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# New whole number classification and the 3 to 2 ratio

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## Abstract and Figures

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.
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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 (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 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 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 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
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 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 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 ()
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
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
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
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
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
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
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
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 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
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
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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 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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
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24
25
26
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30
32
33
34
36
40
44
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48
49
64
81
0
1
2
3
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9
10
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12
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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 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
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
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
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
70
71
72
73
74
75
76
77
78
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
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
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
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
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
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
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
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
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
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
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
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
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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
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
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
0
1
2
3
4
5
6
7
8
9
10
11
12
13