The ultimate numbers and the 3/2 ratio
The Ultimate Numbers and the 3/2 Ratio
Abstract. 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.
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 thus 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
2.3.1 Other definitions
Let n be a whole number (belonging to ℕ*), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.
Let n be a natural whole number (belonging to ℕ*), this one is non-ultimate if at least one divisor (whole number) lower than its value and other than 1 divides it.
Below are listed, to illustration of definition, some of the first ultimate or non-ultimate numbers defined above, especially particular numbers zero (0) and one (1).
- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it. - 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it. - 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it. - 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *. - 6 is non-ultimate: the number 6 admits numbers 2 and 3 (numbers being less than it) as divisors *. - 7 is ultimate: since the division by 0 has no defined result, the number 7 does not admit any divisor* being less than it. The non-trivial divisors 2, 3, 4, 5 and 6 cannot divide it into whole numbers. - 12 is non-ultimate: the number 6 admits numbers 2, 3, 4 and 6 (numbers being less than it) as divisors*.
Thus, by these previous definitions, the set of whole numbers is organized into these two entities:
- the set of ultimate numbers, which is the fusion of the prime numbers sequence with the numbers 0 and 1.
- the set of non-ultimate numbers identifying to the non-prime numbers sequence, deduced from the numbers 0 and 1.
- non-trivial divisor.
2.4 Conventional designations
As "primes" designates prime numbers, it is agree that designation "ultimates" designates ultimate numbers. Also it is agree that designation "non-ultimates" designates non-ultimate numbers. Other conventional designations will be applied to the different classes or types of whole numbers later introduced.
2.5 The ultimate numbers and the decimal system
It turns out that the tenth ultimate number is the number 19, a number located in twentieth place in the sequence of the whole numbers. This peculiarity undeniably links the ultimate numbers and the decimal system. So the first twenty numbers (twice ten numbers) are organized into different 1/1 and 3/2 ratios according to their different attributes.
By the nature of the decimal system, as shown in Figure 1, the ten digit numbers (digits confused as numbers) are opposed to the first ten non-digit numbers by a ratio of 1/1. Also, there are exactly the same quantity of ultimates and non-ultimates among these twenty numbers, so ten entities in each category. In a double 3/2 value ratio, six ultimates versus four are among the ten digit numbers and six non-ultimates versus four are among the first ten non-digit numbers.
As shown in Figure 2, it is also possible to describe this arithmetic phenomenon by crossing criteria. Thus, the first ten ultimates are opposed to the ten non-ultimates by a 1/1 value ratio. Also, there are exactly the same quantity of digit numbers and non-digit numbers among these twenty numbers. In a twice 3/2 ratio, six digits versus four are among the ten ultimates and six non-digits numbers versus four are among the first ten non-ultimates.
Technical remark: due to a certain complexity of the phenomena presented and to clarify their understanding, no figure (table) has a title but just a legend in this paper.
2.6 The twenty fundamental numbers
Whole numbers sequence is therefore initialized by twenty numbers with symmetrically and asymmetrically complementary characteristics of reversible 1/1 and 3/2 ratios. This transcendent entanglement of the first twenty numbers according to their ultimate or non-ultimate nature (ultimate numbers or non-ultimate numbers) and according to their digit or non-digit nature (digits or non-digit numbers) allows, by convention, to qualify them as "fundamental numbers" among the whole numbers set. Figure 3 describes the total entanglement of these twenty fundamental numbers.
Thus, the set of the first twenty whole numbers is simultaneously made up to a set of twenty entities including ten ultimate numbers and ten non-ultimate numbers and to a (same) set of twenty entities including ten digit numbers (10 digits ) and ten non-digit numbers (not digits). Also, each of these four entangled subsets of ten entities with their own properties opposing two by two in 1/1 value ratio is composed of two opposing subsets in 3/2 value ratio according to the mixed properties of its components. This set of the first twenty numbers is defined as the set of fundamental numbers among the whole numbers. So it is agree that designation "fundamentals" designates these twenty fundamental numbers previously defined.
2.7 The thirty initial numbers
Also, according to the progressive consideration of three sets of 10, 20 and then 30 entities (the first thirty whole numbers), the ratio between the ultimate and non-ultimate numbers increases from 3/2 (10 numbers) to 1/1 ( 20 numbers) then switches to 2/3 (30 numbers). Thus (Figure 4), depending on whether we consider the first ten, the first twenty and then the first thirty whole numbers, 6 ultimates are opposed to 4 non-ultimates, then 10 ultimates are opposed to 10 non-ultimates then finally 12 ultimates are opposed to 18 non-ultimates. Beyond this triple set, no similar organization of (consecutive) groups of ten entities takes place. These thirty numbers are therefore here called "initials" among the set of natural numbers.
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.
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