Internet Draft E. Terrell
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Expires October 28th, 2006 April 2006
The Mathematics of Quantification, and the Rudiments Of
the Ternary Logical States of the Binary Systems
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The Ternary Logical States of the Binary System October 28, 2006
Abstract
This paper, opening with the historical that documents the source
of the Binary Enumeration Error, utilizes the proof of 'Fermat's
Last Theorem' (Normative References - [1], [2] and [3]), the
Mathematics of Quantification, and the Logic of Set Theory, to
prove that the Binary System represents a 'Closed and Finite'
Alternate Mathematical Field. That is, using the Elementary Laws
of Algebra, with the Basic Principles from Analytic Geometry,
provides the final clarification simplifying the proof for the
correction of the Counting Errors and the Logical Foundation for
the New Binary System. And more importantly, this also establishes
the basic foundational principles for 3 State Ternary Logic. In
other words, using an askew, or mathematically incorrect Binary
System, defined as the misinterpretation of ZERO, sustains the
Counting Error (an Accumulating Propagation) levying a substantial
loss of IP Addresses in the IPv4 IP Specification, affecting as
well the Address Pool Total for the IPv6 Specification. Hence,
from the foregoing foundation an unquestionable proof concludes;
the Elementary Mathematical 'Resolution of the Counting Error in
the Binary System’ & the 'Fall of Differential Calculus' -
[4. IANA Considerations].
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The Ternary Logical States of the Binary System October 28, 2006
Table of Contents
Abstract
Introduction
1. The Beginnings of Binary Enumeration
1.1 Gottfried Wilhelm von Leibniz's Binary System
1.2 George Boole's Mathematical Logic
1.3 The Arithmetical Error and the flaw in Binary Enumeration
2. The Unary and The Binary Mathematical Systems
2.1 Two Distributive Laws & The Binary System Proves Fermat's
Last Theorem
2.2 The Mathematics of Quantification and Binary Arithmetic
System
2.3 The Binary and Ternary Systems and George Boole's
Mathematical Logic
3. Security Considerations
4. IANA Considerations - 'Resolution of the Counting Error in
the Binary System'
5. References
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Introduction
The investigation of the origin of the Binary System revealed
that Leibniz, its principle author, is responsible for the
askew error, because he never understood or actually developed
a Binary System of counting. And this is clearly shown to be
the handicap that not only resulted in the Loss of available
IP Addresses in the IPv4 Specification, but it contributed to
the difficulties preventing the development of the Binary and
Ternary Relations defined by Boolean Algebra. That is, by
clearly showing that this is a Closed Finite Mathematical
System, which defines an incremental progression using ' 1's '.
This greatly simplified the Boolean Mathematical Relationships
for the 'Theory of Three State Logic', and corrected the error
in Binary Enumeration, which generated the loss of IP Addresses
in the IPv4 Specification. In other words, the proof of
"Fermat's Last Theorem" defines a special case of the
Distributive Law, which is defined in the mathematical logic
of Set Theory, as the Intersection of the two Universal Sets
that represents the Binary and the Unary Systems. And this
conclusively proves, that there are only Two logical
Systems of Counting, which are mathematically viable.
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1. The Beginnings of Binary Enumeration
The History of the Binary System has its recorded beginnings
starting about the 5th century BC. But, there is a problem with
this recorded date, because the historians have not defined, or
established an agreement regarding what they mean jointly, or
independently, when they are referencing the development of the
Binary System. In other words, for many people, specifically
mathematicians, when they speak or make reference to the Binary
System, they are talking about mathematics. The Binary System,
as a Mathematical System actually did not come into fruition
until the 1600. That is, from the 5th Century to the 1600, what
is thought to be a Binary System for Mathematical Enumeration,
was in fact, either a system of Drum Beats for communications,
a system of Open and Closed Bars used for counting, or a system
for distinguishing musical notes in musical compositions. In
any case, each of these so called Binary Systems shared the
same flaw; they skew the counting by the misrepresentation of
the Binary equivalent of '1'.
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1.1 Gottfried Wilhelm von Leibniz's Binary System
The general consensus regarding Leibniz would contend that,
he made significant contributions to the foundations of
Mathematics, Philosophy, and the beginnings of Set Theory.
However, because he was indeed, a man of the times, Leibniz
was occupied by a broad range of subjects. Nonetheless, while
he did make significant contributions to humanity, an
investigation of some of his most noted contributions would
show that he did not completely finish the work for closure
of the proposed subject(s). That is, I am of the opinion that,
for most of his life, Leibniz was looking for the pieces of
his puzzle, the clues or solution to clarify the concerns
involving his ongoing research in the areas of Philosophy,
Logic, and Metaphysics (The Laws and Logic of Critical
Thinking). Needless to say, my opinion is evinced more
clearly by the study of the works from one of his
contemporaries, Perrie de Fermat, and the man most
profoundly influenced by his research in Metaphysics,
George Boole.
Nevertheless, while Leibniz correctly translated the
symbolisms for enumeration, as presented in the book of I
Ching, into a Binary System of counting, which was similar
to the Unary System. However, the reality of this
accomplishment is that, his only achievement was the 'Ø'
and the '1' solution to his problem concerning his
Metaphysical Research, which pertained to the Logical
Analysis for the presentation of 'The Laws and Logic of
Critical Thinking'. In which case, had he either knew, or
fully understood that Numerology, or Number Theory in
general, involved the Logical Analysis of the Elementary
Laws of Mathematics. He probably would have correctly
completed his Numbering System, and 'Fermat's Last
Theorem' would not have become one of the greatest, from
a historical perspective, Mathematical Enigmas of all
times. In any case, since 'Fermat's Last Theorem' was not
solved until November 1979, there was no logical
connection ever established between the works of Fermat
and Leibniz. Hence, in the absence of a logical reason
for a comparable analysis, there was no reason to
question the validity of Leibniz's numerical translation.
In other words, the Modern Binary System, as depicted in
figure 1, is the direct consequence from the work of
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Leibniz, and it remains logically incorrect. This is because,
the discovery of the solution to the problem that qualified
as the logical reason for the comparable analysis questioning
his results, from the mathematical perspective, it violates
the laws from elementary mathematics, the Field Postulates,
the Axioms for Equality, and the logical foundation of
Set Theory.
Modern Primitive
Binary Unary
System System
00 0
01 1
10 11
11 111
100 1111
101 11111
110 111111
111 1111111
1000 11111111
1001 111111111
1010 1111111111
1011 11111111111
1100 111111111111
1101 1111111111111
1110 11111111111111
1111 111111111111111
10000 1111111111111111
figure 1
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1.2 George Boole's Mathematical Logic
The influence of Leibniz upon George Boole is unquestionable,
however, Boole's greatest contribution to mathematics
overshadows considerably, his retake on objectives of
Leibniz's life's work. In other words, Boole's work; "An
investigation of the Laws of Thought on Which are founded
the Mathematical Theories of Logic and Probability", is a
mathematical and logical marvel that clearly renders a
rational demystification of the Metaphysical rhetoric
encompassing the logic of the 'Ø' and the '1' foundation,
which was the hallmark of Leibniz pursuit to resolve 'The
Laws and the Logic Foundation of Critical Thinking'. Still,
George Boole was unaware of the contributions he made to
Mathematics and the Mathematical Sciences, because it was
embedded in his most famous work; "An investigation of
the Laws of Thought on which are founded the Mathematical
Theories of Logic and Probability". Furthermore, while
using the principle foundation of the '0'and the '1'
concepts created by Leibniz, Boole correctly established
an Algebraic and Logical Foundation that was later to
have applications throughout the fields Computer Science
and Electronics. However, the result from Boole's work
was wrongly interpreted as the 'Logic of the Binary
System', when in fact, it is actually 'The Logic of the
Unary System', because only One State Works, or because
only One Stated Condition can be True, as shown in
Figure 2.
The Truth Relation of Two State Logic
Key to the Truth Table
The Table on the Right shows the combinations |A = T |A = T |
of Truth Values for the two Operands, A and B, |B = T |B = F |
in relation to the Truth Function
|A = F |A = F |
|B = T |B = F |
True Or If A Implies B Iff End
|T F| |T F| |T F| |T F| |T F| |T F| |T F| |T F|
|F T| |T F| |T F| |T F| |T F| |T F| |T F| |T F|
Nand X or Not B Nimp Not A Nif Nor False
|T F| |T F| |T F| |T F| |T F| |T F| |T F| |T F|
|F T| |T F| |T F| |T F| |T F| |T F| |T F| |T F|
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|--------------------------------------------------------------|
|Boolean Algebra: Algrbraic and Logical Laws of Two State Logic|
| |
| AND ( . } Associative Law Communative Law |
| 0.0 = 0 (A.B).C = A.(B.C) A + B = B + A |
| A.0 = 0 (A + B) + C = A + (B + C) (A.B) = (B.A) |
| 1.0 = 0 |
| A.1 = A |
| 0.1 = 0 Distributive Law Identity Law |
| A.A = A (A + B) x C = AC + BC A + A = A |
| 1.1 = 1 (AB) + C = (A + C)(B + C) AA = A |
| A.A' = 0 |
| |
| |
| |
| Or ( + ) Precedence Not ( * ) DeMorgan's Theorem |
| 0+0 = 0 AB = A.B 0* = 1 (A.B)' = A' + B* |
| A+0 = A A,B + C = (A.B) + C 1* = 0 (A+B)' = A' + B' |
| 1+0 = 1 A* = A |
| 0+1 = 1 |
| A+1 = A |
| A+A = A |
| A+A'= 1 |
| 1+1 = 1 |
| |
|--------------------------------------------------------------|
figure 2
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Nevertheless, given that an argument can be made claiming
the existence of Two States, '0' and '1'. However, not until
it is realized that Boole's ascribes to a literal usage,
using their actual numeral values, it will then become
understood that a Unary System is a Two State System,
because it is a System of Counting uses '1s' to represent
something and a '0' to represent nothing: 'Hence, A Two
State System'. So, the question of ponder that one might
ask is: 'If the number of States in the Logic of the
Modern Binary System equals that of the Unary System.
How many States defined by Boolean Relationships does the
True Binary System have ...??... Figure 3.
States of the Modern Binary System States of a Unary System
00 \ / 0
2 States
01 / \ 1
figure 3
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1.3 The Arithmetical Error and the flaw in Binary Enumeration
While it should be quite clear that a fundamental knowledge of
Archaeology, Anthropology, and perhaps a knowledge of the early
Languages, should be the perquisite required for the study of any
ancient Civilization. Still, there should never be any doubts,
because if there was a Civilization whose first system of counting
was a True Binary System this would probably be the most advanced
Civilization in the Universe. In other words, because of the
inherent complexities involved in the meaning and the
interpretation of the concept of Zero, the development of a True
Binary System by any Ancient or Primitive Civilization borders on
the Highly Unlikely, or the Impossible. In which case, prior to
Leibniz's discovery of the Two State Logical System for his
Metaphysical Analysis of Critical Thinking, I cannot accept as
being possible, that any Civilization before this time could have
created or fully understood the Mathematical nuances of the Binary
System. The case in point, the mathematical error discovered in
1999, which clearly defined a mathematical discrepancy between two
different Binary Mathematical Systems. However, it is also quite
obvious that know one since Leibniz, could either rationalize this
difference, or understood why a difference occurred. And while the
most notable self-righteous and unspoken claims, under the guise of
Religion, Politics, Racial, or Economic deprivation /
discrimination, for every Civilization since mankind’s beginnings,
has been the horrifyingly torturous control and exploits of its
people. Yet, even with the persistence of these living conditions
today, it is still difficult not wonder, how, or why it is possible
for a blunder having such simple a solution, could have lasted for
so long. ...???...
In other words, the pointed reality of this discrepancy ask
the question: 'Is it possible for a one-to-correspondence of
a Set X, with the Set of Integers, I, which yields the count
of the total number of members the Set X contains, to have
more than one value to represented in the Set I?' {... No!}
That is, it is not possible for any one-to-one pairing
between the members of two Sets, the Set X, and the Set I,
for any member contained in any one of the two Sets to have
more than one pairing with the members of the other Set. And
this is because, such a pairing establishes a count that can
be translated into a equality, when both Sets, given in Table
I, are said to represent the same (Identical) method for
enumeration.
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TABLE I
1 2 3
Modern Modern Primitive
Binary Positive Unary
System Integers System
00 0 0
01 1 1
10 2 11
11 3 111
100 4 1111
101 5 11111
110 6 111111
111 7 1111111
1000 8 11111111
1001 9 111111111
1010 10 1111111111
1011 11 11111111111
1100 12 111111111111
1101 13 1111111111111
1110 14 11111111111111
1111 15 111111111111111
10000 16 1111111111111111
In any case, to say the very least, it should be quite clear
from the examination of Table I, that if a given Binary
Number, say, '11111111', has two Integral Values, '255'
and '256', there is a undeniable problem with the Binary
System when it is used as a System of Counting. Still,
anyone, and with good reason, could quite easily present
the excuse; "It is a Typo-Graphical Error!", as a viable
opposing argument. However, such an argument would easily
fail, because there is absolutely No proof, if
{a, b} = {0, 1}, which would now account for the existence
of the 4 conditions that must clearly represent a number;
Substitution Law for Equality now yields,{a, a}, {a, b},
{b, a}, and {b, b} given in Table II. Especially since, it
is evident in this scenario that Zero cannot be equal to
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either '0', or the Null Set, (Out of Sight, Out of the
Conscience thought ... Does not exist!) because 'a' in the real
sense of reality, references something tangible. Furthermore,
when comparing the three rows from Table I, it is also evident
that there is a common coefficient between different
numerical representations, which are equal to the same number.
But, this assessment is only valid between the members of
columns 2 and 3 in Table I, and conditionally valid between
the members of columns 1, 2, and 3, in Table II.
Note: The unfortunate reality of Table II, is that, the New
Binary System impacts Gregor Mendel's work in Genetics.
In other words, from an 'A a' and 'B b' paring,
{A, a, B, b}, Mendel's results referenced only 6 of
the possible 16* combinations; {A, A}, {B, B}, {A, B},
{B, b} {A, a}, and {a, b}. However, while I have not
wrote the New Foundation representing Finite Chemistry,
the reality of the mathematical results from the
Mathematics of Quantification now questions the
validity of Mendel's claims. In any case, it has been
proven, using the current foundation, that the order
of the addition of Chemicals is a vital consideration
for the determination of the Chemistry of the
resulting Chemical Compound (10 combinations are
missing*). Still, what's alarming? Well.
...considering the 'X' and 'Y' Chromosomes that
represent this relationship. This also suggest the
possibility of an error in the Chromosome Count
defining the Base Pairs; A = adenosine, C = cytosine,
G = guanine, and T = thymine, given that they current
identify 23 + 23 = 46 Chromosomes. That is, from
the Mathematics of Quantification this defines,
2^5 + 2^5 = 2^6 = 64 = 8^2 Chromosomes, four
pairs of 8 Bit Bases Pairs, or 32 + 32 = 64, that
yields about 2^32 = 4,294,967,296 Bases, which
translates into two 8^10 pairs of 8 Bit Bases Pairs
per Cell of human DNA. (et 2004)
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TABLE II
1 2 3
Another Modern Primitive
Binary System Positive Unary
Representation Integers System
0 0 0
aa 1 1
ab 2 11
ba 3 111
bb 4 1111
baa 5 11111
bab 6 111111
bba 7 1111111
bbb 8 11111111
baaa 9 111111111
baab 10 1111111111
baba 11 11111111111
babb 12 111111111111
bbaa 13 1111111111111
bbab 14 11111111111111
bbba 15 111111111111111
bbbb 16 1111111111111111
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Nevertheless, while studying the analysis from Tables III and
IV, recall the former proofs, because it was clearly shown
that if '00 = aa = 1', and '01 = ab = 02', and the Exponent
'F = either a Rational or Irrational Number, then the Binary
Translation could only equal the Binary Representation for
the Number. This meant, the exponent 'F' was not a whole
Number. However, when the result from the sequential variable
of the exponent having a of base '2' equaled the value of a
whole number, and the exponent was also a whole number, then
given that 'Multiplication is the Quantified Sum of Addition',
the the value of the exponent equaled the sum of the Binary
1's and the Product of the Binary 1's equaled the Binary
Number and the Unary Number. That is,
because '2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 = 1111111 = 2^7'
and '2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 = 11111111 = 2^8',
there is clearly a relationship between the columns in Table
IV, and since (2 + 2) = (2 x 2), it shall be proven in Part
II, not only that the established proof for the New Binary
System remains correct. But, that its validity is derived
from the proof of 'Fermat's Last Theorem' and the discovery
of the 'Distributive Law for Exponential Functions'.
Nevertheless, this proves that the differences between
Tables III and IV clearly do not represent a Contradiction,
the necessary requirement as stated by "Chief Executive
Administrator for The Electronic Library of Mathematics",
Aleksandar Perovic, when he said: "Mathematicians do not
accept claims at truth of any possible, non-selfcontradictory
(= consistent) mathematical system". Needless to say, while
this difference is not a Contradiction, it is indeed a
troubling Inconsistency which at the very least, warrants an
investigation.
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TABLE III
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Binary Positive
Enumeration Representation Integer
1. 0^0 = 0 00000000 = 0 0
2. 2^0 = 1 00000001 = 01 1
3. 2^1 = 2 00000010 = 10 2
4. 2^F = 3 00000011 = 11 3
5. 2^2 = 4 00000100 = 100 4
6. 2^F = 5 00000101 = 101 5
7. 2^F = 6 00000110 = 110 6
8. 2^F = 7 00000111 = 111 7
9. 2^3 = 8 00001000 = 1000 8
.
.
.
129. 2^7 = 128 10000000 = 10000000 128
.
.
.
257. 2^8 = 256 100000000 = 100000000 256
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TABLE IV
"The Reality of the New Binary System of Enumeration"
And the Series Generated when Counting, using the
"1's" and "0's", and from the Axioms for Equality, "a's" and "b's"
Exponential Binary Positive
Enumeration Representation Integer
1. 0^0 = 0 0 0
2. 2^0 = 1 00 = aa 1
3. 2^1 = 2 01 = ab 2
4. 2^F = 3 10 = ba 3
5. 2^2 = 4 11 = bb 4
6. 2^F = 5 100 = baa 5
7. 2^F = 6 101 = bab 6
8. 2^F = 7 110 = bba 7
9. 2^3 = 8 111 = bbb 8
.
.
.
129. 2^7 = 128 01111111 = bbbbbbb 128
.
.
.
257. 2^8 = 256 11111111 = bbbbbbbb 256
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2. The Unary and The Binary Mathematical Systems
Throughout mankind's beginnings, there has been several
different Systems of Counting, several different methods
for performing elementary arithmetic, and an equal number
symbols for those that were written, as well as the variety
of sounds for those that were only spoken. However, only one
numbering system, which is nearly complete, survived the
trials of mankind's journey towards civilization; 'The Unary
System'. And while the Laws from the Axioms for Equality,
the Field Postulates, and Logic of Set Theory, which are
an essential part of Unary System, was not developed until
long after its discovery, sometime during the early and mid
1800's. Still, it doubtful that anyone before 1979, tested
the validity of the Unary System. Needless to say, it
should be quite clear now, that every System of Enumeration
must comply with the Laws from the Axioms for Equality, the
Field Postulates, and Logic of Set Theory before it can ever
be accepted as a valid System of Counting, which conforms to
the elementary laws of arithmetic. In other words, the
additional requirement, which any civilization must meet to
claim the creation or the development of a True Binary
System, is one that requires a prior the knowledge of the
Unary System. If not, how could anyone justify the use of
two objects to account for only one material possession...
Hence, to use a Stick to represent the summation of an
arithmetic progression incremented by the addition of 1,
is far simpler than the use or discovery of the 'Stick
and a Rock', which is used to represent the same
incremented addition. Clearly, if this were not the case,
then the Binary System would not have, after its initial
claim of discovery, to wait 2500 years to become a True
Binary System.
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2.1 Proof of Fermat's Last Theorem; and The Two Distributive
Laws
It is extremely amazing that it required more than 300 years
after 'Perrie de Fermat' composed, before his death in 1665,
a riddle involving an elementary algebraic equation, which
eluded everyone, including the greatest mathematicians, to
find, in 1979, a solution. A joke? Perhaps. But, Fermat was
the first to claim, while writing this riddle, that he knew
the simple solution. And clearly, if this were true, which I
believe that it is, then perhaps, "Fermat's Last Theorem"
should rightfully be called; 'the greatest joke of all times'.
However, while I accept Fermat's claim, I do not believe that
he actually knew, or fully understood, the profound
implications of his discovery. Especially since, it may be
concluded, as presented below, there are only 3 logically
viable 'Interconnected Complimentary Solutions' that would
solve the riddle regarding why;
"There are No solutions in Whole Numbers to the Equation,
X^N + Y^N = Z^N, when N > 2".
1. There is no Common Coefficient between the Sum of Two
Exponents, the Exponent equaling their Result, and their
respective Roots, when 'N > 2' , and 'N' defines the
Exponent of the base variables. (Equal Number of Parts
Contained in the Whole.)
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2. Fermat's Solution defines how he interpreted the problem,
which is based upon the current mathematical knowledge
known during his time, Pythagoras Theorem, and the
Analytical Geometric solution(s) explaining the Difference
regarding (the Geometric Shapes of Objects) 'Why', when
'N = 2'; 'The Sum of the Area of two Perfect Squares Equals
the Area of another Perfect Square'.
- Or -
"The Sum of the Area(s) of TWO Squares having equal
Integer Sides, equals the Area of another Square having
equal Sides that are Integers."
And, when 'N = N', 'The Sum of the Areas of two Perfect
Nth Powers is not Equal to the 'ROOT' defining the Area of
a Perfect Nth Power'. Nevertheless, this assumption builds
an explanation that explains this difference, which is
believed to be the foundation for the proof that Fermat
claimed would not fit in the margin of his paper, but
would explain why, when 'N > 2', his theorem is true.
3. In Exponential Operations, there is No equal Distribution
of Multiplication over Addition when 'N > 2', and 'N'
defines the value of the Exponent. (The Discovery of
the Distributive Law for Exponential Functions, and the
Foundation for the Finite Mathematical Field: "The
Rudiments of Finite Algebra; The Results of
Quantification".)
Note: If this is the Text version of this manuscript, then
the imagination of the reader is required to picture
the shapes of the Objects being described. But, if
this is the PDF version, then all of the figures
representing the Objects Shapes and special
Mathematical Symbols are included.
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X Y X Y
+-------+ +---------+ +---+-------+
| | | | | \ |
| | + | | = | \ |X
| | | | | Z +
+-------+ | | + \ |
+---------+ | \ |Y
| \ |
+-------+---+
Figure 4
Nevertheless, proof that it is assumed Fermat was thinking
about, would be something like this, when 'N = 2':
""If the Length of the Side of a Perfect Square inscribing
another Perfect Square is equal to 'X + Y', then the Sum
of the Areas of Two Perfect Squares is equal to the Area
of the Perfect Square inscribing another Perfect Square,
and since the Area of a Square is given by;
1. 'L x W = Area',
the Area of the Inscribing Perfect Square, from the
Mathematics of Quantification is given as;
2. (X + Y) x (X + Y) = (X + Y)^2 = X^2 + 2XY + Y^2
And if the Length of the Side of the inscribed Perfect
Square is equal to 'Z', and the Area of this Perfect
Square is given by equation 1, then from Pythagoras
Theorem, 'Z' is the Root of the equation given by;
3. X^2 + Y^2 = Z^2 = L x W = Z x Z
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Hence, the 'X, Y, and Z' variables, by Pythagoras Theorem now
equals the Sides of the 4 Right Triangles forming, or Creating
the Boarders of the Inscribing Perfect Square and the Perfect
Square it inscribes. That is, if the Length of the Two Sides
joining the 90 degree angle of the Right Triangle equals 1/4
the Length of the Perimeter of the Inscribing Perfect Square,
then the Sum of the X and Y variables defining the Two Sides
of the Right Triangle equals the Length of the Side of the
Inscribing Perfect Square. And given by equation 4, we have;
4. X + Y = Y + X, which means:
If the Sum of the Length of the Two Sides, 'X + Y', of a
Right Triangle forming the Right Angled boarder of any
Perfect Square having Four Equal Sides, is equal to 1/2 the
Length of its Perimeter, then the Sum of the Length of the Two
Equal Sides, which are Integers, of any Right Triangle, is equal
to 1/2 the Length of the Perimeter defining a Perfect Square
having Four Equal Sides that are Integers.
(The Commutative Law for Addition; "X + Y = Y + X". ET 2004)
'And clearly, I can now conclude, Fermat, being the co-discoverer
of Analytic Geometry, only knew of some of the methods of
Euclidian Geometry, and most, if not all of the Algebraic methods
known during his time. Furthermore, the foregoing is evinced more
clearly when it is realized that Fermat never associated the Two
Digit System of Plotting a One Number Point with Binary
Enumeration, yet, he clearly understood the association between
algebraic system for enumeration and the definition of the point
presented by Euclid. In other words, while he clearly understood
the algebra and the geometry defining the shapes of the objects
involved in his proof, he never grasps the connection between
algebra and geometry established by Analytic Geometry.'
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Furthermore, if the Sum of the Length of the Two Sides,
'X + Y' of any Right Triangle forming the boarder of any
Perfect Square equals the Length of 2 of Sides of a Perfect
Square defining the Closed shape of a Rectangular figure
having Perpendicular Sides, then the boarders of the Perfect
Square is defined by Four Equal Right Triangles. Hence, from
Pythagoras Theorem, if of the Two Sides of the Right Triangles
forming the boarders of the Perfect Square join to form the
90 degree Right Angles connecting the 4 Sides of the Perfect
Square, then the Two Sides of the Right Triangles must
respectively Equal the Adjacent Side and the Side Opposite
the Hypotenuse. Therefore, since the Right Triangles join the
Sides of the Perfect Square, the connection of the Side forming
the Hypotenuse of the Right Triangles must also meet, and be
joined at 90 degree angles. And if the Four Right Triangles are
equal, then the Length of Hypotenuse equals the Length of One
Side of an Inscribed Perfect Square.
In other words, this means that; The Sum of the Areas of
Two Perfect Squares equals the Area of the Perfect Square
Inscribing another Perfect Square, if and only if, The Sum
of the Areas of the Four equal Right Triangles forming the
boarders of the Inscribing Perfect Square and the Area of
the Perfect Square it Inscribes, equals the Area of the
Perfect Square Inscribing another Perfect Square. And from
equation 5, the Area of a Triangle is given by;
5. 1/2(b x h)
And given that only the Adjacent Side and the Opposite
Side of the Right Triangles can, respectively equal the
Base, b, and the Height, h, then there are 4 Right Triangles
having equal sides, X and Y, by equation 5, and the Area of the
4 Right Triangles is given by;
6. 4((1/2(X x Y) = 4/2(XY) = 2XY
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And from these results, it is easy to discern the equation
for Sum of Areas of the 4 Right Triangles, as given by the
equation;
7. (X - Y) x (X - Y) = X^2 - 2XY + Y^2, where
8. X^2 + Y^2 = 2XY
Hence, the Area of the Perfect Square Inscribing a Perfect
Square, which is equal to the Sum of the Areas of Two
Perfect Squares, is given by;
9. (X + Y) x (X + Y) = X^2 + 2XY + Y^2 = 2XY + Z^2
Therefore;
10. X^2 + Y^2 = 2XY - 2XY + Z^2 = X^2 + Y^2 = Z^2
Thus, the equation, X^2 + Y^2 = Z^2, which is defined by
Pythagoras Theorem, clearly states that the Sum of the
Areas of Two Perfect Squares is equal to the Area of a
Perfect Square"".
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And clearly, from his analysis, Fermat would have concluded
the X and Y relations:
11. If X = Y, then X and Y are Two equal Perfect
Squares, and If X > Y, or Y > X, then X and Y
are Two different equally Perfect Squares.
X Y X Y
+-------+ +---------+ +---+-------+
/ /| / /| / /|
/ / | / / | / / |
+-------+ | + +---------+ | = +---+-------+ |
| | | | | | | \ | +
| | + | | + | \ |X |
| | / | | / | Z + |
+-------+ | | / + \ | +
+---------+ | \ |Y /
| \ | /
+-------+---+
Figure 5
And from this analysis, Fermat would easily conclude that
if the length of the Sides of a Perfect Cube are equal to
that of a Perfect Square, when 'N = 3', then the Area of
Cube is given by;
12. L x W x T = Area
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Hence, he would have also known, 'if the Area of a Cube, as given
by equation 12, the Sides of the Perfect Cube are equal to that of
a Perfect Square', then when 'N = 3', the Sides of the Perfect Cube
must also be equal when the change in equation 12 is given by
equation 13;
13. L x W x T = Area = X x Y x R = Z^3
In other words, If the Root of Z^3 is equal to (X + Y), then
the Area of a Perfect Cube, which inscribes another Perfect Cube
is equal to the equation given by;
14. (X + Y) x (X + Y) x (X + Y) =
(X + Y) x (X^2 + 2XY + Y^2) =
X^3 + 3YX^2 + 3XY^2 + Y^3
Furthermore, he would have quickly noticed that a Perfect
Cube has 8 90 degree Angles forming its boarders, or 4 pairs
of 3 dimensional Right triangles, Prisms having 5, 2
dimensional face. This he would have reasoned further, meant
that, only a Pyramid could have 4 equal lengths measuring its
sides. In other words, Fermat would have quickly concluded
that, it is not possible for either any one of the 8, or 4
pairs of Right Triangles forming the boarders of a Perfect
Cube, could have equal sides, and still be a Right Triangle.
Needless to say, he would have also known that this did not
mean that the Sum of the Areas of these 3 dimensional Right
Triangles did not equal the Area of a Perfect Cube.
Nevertheless, he would continue to follow the logic from the
conclusions involving 'N = 2' by first, confirming the formula
for the Area of a 3 dimensional Triangle, to determine if the
Sum of the Areas of Two Perfect Cubes is equal to the Area of
another Perfect Cube. However, he would eventually notice,
that while the Volume and the the Area of a Perfect Cube were
equal formulas, the Volume and the Area of a 3 dimensional
Triangle, or Prism, represented 2 different formulas. Where
by, the Area of a 3 dimensional triangles is given by equation
13a, the Volume of the same Triangle is given by equation 13b;
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14a. Area of a Prism = A = 2(b^2) + 3b(h),
where b^2 = Area of base,
3b = b + b + b = Perimeter of base,
and h = Height of the Prism
14b. Volume of Traingle = V =
Area of the Base (B^2) x the Height (h) =
b^2h = b^2(h) = B^2 x h,
V = b^2(h)
Clearly, while an argument can be made regarding the
difference between the formulas in equations 14a and 14b,
which represents the two distinct results that respectively
measure the 'Area of a Prism' and the Volume of 3 dimensional
Triangle. Even still, Fermat would have probably continued to
follow the logical patterns reasoning derived from the
conclusions when 'N = 2', because he could quite easily test
for the conclusions that would verify either one, or both of
these formulas. Thus, following the logical reasoning
concluding equations 6, 7, and 8, in an attempt to derive the
results that would conclude the Perfect Cube, which logically
concludes results similar to that involving equations 9 and 10.
Needless to say, I am hard pressed to imagine, but I seriously
doubt that Fermat was surprised by his discovery, when trying
to confirm equations 14a and 14b, that there are actually 5
different formulas, which must be used in the logical analysis
that would determine the validity of; 'The Sum of the Areas /
Volume of Two Perfect Cubes are equal to the Area, or Volume of
another Perfect Cube'. In any case, it should be understood
that the Cubes of the 'X, Y, and Z' variables must be Positive
Integers, because their respective Cube Roots must be a
Positive Integer. Where by, given below, we have;
15. [(X + Y) x (X + Y)] X (X - Y) = X^3 + X^2y - XY^2 - Y^3
16. [(X - Y) x (X - Y)] X (X + Y) = X^3 - X^2y + XY^2 + Y^3
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17. [(X - Y) x (X + Y)] X (X - Y) = X^3 - X^2y - XY^2 + Y^3
18. [(X - Y) x (X + Y)] X (X + Y) = X^3 + X^2y - XY^2 - Y^3
And since by Definition;
Exponent: Any symbolic representation, 'Q', which
is used in conjunction with the Number, 'X',
representing a Multiplicand, represents
the count of the number of Identical
Multiplicands used in the equation
representing Product of Q Multiplicands;
x^Q = (Xv1 x Xv2 x Xv3 x ... XvQ).
Hence, given by equation 19, we have;
19. [(X - Y) x (X - Y)] X (X - Y) = X^3 - 3X^2y + 3XY^2 - Y^3
Clearly, once Fermat realized, upon inspection of equations
14a through 19, that neither the Sum of the Areas, or the
Volumes of the Right Angled Prisms forming the Perimeter of
the Perfect Cube equaled the factors from equation 12,
'3X^2Y + 3XY^2', whose difference would yield the same
conclusions established by equation 18. He would have
reasoned that, 'The Sum of either the Area, or the Volume of
Two Perfect Cubes did not equal another Perfect Cube',
because there is a divergence diminishing the equality
between factors in noted equations. And further testing, he
would have reasoned, the divergence diminishing the equality
between factors increases for every unit of increase of the
Exponent, 'N'. Hence, he would finally concluded, since
(2 + 2) = (2 x 2), "There are No solutions in Whole Numbers
to the Equation, X^N + Y^N = Z^N, when N > 2", because the
Operation of Multiplication, M, is equal to the Operation of
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Addition, A, M = A, only when the number Variables involved
in each of these operations, is equal to TWO. And the
translation, or interpretation of this conclusion yields;
'The Whole Number sought cannot be equal to the Cube Root of
the Area of a Perfect Cube which is equal to the Sum of the
Areas of Two Perfect Cubes, because then it will equal the
Square Root for the Area of a Perfect Square, when it equals
the Product of Two Equal Whole Numbers'.
And since an equation of Multiplication is equal to an equation
of Addition only when each of these operations involves two
variables, then only an equation equaling Sum of the Two
Variables equal to the Two Perfect Squares can equal the Product
of the Two equal variables that is equal to a Perfect Square'.
In which case, there is no Integer that can equal the Nth
Root of the Nth Power that is equal to the equation of the
Sum of Two Nth Powers. "In other words, since an equivalency
between the Operations of Multiplication and Addition only
exists at Power of 2 (denoting the number of Variables
involved in both of these operations), then only the Sum
of (in this case; Two) Perfect Squares can equal the
product of the two equal multiplicands, which is equal to
another Perfect Square, and still retain an integer
solution for the values of the Variables representing
Power of the Exponent and the respective Roots".
Note: I investigated the same conditions, in the proof
entitled; "The Proof of Fermat's Last Theorem; The
Revolution in Mathematical Thought". However, I
concluded, from the same data, that "If 'N > 2' in
the equation, X^N + Y^N = Z^N, then there are no
Whole Number Solutions for the Nth Power of the
Sum of Two Nth Powers and their respective Nth
Roots". That is, because there is No incremental
(Additive) progression using ' 1's ' defined for
Fermat's / Pythagoras Equation, the Integer
Coefficient, which is the Common Coefficient
between the Powers of N and their respective Nth
Roots do not exist. Nevertheless, this concludes
the rendering of the proof, that I believe, Fermat
understood to be True.
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Nevertheless, from the analysis of the forgoing conclusions,
and the realization that equation 8 and the equation from
"Fermat's Last Theorem", represented a special case defining
the 'Distributive Law', as given by equations 19 through 24.
That I also understood the profound meaning of the proof of
"Fermat's Last Theorem". In other words, I reasoned first;
'Any complete proof of "Fermat's Last Theorem" must be
founded upon the 'Distributive Law', and conclude with the
discovery of a New 'Distributive Property'. And this meant
that when 'N > 2' in the equation, X^N + Y^N = Z^N, the
Multiplication was not equally Distributed over the
operation of Addition. Hence, from the results of equations
19 through 24, it is was easy to conclude, since the
Operation of Multiplication is not equally Distributed over
Addition in the case where 'N > 2': There is no Common
Coefficient between the Nth Power of the Sum of Two Nth
Powers, and their respective Nth Roots. In which case, I
concluded that Fermat was correct, and had the knowledge of
proof I demonstrated above. However, Fermat's mathematical
background lead me to conclude that he did not understand
fully the implication of his riddle, because I believe, if
he did, he would made a pointed reference.
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Special Case of the Distributive Law is the conclusion of
Equation 25:
20. (X - Y)^2 =
(X - Y) × (X - Y) =
X^2 - 2XY + Y^2
21. X^2 + Y^2 =
2XY =
XY + XY
22. (X + Y)^2 =
(X + Y) × (X + Y) =
X^2 + 2XY + Y^2
23. X^2 + 2XY + Y^2 =
2XY + Z^2
24. X2 + Y2 =
Z^2 + 2XY - 2XY =
X^2 + Y^2 = Z^2
25. Z^2 = 2XY:
hence, X^2 + Y^2 = Z^2
X^2 + Y^2 = 2XY
X^2 + Y^2 = XY + XY = X(Y + Y)
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Furthermore, because the conclusion from the proof and the
equation involved in "Fermat's Last Theorem", represented
an Algebraic Expression of the Exponential Function
concluding the existence of the 'Distributive Law for
Exponential / Non-Linear Functions. I knew, or reasoned,
since the Distributive Law is also logically valid in
'Set Theory', that an Exponential Expansion of the
Mathematical Logic of Set Theory must also sustain logical
validity, and conclude the logical support for the
conclusions derived from the foregoing proof: The Discovery
of a New Distributive Property. Still, the clarification and
definition of the Exponent, and the Exponential Operations
employed in the Mathematical Logic of Set Theory, required
more precise definitions of the familiar operations involving
Addition, Subtraction, Multiplication, and Division. In other
words, the Exponential Expansion of Set Theory, which also
logically sustains only the operations of Addition and
Subtraction, nearly mirrors the proof of the 'Distributive
for Exponential / Non-Linear Functions'. And the Exponential
Expansion of the Field Postulates, concluded the existence
of the Mathematics of Quantification, which is defined as a
Finite Mathematical Field, conditionally closed over the Set
'R' for the Operations involving Addition, Subtraction,
Multiplication and Division.
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The Definitions
Multiplication: The Quantified Sum of the equal distribution of
the Multiplicand, which is equal to the Addend that is used in
the Summation of the equal Addends, which are equally
distributed by a factor equal to the other Multiplicand that is
used in the equation representing a product. "Hence,
Multiplication is the Quantified Sum of Addition"
5 x 14 = (14 + 14 + 14 + 14 + 14)
= (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5)
= 70
Division: The Quantified Difference of an ever changing
Dividend, which becomes the Subtrahend that is used in the
repeated Subtractions performed on a Constant, which is the
Divisor the becomes the Minuend in the equation. "Hence,
Division is the Quantified Difference of the Repeated
Subtraction performed on a Constant, which results in the
Count of the Total Number of Parts Contained in the Whole.
18/2 = 9, and Nine Subtractions of 2 from 18 equals;
'(((((((((18 - 2) -2) -2) -2) -2) -2) - 2) - 2) - 2)'
Addition: The mathematical operation representing a
Summation, indicating a growth, or an increase in the
number of the members contained in the Whole, by the
inclusion of new members: The Union of Sets; 'U'.
Subtraction: The mathematical operation representing a
Difference, indicating a depreciation, or a reduction in
the number of the members contained in the Whole, by the
exclusion of members: The Disunion of Sets; 'BarU'.
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The Definitions
Disjoint: If there are two sets, A and B, such that, A and B
share no common members, then the two sets are said
to be Disjoint; A ñ B, (read; A is not connected to
B: 'A ñ B = Ø'.
Dis-Union: If A U B = C and C n A = A is true, then the
Dis-Union of the Set A from the Set C,
C BarU A = B, (read; C dis-union A) is the
exclusion of the members from the Set C, which
are common to the Sets C and A, iff, A ñ B = Ø.
2. If A Not= C and C n A = B, then C U A = E ñ D.
3. If every Set A is a Sub-Set of itself, and
A n A = A, then A U A = Ø.
Exponential Cardinal: If for every X, where X E U, there is a
condition, such that;
X n X = X,
X n X n X = X,
(Xv1 n Xv2 n Xv3 n ... n XvQ) = X, and
X^Q = X is True.
Then there is a Exponential Number, Q, called
the Exponential Cardinal of X, which is the
number that represents the occurrences of X in
the equation representing its Intersection.
Set: If a Unit Whole contains a collection of Objects, and
each Object defines, one and only one, Part belonging
to the Unit Whole, then the Unit Whole defines a Set
as a Collection of Objects, iff, each Object defines
one and only one Element, or Member, that defines the
Part belonging to the Unit Whole.
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The Theorems
Sub-Set: If every element, E, of a Set B is a E of the Set A,
then the Set A is said to contain every E of the Set
B, and the Set B is said to be a Sub-Set of the Set
A. Hence, every Set is a Sub-Set of itself, iff,
A n A = A.
Cardinal Number: If it may be concluded that the Multiplicative
Identity Law is True, and X x 1 = X, where X
does not change, then from Set Theory, X is the
Multiplicative Identity of Itself. And if this
defines X, when X = X^Q, then X defines the
Identity Element as the Unit Base, or the
Cardinal Number = 1 defines the Common
Coefficient as the Multiplicative Identity
Element for all X| X E U.
Therefore, if {Uv1 n Uv2 n Uv3 n ... n UvQ} = U^Q = U^Qv{N} = U,
and given that Multiplication is the Quantified Sum of Addition,
where X^Q = U^Qv{N} is True. Then for all X| X E U = U^Qv{N} = UvN,
the Cardinality of any Set UvN, is the Sum or Union of Cardinal
Numbers, or
UvN = {Xv1 U Xv2 U ... U XvQ} = (1v1 + 1v2 + ... + 1vQ),
iff, for all X| X E U, X = 1 defines the Cardinal Number for the
Element of every Set as a Sub-Set of I | I = Set of Integers.
In which case, the Unary Set, {1}, defines the Cardinal for the
Element X of the Set I for all X| X E I, given that I = {X},
when X = 1, and the Cardinal for every Element X of the Set I
for all X| X E I, when
I = {X, X, X ... X}, and X = 1, I = (1v1 + 1v2 + 1v3 + ... + 1vQ).
Hence, the definition of a Cardinal Number is given by:
Cardinal Number: The Cardinal Number is the Multiplicative
Identity Element for all X| X E I, which
represents the Element of the Unary Set
that is used to determine the Cardinality
of every Set from the Sum or Union of the
Multiplicative Identity Element for
every E X of the Set I: iff X^Q = X.
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Note: This defines the Unit Base X, for all X| X E I as the
Element of the Unary Set, because X is the
Multiplicative Identity of Itself that defines, X = 1.
[The next proof presented, is the interpretation of the Proof,
or implications, that Fermat never understood, or could not
explain. This is the accepted rationalization because Set
Theory, the complete Logical Model of Mathematics, was not
finished for nearly 200 years later. However, because he
Co-Discovered the Cartesian Coordinate System representing
the Mathematics of Analytic Geometry. The mathematical
relationships from the foregoing, he should have maintained
an above average understanding of the foundational theory of
the proof presented. Still, for me, these results initially
implied the existence of: the 'Distributive Law for
Exponential / Non-Linear Functions'; an alternate
Mathematical Field that was Finite and Closed / True as
defined by the Axiom for Equality, the Field Postulates,
and Set Theory. In which, it was later discovered, actually
defined the Binary Set and the {Binary Enumeration &
Mathematics} Mathematics of the Binary System.
e. Terrell 1983]
Nevertheless, since the foregoing conclusions proves that
because the 'Multiplicative Identity Element' defines the
Universal 'Common Coefficient', which is the same for all
Objects, as the element, 1, defined in the Unary Set. And
since it may also be concluded that counting is actually
the assignment of a '1' to every object to be counted, and
then, adding the "1's" that represent the objects,
determines the Cardinality of the Set containing the objects
being counted. Clearly, if the Set I, the Set of Integers
defines the Set of all Symbols used to represent the result
of the addition, inclusion, or incremental progression using
the element, 1, defined in the Unary Set (given by Table II),
then the (Arabic Numerals / Positive Integers) Modern System
of Counting is defined by the Unary Set: As a Unary System.
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In other words, since the Cardinal Number, by definition,
must define the Neutral Multiplicative Identity Element that
represents the Unit Base X of X^Q, then any change in the
Count of the Number of Members contained in the Set X, must
define the Union (or Sum) of the members belonging to the
Disjoint Set representing the Set Xv[2 thru N], iff X = X^Q,
the Cardinality of the Set equals the Sum of the Cardinal
Numbers representing each of the its Members.
In which case:
If the Unit Base X of X^Q is defined ONLY when
X = XvN = X^Q remains valid, and;
I. 2 Members in a Binary Set =
(A U B)^Q = Xv[2 = (A U B)] = X^Q, or
II. 3 Members in a Ternary Set =
(A U B U C)^Q = Xv[3 = (A U B U C)] = X^Q, or
III. 4 Members in a Quaternary Set =
(A U B U C U D)^Q = Xv[4 = (A U B U C U D)] = X^Q, or
IV. N Members in a N-nary Set =
(A U B U ... U NvN)^Q = Xv[N = (A U B U ... U N)] = X^Q,
is TRUE,
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THEN:
I.a 2 Members in a Binary Set =
Xv[2 = (A U B)] = X^2 = X^Q, or
II.a 3 Members in a Ternary Set =
Xv[3 = (A U B U C)] = X^3 = X^Q, or
III.a 4 Members in a Quaternary Set =
Xv[4 = (A U B U C U D)] = X^4 = X^Q, or
IV.a N Members in a N-nary Set =
Xv[N = (A U B U ... U N)] = X^N = X^Q,
Must also be TRUE.
In other words, the Proof for the existence of any Numbering
System involving the Unit Base X of X^Q, would conclude the
definition for the existence of another system of counting.
And this defines a Unit Base X of X^Q containing more Base
elements than Unary System, as the UNION of More than One
Element; Confirms Fermat's Last Theorem only for the Binary
System for all N > 2. That is, given by the foregoing proof
of Fermat's Last Theorem, which is translated into the rigor
from the Mathematical Logic of Set Theory, and confirms the
Conditions for;
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( A^nN U B^nN ) = (A U B)^nN; given below, we have:
X Y X Y
+---+-------+ +---+-------+
| \ | / /|
| \ |X / / |
| Z + +---+-------+ |
+ \ | | \ | +
| \ |Y | \ |X |
| \ | | Z + |
+-------+---+ + \ | +
| \ |Y /
| \ | /
+-------+---+
Figure 10
If for all X | X E I, X = X for every XvU = X^Q, and when
X = XvU there is a XvN | XvN = X^Q, which also True for all
X | X E I for every X = XvN when X = X and
XvN = (A U B U C U ... U N), then XvN = XvU,
if and only if (iff):
X^QV{U} = XvU = X^Q = 'X' = X^Q = XvN = X^Qv{N},
or XvN Not= XvU, because X Not= XvN.
Proof: Since the Theorem concluding the definition for the
Cardinal Number defines the E of Unary Set as the
Unit Base X of X^Q for all X | X E I, then the
Multiplicative Identity Element for all X | X E I
defines XvN = XvU when X = XvU.
Therefore, when XvN = XvU,
and N = 2 = Q, X n X
= X^[Q = 2] = (A U B) n (A U B)
X^[Q = 2} = (A U B) n (A U B)
= (A n A) U [(A n B) U (A n B)] U (B n B)
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If for all X | X E I, X = X for every XvU = X^Q, and when
X = XvU there is a XvN | XvN = X^Q, which also True for all
X | X E I for every X = XvN when X = X and
XvN = (A U B U C U ... U N), then XvN = XvU,
if and only if (iff):
X^QV{U} = XvU = X^Q = 'X' = X^Q = XvN = X^Qv{N},
or XvN Not= XvU, because X Not= XvN.
Proof: Since the Theorem concluding the definition for the
Cardinal Number defines the E of Unary Set as the
Unit Base X of X^Q for all X | X E I, then the
Multiplicative Identity Element for all X | X E I
defines XvN = XvU when X = XvU.
Therefore, when XvN = XvU,
and N = 2 = Q, X n X
= X^[Q = 2] = (A U B) n (A U B)
X^[Q = 2} = (A U B) n (A U B)
= (A n A) U [(A n B) U (A n B)] U (B n B)
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II.a 3 Members in a Ternary Set =
Xv[3 = (A U B U C)] Not= X^2 Not= X^3 Not= X^Q Not= X, or
III.a 4 Members in a Quaternary Set =
Xv[4 = (A U B U C U D)] Not= X^2 Not= X^4 Not= X^Q Not= X, or
IV.a N Members in a N-nary Set =
Xv[N = (A U B U ... U N)] Not= X^2 Not= X^N Not= X^Q Not= X,
The Distributive Law for The Distributive Law for
Non-Linear Functions Linear Functions
Binary Set Unary Set
\ /
+----|+++|----+
| |+++| |
| |+++| |
| |+++| |
| |+++| |
| |+++| |
+----|+++|----+
/ | \
/ | \
+---------------------/ v \--------------------------+
The Position or Point of Intersection between the Two System
of Counting (Number Sets) defines a Special Case of the
Distributive Law (The Intersection between the Binary and
the Unary Sets) for Positive Integers.
+------------------------------------------------------------+
Figure 11
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Nevertheless, these conclusions confirm the existence of the
Two Systems of counting defining; 'The Unary Set' and 'The
Binary Set', they also support the conclusion defining these
Sets, by Figure 11, as; 'The Infinite Set = Unary System'
and 'The Finite Set = Binary System'. Furthermore, it should
be clearly understood:
When X = (A U B), X defines the Binary pair {a, b}
And reasoned further that if either 'a', or 'b' is equal to
the Null Set {Ø}, then the foregoing conclusions would be
invalid. Moreover, since the Cardinal Number, the
Multiplicative Identity Element of the Unary Set, is same
for Binary Set, the Binary pair, {a, b}, must represent, by
Figure 12, a unique combination of the Binary Pair
incrementing in units of '1', which defines the Cardinality
of any Set, also defined by the Unary System.
+---------------------------------------------------------+
| The Combinations generated using the Binary Pair; {A,B} |
+---------------------------------------------------------+
| Binary Set Unary Set Positive Integers |
| |
|{A,A} 0r {a, a} = 1 = 1 |
| |
|{A,B} 0r {a, b} = 11 = 2 |
| |
|{B,A} 0r {b, a} = 111 = 3 |
| |
|{B,B} 0r {b, b} = 1111 = 4 |
| |
+---------------------------------------------------------+
Figure 12
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In other words, from the definition of the Cardinal Number,
the Cardinality of the Unary and the Binary Sets represents a
1 : 11 ratio, which denotes the number of Elements each Set
contains. Nevertheless, the defining expression representing
this relationship given by;
'Unary Set = 1', 'Binary Set = 11', or '1 = 2' - 'Prime Numbers'
Note: A 'Prime Number' or 'Prime Integer', is a positive
integer, 'p is Greater Than or Equal to 1', that has
no positive integer divisors other than itself, 'p',
and '1'.
And if, from the Substitution Law for Equality; {0, 1} = {a, b},
where '1 = {00}, and {00} is Not Equal to {Ø}', then the correct
Binary System and its associated method for enumeration, given by
Table IV, confirms '11111111 = 256 = 2^8, because
2^8 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 11111111 = 256'.
Hence, the definition of the Cardinal Number, by figure 11,
defines the special case of the Distributive Law as the
intersection of the Distributive Properties defining the Binary
and the Unary Sets, for all X | for every Element of I, the
Cardinal Number X, defines the Cardinality of both Sets.
2.2 The Mathematics of Quantification and Binary Arithmetic
System
It should be clearly understood that the forgoing conclusions,
and the new definitions and theorems from the Logic of the
Mathematics of Quantification, defines the closure Laws for
the operations of Subtraction and Division. And this completes
the Set of Laws defining the operations of Addition,
Multiplication, Subtraction, and Division, which governs the
Mathematics and the Mathematical Logic defined by Set Theory,
the Field Postulates, and the Axioms for Equality. That is,
given by Table V, we have:
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TABLE V
+------------------------------------------------------------+
| AXIOMS for EQUALITY |
| |
| |
| Fundamental Law for Equality: A + (-A) = 0 |
| Additive Identity Law for Equality: A + 0 = A |
| Multiplicative Identity Law for Equa;ity: A = 1 = A |
| Common Coefficient Law or Equality: A/1 = A = A x 1 |
| Substitution Law for Equality: If A = B, and B + C = D, |
| then A + C = D |
| Reflective Law for Equality: A = A |
| Transitive Law for Equality: A + B = B + A |
| |
+------------------------------------------------------------+
Table VI (Addition)
111 = 8 1111 = 16 11111 = 32
110 = 7 1010 = 11 10110 = 23
_____ 15 _____ 27 ______ 55
1110 11010 110110
111111 = 64 100 = 5 1000 = 9
101011 = 44 100 = 5 1000 = 9
______ 108 ____ 10 _____ 18
1101011 1001 10001
Table VII (SUBTRACTION)
111 = 8 1111 = 16 11111 = 32
110 = 7 1010 = 11 10110 = 23
____ 1 _____ 5 ______ 9
00 = 1 100 1000
111111 = 64 100 = 5 1000 = 9
101011 = 44 100 = 5 1000 = 9
______ 20 ____ 0 _____ 0
10011 0 0
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2.3 The Binary and Ternary Systems and George Boole's
Mathematical Logic
It should readily be concluded, because it has been mentioned
that the Boolean, or Leibniz, Operators are Unary; they are
both logically valid for the Unary and the Binary Systems.
Furthermore, since Zero, Ø, or the Null Set, is not defined
by the Cardinal Number, which is equal to the Unit Base X of
X^Q for all X| X is an Element of I, then Ø, is not an element
of the Set of Integers, 'I'. Hence, Binary and Ternary Logic,
or 3 State Logic is defined by the Unary Set, and contains the
elements {Ø, +1} and {-1, Ø, +1}, which are governed by the
Closure Laws. Given by Table VIII, we have;
TABLE VIII
Logical States of the Unary and the Binary Systems
+---------------------------+---------------------------------+
| BOOLEAN BINARY STATES | BOOLEAN TERNARY STATES |
+---------------------------+---------------------------------+
| | |
| | |
| { 0 } { + 1 } | { - 1 } { 0 } { + 1 } |
| | |
| No Yes | No Nothing Yes |
| | |
| False True | False Undecided True |
| | |
| Open Closed | Open By-Pass Closed |
| | |
| Stop Forward | Reverse Stop Forward |
| | |
| Two State Switch | Three State Switch |
| | |
| | Or, Flip-Flop 4 state Switch |
| | |
+---------------------------+---------------------------------+
Note: It should be understood nevertheless, these conclusions
confirms that the Binary System is Finite and Closed
over 'R' (not true for all values of the Base Variables
over 'R' - The Logic of Set Theory), and the Unary System
is Infinite, and it is also Closed over 'R'.
[VIMP - e. terrell Nov. 1979 to Aug 1983]
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3. Security Considerations
This document, whose only objective was the deliberation of
the final explanation of the new foundation for the Binary
System, which resulted from the Mathematics of Quantification,
does not directly raise any security issues. Hence, there are
no issues that warrant Security Considerations.
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4. IANA Considerations - 'Resolution of the Counting Error in the
Binary System'
I. IPv4 Address Loss Table
Exponential Binary IPv4 Address
Enumeration Representation Specification
+--------------------+------------------------+------------------+
| |
1. 0^0 = 0 | 0 | 0
| |
2. 2^0 = 1 | 00 = aa | 0
| |
3. 2^1 = 2 | 01 = ab | 1
| |
4. 2^F = 3 | 10 = ba | 2
| |
5. 2^2 = 4 | 11 = bb | 3
| |
6. 2^F = 5 | 100 = baa | 4
| |
7. 2^F = 6 | 101 = bab | 5
| |
8. 2^F = 7 | 110 = bba | 6
| |
9. 2^3 = 8 | 111 = bbb | 7
. | . | .
. | . | .
. | . | .
129. 2^7 = 128 | 01111111 = bbbbbbb | 127
. | . | .
. | . | .
. | . | .
257. 2^8 = 256 | 11111111 = bbbbbbbb | 255
+--------------------+------------------------+------------------+
Totals: 256 | 256 = 256 | 255
+--------------------+------------------------+------------------+
IPv4 Address Loss using an askew Binary System;
256^4 - 255^4 = 66,716,671 IP Addresses
+--------------------+------------------------+------------------+
II. Using Extended ASCII CODE & Binary '00' = 1
In the Extended ASCII CODE character Set, True Zero is
defined as the Null Set Character, ' Ø '. However,
because Binary equivalent of ' 1 ' is ' 00 ', I believe
that it would be easier if the Character Set were changed
to represent the Binary equivalent of ' 1 ' as ' 0 ', as
opposed to '00', because '00' is 2 Bits and '0' is '1' Bit.
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Exponential System
Binary System Zero of Counting
--------------------+-----------------+-----------------------
No Definition 0 0^X = 0 = 0EX
1. 00 = aa No Definition 2^0 = 1 = 2E0
2. 01 = ab No Definition 2^1 = 2 = 2E1
3. 10 = ba No Definition 2^F = 3 = 2EF
4. 11 = bb No Definition 2^2 = 4 = 2E2
: : :
: : :
8. 111 = bbb No Definition 2^3 = 8 = 2E3
9. 1000 = baaa No Definition 2^F = 9 = 2EF
10. 1001 = baab No Definition 2^F = 10 = 2EF
[Given that: E = Exponential Operator; F = Variable Irrational Number;
and X = Any Variable defined as a Member of the Real Number Set]
III. Equating the Exponent from a Base 2 Exponential
Operation to the Binary Translation that
Equals the Result *
More importantly, when rationalizing these conclusions, their
validity becomes even more evident when any mathematical
comparison between the 'Bit-Mapped' Lengths, or Displacement
of an IP Address, is made with the Equation representing the
Total Number of Available IP Addresses - the Address Pool
representing the Addressing Specification; e.g. IPv4, or IPv6.
That is; If the Bit Length is Equal to 32, in the IPv4
Specification, or 128 Bits in the IPv6 Specification,
and their respective Address Pool Totals is given by:
IPv4 = 32 Bit Length (Bit-Mapped Displacement)
32 Bit = 2^32 Address Pool Total
2^32 = 4,294,967,296 IP Addresses
IPv6 = 128 Bit Length (Bit-Mapped Displacement)
128 Bit = 2^128 Address Pool Total
2^128 = 3.4028236692093846346337460743177e+38 IP Addresses
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Then it becomes quite obvious that the Total Number of IP
Addresses available in the Address Pool for either the IPv4,
or the IPv6 Specification, is a function of the Address's
Bit-Mapped Displacement, or Bit Length. In other words, a Bit
Length Regression to Progressively smaller Address Bit-Mapped
Displacement Units, just as the foregoing conclusions revealed,
accounts for the total number of available IP Addresses in the
Address Pool - and this also determines, equals, and represents,
the exact number of Bits equal to the Number representing the
IP Address Pool Total. In other words, this Number or Integer,
which equals the Result from an Exponential Base 2 Operation,
has a Binary Translation that is equal to the value of the
Exponent in the Equation.
Hence, Enumerating, or Counting using only the Exponent reveals:
1) An 8 Bit-Mapped Length = 2^8 = 256 IP Addresses = 256 = 11111111
2) A 7 Bit-Mapped Length = 2^7 = 128 IP Addresses = 128 = 1111111
3) A 6 Bit-Mapped Length = 2^6 = 64 IP Addresses = 64 = 111111
4) A 5 Bit-Mapped Length = 2^5 = 32 IP Addresses = 32 = 11111
5) A 4 Bit-Mapped Length = 2^4 = 16 IP Addresses = 16 = 1111
6) A 3 Bit-Mapped Length = 2^3 = 8 IP Addresses = 8 = 111
7) A 2 Bit-Mapped Length = 2^2 = 4 IP Addresses = 4 = 11
8) A 1 Bit-Mapped Length = 2^1 = 2 IP Addresses = 2 = 01
9) A '0' Bit-Mapped Length = 2^0 = 1 IP Address = 1 = 00
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So, how then is it possible for anyone to use an Askew
Binary System of Counting, when the Exponent representing
the Bit-Mapped Displacement in the Base 2 Exponential
Equation, equals the Binary Translation representing the
" Equation's " Result?
- The Binary Translation Comparison Table -
Computer Operating Systems, Electronic..., and Software is Wrong!
4 = 100 - Binary Translation: How...? When 2^2 = 4 = 11
3 = 11 - Binary Translation: How...? When 2^F = 3 = 10
2 = 10 - Binary Translation: How...? When 2^1 = 2 = 01
1 = 01 - Binary Translation: How...? When 2^0 = 1 = 00
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IV. Binary Zero { 00 } Representing an Irrational Number...??
If every Base 2 Exponential Equation representing the Product
of 2 or more Identical Multiplicands, defines the Result as a
Function of the Square Root of 2 when Binary '00' = 1. Then,
from the "Proof of Fermat's Last Theorem, and the Mathematics
of Quantification; when "00" = 1, "00" defines an Irrational
Number, which is a Member of the "Real Number Set" - Where by;
' IF ' " 00 " = 1 is True, then;
X( 0 + 0 ) = ( 0^2 + 0^2 ) = 1
= (2^1/2)/2 [((2^1/2)/2) + (2^1/2)/2)]
= (2^0.5)/2 [(2^0.5)/2) + (2^0.5)/2)]
= (2^0.5)/2)^2 + (2^0.5)/2)^2
" 1 " = (0.707106)^2 + (0.707106)^2
" 1 " = 0.5 + 0.5 = X( 0 + 0 )
Where, if " 1^0.5 " = " 1 ", and " F = 0 "; then
" F = Variable Irrational Number ". Hence;
(2^0.5)/2 = 0.70710678118654752440084436210485
0.5 = (0.70710678118654752440084436210485)^2
" 1 " = ( F^2 + F^2 ) = 2^0 = " 00 "
[ * - See page 41; Figure 12; [12]; Exponential Cardinal page 32
- Note; 2^0.5 = ' The Square Root of 2 ' ~ 1.4142135623731 ]
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V. ‘Obsolescence’ of the 'HEX' System with the 'Base
2 Binary Exponential' System of Counting; 2EX
Aides from (also) being an 'askew' system of counting, the
inefficiency of the HEX System of counting becomes quite obvious
when using the 'CIDR Network Descriptor', as outlined by the
"Work(s) in Progress" [12]; 'The CIDR Network Descriptor expands
the size of the IPtX Address Space beyond the IPv6 IP Addressing
Specification". In other words, the ancillary discrepancy is that,
it cannot be used in performing Mathematical Calculations, because
it 'Can-Not' accurately represent the 'Exponent' (the Bit-Mapped
Displacement), nor define the Numeral equaling the Bit-Mapped
Length; 'Exponential System of Counting' [page 47]. That is, the
HEX System of counting can only be used to depict (or represent)
the Numeral prior to converting it to the equivalent Binary
Representation.
And more importantly, because the Binary Base 2 Exponential System
can represent any Irrational Number(s), which includes Decimal
Fractions, it can be used in All Mathematical Calculations. In
other words, Binary Zero has a numerical value (as noted - See
[IV.]), which is not defined by the conversion from the HEX
System of Counting, rendering Bit-Mapped Displacement for a Binary
Numeral. Hence, the Binary Base 2 Exponential System of Counting,
is not only the suitable replacement for the HEX System, but it
is the appropriate system, which should be used to represent
[every 2 State System defined in Nature as representing the
Binary Pair, {0,1}] the Binary operations defining the Computer.
And this, more notably means it can Bit-Map exactly the Frequency
of any Transmission Signal, and every Frequency defined by the
Electromagnetic Spectrum.
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Note: Microsoft's Windows Calculator, and others, is wrong.
- The Operating Systems and Software of Microsoft, Cisco,
IBM, Wolfram, and others, who use the HEX System of
Counting, are also wrong; there is No Conversion with
the Base 2 Exponential Equations defining the Binary
System.
- And this includes every Electronic Device / Component
- In other words, using the HEX System of Counting does
not change anything, because it maps to the current
Binary System- it is also an Askew System of Counting.
In which case, any measurements derived from its use in
any Calculation(s) will be Wrong... And if, the Trial
and Error Tests cannot be performed, or fail, lives
could be Lost as a direct result...
- The Irony? Today’s Authority in Mathematics maintains;
Isaac Newton was a great Mathematician who invented
Calculus. The truth however, is that; 'There was never
a Conflict of Plagiarism, between Newton and Leibniz,
which involved the discovery of Calculus: A Ruse.
Newton hated Leibniz, because Leibniz proved that the
Mathematics involving Newton's Laws of Motion was
wrong!' A fact, nearly a 100 years later, that was
proven to be true by several noted mathematicians,
which includes "Emilie de Breteuil du Chatelet".
Even still, this marks only the beginning of Newton's
failures, because not all of his Mathematical and
Scientific Research could be interpreted by the
Mathematicians and Physicists during this period. In
other words, there are additional flaws, not only with
his interpretation of Galileo’s Research, but, in the
Logical Foundation of Calculus, the Mathematical System
he is accredited for inventing.
Consider, for example, Newton's Third Law of Motion,
which states;
"For Every Action there is an Equal and Opposite
Reaction."
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The problem however, is that, this is a Law defining an
'Action Reaction Event' that is not (as such) defined in
Nature. That is, "Unless Acted on by an Additional Force"
(i.e. "Acted Upon" by another Force), this represents an
'Action Reaction Event' that does not occur normally in
Nature (or anywhere in the Universe).
Now consider the Opposing Argument:
'When a Ball, a Rock, or Rain for that matter, falls in
a Pool of Water - What happens? A Wave Front is formed,
which travels in all directions, forming a Circle of
Propagating Waves that diminishes in size, over time,
until the Wave Front vanishes [fig c.]. So - For
Newton to be correct, as given by fig a., a Ball
rolling Down, then Up an Inclined Plane, must have
Equal displacement(s) for both Planes, and Frames of
Reference. In fact, it does not matter whether or not
the Ball's Motion is on an Inclined Plane, dropped into
a Pool of Water, or it is being Bounced, as a Child's
play toy, because the Principles of Physics are still
the same'.
In other words, from the Logic of the 'Mathematics of
Quantification', "Equal and Opposite" means Balance, or
Equilibrium; i.e. 'No ability to Change', or 'Continuous
without Change'. And for this to exist in Nature -
Well... to put it in another way; 'No Life can exist'.
And clearly, if there were no difference, the comparison
of 'fig a.' and 'fig b.', then the Ball's Motion on the
Incline Plane would continue indefinitely, and never
stop; 'The Perpetual Motion Machine'.
In any case, while 'Vector Mathematics' concludes that
the Measurements for the 'A' and 'B' [fig a.] Planes
are Equal; it does not mean Newton was correct. In other
words, however small of a difference, the interacting
Forces involved in the Mathematical Relationship define
an 'Action Reaction Event', which measures the Interaction
between the Forces that defines the Dimensional
Measurements of the resulting Frame of Reference (Today,
it would be called; 'The Magnitude and Directional
Difference between Vector Quantities'). In which case, if
Newton were correct, then the Height of a bouncing Ball
would never change; it would be a Constant, and the Ball
would never stop bouncing [Normative References
[Physics 1.].
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(Ball Rolling Down Hill) (Ball Rolling Up Hill)
/ /
(a) / (a') / Final
Start - o O Final - o (e) / o / Position
Position |\ Position /| Start | \ \ o (e')
| \ / | Position | \ O /|
| \ / | | \ / |
| \ / | | \ / |
| A \ / B | | A' \ / B'|
| \ / | | \ / |
o-------o-------o o-------o-----o
(b) (c) (d) (b') (c') (d')
A = B A' /= B'
Not Equal to
fig a. fig b.
(The Ball's Downward Motion)
\ /
|||||
o (Wave Front) ||| o (Wave Front)
o o o o
o o | o o
o A o o B o
o o v o o
o o-----o (Water Surface Level) o
\ O /(Dropped Ball)
v
Harmonic Wave Pattern of a Ball Dropped into a Pool of Water -
yielding Equal Wave Fronts in a Circular Pattern surrounding the
Point of Impact, that diminish as the Wave Propagates outward
from this central Point.
fig c.
'Thus, the Ball's slower Upward Motion [fig b.] must define
the Loss of Acceleration as the Energy (Heat) Dissipation
(related Force) resulting from the opposing Resistance
Force (the effects from the downward Force of Gravity
'Relative to the Ball's Mass, or Mass Displacement Unit'
[the "Binary Base 2 Exponential" conversion for the Growth
in the changing values of the Ball's Opposing Inertia Rest
Mass]'), which slows the Ball's Motion.'
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- ACTION REACTION EVENT -
Action "Event" Force - (b) (y) - Action Force
| |/
| |
RF = Reaction Force | /|
| / |
RE = Reaction Event | / |
| / |
\ | / |
\ | / |
\ | / |
\ | / |
_ _ _ _\|/ (Z)
/ / Reaction "Event" Force
Action "Event" Force (a) /|\ |
| \ |
\ |
\ |
Resistance Force - (x) |
\ |
\ |
\ |
\|
fig d.
In other words, the resulting Forces defining the 'Action
Reaction Event', or the Interaction between 2 or more
Vector Quantities, must clearly define at least 3 Forces;
i.e. the Objects are 3-D, and not a 2-D Paper Drawing –
Consider for example, the 3 Dimensional Perpendicular
Relationship between Electricity and Magnetism; where the
Magnetic Force defines a Common Phenomenon resulting from
the 'Action Reaction Event' involving the creation or
occurrence of Electricity. That is, Magnetism defines a
unique Electromagnetic Frequency having a Perpendicular
flow direction that Propagates simultaneously only with an
Electric Current. However, this accompanying Perpendicular
Frequency, it should also be noted, is a Common
(Non-Magnetic) Attribute having a Frequency of Vibration
that defines the Phenomenon resulting from an 'Action
Reaction Event', which is Uniquely associated with every
Frequency defined by the Electromagnetic Spectrum.
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Proof: Defining the Problem -
Let the Ball's Mass, and the Force of Gravity, define the
Ball's Accelerated Motion; first Down, then Up an Inclined
Plane. And let the 'Action Reaction Event' define the
Ball's 'Rest Mass' as a function of its 'Rest Mass
Displacement Unit'. The measure of the Dimensional
Displacement Ball's Mass measures the Motion of the Mass in
terms of the Distance Traveled in a Unit of Time; Velocity
as a function of its Mass, or Velocity of the Mass equals
Force of the Mass - where;
MassDistance = (M x D) = X 'GramInches' = 'X MassDistance').
Then the Equation of the Ball's 'Final Position', (x), is
given by (the Variables - [fig e.]);
1.1) (y) - (c) = (x)
'Resistance Force (the Ball's)
~ Gravitational Inertia Mass'
| |
Starting Y v
Position (y) - O |
| \ | / Final
| \ | O - Position (x)
| \ | / |
| \ | / |
| \ | / |
| \45 | / |
Action "Event" Force ---> \ | / |
___________|________\ o /______|______________X__
(c)\-- Rest Mass Displacement (z)
|
fig e.
[5. Normative References - [6.] The Rudiments of Finite Physics]
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Given that;
1) the optimum 'Angle of Separation' of 45 degrees, defines
the balance of the Forces acting upon the Ball's 'Downward
Motion' on the Incline Plane also maintains a Result, when
all Variables and Parameters are Equal, which is Equal to the
Result of the Equation defining a Linear 'Action Reaction
Event'; i.e. when the given 'Angle of Separation' is Equal
to 180 degrees. ('As in a Game of Billiards, when the 'Cue
Ball' is used to HIT another Ball into a Pocket) [fig f.].'
| 'Rest Mass Displacement'
| Position of the Billiard Ball
| <--- 'Resistance Force'
'Action Force' ---> | / (x) - Direction -
- Direction - (y) O - - - - >(c) -> O - - - - - - > O
Starting Position /| (z)
of the Cue Ball / | 'Reaction Force' --->
/ - Direction -
/ Final Position of
/ the Billiard Ball
|
v
- Straight Line -
'Angle of Separation' = 180 Degree
fig f.
2) the value of the 'Unit Time', which used to determine the
Ball's 'Rest Mass' Velocity [the Ball's 'Rest Mass Velocity'
measures the Distance Ball's Mass Traveled, 'Rest Mass
Displacement', in a Unit of Time] is equal to '1'.
3) the Amount of Resistance, which defines the Opposing Force
measuring the Ball's Resistance to Motion, is equal to the
Force defining the Ball's 'Rest Mass Velocity'; where the
'Rest Mass Displacement' Position is denoted by (z). [And
it should also be noted, the conclusions derived from this
argument applies to the 2 Dimensional Perspective measuring
the displacement made by the Ball's Mass in the UP and Down
Motion on the Incline Plane. Noting more specifically, that
the Ball is Not a 'Point-Mass', its Shape, the measurement
of the 'Rest Mass Displacement Unit' has the Dimensions,
which resolve its Geometry; given that a 3 Dimensional 'Rest
Mass Displacement' equals the Ball's (3-D Mass) 'Density
Displacement'.
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where, the Ball's 'Rest Mass Displacement' = MD
MD = Mass Distance = the Ball's Mass = M
and M/1 = MD/1 = M/t = MD/t = Mass
where t = Unit of Time = 1
Thus, from the foregoing;
4) if the Ball's Rest Mass Position represents its Potential
Energy, then its Kinetic Energy defines the Resistance Force
derived from the Mass of the Ball, which defines the Minimum
amount of Force required to move the Ball a distance
equivalent equal to the Dimensions Displaced by its Mass, or
'One Mass Displacement Unit'.
5) if the Ball's Rest Mass Position defines 'Potential Energy'
as 'Static Equilibrium', then 'Dynamic Equilibrium' defines
'Kinetic Energy'.
6) if the optimum 'Angle of Separation' of 45 degrees, defines
the balance of the Forces acting upon the Ball's 'Down and
Upward Motion' on the Incline Plane, also maintains a Result
that is Equal to the Result of the Equation defining the
Vector Quantities involved in a Linear 'Action Reaction
Event' (when the given 'Angle of Separation' is Equal
to 180 degrees); then the 'Resistance Force' defined by the
Ball's 'Rest Mass Displacement Unit' is a Constant, because
the only difference between the Lines is their 'Angle of
Separation'.
And given the conclusions from the 'Mathematics of
Quantification' and the proof of 'Fermat's Last Theorem',
which provides for the condition of Equality to exist
between the Results from the 2 equations defining the
effects from the Force of Gravity on a Ball traveling on
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a Straight (Horizontal Line) Path and a Sloped
(Non-Horizontal Line) Path, as equal. [- true as well for
the Linear and the Non-Linear Lines, and the Binary and
Unary Sets.] Then, if the Slope of the Line through any
Point along the Downward Incline of the Ball's Path equals
Zero, the Return or Upward Path. The 'Zero Position', (c)
in 'fig e.', defines a 'Force of Resistance', which is
equal to the position' (c) of 'fig f.', that defines the
Ball's 'Rest Mass', or 'Rest Mass Displacement Unit' along
a Horizontal Path. - as given by 'fig g.', where the Lines
are given by;
ab = ac = bc = cb = de = ce
Now let;
the 'Cue Ball' = (a), have a 'Rest Mass' equal to the
'Rest Mass' of the 'Billiard Ball' = (c) - which is
equal to the 'Rest Mass' of the Ball = (b) rolling Down,
then Up an Incline Plane -
Y
(b) | (d)
|___________o______|_______o___________|
| /|\ (k) = (k') /|\ |
| /45 o | /|45\ |
| / | \ (k') /||| \\ |
| / | \ | / ||| |\\ |
| / | \ | / |(k)| \\ |
| / | \| / ||| | \\ |
|____/45____|___45_\/45__|||_|__45\____|__X
| (a) + /(c) + \ + \ |
(x3) /| (x2) \ (e)
/ | \
/ | (k) = (x1)
Point of Resitance
fig g.
Clearly, since the Forces and Displacement of the Balls
are Equal before an identical Mass Equivalent Force of
Resistance causes a Velocity Reduction, or a Decrease in
Acceleration of both Balls at the Position (c), then;
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(a) - (c) = (x1) and G(MD(a))/t - MD(c)/t
(b) - (c) = (k) and G(MD(b))/t - MD/t
(k) = (k'); where k' - [Read; the Compliment of k: k']
Hence; if (a) = (b),
when; ' (c) = MD/t = MD/1 = MD = M = Mass '
then; (k)^2 + (c)^2 = (a)^2; or X^2 + Y^2 = Z^2
Hence, figure b [fig b.] is the correct depiction, and
Newton's Third Law should have been written as;
If "For every Action there is a Reaction, then the
Interaction between these Forces defines an 'Action
Reaction Event', which is a Natural occurrence in Nature".
- Or - more appropriately as;
"For every Force of Action there is an Equal Force of
Reaction, 'If and only If', the 'Quantified Sum of the
Reaction Forces' is equal to the 'Force of Action',
which initiates the 'Action Reaction Event'".
In other words, Newton's Third Law of Motion defines the 'Action
Reaction Event' of 'Equal Opposites' as either;
X = Y, or X - Y = 0
And this is clearly wrong, because there are at least 3 Forces;
1) Force of Action,
2) Force of Resistance,
3) Force of Reaction - from the Mathematics of
Quantification, the equation is given by;
X + Y = Z, or X^2 + Y^2 = Z^2
["...'1 = 2' - 'Prime Numbers'" - Page [43] -
[Normative References -
[Physics 1.] The Rudiments of Finite Physics]
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Furthermore, the Foundations of the Calculus, which
Newton is accredited for inventing, becomes questionable
with the introduction of an Alternate Mathematical Field.
Especially since, the New Field represents the development
of a New System of Counting, or more specifically, a
different (definition) way of representing a Number.
In other words, the point to be made in this case, is
that; the 'Derivative of a Constant' 'Is Not Equal to
Zero'. Especially since, if the 'Constant' is unknown,
then it's Derivative, using the New representation for
a Numeral [page 47], is given by (the 'Power Rule');
2EX = C ;
where C = Constant,
and 2EX defines any Numeral in Real Number Set
So.. - What's the Derivative of '2EX'?
Using the 'Power Rule', Let 2 = N, and 2EX = 2^X, then;
d
-- (NEX) = XNE(X - 1) = XN^X - 1
dx
- Or -
d
-- (2EX) = 2XE(X - 1)
dx
Given that;
XN^X - 1 = 2XE(X -1 ) = 2XEX - 1 = 2X^X - 1
{This represents; 'The Fall of Differential Calculus'-
'The Rudiments of Finite Algebra' - [2.]}
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- As for 'Time-Travel' and 'Parallel (Nested) Universes':
the thoughts of Science Fiction writers, the Beliefs of
World renowned Physicists, or the utterances of the
disassociated - those who are believed to be Insane,
because they do not have a University Affiliation.
It does not matter who believes 'what', because;
1) 'Time-Travel' is an impossibility, which would
violate the Conservation Laws. In other words,
Matter and Energy Cannot be Re-Animated; Created
or Destroyed.
2) 'Parallel Universe(s)', just like the existence of
more than 3 Dimensions, or any claim that Empty
Space defines a 'VOID of Nothingness' having
Material Properties: a Physical Impossibility,
because it violates the Conservation Laws of
Physics.
- Clearly, in a Supercilious world controlled by Posturing
Charlatan(s), mired by the allegories of Buffoons, only
the Insane is believed to be Intelligent...
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Work(s) in Progress;
Computer Science / Internet Technology:
These drafts represent the twelve chapters of the Networking
Bible, designing a Network IP Addressing Specification that
maintains a 100 Percent backward compatibility with the IPv4
Specification. In other words, this is a design specification
developed from the Theory of the Expansion of the IPv4 IP
Addressing Specification, which allowed the representation of
the Network for the entire World on paper, and the possibility
of an Infinite IP Address Pool. Nevertheless, the
Internet-Drafts listed below, "Cited as Work(s) in Progress",
explain the design Specification for the development of the
IPtX (IP Telecommunications Specification) Protocol Addressing
System and the correction of the Mathematical Error in the
Binary System.
1. http://www.ietf.org/internet-drafts/draft-terrell-logic
-analy-bin-ip-spec-ipv7-ipv8-10.txt "Work(s) in Progress"
(Foundational Theory for the New IPtX family IP Addressing
Specification, and the Binary Enumeration correction)
2. http://www.ietf.org/internet-drafts/draft-terrell-simple
-proof-support-logic-analy-bin-02.txt "Work(s) in Progress"
(The completion of the 2nd Proof correcting the error in
Binary Enumeration)
3. http://www.ietf.org/internet-drafts/draft-terrell-visual
-change-redefining-role-ipv6-01.pdf "Work(s) in Progress"
(Argument against the deployment of IPv6)
4. http://www.ietf.org/internet-drafts/draft-terrell-schem
-desgn-ipt1-ipt2-cmput-tel-numb-02.pdf
"Work(s) in Progress" (The foundation of the New IPtX
IP Addressing Spec now simular to the Telephone
Numbering System)
5. http://www.ietf.org/internet-drafts/draft-terrell-internet
-protocol-t1-t2-ad-sp-06.pdf - "Work(s) in Progress"
(The IPtX IP Addressing Specification Address Space / IP
Address Allocation Table; establishes the visual perspective
that actually represents Networking Schematic of the entire
World.)
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6. http://www.ietf.org/internet-drafts/draft-terrell-iptx-spec
-def-cidr-ach-net-descrip-01.pdf - "Work(s) in Progress"
(Re-Defining 'CIDR' {Classless Inter-Domain Routing
Architecture} for the IPtX Addressing Standard)
7. http://www.ietf.org/internet-drafts/draft-terrell-math
-quant-new-para-redefi-bin-math-04.pdf "Work(s) in Progress"
(The completion of the 3rd Proof correcting the error in
Binary Enumeration)
8. http://www.ietf.org/internet-drafts/draft-terrell-gwebs
-vs-ieps-00.pdf - "Work(s) in Progress"
Global Wide Emergency Broadcast System)
9. http://www.ietf.org/internet-drafts/draft-terrell-iptx
-dhcp-req-iptx-ip-add-spec-00.pdf "Work(s) in Progress"
(The development of DHCP {Dynamic Host Configuration
Protocol} for the IPTX IP Addressing Spec)
10. http://www.ietf.org/internet-drafts/draft-terrell-iptx
-dns-req-iptx-ip-add-spec-03.pdf "Work(s) in Progress"
(The development of DNS {Domain Naming Specification} for
IPTX IP Addressing Spec)
11. http://www.ietf.org/internet-drafts/draft-terrell-math-quant
-ternary-logic-of-binary-sys-06.pdf(Derived the Binary System
from the proof of "Fermat's Last Theorem", and Developed the
Ternary Logic for the Binary System)
12. http://www.ietf.org/internet-drafts/draft-terrell-cidr-net
-descrpt-expands-iptx-add-spc-16.pdf- "Work(s) in Progress"
(An application of Quantum Scale Theory, the 2^X : 1
Compression Ratio, the Expansion derived from the 'CIDR
Network Descriptor, and the Mathematics of Quantification
provided the foundation for the development of the
"Intelligent Quantum Tunneling Worm Protocol"; A Routable
Mathematical Exponential Expression, BackEnd IP Addressing
Protocol that provides an (nearly) Unlimited IP Address
Space using the Compression Ratio 2^X : 1.)
NOTE: These Drafts has Expired at www.ietf.org Web Site.
However, you can still find copies of these Manuscripts
posted at Web Sites all over the World. Suggestion;
Perform Internet Search using either Yahoo or Google.
Keyword: "ETT-R&D Publications"}.
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5. Normative References
Pure Mathematics:
1. The Proof of Fermat's Last Theorem; The Revolution in
Mathematical Thought {Nov 1979} E. Terrell
2. The Rudiments of Finite Algebra; The Results of
Quantification {July 1983} E. Terrell
3. The Rudiments of Finite Geometry; The Results of Quantification
{June 2003} E. Terrell
4. The Rudiments of Finite Trigonometry; The Results of
Quantification {July 2004} E. Terrell
5. The Mathematics of Quantification and the Metamorphosis of Pi:Tau
{October 200} E. Terrell
6. The Mathematics of Quantification & The Rudiments of Finite
Physics The Analysis of Newton's Laws of Motion...the Graviton'
{December 2004) E. Terrell
7. Squaring the Circle? First! What is the Circle's Area?
{January 2005}
The Rhind Papyrus Tale and the 10,000 year old quest involving
"Squaring the Circle"; derivation of the equation resolving the
Area of the Circle. An illusion perplexing the Sight and Mind
of the greatest mathematicians for about 10,000 years, which
maintains an elementary algebraic solution:
(Pi(r)/2)^2 = Area of Circle.
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Informative References
1. G Boole ( Dover publication, 1958 ) "An Investigation of
The Laws of Thought" On which is founded The Mathematical
Theories of Logic and Probabilities; and the Logic of
Computer Mathematics.
2. R Carnap ( University of Chicago Press, 1947 / 1958 )
"Meaning and Necessity" A study in Semantics and
Modal Logic.
3. R Carnap ( Dover Publications, 1958 ) " Introduction to
Symbolic Logic and its Applications"
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Author:
Eugene Terrell
Principle Director
Research & Development
Engineering Theoretical Technologies
Research & Development Publications
(ETT-R&D Publications)
3312 64th Avenue Place
Oakland, CA. 94605
Voice: 510-636-9885
E-Mail: eterrell00@netzero.net
"This work is Dedicated to my first and only child, 'Princess
Yahnay', because she is the gift of Dreams, the true treasure
of my reality, and the 'Princess of the Universe'. (E.T. 2007)"
Note: Illinois Institute of Technology, University of Chicago,
Northeastern Illinois University, University of Illinois
Chicago Circle Campus, Stanford University, UCLA,
Kennedy-King College, Canada, United States, Russia,
Germany, France, Scientific American, and several other
popular magazines received a copy of one, or both, of
the proofs are listed above; 1 and 2, the notarized
proofs that were sent for review between, 1980 and 1983
(to name, just only a few recipients).
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Copyright (C) The IETF Trust (2007).
This document is subject to the rights, licenses and restrictions
contained in BCP 78, and except as set forth therein, the authors
retain all their rights.
This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST,
AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES,
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The IETF invites any interested party to bring to its attention any
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Acknowledgement
Funding for the RFC Editor function is provided by the IETF
Administrative Support Activity (IASA).
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