[Alberto]

Can someone explain to me what exactly an expression (they call it "division") like this means?

	a/b

[Latro]

If you're serious, the algebraic definition is that a/b is the solution c to the equation
c*b = a
where * is multiplication.

[CrudeHypothesis]

^that, or:

a/b means how many of b fit into a

[Monte314]

More precisely, the term Antanairesis (variously spelled by the ancients) refers to the method of successive subtraction for computing the ratio of two integers. Today we refer to this as the Euclidean Algorithm (though it was developed earlier by the Pythagoreans).

Modern Abstract Algebra has an area of study called Ring Theory, which includes a collection of structures called Euclidean Domains (ED's) in which this algorithm can be employed (due to the existence of a "Euclidean function").

The Euclidean Algorithm can be used to show that all Ideals in an ED are Principal, so that factorization into primes is unique up to the ordering of the factors. This is essential to the existence of a rich "number theory" in the Ring. These ideas are theoretically important, and also have significant applications in cryptography, coding and compression, and signal processing.

[PlatoHagel]

I found
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quite interesting.

[Monte314]

"this" is OK, I suppose. I'm into "that" myself.

[Alberto]

3*2
***
***

2*3
**
**
**

The two expressions are not really equivalent, though we may be used to deem them such: the former may mean we distribute 3 by 2 dimensions.
The latter we distribute 2 by 3 dimensions.

Division means once again to distribute a number N of items along as many Dimensions as number D prescribes:
N\D

The operation has the same intention of a multiplication, but is more ambiguous.

N/D
does that mean
(1) overall/items=dimensions to allocate
or
(2) overall/dimensions to allocate=items?

These operations are implicitly creating dimensions to allocate.
At this point, if you see the thing in this perspective, you can envision dozens of new mathematical operations (of which the "standard" division and multiplication are just one instance, I have no idea whatsoever why the one preferred), depending on how the starting setting "decides" to create and allocate such dimensions.

If you consider X as the amount of elements, Y as the amount of Dimensions, and R as the rounds you repeat the process, for each round you could have different ways to calculate the current X and Y:

(1) You derive the next X (elements) from the current Y (dimensions) amount
taking in only the first entry of the whole Y process
taking in only the last produced entry of the whole Y process
taking in the sum of all the previous entries of the whole Y process
taking in only 1
----------------------------------------------
(2) You derive the next X (elements) from the current X (elements) amount
taking in only the first entry of the whole X process
taking in only the last produced entry of the whole X process
taking in the sum of all the previous entries of the whole X process
taking in only 1
----------------------------------------------
(3) You derive the next Y (dimensions) from the current Y (dimensions) amount
taking in only the first entry of the whole Y process
taking in only the last produced entry of the whole Y process
taking in the sum of all the previous entries of the whole Y process
taking in only 1
----------------------------------------------
(4) You derive the next Y (dimensions) from the current X (elements) amount
taking in only the first entry of the whole X process
taking in only the last produced entry of the whole X process
taking in the sum of all the previous entries of the whole X process
taking in only 1

---------- Post added 06-09-2012 at 11:37 AM ----------

ps please keep in mind that a division is no nuisance: it is the first mathematical operation that the Biblical demiurge used. And, indeed, it did not divide, but it... multiplicated!
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[PlatoHagel]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	"this" is OK, I suppose. I'm into "that" myself.

okay, okay what you wrote lead me to "that"....."this" could be the subject in context of the history of and the question in OP. So with your full monte you count.
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