infinitesimals are basically the space of (laurent) series in one variable with coeffs in R with the natural lexicographical ordering.
 
  • #5
matt grime said:
Your topological interpretation really isn't correct. The intersection of all open sets that contain x is the (closed) one point set that contains x.

yes, evident for the limit! (I should have deduced it before posting it , it is a shame :blushing: ). So I need to change the definition is case of the topology context.

matt grime said:
The infinitesimals are basically the space of (laurent) series in one variable with coeffs in R with the natural lexicographical ordering.

So do you mean (I am afraid with the "basically") that I have a bijection between the points of the vectorial space (complete space? if there is a norm and what norm?) of the Laurent series and the infinitesimals that comply with the definition (e>o 0<|dx|<e, for any inifinitesimal dx )?
(your lexicographic ordering is first i ordering of zi (i belongs to Z) then the ordering of coefficent ai? - that reminds me something about the ordering pb of the infinitesimals).

Thanks in advance, It helps me a lot.

Seratend.

Now, rediscovering the laurent series.
 
  • #6
I never mentioned anything about convergence or topology or norms - the series are the formal power series (in dx) (with only finitely many non-zero coeffs) so a typical element looks something like:

a+ b(dx) +c(dx)^2

and we know dx<r for all real r

c(dx)^2<b(dx) for all real c and b.

but this is a nonstandard way of thinking about nonstandard analysis (something that I know less than zip about as is rather clear from my posts here) and was just an attempt to describe the underlying set of the surreal numbers, and approximately how they are ordered.
 
  • #7
matt grime said:
I never mentioned anything about convergence or topology or norms - the series are the formal power series (in dx) (with only finitely many non-zero coeffs) so a typical element looks something like:

a+ b(dx) +c(dx)^2

and we know dx<r for all real r

c(dx)^2<b(dx) for all real c and b.

but this is a nonstandard way of thinking about nonstandard analysis (something that I know less than zip about as is rather clear from my posts here) and was just an attempt to describe the underlying set of the surreal numbers, and approximately how they are ordered.

I begin to understand the topic, even if I still not have found a good/simple paper on it. I have only speaken about topology and norms as a possible mapping to introduce this object (through a kind of bijection). However, this is not important, as I have seen, thanks to your initial input, that there are several possibilities to introduce these amazing objects (in order to keep a set).

Ok, you were speaking about the Laurent series as one possible representation of the hyper real numbers (i.e. an hyper real number represented by a countable set of real numbers may be represented by a generic Laurent serie – currently I don’t know if it is a single or a class of series).
I also begin to understand that this extension of the |R set plays with the boundaries of the zermello-fraenkel set theory axioms (we are near the set of all sets).

So I think I can say that my initial definition is somewhat correct (e>0, 0<|dx|<e where e is a real number, < is the ordering relation that is the usual order relation for real numbers). Thus, I have no solution if I consider that dx is a real number, but if I take a "larger" set, well constructed in order to get a set (ZF axioms), I can have a solution (in fact I understand an infinite, non countable number of solutions).

Do you know some interesting properties about the topology and the norm we can build on this new set (as an extension of the usual topology and norm on the |R set)? (I am still looking for a good and compact lecture about this topic).

Seratend.

Trying to swim across the hyper real number space :eek: .
 

What is infinitesimal calculation?

Infinitesimal calculation is a mathematical technique used to study and analyze infinitesimally small quantities. It involves using limits and approximations to find the behavior of a function as its input approaches a certain value.

What are some common applications of infinitesimal calculation?

Infinitesimal calculation is commonly used in physics, engineering, and economics to model and analyze systems with continuously changing variables. It is also used in calculus to find derivatives and integrals of functions.

How does infinitesimal calculation differ from regular calculus?

Infinitesimal calculation is a subset of calculus that focuses on the study of infinitesimally small quantities, whereas regular calculus deals with larger quantities. Infinitesimal calculation requires the use of limits and approximations, while regular calculus involves finding exact values.

What are some challenges in performing infinitesimal calculation?

One of the main challenges in infinitesimal calculation is dealing with the concept of infinity and infinitesimals, which can be difficult to visualize and comprehend. Another challenge is finding the appropriate limits and approximations to accurately model a given system.

Can infinitesimal calculation be applied to real-world problems?

Yes, infinitesimal calculation is commonly used to solve real-world problems in physics, engineering, and economics. It allows for more accurate and precise modeling of continuously changing systems, making it a valuable tool in many fields of study.

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