THE ULTIMATE NUMBERS AND THE 3/2 RATIO
Jean-Yves BOULAY
According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.
1. Introduction
This study invests the whole numbers* set and proposes a mathematical definition to integrate the number zero (0) and the number one (1) into the so-called prime numbers sequence. This set is called the set of ultimate numbers. The study of many matrices of numbers such as, for example, the table of cross additions of the ten digit-numbers (from 0 to 9) highlights a non-random arithmetic and geographic organization of these ultimate numbers. It also appears that this distinction between ultimate and non-ultimate numbers (like also other proposed distinctions of different classes of whole numbers) is intimately linked to the decimal system, in particular and mainly by an almost systematic opposition of the entities in a ratio to 3/2. Indeed this ratio can only manifest itself in the presence of multiples of five (10/2) entities. Also, it is within matrices of ten times ten numbers that the majority of demonstrations validating an opposition of entities in ratios to value 3/2 or /and value to 1/1 are made.
* In statements, when this is not specified, the term "number" always implies a "whole number". Also, It is agreed that the number zero (0) is well integrated into the set of whole numbers.
2. The ultimate numbers
2.1 Definition of an ultimate number
Considering the set of whole numbers, these are organized into two sets: ultimate numbers and non-ultimate numbers.
Ultimate numbers definition:
An ultimate number not admits any non-trivial divisor (whole number) being less than it.
Non-ultimate numbers definition:
A non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Note: a non-trivial divisor of a whole number n is a whole number which is a divisor of n but distinct from n and from 1 (which are its trivial divisors).
2.2. The first ten ultimate numbers and the first ten non-ultimate numbers
Considering the previous double definition, the sequence of ultimate numbers is initialized by these ten numbers:
|
0 |
1 |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 |
Considering the previous double definition, the sequence of non-ultimate numbers is initialized by these ten numbers:
|
4 |
6 |
8 |
9 |
10 |
12 |
14 |
15 |
16 |
18 |
3. Addition matrix of the ten digits
The table in Figure 5 represents the matrix of the hundred different possible sums of additions (crossed) of the ten digit numbers (from 0 to 9) of the decimal system (ie the first ten whole numbers). Within this table operate multiple singular arithmetic phenomena depending on the ultimate or non-ultimate nature of the values of these hundred sums and their geographic distribution including mainly various 3/2 value ratios often transcendent.
3.1 Sixty versus forty numbers: 3/2 ratio
Among these hundred values, there are 40 ultimate numbers (5x → x = 8) and consecutively 60 non-ultimate numbers (5y → y = 12). These two sets therefore oppose each other in an exact 2/3 value ratio.
This page is in work, please consult the complet article here:
The ultimate numbers
The ultimate numbers (788.7 Ko)
https://www.researchgate.net/publication/339943634_The_ultimate_numbers_and_the_32_ratio
Site officiel Les nombres ultimes et le ratio 3/2
