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New whole number classification and the 3 to 2 ratio

The four whole number subsets

Jean-Yves Boulay

National Education - FRANCE

jean-yves.boulay2@ac-nantes.fr, https://www.researchgate.net/profile/Jean-Yves-Boulay .

ORCID: 0000-0001-5636-2375

MSC classification: 11A41-11R29-11R21-11C20-11A67

Keywords: whole numbers, prime numbers, set theory, inclusion, triangular number.

Abstract: According to new mathematical definitions, the set (ℕ) of whole numbers is subdivided into four subsets (classes of

numbers), one of which is the fusion of the prime numbers sequence with number zero and number one. This subset, at the first

level of complexity, is called the set of ultimate numbers. Three other subsets, of progressive level of complexity, are defined

since the initial definition isolating the ultimate numbers and the non-ultimate numbers inside the set ℕ. The interactivity of

these four classes of whole numbers generates singular arithmetic arrangements in their initial distribution, including exact

ratios of 3 to 2 or 1 to 1 as value.

1. Introduction

The concept of number ultimity has already been introduced in the article "The ultimate numbers and the 3/2 ratio, just two

primary sets of whole number" [1] where many singular arithmetic phenomena are presented in relation to two primary sets of

numbers: ultimate numbers and non-ultimates numbers. This new article describes more fully how the set ℕ (of whole

numbers) can be organized into subsets with arithmetic properties proper and unique but also, simultaneously interactive and

inclusive.

From the concept of ultimate number, we propose a classification of all whole numbers into four sets. Thus, each of the whole

numbers, according to its arithmetic properties of a more or less high level of complexity, is clearly classified in one of these

four sets, subsets of the set ℕ. This classification unambiguously includes the exotic numbers 0 (zero) and 1 (one).

This new classification also introduces new mathematical notions: fundamental numbers, primordial numbers, extreme or

median class of number. These different concepts are widely demonstrated by the study of numerous closed matrices equal to

5x entities as size. These matrices are always composed of first sequences of numbers from the set ℕ since it is in its initial

constitution that singular arithmetic phenomena are revealed regarding the distribution of the different types of numbers

considered.

According to multiple approaches and demonstrations, it turns out that in its initial part, the different identified components of

the set ℕ are singularly organized in a 3/2 ratio (or reverse) as reciprocal and/or transcendent magnitude. These transcendences

often operate by integrating the concepts of remarkable identity and triangular number.

At the end of the article, a glossary lists the new mathematics concepts introduced in this paper.

In statements, when this is not specified, the term "number" always means "whole number". It is therefore agreed that the

number zero (0) is well integrated into the set of whole numbers (ℕ) .

2. Ultimity concept depiction

Using a definition with inescapable characters and not subject to any ambiguity, we introduce here the concept of ultimity that

a whole number can possess. Recalling the notion of prime number, we then introduce a compact and absolute definition

splitting the set ℕ into two primary sets. A development, taking the very first numbers as an example, clarifies this innovative

concept which turns out to be of a strong simplicity.

2.1. Prime number definition

In mathematical literature the definition of primes looks like this:

“Prime numbers are numbers greater than 1. They only have two factors, 1 and the number itself. This means these numbers

cannot be divided by any number other than 1 and the number itself without leaving a remainder.”

This is often supplemented by:

“Numbers that have more than 2 factors are known as composite numbers.”

These definitions supposed to describe and classify the whole numbers immediately present two great ambiguities since they

are embarrassed by the two singular numbers that are zero (0) and one (1).

2.2. New definition approach

The conventional definition applied to define whether a number is prime or not does not specify the character of inferiority of

the divisors. This seems obvious, trivial. However, with regard to the particular numbers that are zero and one, this notion has

all its importance. Indeed, the number zero, the first of the whole numbers, has many divisors. But these dividers are all

superior to it in value. Also, number one, second whole number, has not divisor inferior to it because it cannot be divided by

zero, the one which is inferior to it. This novel approach to the concept of divisibility of numbers which includes the notion of

inferiority (and consequently the notion of superiority) of divisors allows the creation of two unique sets where all whole

numbers can be referenced.

2.3. Absolute definitions

Considering the set of all whole numbers (ℕ), these are organized into two primary sets: ultimate numbers and non-ultimate

numbers.

Ultimate number definition: An ultimate number admits at most one divisor being inferior to it in value.

Non-ultimate number definition:

A non-ultimate number admits more than one divisor being inferior to it in value.

2.4. Conventional designations

As "primes" designates prime numbers, it is agree that appellation "ultimates" designates ultimate numbers. Also it is agree

that appellation "non-ultimates" designates non-ultimate numbers. So, the concept introduced here is therefore called that of

ultimity of whole numbers. Otherwise transcendent conventional appellations will be applied from these two different classes

of whole number just defined, in particular with introduction of four subsets of whole numbers. At the end of this paper, a

glossary lists all the main number concepts introduced in this article about the whole number set.

2.5. Expended definitions

Let n be a whole number (belonging to ℕ), this one is ultimate if at most one divisor being inferior to it in value

divides it.

Let n be a whole number (belonging to ℕ), this one is non-ultimate if more than one divisor being inferior to it in

value divides it.

2.6. Abbreviated definition

It is therefore possible to classify very clearly and unequivocally all whole numbers according to ultimity concept. Figure 1

summarizes the process of identifying any whole number that can only be either ultimate or non-ultimate.

Let n be a whole number

⬋ ⬊

if this one admits

at most one divisor

being inferior to it…

if this one admits

more than one divisor

being inferior to it…

⬋

⬊

this one is an ultimate

this one is a non-ultimate

Figure 1: Process of identifying any whole number according to ultimity concept.

This ultimity or non-ultimity identification mechanism is universal for all the sequence of whole numbers starting with the

number zero.

3. The four classes of whole numbers

The segregation of whole numbers into two sets of entities qualified as ultimate and non-ultimate is only a first step in the

investigation of this type of numbers. Here now is a further exploration of this set of numbers revealing its organization into

four subsets of mathematics entities with their own but interactive properties.

3.1. Four unlike types of whole numbers

From the definition of ultimate numbers introduced above, it is possible to differentiate the whole number set into four final

classes, inferred from the two primary classes and progressively defined according to these criteria:

Whole numbers are subdivided into these two categories:

- ultimates: an ultimate number admits at most one divisor being inferior to it in value.

- non-ultimates: a non-ultimate number admits more than one divisor being inferior to it in value.

Non-ultimate numbers are subdivided into these two categories:

- raiseds: a raised number is a non-ultimate number, power* of an ultimate number.

- composites: a composite number is a non-ultimate and non-raised number.

Composite numbers are subdivided into these two categories:

- pure composites: a pure composite number is a non-ultimate and non-raised number not admitting raised number as

divisor.

- mixed composites: a mixed composite number is a non-ultimate and non-raised number admitting at least one raised

number as divisor.

*It is implied an integral power and greater than 1.

3.1. Degree of complexity of number classes

The table in Figure 2 summarizes these different definitions. It is more fully developed in Figure 7 in Chapter 6 where the

interactions of the four classes of whole numbers are highlighted.

The whole numbers:

ultimates:

non-ultimates:

an ultimate number

admits at most one

divisor being inferior

to it in value

a non-ultimate number admits more than one divisor being inferior to it in value

raiseds:

composites:

a raised number is

a non-ultimate number,

power of

an ultimate number

a composite number is a non-ultimate and non-raised number

pure composites:

mixed composites:

a pure composite number is a

non-ultimate and non-raised

number not admitting raised

number as divisor

a mixed composite number is a

non-ultimate and non-raised

number admitting at least one

raised number as divisor

level 1

level 2

level 3

level 4

degree of complexity of the final four classes of numbers

Figure 2: Classification of whole numbers from the definition of ultimate numbers (see Figures 5 and 7 also).

4. New whole number classification

We now propose a clear differentiation of all components of the set ℕ into four well-defined number classes. Also, this

classification unequivocally ranks the exotic numbers zero (0) and one (1).

4.1. The four subsets of whole numbers

By the previous definitions and demonstrations, we propose the classification of the set of whole numbers into four subsets or

classes of numbers:

- the ultimate numbers called ultimates (u),

- the raised numbers called raiseds (r),

- the pure composite numbers called composites (c),

- the mixed composite numbers called mixes (m).

4.1.1. Conventional denominations

So it is agree that designation "ultimates" designates ultimate numbers (as "primes" designates prime numbers). Also it is agree

that designation "raiseds" designates raised numbers, designation "composites" designates pure composite numbers and

designation "mixes" designates mixed composite numbers. It is also agreed that is called u an ultimate number, r a raised

number, c a pure composite and m a mixed composite number.

4.2. Organization charts of whole numbers

This new classification of whole numbers requires some other illustrations of the organization of the ℕ set.

4.2.1. Hierarchical organizational chart

Thus this set ℕ can be described by a hierarchical organization of its components. At the end of the hierarchy are the four new

classes of numbers previously introduced. Figure 3 illustrates this organization.

whole numbers

ultimates

non-ultimates

raiseds

composites

pure

composites

mixed

composites

Figure 3: Hierarchical classification of whole numbers since the definition of ultimate numbers.

4.2.2. Inclusive diagram

Also, as illustrated in Figure 4, an inclusive structure is revealed in the organization of the set ℕ.

Thus the set of whole numbers contains the set of ultimates and that of non-ultimates, the set of non-ultimates contains the set

of raiseds and that of composites, this latter set contains the one of pure composites and that of mixed composites.

Figure 4: Inclusive (Euler's) diagram of the classification of whole numbers.

Conversely, can we conclude that set of the mixed composites is therefore included in that of the composites, this one latter

being included in that of the non-ultimates, itself included in set of the whole numbers. Set of the pure composites is found in

the same inclusions. Set of the raiseds is included in that of the non-ultimates, this one latter being included in set of the whole

numbers. Finally, set of ultimates is only included in that of whole numbers. The table in Figure 5 summarizes this inclusive

organization of the set of whole numbers.

whole numbers set (ℕ)

⊇

ultimates set

⊇

raiseds set

⊇

pure composites set

non-ultimates set

composites set

mixed composites set

Figure 5: Inclusion of the seven sets of numbers constituting the set of whole numbers.

5. Ultimate divisor

Distinction of whole numbers into different classes deduced from the definition of ultimate numbers allows us to propose the

double concept of ultimate divisor and ultimate algebra.

5.1 Ultimate divisor: definition

An ultimate divisor of a whole number is an ultimate number less than this whole number and non-trivial divisor of

this whole number.

For example the number 12 has six divisors, the numbers 1-2-3-4-6-12, but only two ultimate divisors: 2 and 3. Also, the

numbers zero (0) and one (1), although definite numbers as ultimate, are never ultimate divisors. As a reminder, the division by

zero (0) is not defined and therefore this number is not an ultimate divisor. The number one (1) is a trivial divisor, it does not

divide a number into some smaller part.

5.2 Concept of ultimate algebra

The ultimate algebra applies only to the set of whole numbers and is organized, on the one hand, around the definition of

ultimate divisor (previously introduced), on the other hand around the definition of ultimate number (previously introduced).

This algebra states that any whole number is either an ultimate number having no ultimate divisor, or a non-ultimate number

(which can be either a raised, or a pure composite, or a mixed composite) breaking down into several ultimate divisors. In this

algebra, no whole number x can be written in the form x = x × 1 but only in the form x = x (ultimate) or in the form x = y × y

×… (raised) or x = y × z ×… (composite) or x = (y × y ×…) × z ×… (mixed). Also in this algebra, it is not allowed to write for

example 0 = 0 × y × z × ... but only 0 = 0.

5.3. Ultimate divisors and number classes

The table in Figure 6 synthesizes the four interactive definitions of the four classes of whole numbers by incorporating the

double concept of ultimate divisor and ultimate algebra.

An ultimate number (u) admits at most one

divisor being inferior to it in value.

Can be written in the form:

A raised number (r) is a non-ultimate

number, power of an ultimate number.

Can be written in the form:

u = u

→ without

ultimate divisors

u

→

r

r = u × u

→ only with

identical ultimate divisors

↓

↓

A pure composite number (c) is a non-

ultimate and not raised number admitting no

raised number as divisor.

Can be written in the form:

A mixed composite number (m) is a non-

ultimate and not raised number admitting at

least one raised number as divisor

Can be written in the form:

c = u × u’

→ only at different ultimate divisors

c

→

m

m = u × u × u’ = r × u’ = c × u

→ simultaneously at

identical ultimate divisors and at

different ultimate divisors

Figure 6: Interactions of the four classes of whole numbers. See Figures 2, 3 and 7 also.

Crosswise to the hierarchical or inclusive organizations (illustrated in Figures 2 and 3) of the different sets of whole numbers,

the four final natures of numbers therefore also have a linear and semi-circular interaction. Thus, illustrated in Figure 7, is it

possible to oppose the two classes of ultimate (u) and mixed (m) numbers to the two classes of raised (r) and composite (c)

numbers and to qualify respectively these two groups as classes extreme and median.

extreme nature class

of ultimates (u)

median nature class

of raiseds (r)

u = u

u

→

r

r = u × u

↓

↓

c = u × u’

c

→

m

m = u × u × u’

= r × u’ = c × u

median nature class

of composites (c)

extreme nature class

of mixes (m)

Figure 7: Nature and interactions of the four classes of whole numbers. See Figure 6 also.

Remark: the possible writing composition of the number classes in the tables of Figures 6 and 7 is an sample of minimum

writing. As example, for composites, it can be written c = u × u’ × u’’, for mixes it can be written m = u × u × u’ × u’.

5.4. Specific features of the numbers zero and one

By these postulates proposing a concept of ultimate algebra, it is agreed and recalled that although defined as ultimate

numbers, the numbers zero (0) and one (1) are neither ultimate divisors, nor composed of ultimate divisors. Thus we can say

that although these two notions (concepts) are defined differently, the set of ultimate divisors and that of prime numbers are

therefore confused.

→ In appendix, the concept of ultimate divisor is developed in the study of the matrix of the first hundred numbers.

5.4.1. Prime number definition

From the definition of ultimate number and the concept of ultimate divisor previously introduced, we propose a compressed

definition of called prime numbers:

a prime number is an ultimate number, which can be a ultimate divisor of a whole number.

6. The forty primordial numbers

The new classification of whole numbers in four subsets generates singular arithmetic phenomena in the initial distribution of

the different sets of numbers considered. These phenomena result into varied and very often transcendent ratios of exact value

3/2 (or / and reversibly of value 2/3). The prime and initial organization of the set of whole numbers highlights four times ten

numbers that we will qualify as "primordial".

The first 10 whole numbers: 0 1 2 3 4 5 6 7 8 9

6 ultimate

0 1 2 3 5 7

← 3/2 ratio →

4 non-ultimate:

4 6 8 9

↓

The first 10 non-ultimates: 4 6 8 9 10 12 14 15 16 18

4 raised:

4 8 9 16

← 2/3 ratio →

6 composite:

6 10 12 14 15 18

↓

The first 10 composites: 6 10 12 14 15 18 20 21 22 24

6 pure:

← 3/2 ratio →

4 mixed:

0 1 2 3 5 7

← 3/2 ratio →

4 8 9 16

← 2/3 ratio →

6 10 14 15 21 22

← 3/2 ratio →

12 18 20 24

11 13 17 19

← 2/3 ratio →

25 27 32 49 64 81

← 3/2 ratio →

26 30 33 34

← 2/3 ratio →

28 36 40 44 45 48

3/2 ratio

2/3 ratio

3/2 ratio

2/3 ratio

The first 10

ultimates

The first 10

raiseds

The first 10

pure composites

The first 10

mixed composites

10 mixed composites

20 composites (pure or mixed)

30 non-ultimates

The 40 primordial numbers

Figure 8: From the first ten numbers of the three source classes of whole numbers, generation inside 3/2 ratios of the first ten numbers of

each of the four final number classes: the 40 primordials. See Figure 2, Figures 6 and 9 also.

6.1. Whole number classes and ratio 3 to 2

The progressive differentiation of source classes and final classes of whole numbers is organized (Figure 8) into a powerful

arithmetic arrangement generating transcendent ratios of value 3/2. Thus, the source set of whole numbers includes, among its

first ten numbers, 6 ultimate numbers against 4 non-ultimate numbers. The next source set, that of the non-ultimates, includes,

among its first ten numbers, 4 raised numbers against 6 composite numbers. Finally, the source set of composites includes,

among its first ten numbers, 6 pure composites against 4 mixed composites.

A very strong entanglement links all these sets of numbers which oppose in multiple ways in ratios of value 3/2 (or reversibly

of ratios 2/3). For example, the first 6 ultimates (0-1-2-3-5-7) are simultaneously opposed to the 4 non-ultimates (4-6-8-9)

among the first 10 whole numbers, to the 4 raiseds of the first 10 non-ultimates (4-8-9-16) and to the follow 4 ultimates among

the first 10 ultimates (11-13-17-19).

6.2. The forty primordial numbers

This entangled classification of whole numbers makes it possible to define (Figures 8 and 9) a set of forty primordial numbers.

These forty primordial numbers are the set of first ten numbers in each of the four final classes of whole numbers. It is

understood that the term "primordials" designates these forty primordial numbers.

Thus, by convention, these forty numbers qualified as primordial and called “primordials” are:

- 0-1-2-3-5-7-11-13-17-19 → the first ten ultimates,

- 4-8-9-16-25-27-32-49-64-81 → the first ten raiseds,

- 6-10-14-15-21-22-26-30-33-34 → the first ten composites,

- 12-18-20-24-28-36-40-44-45-48 → the first ten mixes.

6.3. Primordial numbers and 3/2 ratio

Also, as shown in Figure 8 and in other viewing angle Figure 9, these four sets of ten numbers are all made up of subgroups of

always four and six entities according to their respective initial formation and, depending of this initial formation, a value ratio

3/2 (or reversibly of 2/3) always exists between adjacent complexity level subgroups (see Figure 2).

0 1 2 3 4 5 6 7 8 9

whole numbers

← 6 numbers

4 numbers →

0 1 2 3 5 7

11 13 17 19

4 6 8 9

10 12 14 15 16 18

non-ultimates

ultimates

← 4 numbers

6 numbers →

raiseds

4 8 9 16

25 27 32 49 64 81

6 10 12 14 15 18

20 21 22 24

composites

← 6 numbers

4 numbers →

pure composites

mixed composites

6 10 14 15 21 22

26 30 33 34

12 18 20 24

28 36 40 44 45 48

Figure 9: Initial arithmetic arrangements in 3/2 ratios inside hierarchical classification of whole numbers.

See Figures 2 and 8 also.

6.4. Two sets of primordial numbers

Thus, according to their appearance in the four final subsets and the origin of their respective source set (see Figures 8 and 9),

the forty primordial numbers can be distinguished in two groups:

- the primo primordials,

- the secondary primordials.

As illustrated in Figure 10, there are 20 primary primordials and 20 secondary primordials. This last table summarizes the

initial distribution of the first ten numbers of each of the four categories of whole numbers. In this table the ratio 3 to 2 (or 2 to

3) is very highlighted. This demonstrates a very powerful arithmetic organization of the initialization of the four whole number

subsets just defined and unequivocally identified.

first

10 ultimates

first

10 raiseds

first

10 composites

first

10 mixes

20 primo

primordials

➡

6 ultimates

← 3/2 →

4 raiseds

← 2/3 →

6 composites

← 3/2 →

4 mixes

0-1-2-3-5-7

4-8-9-16

6-10-14-15-21-22

12-18-20-24

↑ 3/2 ↓

↑ 2/3 ↓

↑ 3/2 ↓

↑ 2/3 ↓

20 secondary

primordials

➡

11-13-17-19

25-27-32-49-64-81

26-30-33-34

28-36-40-44-45-48

4 ultimates

← 2/3 →

6 raiseds

← 3/2 →

4 composites

← 2/3 →

6 mixes

Figure 10: Distinction of primo primordials and secondary primordials in the 4 final subsets of whole numbers.

See Figures 8 and 9 also.

7. Matrix of the forty primordials

The ranking in order of their magnitude, of the forty primordials within a matrix of 4 rows by 10 columns and their distinction

into primo primordials and secondary primordials as defined above reveals a non-random distribution of these two groups of

numbers.

7.1. Symmetrical arrangements

Thus, as illustrated in Figure 11, in this matrix, these two types of numbers are always distributed in equal quantity in each of

the five zones of two symmetrically opposite columns. In each of these five areas are 4 primo primordials and 4 secondary

primordials.

4 primo primordials

4 secondary primordials

4 primo primordials

4 secondary primordials

0

1

2

3

4

5

6

7

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48

49

64

81

4 and 4

4 and 4

4 primo primordials

4 secondary primordials

Figure 11: Symmetrical distribution of primo primordials and secondary primordials in the matrix of the

forty primordials. See Figure 10 also.

The particular distribution of primo primordials and secondary primordials in the matrix of the forty primordials generates a

lot of singular arithmetic arrangements of which, illustrated in Figure 12, oppositions in 3/2 or 2/3 value ratios depending on

the different considered geometric configurations.

12 primo primordials

↑ 3/2 ↓

8 secondary primordials

0

1

2

3

4

5

6

7

8

9

8 primo primordials

↑ 2/3 ↓

12 secondary primordials

10

11

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45

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49

64

81

Figure 12: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10 and 11 also.

In Figure 12, is only an outline of a very sophisticated entanglement of how the twenty numbers qualified as primo primordials

and secondary primordials are organized within the matrix of the forty primordials previously defined.

8 primo

primordials

↑ 2/3 ↓

12 secondary

primordials

0

1

2

3

4

5

6

7

8

9

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81

12 primo

primordials

↑ 3/2 ↓

8 secondary

primordials

Figure 13: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10 and 11 also.

Indeed, as it clearly appears in the tables of Figures 13 to 17, in each symmetric sub matrices of always 20 entities, in a ratio of

value 3 to 2 or reversibly from 2 to 3, there are always opposed 12 primo primordials to 8 secondary primordials or vice versa.

12 primo

primordials

↑ 3/2 ↓

8 secondary

primordials

0

1

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3

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81

8 primo

primordials

↑ 2/3 ↓

12 secondary

primordials

Figure 14: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10, 12 and 13 also.

So Figure 15, other oppositions of primo and secondary primordials in 3/2 ratios.

12 primo

primordials

↑ 3/2 ↓

8 secondary

primordials

0

1

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81

8 primo

primordials

↑ 2/3 ↓

12 secondary

primordials

Figure 15: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10 and 11 also.

So Figure 16, other oppositions of primo and secondary primordials in 3/2 ratios.

12 primo

primordials

↑ 3/2 ↓

8 secondary

primordials

0

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8 primo

primordials

↑ 2/3 ↓

12 secondary

primordials

Figure 16: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10 and 11 also.

So Figure 17, still other oppositions of primo and secondary primordials in 3/2 ratios.

12 primo

primordials

↑ 3/2 ↓

8 secondary

primordials

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

24

25

26

27

28

30

32

33

34

36

40

44

45

48

49

64

81

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

24

25

26

27

28

30

32

33

34

36

40

44

45

48

49

64

81

8 primo

primordials

↑ 2/3 ↓

12 secondary

primordials

Figure 17: Distribution of primo primordials and secondary primordials in the matrix of the forty primordials.

See Figures 10 and 11 also.

7.2. Dissymmetric arrangements

Other singular arithmetic arrangements are revealed in this matrix of the forty primordials always by the distinctions between

the twenty primo primordials and the twenty secondary primordials.

Thus, Figure 18, the alternative and dissymmetric isolation of 3 and 2 numbers in each of the four lines of this matrix allows

the creation of four sub matrices always opposing in value ratio 3/2 or in value ratio 1/1 according to the configurations

considered and the nature (primary or secondary) of the forty primordial numbers previously defined.

alternate sub-matrices to 24 primordials

(8 times 3 entities)

← 3/2 ratio →

alternate sub-matrices to 16 primordials

(8 times 2 entities)

12 primo

primordials

0

1

2

5

6

7

12

13

14

17

18

19

20

21

22

26

27

28

36

40

44

49

64

81

← 3/2 ratio →

3

4

8

9

10

11

15

16

24

25

30

32

33

34

45

48

8 primo

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 secondary

primordials

8 secondary

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 primo

primordials

2

3

4

7

8

9

10

11

12

15

16

17

22

24

25

28

30

32

33

34

36

45

48

49

← 3/2 ratio →

0

1

5

6

13

14

18

19

20

21

26

27

40

44

64

81

8 primo

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 secondary

primordials

8 secondary

primordials

Figure 18: Distribution in 3/2 and 1/1 ratios of primo primordials and secondary primordials in alternative sub-matrices of the

forty primordials. See Figure 10 also.

Also, in Figure 19, the same phenomena are generated, in the constitution of four other sub-matrices where the alternation 3 to

2 of the entities is made two lines by two lines.

alternate sub-matrices to 24 primordials

(4 times 6 entities)

← 3/2 ratio →

alternate sub-matrices to 16 primordials

(8 times 4 entities)

12 primo

primordials

0

1

2

5

6

7

10

11

12

15

16

17

22

24

25

28

30

32

36

40

44

49

64

81

← 3/2 ratio →

3

4

8

9

13

14

18

19

20

21

26

27

33

34

45

48

8 primo

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 secondary

primordials

8 secondary

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 primo

primordials

2

3

4

7

8

9

12

13

14

17

18

19

20

21

22

26

27

28

33

34

36

45

48

49

← 3/2 ratio →

0

1

5

6

10

11

15

16

24

25

30

32

40

44

64

81

8 primo

primordials

↑ 1/1 ratio ↓

↑ 1/1 ratio ↓

12 secondary

primordials

8 secondary

primordials

Figure 19: Distribution in 3/2 and 1/1 ratios of primo primordials and secondary primordials in alternative sub-matrices of the

forty primordials. See Figure 10 also.

8. Matrix of the first hundred numbers and primordial numbers

Therefore, in the matrix of the first 100 whole numbers and in a ratio of 3/2, 60 non-primordial numbers oppose to the 40

primordials previously defined in Chapter 6.2.

36 non-primordial numbers

↑ 3/2 ratio ↓

24 primordial numbers

0

1

2

3

4

5

6

7

8

9

sub-matrix of twice

3 columns

(60 numbers)

← 3/2 ratio →

sub-matrix of twice

2 columns

(40 numbers)

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

36 non-primordials

↑ 3/2 ratio ↓

24 primordials

← 3/2 ratio →

← 3/2 ratio →

24 non-primordials

↑ 3/2 ratio ↓

16 primordials

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

24 non-primordial numbers

↑ 3/2 ratio ↓

16 primordial numbers

Figure 20: Distinction and distribution of the 40 primordial and 60 non-primordial numbers in the matrix of the first

hundred numbers.

In the matrix of the first hundred numbers, the position differentiation of the 40 primordials generates singular phenomena of

3/2 ratio depending on the different areas considered to 60 versus 40 entities or to 50 versus 50 entities.

Thus, in this matrix, it turns out in Figure 20, that the distinction of two sub-matrices of twice 3 columns against twice 2

columns generates sets of primordial numbers and non-primordial numbers which are opposed in 3/2 transcendent ratios to 36

versus 24 entities and 24 versus 16 entities.

In the sub-matrices of equal sizes and alternately made up of the upper and lower quarters of the complete matrix of the first

hundred numbers as illustrated in Figure 21, the 60 non-primordial numbers are distributed in values of equal quantities and

these sets of twice 30 non-primordial oppose in 3/2 value ratios to the 40 primordials also distributed in two equal sets of 20

entities.

Matrix to 2 times 25 numbers (50 numbers)

← 1/1 ratio →

Matrix to 2 times 25 numbers (50 numbers)

0

1

2

3

4

10

11

12

13

14

20

21

22

23

24

30

31

32

33

34

40

41

42

43

44

55

56

57

58

59

65

66

67

68

69

75

76

77

78

79

85

86

87

88

89

95

96

97

98

99

← 1/1 ratio →

5

6

7

8

9

15

16

17

18

19

25

26

27

28

29

35

36

37

38

39

45

46

47

48

49

50

51

52

53

54

60

61

62

63

64

70

71

72

73

74

80

81

82

83

84

90

91

92

93

94

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

← 1/1 ratio →

← 1/1 ratio →

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

Figure 21: Equal distribution of the 60 non-primordials and 40 primordials in two sub-matrices of the first

hundred numbers.

8.1. Linear sub-matrices of sixty and forty numbers

In the sub-matrix of 60 entities made up alternately of the first six numbers then of the last six numbers of each of the ten lines

of the matrix of the first hundred numbers introduced Figure 20, the non-primordial and primordial numbers are opposed, left

part of Figure 21, into two sets of 3/2 value ratio and these sets are themselves opposed to the two reciprocal sets of the

complementary sub-matrix of 40 entities in 3/2 value transcendent ratios.

Also, exactly the same phenomena occur inside and between the two sub-matrices, of 60 versus 40 entities where the

alternation of the considered numbers applies two by two lines as illustrated in the right part of Figure 22.

Sub-matrix to 10 times

6 numbers (60 numbers)

← 3/2 ratio →

Sub-matrix to 10 times

4 numbers (40 numbers)

Sub-matrix to 5 times

12 numbers (60 numbers)

← 3/2 ratio →

Sub-matrix to 5 times

8 numbers (40 numbers)

0

1

2

3

4

5

14

15

16

17

18

19

20

21

22

23

24

25

34

35

36

37

38

39

40

41

42

43

44

45

54

55

56

57

58

59

60

61

62

63

64

65

74

75

76

77

78

79

80

81

82

83

84

85

94

95

96

97

98

99

6

7

8

9

10

11

12

13

26

27

28

29

30

31

32

33

46

47

48

49

50

51

52

53

66

67

68

69

70

71

72

73

86

87

88

89

90

91

92

93

0

1

2

3

4

5

10

11

12

13

14

15

24

25

26

27

28

29

34

35

36

37

38

39

40

41

42

43

44

45

50

51

52

53

54

55

64

65

66

67

68

69

74

75

76

77

78

79

80

81

82

83

84

85

90

91

92

93

94

95

6

7

8

9

16

17

18

19

20

21

22

23

30

31

32

33

46

47

48

49

56

57

58

59

60

61

62

63

70

71

72

73

86

87

88

89

96

97

98

99

36 non-primordials

↑ 3/2 ratio ↓

24 primordials

← 3/2 ratio →

← 3/2 ratio →

24 non-primordials

↑ 3/2 ratio ↓

16 primordials

36 non-primordials

↑ 3/2 ratio ↓

24 primordials

← 3/2 ratio →

← 3/2 ratio →

24 non-primordials

↑ 3/2 ratio ↓

16 primordials

Figure 22: According to different alternatively linear sub-matrices: distribution of the 60 non-primordials and the 40 primordials in

opposing sets to 3/2 transcendent ratios.

8.2. Concentric and eccentric sub-matrices

In this matrix of the first hundred numbers, more sophisticated arrangements bring into opposition sets of non-primordial and

of primordial numbers in exact 3/2 ratios. Thus, as described in the left part of Figure 23, five concentric zones are opposed,

three versus two, in the distribution of their non-primordial and primordial numbers in 3/2 ratios. The same phenomenon is

reproduced by considering the five eccentric zones presented in the right part of this Figure 23.

Concentric configurations:

Eccentric configurations:

sub-matrix to 3

concentric zones

(60 numbers)

← 3/2 ratio →

sub-matrix to 2

concentric zones

(40 numbers)

sub-matrix to 3

eccentric zones

(60 numbers)

← 3/2 ratio →

sub-matrix to 2

eccentric zones

(40 numbers)

0

1

2

3

4

5

6

7

8

9

10

19

20

22

23

24

25

26

27

29

30

32

37

39

40

42

44

45

47

49

50

52

54

55

57

59

60

62

67

69

70

72

73

74

75

76

77

79

80

89

90

91

92

93

94

95

96

97

98

99

11

12

13

14

15

16

17

18

21

28

31

33

34

35

36

38

41

43

46

48

51

53

56

58

61

63

64

65

66

68

71

78

81

82

83

84

85

86

87

88

0

2

4

5

7

9

12

14

15

17

20

21

22

24

25

27

28

29

34

35

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

64

65

70

71

72

74

75

77

78

79

82

84

85

87

90

92

94

95

97

99

1

3

6

8

10

11

13

16

18

19

23

26

30

31

32

33

36

37

38

39

60

61

62

63

66

67

68

69

73

76

80

81

83

86

88

89

91

93

96

98

36 non-primordials

↑ 3/2 ratio ↓

24 primordials

← 3/2 ratio →

← 3/2 ratio →

24 non-primordials

↑ 3/2 ratio ↓

16 primordials

36 non-primordials

↑ 3/2 ratio ↓

24 primordials

← 3/2 ratio →

← 3/2 ratio →

24 non-primordials

↑ 3/2 ratio ↓

16 primordials

Figure 22: From the matrix of the first hundred numbers, concentric and eccentric configurations of sub-matrices to 60 and 40 entities

opposing their non-primordials and their primordials in 3/2 ratios.

Also, in configurations identical to those introduced, by mixing these sub-matrices of 40 and 60 entities (presented in Figure

22) and after having each split them vertically into two equal parts of 30 and 20 entities, we obtain, Figure 23, new matrices of

50 entities each. In these mixed configurations, the non-primordials and the primordials are divided into exact ratios of value

1/1 with always 30 non-primordials versus 30 and always 20 non-primordials versus 20.

Mixed concentric configurations:

Mixed eccentric configurations:

5 half concentric mixed

zones (50 numbers)

← 1/1 ratio →

5 half concentric mixed

zones (50 numbers)

5 half eccentric mixed

zones (50 numbers)

← 1/1 ratio →

5 half eccentric mixed

zones (50 numbers)

0

1

2

3

4

10

15

16

17

18

20

22

23

24

28

30

32

35

36

38

40

42

44

46

48

50

52

54

56

58

60

62

65

66

68

70

72

73

74

78

80

85

86

87

88

90

91

92

93

94

5

6

7

8

9

11

12

13

14

19

21

25

26

27

29

31

33

34

37

39

41

43

45

47

49

51

53

55

57

59

61

63

64

67

69

71

75

76

77

79

81

82

83

84

89

95

96

97

98

99

0

2

4

6

8

12

14

16

18

19

20

21

22

24

26

34

36

37

38

39

40

41

42

43

44

50

51

52

53

54

64

66

67

68

69

70

71

72

74

76

82

84

86

88

89

90

92

94

96

98

1

3

5

7

9

10

11

13

15

17

23

25

27

28

29

30

31

32

33

35

45

46

47

48

49

55

56

57

58

59

60

61

62

63

65

73

75

77

78

79

80

81

83

85

87

91

93

95

97

99

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

← 1/1 ratio →

← 1/1 ratio →

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

← 1/1 ratio →

← 1/1 ratio →

30 non-primordials

↑ 3/2 ratio ↓

20 primordials

Figure 23: From the matrix of the first hundred numbers, concentric and eccentric mixed configurations of sub- matrices to 50 entities

each opposing their non-primordials and their primordials in 3/2 ratios.

It is important to underline here the total similarity of these arithmetic phenomena with those operating in the similarly sub-

matrices introduced in preview paper "The ultimate numbers and the 3/2 ratio, just two primary sets of whole number" [1]

where data source (table of crossed additions of the twenty fundamentals) are however completely different.

9. Association of opposing classes

Depending on their degree of complexity as introduced Figure 2 in Chapter 3, the four classes of numbers can be grouped into

two sets of extreme or median classes. Thus, the ultimate numbers, of level 1 complexity and the mixed numbers, of level 4

complexity form a set of entities of extreme classes and the raised and composite numbers, of level 2 and 3 complexity, form a

second set of median classes. So it is agree that designation "extremes" designates numbers of extreme classes and designation

"medians" designates numbers of medians classes.

9.1. 100-number matrix and number class.

Figure 24, in the matrix of the first hundred numbers, these two sets are made up of 55 numbers to extreme classes and 45

numbers to median classes. So 5x extremes (→ x = 11) and 5x medians (→ x = 9) constitute this closed matrix. In increasingly

diluted sub-matrices of 60 versus 40 entities, these two families of numbers are always distributed in 3/2 value ratios.

27 ultimates

(level 1 complexity)

u

→

r

10 raiseds

(level 2 complexity)

27 ultimates

+

28 mixes

=

55 numbers

of extreme classes

↓

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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55

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59

60

61

62

63

64

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66

67

68

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70

71

72

73

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76

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79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

↓

10 raiseds

+

35 composites

=

45 numbers

of median classes

35 composites

(level 3 complexity)

c

→

m

28 mixes

(level 4 complexity)

Figure 24: Counting of the four classes of numbers in the matrix of the first 100 numbers according to their degree

of complexity (see Figures 2 and 3 also).

9.1.1. Dilution of sub-matrices

Thus, in the left part of Figure 25, in two compact blocks of 60 versus 40 entities made up respectively of the first 60 and the

following 40, the extreme numbers and the median numbers are distributed in ratios of values 3/2 with, respectively for each

set of numbers, 33 extremes versus 22 and 27 medians versus 18. Splitting the matrix of the first hundred numbers into 10

blocks of 5 times 12 versus 5 times 8 entities as illustrated in the right part of Figure 25 generates the same arithmetic

phenomena.

Sub-matrix to (1 time)

60 numbers

← 3/2 ratio →

Sub-matrix to (1 time)

40 numbers

Sub-matrix to 5 times

12 numbers (60 numbers)

← 3/2 ratio →

Sub-matrix to 5 times

8 numbers (40 numbers)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

0

1

4

5

8

9

10

11

14

15

18

19

20

21

24

25

28

29

30

31

34

35

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

62

63

66

67

72

73

76

77

82

83

86

87

92

93

96

97

2

3

6

7

12

13

16

17

22

23

26

27

32

33

36

37

60

61

64

65

68

69

70

71

74

75

78

79

80

81

84

85

88

89

90

91

94

95

98

99

33 extremes

27 medians

← 3/2 ratio →

← 3/2 ratio →

22 extremes

18 medians

33 extremes

27 medians

← 3/2 ratio →

← 3/2 ratio →

22 extremes

18 medians

Figure 25: Distribution of the numbers to extreme and median classes in two little diluted and more diluted double sub-matrices of the

first 100 numbers.

Also, the two sets of extreme and median numbers are further divided into 3/2 value ratios in a more diluted fractionation of

this matrix into 20 blocks of 10 times 6 and 10 times 4 entities as described in the left part of Figure 26. Lastly, on the right

side of Figure 26, in a final fractionation of this matrix into 40 blocks of 20 times 3 entities versus 20 times 2 entities, the same

partitions of the two families of numbers in 3/2 ratios are still observed with always 33 extremes versus 22 and 27 medians

versus 18.

Sub-matrix to 10 times

6 numbers (60 numbers)

← 3/2 ratio →

Sub-matrix to 10 times

4 numbers (40 numbers)

Sub-matrix to 20 times

3 numbers (60 numbers)

← 3/2 ratio →

Sub-matrix to 20 times

2 numbers (40 numbers)

0

1

4

5

8

9

10

11

14

15

18

19

20

21

22

23

24

25

26

27

28

29

32

33

36

37

42

43

46

47

52

53

56

57

62

63

66

67

70

71

72

73

74

75

76

77

78

79

80

81

84

85

88

89

90

91

94

95

98

99

2

3

6

7

12

13

16

17

30

31

34

35

38

39

40

41

44

45

48

49

50

51

54

55

58

59

60

61

64

65

68

69

82

83

86

87

92

93

96

97

0

2

4

6

8

10

12

14

16

18

20

21

22

23

24

25

26

27

28

29

31

33

35

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69

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73

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75

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77

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79

80

82

84

86

88

90

92

94

96

98

1

3

5

7

9

11

13

15

17

19

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

62

64

66

68

81

83

85

87

89

91

93

95

97

99

33 extremes

27 medians

← 3/2 ratio →

← 3/2 ratio →

22 extremes

18 medians

33 extremes

27 medians

← 3/2 ratio →

← 3/2 ratio →

22 extremes

18 medians

Figure 26: Distribution of the numbers to extreme and median classes in two diluted and very diluted double sub-matrices of the first

100 numbers.

9.1.2. Triangular numbers and triangular sub-matrices

There are therefore 55 extremes and 45 medians among the first 100 whole numbers. These two values are actually triangular

numbers: 55 being the sum of the numbers 1 to 10 (T10 = 55) and 45 being the sum of the numbers 1 to 9 (T9 = 45).

As it appears in Figure 27, in the symmetric and triangular sub-matrices of 55 and 45 entities, it turns out that the extremes and

medians are always 5x in number. Also, the same values of these two categories of numbers are identical in the configurations

symmetrically considered.

55-number

triangular sub-matrix

→ T10

45-number

triangular sub-matrix

→ T9

45-number

triangular sub-matrix

→ T9

55-number

triangular sub-matrix

→ T10

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30 extremes

↑ 6/5 ratio ↓

25 medians

← 6/5 ratio →

← 5/4 ratio →

25 extremes

↑ 5/4 ratio ↓

20 medians

25 extremes

↑ 5/4 ratio ↓

20 medians

← 5/6 ratio →

← 4/5 ratio →

30 extremes

↑ 6/5 ratio ↓

25 medians

Figure 27: Distribution of the numbers to extreme and median classes in triangular sub-matrices of the first 100-number matrix.

The different values observed are therefore equal to 30, 25 and 20 entities. These values actually correspond to combinations

of two other triangular numbers: 10 which is equal to T4 and 15 which corresponds to T5. We therefore note that these two

values are also equal to 5x entities and respectively opposite in a 2/3 ratio with 2 and 3 as the value of x. So, 30 is equal to 2

times T5 (2 times 15), 20 is equal to 2 times T4 (2 times 10). Also the value 25 is equal to one time T5 + one time T4 (15 + 10).

It therefore turns out that in the matrix of the first 100 whole numbers, the two classes of numbers that we have described as

extreme and median are subtly distributed in sets always equal to 5x and in combinations of triangular numbers, which are also

equal to 5x in number.

We now demonstrate, Figure 28, that the four triangular configurations introduced in Figure 27, correspond to assemblies of

four other triangular sub-matrices whose sizes are also arrangements of the triangular numbers T4 (→ 10) and T5 (→ 15).

In these four source sub-matrices, it turns out that the respective number of extremes and medians is identical to the triangular

matrix sub-dimensions which constitute them. Thus, 15 extremes and 10 medians constitute the two sub-matrices of triangular

size T4 (→ 10) + T5 (→ 15). Also, 10 extremes and 10 medians constitute the sub-matrix of triangular size T4 (→ 10) + T5 (→

10). Finally, 15 extremes and 15 medians constitute the sub-matrix of triangular size T5 (→ 15) + T5 (→ 15).

10 + 15 numbers

triangular sub-matrix

→ T4 + T5

← 1/1 ratio →

10 + 15 numbers

triangular sub-matrix

→ T4 + T5

10 + 10 numbers

triangular sub-matrix

→ T4 + T4

← 2/3 ratio →

15 + 15 numbers

triangular sub-matrix

→ T5 + T5

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15 extremes

↑ 3/2 ratio ↓

10 medians

← 1/1 ratio →

← 1/1 ratio →

15 extremes

↑ 3/2 ratio ↓

10 medians

10 extremes

↑ 1/1 ratio ↓

10 medians

← 2/3 ratio →

← 2/3 ratio →

15 extremes

↑ 1/1 ratio ↓

15 medians

Figure 28: Distribution of the numbers to extreme and median classes in triangular source sub-matrices of the first 100-number

matrix. See Figures 27 and 29 also.

From these four source sub-matrices introduced in Figure 28, other symmetric configurations deserve attention due to their

strong singularity. Thus, as illustrated in Figure 29, the assembly of the symmetric matrices allows the creation of two new

sub-matrices of 2 times T4 and 2 times T5 as size. As well as two others of T4 + T5 as another size.

25 + 25 numbers recombined triangular sub-matrix

→ (T4 + T5) + (T4 + T5)

20 + 30 numbers recombined triangular sub-matrix

→ 2T4 + 2T5

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⬋

⬊

⬋

⬊

10 + 10 numbers

triangular sub-matrix

→ T4 + T4

← 2/3 ratio →

15 + 15 numbers

triangular sub-matrix

→ T5 + T5

10 + 15 numbers

triangular sub-matrix

→ T4 + T5

← 1/1 ratio →

10 + 15 numbers

triangular sub-matrix

→ T4 + T5

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