Revision 8140777 of "Term grouping criteria" on simplewiki

[[File:Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg|thumb|A notebook of [[Ramanujan]] talking about the sum of infinite numbers]]
The '''term grouping criterion''' is a mathematical [[slang]] that says that if a [[series]] of infinite [[integer]] terms is written in no particular order, each term could be rearranged or decomposed to form a new series with the terms grouped "in order".

== Uses ==
The criterion was first described following a published article on the series of the sum of infinite [[natural numbers]] and on the relation to the sum of infinite addends one<ref>{{Cite journal|last=Atanes|first=Nicolás|date=2022|title=Casimir effect and sum of infinite natural numbers|journal=Diario 16}}</ref>. 

When "in order" is mentioned, it means that there is a relationship between the terms, either finite or infinite, for example that all the terms are at a [[distance]] of one from the previous and next [[summand]], or that each summand is the [[Mean|mean value]] between the two summands that surround it.

According to this criterion, suppose the following sum: <math>3+9+6=18</math>. Since all the terms are integers, although in disorder, we can apply the criterion to write it "in order". If we add 2 to 3 and subtract two from 9, the terms will be at a distance of 1 from the nearest summand: <math>5+6+7=18</math>.

In the most complex cases, Atanes showed that infinite chaos is impossible, and that the sum of a random distribution of numbers can be rearranged by distancing each number one unit from the nearest addend. You can start from a known sum (the sum of the infinite natural numbers) to modify it, but depending on the criteria, you can always rearrange them.

== In popular culture ==

* The presenter of the Spanish television program Pablo Motos mentioned this topic in a program saying if "[[Order theory|order]] in the abstract is important".

== References ==
<references />
[[Category:Number theory]]