Question regarding iteration and operations

[Nano-Passion]

Just wanted to clarify..

The operation of the Riemann Sum is addition. It can also be used to signify subtraction by adding negative numbers. We also have an iteration of multiplication by using exponents. For example, 2^3 = 2 * 2 * 2. But even then the complexity doesn't reach to that accomplished behind the formalism of the Riemann Sum.

I was thinking.. do we have something akin to the Riemman sum for the operation of division? I know it is tempting to say negative exponents, but is there research regarding the notation akin to the complexity reached by the Riemann sum?

 

[Char. Limit]

...are you and I thinking about the same Riemann Sum?

 

[DivisionByZro]

As hard as I tried, I couldn't make sense of your post.

 

[Nano-Passion]

Sorry, I don't mean the same Riemann Sum used for practical purposes [finding the area of something.] I was talking about something more abstract and with less practical importance; That is to use iteration of multiplication or division instead of iteration of additions.

Again, this has nothing to do with the Riemann Integral.

Did that clear things up?

 

[Char. Limit]

Well, you ask about division, but since division is really just inverse multiplication, you can do an iteration of division of numbers by just doing an iteration of reciprocals of those numbers. I think.

 

[Nano-Passion]

Well, you ask about division, but since division is really just inverse multiplication, you can do an iteration of division of numbers by just doing an iteration of reciprocals of those numbers. I think.

Well I was wondering if there was any research behind this. I'm not speaking of a simple iteration, but something building up to the complexity and prowess of "summation formulas." Again, purely for abstract purposes. I can't think of any practical importance to this, which might explain if there was no work or effort behind this.

 

[DivisionByZro]

I still can't make much sense of your post. Are you referring to the product/sum of sequences? As Char said, you can always divide by multiplying by the inverse (if it exists). Perhaps by iterated you mean the double product of a sequence? Something else?

 

[Nano-Passion]

I still can't make much sense of your post. Are you referring to the product/sum of sequences? As Char said, you can always divide by multiplying by the inverse (if it exists). Perhaps by iterated you mean the double product of a sequence? Something else?

Iterated means to do something repeatedly. Iteration is the repetition of something.
I was wondering if someone ever researched the use of repeated division instead of repeated sum and put it into a rigorous framework such as Riemann once did. Let me give you an example:

Iteration from i = 1 to n

Ʃc = cn

Here in the most basic sense, the repeated addition can be simply thought of as multiplication. For example, 1(3) = 1+1+1

But here is where some complexity starts taking shape.

Ʃi = $\frac{n(n+1)}{2}$

Here we have the addition of numbers increasing by one each time.

Now I was wondering if there was a framework for repeated division. Again, don't look at it from any practical application point of view. This is probably where the confusion stemmed.