[talanum1 (OP)]

The title says it all.

[evan_au]

Yes. That's what Integration does (if the area under the curve is finite).
See: https://en.wikipedia.org/wiki/Integral

[alancalverd]

Or for non-mathematicians, a small number of little bits can't add up to very much.

[Bored chemist]

Do a Finite Amount of Infinitesimals Sum to a Finite Amount?
Do you mean a finite amount of them, or a finite number?

[evan_au]

Do you mean a finite amount of them, or a finite number?

Oops! I read it as "Infinite Amount of Infinitesimals".

Adding a "finite number of Infinitesimals" approximates 0 (which is a finite number, last time I checked).

[alancalverd]

Surely an infinitesimal (or at least its modulus) is greater than zero? Not by much, admittedly, but the sum of a lot of them will diverge from zero.

[Deecart]

I am pretty sure that a finite sum of infinitesimal lead to an infinitesimal.

[Eternal Student]

Hi all,

There are  (I think)  2 answers that are reasonable mainstream mathematics.

1.   What do you mean by an "infinitesimal"?    The usual definition is that this is a number with a modulus (size) which is less than any strictly positive number.    This may need some clarification:
   An infinitesimal is a number    x    such that   |x|   <     1/n      for every  natural number n.
     <=>    x is an infinitesimal      if and only if        |x| is less than     1 ,  1/2 ,   1/3,   1/4,  1/5,    1/6,   .... etc....       
   
    If you examine this definition carefully then the only Real number that is an infinitesimal is 0.
   

If you add together a finite number of  zero's   then you do always end up with  0.
As previously mentioned,  there is some ambiguity in your (talanum1) use of the phrase   "a finite amount".   I assume this means a finite number of infinitesimals are added together.
   Anyway, keeping it simple....   In the Real numbers.... there is only one infinitesimal and it's 0.  Add together any finite number of them and you still get 0.    An infinite sum is not defined in the basic axioms of the field of Real numbers.   So the overall answer is quite dull....   there is only one infinitesimal in the ordinary Real numbers and it's 0.    You can't get anything bigger than this by just adding a finite number of them together.


2.      Obviously then 0 is not a very interesting or useful infinitesimal.   Leaving the ordinary Real numbers behind we can construct a new field.   The most common one which turns out to be useful is the field of Hyperreal numbers.   In that you can have infinitesimals that are not just 0.    More information can be found in various places.   Here's Wikipedia's entry:
   https://en.wikipedia.org/wiki/Hyperreal_number

   Using Hyperreal numbers and the infinitesimals they provide will gives us "infinitesimal calculus" exactly as it is commonly used to explain or think about calculus  (integration and differentiation).  It's these sort of ideas they (teachers) would have used to motivate and explain calculus at school.  Non-mathematicians don't need to worry too much about the formal details of the Hyperreal numbers, it just does turn out to let you develop and explain calculus in a very intuitive way.  Developing calculus with the Hyperreal numbers is called "Non-standard Analysis"  even though it is actually the more intuitive way of developing calculus and historically this was how calculus was first thought about.

    As an alternative you can stick solidly and rigorously with the Real numbers and use standard Real Analysis to develop the idea of limits - from which calculus will follow.   This "Standard Analysis" was developed precisely because Mathematicians were never happy about using or assuming the existence of infinitesimals.   So it was developed almost immediately with the public acceptance of calculus.   Non-standard Analysis was pushed to the back and frowned upon as if it was a dumbed-down and good enough for school only explanation of what is happening.  Non-standard Analysis regained some recognition much later, especially due to work of people like Robinson who laid down most of the formal mathematics for the Hyperreal numbers.   At the current time, you have a choice about how you explain calculus, both standard and non-standard analysis are equally valid.

   Anyway.... your (talanum1) question only gets interesting when you use something more than just the ordinary Real numbers.  If you keep everything Real then the only number which is "infinitely small" is 0 and precisely just 0.

Best Wishes to everyone.

[wolfekeeper]

Infinitesimals are generally defined using limits. The limit of a finite sum of infinitesimals tends to zero which is finite.

[paul cotter]

Completely off topic: Eternal Student it's a great pleasure to see your return, I really missed your mathematical exactitudes. I appreciate their was a lack of suitable material for you to intervene.

[Eternal Student]

Thanks Paul.
  I've been doing stuff and things elsewhere.  I hope you and everyone else are well.

[talanum1 (OP)]

Now the question is: do a countably infinite amount of infinitesimals sum to a finite amount not = 0? The area under a curve, as used in integrals, sum over an uncountably infinite amount of infinitesimals.

[Eternal Student]

Hi.

Now the question is: do a countably infinite amount of infinitesimals sum to a finite amount not = 0?

     The word  "sum"  implies the use of the addition + operation.     This is a binary operation defined on the Field.    The phrase "Binary" means that we can add any two numbers and this does map to another number.   It's then quite simple to extend the idea of addition to multiple numbers   1 + 2 + 3     is adding 3 numbers together but we understand it to mean   1+2  (just add two numbers to generate another number)  and then  just  + 3  to that number.     
   Sadly, the basic axioms of a Field   (like the Field of Real numbers) do not give any definition or method to add an infinite set of numbers together.
   We can assign meaning and value to an infinite series (or the sum of infinitely many numbers, if you prefer) but this requires the use of limits.  As such it's in the territory of a branch of mathematics called Real Analysis and not Algebra.

    So, using only algebra, we can say that there's a problem with the question:   You can't add an infinite amount of numbers, such an operation is not defined.

    If we do allow ourselves to use Real Analysis and consider infinite series in the usual way then the answer is - dull if we restrict ourselves to Real numbers only.   If we confine our attention  to the Real numbers only then the only infinitessimal number in the Reals is 0    (discussed in a previous post).  An infinite series of  0 's   added together does converge and it converges to 0 every time.   Sorry, I know that makes for a dull answer.
   When people talk about an "infinitessimal" they are generally implying something infintely small but definitely bigger than 0.  Such a number does NOT formally exist in the ordinary Real Numbers but there are other fields like the Hyperreal numbers where such things do exist and so the topic isn't just abstract philosophy about how numbers should or could be.   The answer above is merely restricting attention to the ordinary Real numbers as they are formally defined and constructed in modern mathematics.

The area under a curve, as used in integrals, sum over an uncountably infinite amount of infinitesimals.

      I'm not sure about that.
      Riemann integration requires and permits only countable (not uncountable) partitions of the function to be considered.   Riemann integration is the simpler method and is probably all that would be taught at school level   (when I was at school we wouldn't even have been told it was Riemann integration we were doing).   However, Lebesgue integration would allow more arbitary partitions including an uncountable collection.   Most simple functions are Riemann integrable and can give us definite integrals that are greater than 0 (so it can't be essential to have an un-countable collection of partitions).

    Anyway, I can see where you were trying to go.  Is it that the sum is over an Un-countable collection of terms that makes the entire sum more than just 0?   No, I don't think so.   Thinking of integration as an infinte sum of infinitessimals is either:   (i)  Just a bit informal,    or else   (ii)  Making use of the Hyperreal numbers.

Best Wishes.
 

[paul cotter]

Indeed, Eternal Student, I only found out this year that "my" integration was Riemann integration, most likely around the time you informed me about Lebesgue integration.