the law of digit balance
DESCRIPTION
The Law of Digit Balance reveals a new law revealing hidden equilibrium balancing to zero in ALL number sequences.TRANSCRIPT
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A NEW LAW REVEALING HIDDEN EQUILIBRIUM IN ALL NUMBER SEQUENCES
Σ ∆n = 0
JOYCE P. BOWEN, Ph.D.
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©2003 Joyce P. Bowen revised 2004 revised 2008
All Rights Reserved
ISBN 0-9615454-2-9
Cover Art by Yaounde Olu
The author may be contacted at [email protected]
Astropoint Research Associates Chicago, Illinois
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ACKNOWLEDGEMENTS I am grateful for constructive criticism, advice and/or philosophical support from Emanuel Gugwor, Dr. Sow Aboubacar Sidy, Christine Bowen, Pamela Dominguez, Charmaine Anderson, Jeanne Ricks, Lerone Bennett, Jr., and Jeffrey Shallit. Thank you all.
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TABLE OF CONTENTS Page Introduction……………………………………………………………………….7 Chapter One ………………………………………………………………….…..8
Rules of Analysis
Chapter Two……………………………………………………………………. 12 Circular and Periodic Numbers Chapter Three………………………………………………………………..…14
Other Numbers
Chapter Four………………………………………………………………….... 22 Pi Patterns
Chapter Five…………………………………………………………………..…31
Whole Number Increments Chapter Six………………………………………………………………….…..36 Irrational and Transcendental Numbers Chapter Seven……………………………………………………………….....41
Base Two Chapter Eight……………………………………………………………….. …42
Fibonacci and Lucas Numbers Chapter Nine………………………………………………………………...….46
Pascal’s Triangle Chapter Ten………………………………………………………………….….48
Prime Numbers Chapter Eleven………………………………………………………………....53
Square Business Chapter Twelve………………………………………………………………....63
Numbers of Proton and Neutron Stability Chapter Thirteen……………………………………………………..……..….66
Notions about Nothing and Everything
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Chapter Fourteen…………………………………………………………………….69 Additive Assessment
Chapter Fifteen……………………………………………………………………….74
Multiplicative Assessment
Chapter Sixteen…………………………………………………………………..….78 Factorials
Chapter Seventeen……………………………………………………………..……79 Conclusions and Implications/The end of randomness
Analysis Catalogue…………………………………………………………………..82
Natural Numbers – 1 – 100…………………………………………………..83 Reciprocals of 1 – 100…………………………………………………..……98 Mirrors……………………………………………………………………..….112 Other Number Sequences……………………………………….……..…..118
Bibliography……………………………………..……………………………….…125
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For every continuous symmetry of the laws of physics, there must exist a
conservation law.
For every conservation law, there must exist a continuous symmetry.
(Proved by Emmy Noether in 1905)
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INTRODUCTION
In today=s society numbers are merely seen as tools for calculation. The notion
that numbers are things in themselves@ and have deeper import is not always
considered. In this book I will examine numbers with the goal of uncovering
deeper meanings and evidence of an underlying order to seemingly random
numerical systems. Increment Analysis, assessing number relationships between
adjacent numbers in numerical sequences, is a technique used to identify and
synthesize number relationships.
Increment Analysis differs from the calculus of sequence differences in
that the result of analysis is a charged number line. Increment Analysis
reveals previously unobserved patterns in Fibonacci numbers, Lucas
numbers and prime numbers. In a nutshell, Increment Analysis reveals
obvious, arithmetical characteristics of number sequences that were
heretofore hidden, and where the increments of all whole numbers, or
fragments of irrational or transcendental numbers sum to zero while
exhibiting beautiful symmetries.
There are mirror image groupings that cancel each other with a resultant of zero
in circular numbers. In some cases, 58 and 60 element mirror image
sequences are revealed. In addition, there are oscillating digits denoting a wave
pattern whose sum tends to converge to zero in non-circular number sequences.
When zeros are added before and after any sequence, the result is digits which
cancel to zero. Increment Analysis, therefore, may uncover a type of numerical
homeostasis with zero as a focal point. In other words, there seems to be a
Law of Digit Balance wherein the digits of all numbers in all number sequences
have increments that sum to zero in beautifully arranged symmetries.
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CHAPTER ONE
CONJECTURE
RULES OF ANALYSIS 1. Select a number sequence and evaluate the increments between
adjacent numbers so that they become positive or negative in relation to each other. Ignore decimal points and treat the number as a continuous sequence. For example, the reciprocal of seven is a series of repeating numbers 0.01428571…:
1 4 2 8 5 7 1…: The increment between 1 and 4 is +3 “ “ “ 4 and 2 is –2 “ “ “ 2 and 8 is +6 “ “ “ 8 and 5 is –3 “ “ “ 5 and 7 is +2 “ “ “ 7 and 1 is -6 The resulting number string, therefore, is
+3 -2 +6 -3 +2 -6… 2. Isolate the resulting binary patterns and cancel them to zero,
leaving one or more remainders. In the above case, +3 -2 and –3 +2 cancel each other, and the +6 and –6 also cancel each other, resulting in 0.
3. To see the underlying patterns of any sequence or sequence
segment that is not a circular number, you add zeros before and after the sequence and proceed with the methodology outlined in number 1 above. The digits will always cancel to zero.
4. Though it is important to keep in mind that each digit is an
increment that is ten times more precise than the last in the decimal expansion of numbers, the numbers are taken at face value.
5. For further details on analysis and extracting 2nd, 3rd, or N order
sequences from non-circular number segments without adding zeros, see the chapter on Pi Patterns.
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CONJECTURE: The increments between adjacent numbers in circular,1 or revolving number sequences generate oscillating mirror image number groupings that always cancel to 0.
EXPERIMENTAL DATA
Let nC = digits in a circular number sequence Let ∆ = the increments between adjacent numbers in the sequence If nc is extended indefinitely,
Then: ΣΣΣΣ ∆∆∆∆ nC = 0
The Reciprocal of Number Seven
The sum of the increments between the digits of the circular number, the
reciprocal of seven, yields oscillating digits that cancel to 0.
The reciprocal of the number seven, 1/7, = 0.01428571428571…
Analysis: the increment between 1 and 4 is +3; between 4 and 2 is –2; between
2 and 8 is +6; between 8 and 5, -3; between 5 and 7 is +2, etc. The resultant
number string is +3 –2 +6 –3 +2 –6 +3 –2 +6 –3 +2 –6 …∞∞∞∞ = 0
Hence, the outcome yields 6-element mirror image groupings that cancel to zero.
1 Circular numbers produce the same arithmetic sequence of numbers, but starting in different positions, when
multiplied or divided by given numbers.
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The Reciprocal of Number Thirteen
The sum of the increments between the digits of the reciprocal of thirteen yields
oscillating mirror-image digits that cancel to 0.
The reciprocal of the number thirteen, 1/13, is
0.0769230769230…
The resultant number string after increment analysis is:
+7 -1 +3 -7 +1 –3 +7 -1 +3 -7 +1 - 3…∞∞∞∞ = 0
Again, the outcome yields 6-element mirror image groupings that cancel to zero.
The number thirteen, when divided by any number between one and nine, yields
two cyclic variations based on the numbers 0.076923 and 0.1538461. The
0.076923 pattern was analyzed above. Let’s look at 0.15384615…
15384615… after analysis yields
+4 - 2 +5 -4 +2 - 5 +4 -2 +5 - 4 +2 - 5…∞∞∞∞ = 0
Once again, we see 6-element mirror image groupings that cancel to zero.
The Reciprocal of Number 17
The sum of the increments between the digits of the cyclic number, the reciprocal
of seventeen, 1/17, yield oscillating digits that cancel to 0.
The reciprocal of the number seventeen = 1/17=0.05882352941176470…
After synthesis:
+5 +3 0 - 6 +1 +2 - 3 +7 - 5 - 3 0 +6 -1 -2 +3 -7…∞∞∞∞ = 0
This is a 16-element mirror image grouping that cancels to zero.
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The Reciprocal of Seven divided by its mirror image and then divided
by the result:
Dividing the reciprocal of seven (142857) by its mirror image (758241) and then
dividing the result (5.30769203…) by seven results in oscillating digits that cancel
to zero.
5.30769203 ÷÷÷÷ 7 = 0.7582417…
7 5 8 2 4 1 7…
The resultant number string, after synthesis, is
-2 +3 -6 +2 -3 +6…∞∞∞∞ = 0
…a 6-element mirror image grouping that cancels to zero.
The number seven divided by thirteen results in oscillating digits that
cancel to zero.
7/13 = 0.538461538461538461…:
5 3 8 4 6 1 5 …
After synthesis yields
-2 +5 -4 +2 -5 +4… ∞∞∞∞ = 0
6-element mirror images that cancel to zero.
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CHAPTER TWO
CIRCULAR NUMBERS AND THE NUMBER NINE
All circular numbers are related to the number nine, in that they produce series of nines
when they are divided in half. It may be surmised, therefore, that the number nine is related
to zero, since the increments between the digits of circular numbers cancel to zero.
Consider the following results when the digits of circular numbers are added together to
produce a final number:
1/7=0.142857; 1 + 4 + 2 + 8 + 5 + 7 = 27; 2 + 7 = 9
1/13 = 0.076923; 7 + 6 + 9 + 2 + 3 = 27; 2 + 7 = 9
1/17 = 0.0588235294117647; 5+8 + 8 +3 + 3 + 5 + 2 + 9 + 4 + 1 + 1 + 7 + 6 + 4 + 7 = 72;
7 + 2 = 9
2/13 = 153846; 1 + 5 + 3 + 8 + 4 + 6 = 27; 2 + 7 = 9
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Number Pattern after synthesis
1/7=0.142857; +3 -2 +6 -3 +2 -6…. ∞∞∞∞ = 0
1/13 = 0.076923; +7 -1 +3 -7 +1 -3… ∞∞∞∞ = 0
1/17 = 0.0588235294117647; +5 +3 0 -6 +1 +2 -3 +7 -5 -3 0 +6 -1 -2 +3 -7…∞∞∞∞
= 0
2/13 = 153846; +4 -2 +5 -4 +2 -5…∞∞∞∞ = 0
The mirror image symmetry produced by increment analysis of these numbers is sheer
beauty. Consider the reciprocal of 17 where a 16-element mirror image grouping is
generated!
THE NUMBER NINE CYCLES TO ZERO
In a related matter, whenever nine divides any number between 1 and 8 the result is
repeating digits (repdigits) whose increments equal zero. For example:
1/9 = 0.111111111 = 00000000........∞∞∞∞ = 0
2/9 = 0.222222222 = 00000000........∞∞∞∞ = 0
3/9 = 0.333333333 = 00000000........∞∞∞∞ = 0
4/9 = 0.444444444 = 00000000........∞∞∞∞ = 0
5/9 = 0.555555555 = 00000000........∞∞∞∞ = 0
6/9 = 0.666666666 = 00000000........∞∞∞∞ = 0
7/9 = 0.777777777 = 00000000........∞∞∞∞ = 0
8/9 = 0.888888888 = 00000000........∞∞∞∞ = 0
All of the foregoing implies a “zero-ness” of cycles (circles).
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CHAPTER THREE
NON-CIRCULAR NUMBERS
The previous chapters illustrated the tendency of circular numbers to generate
mirror image oscillating digits that cancel to zero. The case will be made in this
chapter that non-monotonic increasing or decreasing number sequences
generate number patterns with adjacent binary pairs and mirror image
symmetries. Sometimes there is a remainder, a Point of Synthesis, and a mode
of 0. Increment Analysis reveals hidden patterns in all number sequences.
CONJECTURE
All number sequences that are non-monotonic increasing or decreasing generate
non-random number patterns with oscillating digits that result in a residual Point
of Synthesis and a mode of 0.
Let n = digits in a number sequence. Let ∆ = increments between adjacent digits. Let Ps = Point of Synthesis If n is a non- monotonic increasing or decreasing number sequence,
Then: ΣΣΣΣ ∆∆∆∆ n = Ps ≥ 0
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EXPERIMENTAL DATA
The reciprocal of a rough approximation of the fine structure constant (1/137) produces digits that cancel to zero.
1/137 = 0.007299270…
= 0 +7 -5 +7 0 -7 +5 -7 0 ∞∞∞∞ = 0
The result is 6-element mirrors that cancel to zero, the Point of Synthesis.
(1/137 is a natural palindrome, i.e., 729 927, which forms a palindrome mirror)
RECIPROCAL OF # 22
The reciprocal of the number 22 is 0.0454545454545…
4 5 4 5 4 5 4 5 4 5 4 5
The resultant number string after analysis is:
+1 –1 + 1 –1 +1 –1 +1 –1 +1 –1…
= 0
After canceling the binaries, the remainder is 0. Zero, therefore, is the Ps.
THE NUMBER 4 ÷÷÷÷ 22
= 0.181818181818181818…
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1 8 1 8 1 8 1 8 1…
After synthesis yields
+7 –7 +7 –7 +7 –7 +7 –7…∞∞∞∞= 0
The Ps is 0.
The reciprocal of thirteen subtracted from the reciprocal of seven is the same as the reciprocal of seven subtracted from the reciprocal of thirteen, only negative.
0.142857142 (carried out to nine digits)
-0.076923076
0.065934066
and interestingly,
0.076923076
-0.142857142
- 0.065934066
The resulting number sequence in both cases is
0 6 5 9 3 4 0 6 6
After analysis we get
+6 -1 +4 -6 +1 -4 +6 0
After canceling the 6-element mirror-image grouping of +6 -1 +4, -6 +1 -4, we are left
with a Ps of +6, or simply 6. It must be kept in mind that this is carried out to nine digits; the
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remainder of +6 will be replaced by another digit if more decimal places are included. If a
final zero is added to the original sequence, it cancels to zero.
THE RECIPROCAL OF SEVEN DIVIDED BY ITS MIRROR IMAGE
The reciprocal of the number seven divided by its mirror image yields a seeming
variant of the reciprocal of thirteen:
758241 ÷ 142857 = 5.30769203769203769203…;
5 3 0 7 6 9 2 0 3 7 6 9 2 0 3 7 6 9 2 0 3…
The result after synthesis yields:
-2 –3 +7 –1 +3 –7 – 2 +3 +4 -1 +3 – 7 –2 +3 +4 –1 +3 –7 –2 +1…; ↓↓↓↓ ↓↓↓↓ +7 +7 results in the following second order sequence:
-2 –3 +7 –1 +3 - 7 – 2 +7 -1 +3 – 7 –2 +7 –1 +3… ;
A third order sequence reveals:
-2 -3 +7 -1 +3 -7 -2 +7 -1 +3 -7 -2 +7 +2 ↓↓↓↓ ↓↓↓↓ ↓↓↓↓ - 5 +7 +2 -7 -2 +7 +2 -7 -2 +7 +2;
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After canceling the 4-element oscillating mirrors (+7 +2, -7 -2) the resulting Ps is –5. RECIPROCAL OF 82.9
1 ÷÷÷÷ 82.9 =
0.012062726176115…
After synthesis, this yields:
+1 +1 -2 +6 -4 +5 -5 +4 -5 +6 -1 -5 0 +4
second order sequence:
+1 +1 -2 +6 -4 +5 -5 +4 -5 +6 -1 -5 0 +4
↓↓↓↓ ↓↓↓↓ ↓↓↓↓
+2 -2 +2 +5 -5 +4 -5 +5 -5 0 +4
third order sequence:
+2 -2 +2 +5 -5 +4 -5 +5 -5 0 +4
fourth order sequence:
+4 +4
Ps = +8
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SQUARE ROOT OF THE RECIPROCAL OF THIRTEEN
In some number sequences such as those in the last two examples, the mirrors
and oscillating digits are not obvious, though they exist nevertheless. Consider
the following example.
1 ÷÷÷÷ √√√√13 =
1 ÷÷÷÷ √√√√769230
= 877.05758
8 7 7 0 5 7 5 8:
After analysis we get:
-1 0 -7 +5 +2 -2 +3
-1 0 –7 = --8; +5 +3= +8;
--8 +8 = 0
The Ps, therefore, is 0.
FIVE DIVIDED BY TWENTY-ONE:
5 ÷÷÷÷21 =0.2380952;
2 3 8 0 9 5 2
Yields
+ 1 +5 -8 +9 -4 -3;
+1 +5 = +6; -8 +9 =+1; -4 -3=-7;
+6 +1 = +7; -4 -3 = -7
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+7 -7 = 0
The Ps = 0.
SIX DIVIDED BY TWENTY-ONE
6 /21 = 0.2857142;
2 8 5 7 1 4 2
After analysis we get:
+6 -3 +2 -6 +3 -2
6-element mirror with the Ps = 0.
EIGHT DIVIDED BY 21 ( 8/21)
= 0.3809523
= 3 8 0 9 5 2 3
5 -8 +9 -4 -3 +1
↓ ↓ ↓ -3 +9 -7 +1 second order sequence
↓ ↓ +6 -6; +6 - 6 + third order sequence ↓ 0
The Ps of 8/21 is, therefore, 0.
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NINE DIVIDED BY TWENTY-ONE ( 9/21):
= 0.4285714…
4 2 8 5 7 1 4
-2 +6 -3 +2 -6 +3 0
The Ps = 0 THE NUMBER 13492 --
= 1 8 1 9 8 0 1
+7 –7 +8 -1 -8 +1
+7 -7 +8 -1 -8 +1
↓↓↓↓
0 0
Ps = 0
Note: The Ps of 1349 squared is, therefore, 0. The number 1349, by the way, was
derived in the following manner: The reciprocal of the number thirteen is a circular number,
as is the reciprocal of the number seven. Unlike seven, however, thirteen cycles two
groups of numbers when a number between one and nine is divided by thirteen: 0.076923
and 0.153846. The numbers 1, 3, 4, and 9 cycle on the digits 076923, hence the #1349.
The remaining numbers cycle the digits 153846.
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CHAPTER FOUR
PI PATTERNS
ORGANIZED PATTERNS IN THE DIGITS OF PI
Pi, the number that expresses the relationship between the diameter and
circumference of a circle, has fascinated people for centuries. Around 2000 B.C.
the Babylonians used pi as 3 1/8. During that same period the ancient Egyptians
identified the relationship as (16/9)2 =3.1605. In the 12th century B.C. the
Chinese used pi = 3, and in the Bible in 550 B.C., I Kings vii, 23 implies pi = 3. In
the 3rd century B.C. Archimedes establishes pi using a geometric method as pi =
3.14163. These earlier researchers did not use the Greek letter as a symbol for
pi, however. It has only been used for the past 250 years.
In 1766 Johann Heinrich Lambert proves the irrationality of pi and in 1882 F.
Lindemann proves the transcendence of pi.
Digit hunters throughout the centuries have tried to determine the number of
digits in pi. To date, more than 51.5 billion digits have been identified, and no
identifiable pattern has emerged. Until now, that is. David Blatner in his
delightful book entitled The "Joy of Pi" (Walker Publishing Co., 1997) states
"There's little doubt that if we understood this number better--if we could find a
pattern in its digits or a deeper awareness of why it appears in so many
seemingly unrelated equations--we'd have a deeper understanding of
mathematics and the physics of our universe." He goes on to say "The digits of
pi appear so random that if there were a rule to the sequence, it may require
billions--or trillions--of digits to begin to see it." Perhaps not.
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The following number experiment uses a quaternary, reiterative skip sequence
on the first 218 digits of pi to create new digits for increment analysis.
THE FIRST 218 DIGITS OF PI 3.14159265358979323846264338327950288419716939937510582097494459230781 640628620899862803482534211706798214808651328230664709384460955058223 172535940812848111745028410270193852110555964462294895493038196442881 097566593344 QUATERNARY REITERATIVE SKIP SEQUENCE A quaternary reiterative skip sequence was used. This is done in the following manner:
1 - 2 - 3 - 4 1 - 2 - 3 - 4 1 - 2 - 3 - 4 1 - 2 - etc. etc.
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In this sequence 1 reiterates 4, i.e., the one in the next sequence is the
same as the 4 in the last sequence. The following digits are the result of
using this quaternary, reiterative skip sequence on the first 218 digits of pi:
312393823398113312442848888831684632694952734281547 3259428439487634 Analysis: First Order Sequence
Once these skip sequence digits were identified, a relationship between
them was assessed. The following first order sequence resulted from
determining the positive or negative increments between digits of the
resultant skip sequence. (It is important to keep in mind that though each
digit is an increment that is ten times more precise than the last in this
decimal expansion of pi, the numbers are taken at face value.
For example, the first numbers in the resulting sequence are 3 1 2 3 9
3....therefore, from 3 to 1 is -2; from 1 to 2 is +1; from 2 to 3 is +1; from 3 to
9 is +6; from 9 to 3 is -6; etc. etc.
Resultant Patterns—First Order Sequence
-2 +1 +1 +6 -6 +5 -6 +1 0 +6 -1 -7 0 +2 0 -2 +1
+2 0 -2 +6 -4 +4 0 0 0 0 -5 -2 +5 +2 -4 +2 -3 -1
+4 +3 -5 +5 -4 -3 +5 -4 +1 -2 +6 -7 +4 -1 +3
-4 -1 +3 +4 -5 -2 +6 -4 -1 +6 -5 +4 -1 -1 -3 +1
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The following mirrors and oscillating digits were found in the previous
number string.
-2 +1 +1 +6 -6 +5 -6 +1 0 +6 -1 -7 0 +2 0 -2 +1
+2 0 -2 +6 -4 +4 0 0 0 0 -5 -2 +5 +2 -4 +2 -3 -1
+4 +3 -5 +5 -4 -3 +5 -4 +1 -2 +6 -7 +4 -1 +3
-4 -1 +3 +4 -5 -2 +6 -4 -1 +6 -5 +4 -1 -1 -3 +1
Binary patterns emerge in the first 218 digits of pi. A remarkable symmetry
seems evident where positive and negative numbers in reciprocal positions occur
in a seeming non-random manner. In addition, without exception, numbers or
number groupings on the opposite sides of zeroes are have opposite or
balancing polarities in the first order sequence. Observe the following while
keeping in mind the fact that this is the exact order in which they appeared in the
number string:
+6 -6
-6 +1 0 +6 -1
+2 0 -2 + 1 +2 0 -2
-4 +4
0 0 0 0
-5 -2, +5 +2
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+4 +3, -5 +5, -4 -3
+4 -1 +3, -4 -1 +3
In addition, note the fact that +5 -4 or +4 -5 repeat five times in various places
in the string. Also note that there are no 8s or 9s in the string.
The numbers not included in the number groupings identified in the first order
sequence noted above were found to also have binary relationships when a
synthesis methodology was employed.
Synthesis of the First Order Sequence to Create a Second Order Sequence
A synthesis methodology was devised to test the apparent binary nature of the
digits. Basically, synthesis is the resolution of binaries. The following "rules" were
utilized:
1. only adjacent numbers or number groupings were used, or those that are
separated by one or more zeroes; 2. these are subsequently added until a final
result, i.e., resolution of binaries, is achieved. For the first 218 digits, it took four
sequence generations to reach neutralization. The italicized numbers below
result from the number groupings beneath them and represent a subsequent
order.
-2 +2 0 +5 -5 0 +5 -7 (second order)
| / \ / \ | / \ | / \ |
-2 [+1 +1] [+6 -6] +5 [-6 +1] 0 [+6 -1] -7
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0 0 +1 0 +6 0 0 (second order)
| / \ | / \ | / \ / \
0 [+2 0 -2] +1 [+2 0 -2] +6 [-4 +4] [0 0 0 0]
-7 +7 -6 (second order)
/ \ / \ / \
[-5 -2] [+5 +2] [-4 +2 -3 -1]
+7 0 -7 +1 -1 -1 (second order.)
/ \ / \ / \ / \ / \ / \
[+4 +3] [-5 +5] [-4 -3] [+5 -4] [+1 -2] [+6 -7]
+6 (second order)
/ \
[+4 -1 +3]
-2 -1 +4 -4 -1 +1 +4 (second order)
/ \ / \ / \ | | / \ |
[-4 -1 +3] [+4 -5] [-2 +6] -4 -1 [+6 -5] +4
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-1 -4 +1 (second order.)
| / \ |
-1 [ -1 -3] +1
Third Order Sequence
0 0 0 0 -7 0 0 +7 (third order)
/ \ | / \ / \ | | | / \
[-2 +2] 0 [+5 -5 0 +5] -7 0 0 +1 0 +6
0 0 0 -6 0 0 -1 +6 -2 (third order)
| | / \ | / \ / \ | | |
0 0 -7 +7 -6 +7 0 -7 +1 -1 -1 +6 -2
-1 0 0 +3 -3 (third order)
| / \ / \ / \ / \
-1 +4 -4 -1 +1 +4 -1 -4 +1
Fourth Order Sequence
0 0 0 0 0 0 +6 (fourth order)
| | | | / \ / \ |
0 0 0 0 -7 0 0 +7 [-6 0 0 -1] +6
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(-7 0 0 +7 0 0 0 -7 )
0 0 (fourth order)
/ \ / \
[-2 -1] 0 0 +3 -3
(-3 0 0 +3 -3)
A resolution of increments in the first 218 digits of pi, therefore, results in
the following sequence:
0 0 0 0 0 0 +6 0 0 ; where the mode is zero and the Point of Synthesis
is +6.
SUMMARY OF 2ND THROUGH 4TH ORDER SEQUENCES
SECOND ORDER:
-2 +2 0 +5 - 5 0 +5 -7 0 0 +1 0 +6 0 0 -7 +7
-6 +7 0 -7 +1 -1 -1 +6 -2 -1 +4 -4 -1 +1
[+4 -1] [ -4 +1]
THIRD ORDER:
0 0 0 0 -7 0 0 +7 0 0 0 -7 +6
-3 0 0 +3 -3
FOURTH ORDER:
0 0 0 0 0 0 +6 0 0
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Based on the foregoing number experiment there are mirrors and binary
patterns underlying the first 218 digits of pi where the increments tend
toward zero. More research would have to be done, of course, to determine
whether or not this pattern is ongoing up to and beyond the more than 51.5 billion
digits of pi that have been identified thus far. What is definite, however, is that
any sequence extracted from the digits of pi have increments that sum to zero,
as will be demonstrated later.
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CHAPTER FIVE
WHOLE NUMBER INCREMENTS
The major focus thus far has been on reciprocals of numbers and their
increments that sum to zero. Now I will examine whole number increments. It
will be found that they always sum to zero.
With whole numbers, to assess the increments, you must add a zero to the
beginning of the number and a zero to the end (unless the number ends in zero.
In that case, you don’t add an extra zero). The rationale for this is that there is
an understood zero, or absence of a number at the beginning and ending of a
whole number.
CONJECTURE;
The increments of any whole number sum to zero.
EXPERIMENTAL DATA:
Let n = a whole (natural) number
Let ∈ = increment
Then, ∞ Σ ∆ = 0 0…n
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EXPERIMENTAL DATA
Number sample Synthesis
1 0 1 0 =
+1 -1 = 0
2 0 2 0 =
+2 -2 = 0
13 0 1 3 0 =
+ 1 +2 –3 =
+3 -3 = 0
477 0 4 7 7 0 =
+4 +3 0 -7 =
+7 -7 = 0
4579 045790 =
+4 +1 +2 +2 –9 =
+9 -9 = 0
36920 036920=
+3 +3 +3 –7 –2=
+9 -9 = 0
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7787537 077875370
+7 0 +1 –1 –2 –2 +4 –
7 =
+8 –5 +4 -7
+8 -1 -7 =
+8 -8 = 0
3123 031230 =
+3 – 2 +1 +1 –3 =
+3 -2 +2 -3 = 0
2138 021380 =
+2 -1 +2 +5 –8 =
+ 1 +7 -8 =
+8 -8 = 0
146 01460 =
+1 +3 +2 -6 =
+6 -6 = 0
175 01750 =
+1 +6 -2 -5 =
+7 -7 = 0
34
2903 029030
+2 +7 –9 +3 –3 =
+9 -9 +3 -3 = 0
3167 031670 =
+3 –2 +5 +1 -7 =
+1 +6 -7 =
+7 -7 = 0
1699 016990 =
+1 +5 +3 0 -9 =
+9 0 -9 = 0
3851 038510 =
+3 +5 -3 -4 -1 =
+8 -8 = 0
797 0790 =
+7 +2 –2 –7 = 0
4049 040490 =
+4 -4 +4 +5 -9 =
0 +9 -9
35
877 08770 =
+8 –1 0 -7 =
+8 -8 = 0
953 09530 =
+9 -4 -2 -3 =
+9 -9 = 0
2459 024590 =
+2 +2 +1 +4 -9 =
+9 -9 = 0
382 03820 =
+3 +5 -6 -2 =
+8 -8 = 0
267 02670 =
+2 +4 +1 -7 =
+7 -7 = 0
36
CHAPTER SIX
IRRATIONAL AND TRANSCENDENTAL NUMBERS
An irrational number is a number that is not capable of being expressed as an
integer or as a quotient of an integer. A transcendental number is a number,
such as pi, which is not the root of any polynomial equation with integer
coefficients; i.e., it is not an algebraic number. Pi is also irrational. When any
series of consecutive digits is lifted from any irrational or transcendental
number and zero is placed in front of and behind the fragment, the sum of
the increments is always 0!
Conjecture:
The sum of increments in number sequences extracted from irrational and
transcendental numbers always equals 0.
Pi, to 24 digits = 3.14159265358979323846264….
Extract the sequence: 6535897;
37
Add zeros before and after the fragment:
065358970 =
+6 -1 -2 +2 +3 +1 -2 -7 =
+6 -3 +5 -1 -7=
+6 +2 -8 =
+8 -8 = 0
Extracting another sequence from the pi expansion:
8462=
084620 =
+8 -4 +2 -4 -2 =
+8 -2 -6
+6 -6 = 0
38
Feigenbaum numbers have not proven to be transcendental, but are generally
believed to be. They are related to properties of dynamical systems with period
doubling. The ratio of successive differences between period doubling bifurcation
parameters approaches the number 4.669…..
Feigenbaum constant to the 30th digit:
4.66920160910299067185320388204…
Extracting sequence :
6718532:
Adding zeros:
067185320 =
+6 +1 -6 +7 -3 -2 -1 -2 =
+8 -8 = 0
39
The irrational square root of 2 =
√2 = 1.4142135623730950488016887242097…;
Extract sequence:
73095:
add zeros:
0730950 =
+7 -4 -3 +9 -4 -5 =
+7 -7 +9 -9 = 0
40
A “PROOF” THAT ALL OF THE INCREMENTS BETWEEN DIGITS IN TRANCENDENTAL NUMBERS SUM TO ZERO
If it can be proven that the increments between all digits of transcendental
numbers sum to zero, it may lend credence to the position that the digits in
transcendental numbers are not random. To demonstrate this point, I will use a
variation of the proof that the late Middle Ages French scholar Nicole d’Oresme
(ca.1323 – 1382) used to prove the divergence of the harmonic series.
D’Oresme pointed out that 1/3 + ¼ is greater than ½; so is 1/5 + 1/6+ 1/7 + 1/8;
so is 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/ 16 ; and so on. In other
words, by taking 2 terms, then 4 terms, then 8, then 16 terms, and so on, you can
group the series into an infinite number of blocks, every one of which is bigger
than one-half. The entire sum must, therefore, be infinite. No matter what, there
is always another one-half to be added; and that means that the total increases
without limit.2
In the following proof, larger and larger blocks of the decimal expansion of a
transcendental number are considered utilizing a process called additive
assessment (see chapter 14). If the digits in transcendental numbers are taken
in larger and larger blocks with zeroes added before and after the blocks, the
sum of increments is always zero, therefore, the sum of increments in the digit
expansion will always be zero through infinity.
Let n = integer Let nts = any number segment in a transcendental number Let 02 = zeroes before and after any segment Let ∆ = increment segments through infinity If ∆ nts +n + 02 = 0; Then Σ ∆ nts + n + 02 ----- ∞ = 0 Q.E.D. 2 John Derbyshire. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in
Mathematics. Joseph Henry Press. Washington, D.C. 2003
41
CHAPTER SEVEN
BASE TWO
When assessing numbers in base two, we get a similar result when assessing
the face value of the numbers. For example, #7 in Base 10 equals 111 in base
2.
111 =
01110 =
+1 0 0 -1 = 0
#13 in base 2 equals 1101:
1101 =
011010 =
+1 0 -1 +1 -1 = 0
#17 in base 2 equals 10001:
10001 =
0100010 =
+1 -1 00 +1 -1 = 0
42
CHAPTER EIGHT
FIBONACCI AND LUCAS NUMBERS
The Fibonacci Numbers
Fibonacci numbers were first demonstrated by Leonardo of Pisa, better known
as Fibonacci, in his book titled Liber Abaci published in 1202. In it he discussed
the number of rabbits that would be born from a pair of rabbits at the beginning of
every month. The resulting number of rabbits revealed the following pattern:
1,1,2,3,5,8,13,21,34,55, etc. This series converges to a number called the
“golden ratio,” which is rounded off here to 1.6180339887….. Basically, you take
a term, divide it by the previous term, and the number converges to the golden
ratio. This number is found in many places in nature. The formula that
generates this number is:
( 1 + √5) = 1.6180339887….. 2
In an examination of the first 500 Fibonacci numbers, Increment Analysis has
revealed an interesting pattern– when the digits are summed modulo nine, a
recurring sequence with a mirror pattern emerges. The sequence is:
112358437189887641562819…1…….
This series of numbers repeats continually beginning again at every 24th digit.
The 24th digit always sums to nine. The increment pattern that emerges is:
0 +1 +1 +2 +3 -4 -1 +4 -6 +7 +1 -1
0 –1 -1 -2 -3 +4 +1 -4 +6 -7 +8 -8
The series sums to a perfect mirror with an anomaly, +1 +8, -1 –8, which sums to
+9 and –9, which sums to zero.
43
The Lucas Numbers
The Lucas Numbers are similar to the Fibonacci numbers, but they start with 2
and 1 instead of the Fibonacci’s 0 and 1. The series is named after Edouard
Lucas (1842 – 1891), professor of mathematics at the Lycee Charlemagne in
Paris who developed a well-known formula for the Fibonacci numbers as well as
the numbers that bear his name. The first few Lucas numbers are:
2,1,3,4,7,11,18,29,47,76,123,199,322, etc.
When the first 200 Lucas numbers are summed modulo nine, as with the
Fibonacci numbers, a recurring series emerges which also repeats at every 24th
number. In every case, the pivot number sums to eight. The series is:
213472922461786527977538…2…
The resulting mirror becomes evident:
-1 +2 +1 +3 -5 +7 -7 0 +2 +2 -5 +6
+1 -2 -1 -3 +5 +2 -2 0 -2 -2 +5 -6
This is a perfect mirror with an anomaly, +7 +2, -7 -2, that sums to +9 -9, which
= 0.
44
The first 25 Fibonacci numbers
Number ∑ mod 9 1 1 1 1 2 2 3 3 5 5 8 8 13 4 21 3 34 7 55 1 89 8 144 9 233 8 377 8 610 7 987 6 1597 4 2584 1 4181 5 6765 6 10946 2 17711 8 28657 1 46368 (F#24) 9 75025 1
45
The first twenty-five Lucas numbers
Number ∑ mod 9
2 2 1 1 3 3 4 4 7 7 11 2 18 9 29 2 47 2 76 4 123 6 199 1 322 7 521 8 843 6 1364 5 2207 2 3571 7 5778 9 9349 7 15127 7 24476 5 39603 3 64079 (L#24) 8 103682 2
46
CHAPTER NINE
PASCAL’S TRIANGLE
Pascal’s Triangle was designed by Blaise Pascal in the 17th century. The
Chinese, however, knew of this triangle centuries before Pascal. It has been
found that the permutations relate directly to terms of the Binomial Theorem. It
can also be used in the analysis of probabilities. From an Increment Analysis
standpoint, Pascal’s Triangle creates a series of palindrome mirrors. (A
palindrome is a number that reads the same way forward or backward).
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
47
The previous triangle, completed through the seventh row, demonstrates
the following pattern:
0
+1
+1 -1
+2 0 -2
+3 +2 -2 -3
+4 +5 0 -5 -4
+5 +9 +5 -5 -9 -5
The sum of the increments in this triangle, and to one of any power, is
equal to one, i.e., unity. An intriguing question: would the sum of
increments be 1 in a Pascal Pyramid?.
48
CHAPTER TEN
PRIME NUMBERS
Prime numbers are numbers that have only factor of themselves and one. They
are enigmatic, in that no discernable pattern of organization is evident. When
Einstein said God doesn’t play dice when expressing exasperation about
quantum physics, he could have said the same thing about prime numbers.
There have been some advances in understanding them. Researchers have
been able to make progress regarding the conjecture that there are an infinite
number of pairs of prime numbers that differ only by two. It has also been found
that the frequency of twin primes decreases as one gets to larger numbers.
Brun’s constant, another revealing aspect of their nature, is found by adding the
reciprocals of successive twin primes. When this is done, the sum converges to
a specific numerical value which is 1.902160582310… The reciprocals of all of
the primes, on the other hand, diverge.
The Goldbach Conjecture, that every even number is the sum of two primes has
not yet been proven. It is known, however, that all twin primes are of the multiple
6k +/- 1.
Other mathematicians who have thrown their hats into the prime ring are
Mersenne (Mersenne Primes), Sophie Germaine ( Sophie Germaine Primes) and
Fermat (Fermat Primes) just to name a few. One of the most intriguing of prime
conjectures, however, is the Riemann Hypothesis. Briefly stated, when studying
the distribution of prime numbers, Riemann extended Euler’s zeta function,
defined for real part greater than one, to the entire complex plane. When he did,
he noted that his zeta function had trivial zeros at –2, -4, -6,… and that all
49
nontrivial zeros were symmetric about the line Re(s) + ½. The hypothesis is that
all nontrivial zeros are on this line. The Clay Mathematics Institute has offered to
pay $1 million dollars to anyone that can prove this hypothesis to be true.
Increment Analysis has uncovered some peculiarities about primes. When the
primes are assessed as sums modulo nine, they all resolve to digits of the
reciprocal of the number seven. Those digits are 0.142857 142857…..The
primes, however, do not follow in a particular order that has yet been noted.
What has been revealed by Increment Analysis is:
• No prime number with two or more digits sums to 6, 3, or 9.
• No prime with two or more digits ends with the number 5.
• The final digit of any prime number of two or more digits can only be 1, 3,
7, or 9.
• There is a yin yang configuration of the primes. The only even prime,
number 2, is separated from every other prime by odd numbered
increments. Every odd prime is separated from every other prime by an
even numbered increment.
• Though it has been shown that the frequency of twin primes decreases as
one gets to larger numbers, Increment Analysis demonstrates the fact that
the frequency of larger increments increases as one gets to larger
numbers.
50
FREQUENCY OF ALL ADJACENT NUMBER INCREMENTS IN THE FIRST 101 PRIMES:
♦ # 2 appears 25 x
♦ # 4 appears 25 x
♦ # 6 appears 25 x
♦ # 8 appears 7 x
♦ # 10 appears 8 x
♦ # 12 appears 4 x
♦ # 14 appears 3 x
♦ # 18 appears once
In addition, the increment between the 20,861st and 20,862nd prime (235,397
and 235,439) is 42.
THE FIRST 101 PRIMES SUMMED MODUL0 9 DEMONSTRATING DIGITS FROM THE RECIPROCAL OF THE NUMBER SEVEN
(1)23 = The Supernal Triad
5724815241572857481287248151524574152812481472485785281724515727
4572874152154578172872482154858117
0
5
10
15
20
25
#2 #6 #10
#14
3-D Column 1
3-D Column 2
3-D Column 3
51
PRIME EXCLUSION PRINCIPLES
Based on the foregoing, it is safe to assume that, when assessing a number with
two or more digits, you know it is not prime if the final digit is a 5. It is also not
prime if the sum modulo nine is not one of the digits of the reciprocal of seven.
PRIMAL ELEMENTS
When considering the Periodic Table of the Elements, the prime number
elements are arranged in interesting patterns. The periodic table is divided into
classes numbered from 1 through 18. The 18th division is comprised of the “noble
gases.”
I II IIIb IVb Vb Vib VIIb VIIIb Ib IIb III IV V VI VII 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Li none Ac none V none Tc noneMt none Cu none B none N none Cl He
Na Nb Bh Ag Al Bi I
K Ta Pm Au Ga Md Lu
Rb Pr Bk Ho Lr
Totals
4 0 1 0 4 0 3 0 1 0 4 0 4 0 3 0 4 1
The “prime” elements, i.e., the elements with the same number of protons as the
prime numbers, tend to alternate, and they tend to fall in prime numbered
columns, with a few exceptions. When the total number of prime elements in
each group is reduced to a number, the following sequence results:
4 0 1 0 4 0 3 0 1 0 4 0 4 0 3 0 4 1;
401040301040403041 =
04010403010404030410 =
52
+4 -4 +1 -1 +4 -4 +3 -3 +1 -1 +4 -4 +4 -4 +3 -3 +4 -3 -1 =
+28 -28;
Sum modulo 9 =
+10 -10 =
+1 -1
This last section, the analysis of the resulting number sequence reveals a
pattern of primes in the natural elements that resolves to ±1.
53
CHAPTER ELEVEN
SQUARE BUSINESS
In this chapter we will examine the increments in “squares”-- zero sum squares,
magic squares, and Latin squares.
2 x 2 ZERO SUM SQUARES
I have discovered a new class of 2 x 2 squares that utilizes zero and a negative
number to form Zero Sum squares. Zero Sum squares generate series of
positive and negative numbers that sum to zero. The following is a Zero Sum 2
x 2 square comprised of the first three primes and zero wherein each row,
column or diagonal sums to positive and negative prime numbers that together
sum to zero.
3
0
2
-5
54
The previous Zero Sum 2 x 2 prime square generates the following
patterns:
Horizontals : +3 -3
Verticals: +5 -5
Diagonals: +2 -2
Another Zero Sum Prime Number square:
5
0
2
-7
This Zero Sum square generates:
Horizontals: +5 -5
Verticals: +7 -7
Diagonals: +2 -2
55
Zero Sum Prime Number Square
17
0
2
-19
This square generates:
Horizontals: +17 -17
Verticals: +19 -19
Diagonals: +2 -2
56
Zero Sum Prime Number Square:
29
0
2
-31
This generates:
Horizontals: +29 -29
Verticals: +31 -31
Diagonals: +2 -2
57
Zero Sum 2 x 2 non-prime Square
27
0
2
-29
This generates:
Horizontals: +27 -27
Verticals: +29 -29
Diagonals: +2 -2
58
Another Zero Sum Square:
79
0
2
-81
This generates:
Horizontals: +79 -79
Verticals: +81 -81
Diagonals: +2 -2
59
MAGIC SQUARES
At this point a magic square will be examined using increment analysis. A magic
square is one wherein the positive integers of the verticals, horizontals and
diagonals all sum to the same number, the magic constant. There are no 2 x 2
magic squares. The formula for generating such a square is:
n2 M2(n) = 1/n Σ =1/2n (n
2 +1) k=1
If every number in a magic square is subtracted from n2 +1, another magic
square is obtained called the complementary magic square.3
The following third order magic square has integers that all sum to the number
15.
Third Order Magic Square
8
1
6
3
5
7
4
9
2
3 http://mathworld.wolfram.com/MagicSquare.html
60
This generates:
Horizontals: 816 = -7 +5 Verticals: 834 = -5 +1
159 = +4 +4 357 = +2 +2
492 = +5 -7 672 = +1 -5
Diagonals: 852 = -3 -3
456 = +1 +1
When added together, the sum of all increments in this magic square is negative
four. The horizontals and verticals sum to zero, but the diagonals, always
homogenous, tilt the square toward a remainder.
Fourth Order Magic Square
4
14
15
1
9
7
6
12
5
11
10
8
16
2
3
13
61
The above Fourth Order Magic square generates:
Horizontals Verticals Diagonals
+10 +1 -14 +5 -4 +11 +3 +3 +3
-2 -1 +6 -7 +4 -9 +5 +5 +5
+6 -1 -2 -9 +4 -7
-14 +1 +10 +11 -4 +5
As in the third order magic square, this one has horizontals and verticals that
sum to zero, and homogenous diagonals. The resulting +8 +8 +8 = +24, the
remainder.
LATIN SQUARES
A Latin square is an array of symbols wherein every symbol occurs
exactly once in each row and column of the array. Leonhard Euler,
the great mathematician, introduced them in 1783 as a “nouveau
espece de carres magiques”, a new kind of magic square.4
The following is a Second Order Latin square
1
2
2
1
This Second Order Latin square generates:
Horizontals Verticals Diagonals
+1 +1 0
-1 -1 0
This second order Latin Square is a zero sum square.
4 http://www.cut-the-knot.org/arithmetic/latin.shtml
62
The following is a Fourth Order Latin Square
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
This Fourth Order Latin Square
Horizontals Verticals Diagonals
+1 +1 +1 +1 +1 +1 +2 -2 +2
+1 +1 -3 +1 +1 -3 0 0 0
+1 -3 +1 +1 -3 +1
-3 +1 +1 -3 +1 +1
This Latin square exhibits the same type of pattern seen in the magic squares –
the horizontals and vertical increments sum to zero, and homogeneous diagonals
result in a remainder of +2.
63
CHAPTER TWELVE
“MAGIC” NUMBERS (of proton and neutron stability)
2
8
20
28
50
82
126; the sum of these numbers =
316
Adding zeros to this number yields:
03160 =
+3 -2 +5 -6 =
+8 -8
There is, therefore, and “eightness” connected with these numbers.
When 316 is divided by eight, the result is the repeating sequence
0.0253164556962 0253164556962 0253164… =
+15 -15
64
Other interesting features:
√316 = 17.776388834631177700182883350817
The reciprocal of 316 = 0.003164556962025 3164556962025
3164556….seems to say its own name.
Extracting any sequence from this number string sums to zero when zeroes are
added before and after the sequence.
An interesting thing happens when you flip these numbers and line them up next
to each other sideways creating the following number string:
2820285082126
without adding zeroes, the first generation yields:
+6 -6 -2 +2 +6 -3 -5 +8 -6 -1 +1 +4
-8 +8 -8
second generation:
+6 -8 +8 -8 +8 -6 -1 +1 +4
65
Now, when adding zeros to the original number sequence we get the following result.
2820285082126 =
028202850821260 =
+2 +6 -6 -2 +2 +6 -3 -5 +8 -6 -1 +1 +4 -6 =
+29 -29
66
CHAPTER THIRTEEN
NOTIONS ABOUT NOTHING AND EVERYTHING : A LOOK AT
ZERO AND INFINITY
Based on the evidence in this paper that all number sequence increments sum to
zero, zero becomes quite important and deserves closer scrutiny. The number
zero is enigmatic. Not always a member of the pantheon of numbers, the
concept of zero was introduced by the Babylonians around 300 BC. They used
two slanted wedges to represent an empty space. Indian mathematicians later
transformed zero from placeholder to a number, and later, when the Arabs
adopted Hindu-Arabic numerals, they also adopted zero. 5 The Mayans also had
a zero in their counting system. Christianity opposed it, and, in fact, zero and
infinity (nothing and everything) destroyed the Aristotelian philosophy. The void
and infinity, two sides of a coin, threatened the existence of God. For these
reasons, it took some time before zero was accepted by the West. Today, we
take zero for granted, even though it poses paradoxes. It is far more than a
placeholder, and I think the jury is still out regarding its true nature.
Rules regarding zero today are:
� Zero divided by any number is zero
� Any number divided by zero is undefined
� Any number multiplied by zero is zero
� Add zero to any number and it remains unchanged
� One divided by zero equals infinity (non-traditional)
� Dividing a number by infinity yields zero (non-traditional)
� Dividing a number by zero yields infinity (non-traditional)
5 Zero, The Biography of a Dangerous Idea. Seife, Charles. Penguin Books. NY. 2000.
67
At this point I will take a departure from traditional mathematics theory and let
zero speak for itself to get a clearer understanding of its true nature. For one,
zero seems to exhibit three different modes: Positive, Negative and Neutral.
Positive zero would be the concept of “everything”, Negative zero would be the
concept of nothing (no-thing), and Neutral zero would be the concept of
“balance” or ‘‘fullness.” Zero is not just an “emptiness,” but a “fullness” that
tends to displace itself in greater and greater increments in the expansion of
numbers. In this sense, it brings to mind Cantor’s Continuum Hypothesis with its
n. A question can be asked about what happens when the positiveא…2א 1א ,0א
and negative increments between numbers are squared, cubed, quadrupled, etc.
Would we witness an expansion of the complex plane? Are we witnessing larger
and larger zeros, and hence, larger and larger infinities?
ZERO DOMAIN DYNAMICS (ZDD)
Tradition tells us that we can’t divide by zero because we consider the result
“undefined.” When zero speaks for itself, however, an interesting picture
emerges. For instance, If it were feasible to perform the following operations, the
results would be :
(2x0) = 2 (3x0) = 3 (4 x 0) = 4 0 0 0
or 1/0 x 0 = 1; 5/0 x 0 = 5; 6/0 x 0 = 6
The foregoing is considered illogical based on current mathematical philosophy.
It is not possible, the thinking goes, for zero to equal all numbers. This is
precisely what zero is saying, however, and since this is the case, zero is
68
demonstrating equality with infinity. Indeed, on a Riemann Sphere, zero and
infinity are at opposite poles. If we could imagine a stacking of Riemann Spheres,
zero and infinity would be indistinguishable.
When adding, subtracting, multiplying or dividing numbers in the zero realm,
unexpected and “illogical” results prevail. In the zero domain, almost anything
goes. This is true when considering the “phantom values” of numbers, i.e., the
balanced binaries that result when zeroes are added in front of, and behind,
numbers. For example:
31 x 45 = 1395;
0310 x 0450 = 013950
(Phantom Values) (+3 -3) ( +5 -5) = +9 -9
In the zero domain, therefore, ( ∀∀∀∀3 )( ∀∀∀∀5) = ∀∀∀∀9
Technically, though, the result of the above operation is 0, since we are
multiplying 0 x 0.
Another example:
1849673 /14 = 132119.5;
018496730 / 0140 = 013211950;
+14 -14 / +4 - 4 = +11 -11,
therefore, ±14 divided by ±4 = ±11,
but again, we are dividing zero by zero to get zero.
69
CHAPTER FOURTEEN
ADDITIVE ASSESSMENT The additive assessment of a number sequence or series is the process of
sampling bits of information in bigger and bigger pieces and assessing the
samples. For example, take a number sequence and analyze the first two
numbers of the sequence, and then the first three, the first four, etc. until the
entire sequence has been assessed. In increment analysis, additive assessment
is used to assess the increments of larger and larger blocks of a number
sequence (or series). Zeroes are added before and after the segment. When this
is done, the segment always sums to zero, with a graduated pattern of positive
and negative numbers. Following, the additive assessment technique is
demonstrated for the number e (the basis of natural logarithms) up to the 25th
integer, pi, primes (up to the 13th prime) the first 10 Fibonacci numbers, the
reciprocal of the number seven, the reciprocal of number thirteen and phi.
e = 2.718281828449045235360287… 020 = + 2 - 2 0270 = + 7 - 7 02710 = + 7 - 7 027180 = +14 -14 0271820 = +14 -14 02718280 = +20 -20 027182810 = +20 -20 0271828180 = +27 -27 02718281820 = +27 -27 027182818280 = +33 -33 0271828182840 = +34 -34 02718281828450 = +38 -38 027182818284590 = +38 -38 0271828182845900 = +38 -38 02718281828459040 = +42 -42 027182818284590450 = +43 -43 0271828182845904520 = +43 -43 02718281828459045230 = +44 -44 027182818284590452350 = +46 -46
70
0271828182845904523530 = +46 -46 02718281828459045235360 = +49 -49 027182818284590452353600 = +49 -49 0271828182845904523536020 = +51 -51 02718281828459045235360280 = +57 -57 027182818284590452353602870 = +57 -57
Π (Pi) (to 31 digits)
3.141592653589793238462643327950…
030 = + 3 - 3 0310 = + 3 - 3 03140 = + 6 - 6 031410 = + 6 - 6 0314150 = +10 -10 03141590 = +14 -14 031415920 = +14 -14 0314159260 = +18 -18 03141592650 = +18 -18 031415926530 = +18 -18 0314159265350 = +20 -20 03131592653580 = +23 -23 031415926535890 = +24 -24 0314159265358970 = +24 -24 03141592653589790 = +26 -26 031415926535897930 = +26 -26 0314159265358979320 = +26 -26 03141592653589793230 = +27 -27 031415926535897932380 = +32 -32 0314159265358979323840 = +32 -32 03141592653589793238460 = +34 -34 031415926535897932384620 = +34 -34 0314159265358979323846260 = +38 -38 03141592653589793238462640 = +38 -38 031415926535897932384626430 = +38 -38 0314159265358979323846264330 = +38 -38 03141592653589793238462643320 = +38 -38 031415926535897932384626433270 = +43 -43 0314159265358979323846264332790 = +45 -45 03141592653589793238462643327950 = +45 -45 031415926535897932384626433279500 = +45 -45
71
PRIMES
(The first thirteen) 2 3 5 7 11 13 17 19 23 29 31 37 41
020 = +2 - 2 0230 = +3 - 3 02350 = +5 - 5 023570 = +7 - 7 02357 110 = +11 -11 02357 11 130 = +13 -13 02357 11 13 170 = +17 -17 02357 11 13 17 190 = +19 -19 02357 11 13 17 19 230 = +23 - 23 02357 11 13 17 19 23 290 = +29 -29 02357 11 13 17 19 23 29 310 = +31 -31 02357 11 13 17 19 23 29 31 370 = +37 -37 02357 11 13 17 19 23 29 31 37 410 = +41 -41 FIBONACCI NUMBERS
(The first ten)
1 1 2 3 5 8 13 21 34 55 010 = +1 -1 0110 = +1 -1 01120 = +2 -2 011230 = +3 -3 0112350 = +5 -5 01123580 = +8 -8 0112358 13 0 = +13 -13 0112358 13 21 0 = +21 -21 01123358 13 21 34 0 = +34 -34 0112358 13 21 34 55 0 = +55 -55
72
THE RECIPROCAL OF THE NUMBER SEVEN –0.142857142…. 010 +1 -1 0140 +4 -4 01420 +4 -4 014280 +10 -10 0142850 +10 -10 01428570 +12 -12 014285710 +12 -12 0142857140 +15 -15 01428571420 +15 -15 014285714280 +21 -21 0142857142850 +21 -21 01428571428570 +23 -23 014285714285710 +23 -23 0142857142857140 +26 -26 01428571428571420 +26 -26 014285714285714280 +32 -32 0142857142857142850 +34 -34 01428571428571428570 +34 -34 014285714285714285710 +34 -34 THE RECIPROCAL OF NUMBER THIRTEEN…0.076923076…. O70 +7 -7 0760 +7 -7 07690 +10 -10 076920 +10 -10 0769230 +11 -11 07692300 +11 -11 076923070 +18 -18 0769230760 +18 -18 07692307690 +21 -21 076923076920 +21 -21 0769230769230 +22 -22 07692307692300 +22 -22 076923076923070 +29 -29
73
Φ (PHI) (THE GOLDEN MEAN) – 1.6180339… 010 +1 -1 0160 +6 -6 01610 +6 -6 016180 +13 -13 0161800 +13 -13 01618030 +16 -16 016180330 +16 -16 0161803390 +22 -22 The prime and Fibonacci numbers “say their own names.” In other words, the
last prime or Fibonacci of each segment gives the name of the segment, which
follows the natural order of each sequence with no repetition characteristic of the
other numbers assessed thus far. This might be true for every uniformly
monotonic increasing number sequence.
ADDITIVE SIGNATURES As demonstrated in the previous examples, every number sequence has an
additive signature that is characteristic of its assessment. The signatures of the
numbers thus far assessed are shown below.
e ±
2,7,7,14,14,20,20,27,27,33,34,38,38,38,42,43,43,44,46,46,49,49,51,57,57
Primes ± 2,3,5,7,11,13,17,19,23,29,31,37,41
π
± 3,3,6,6,10,14,14,18,18,18,20,23,24,24,26,26,26,27,32,32,34,34,38,38,38,38,38,43,45,45,45
Fibonacci #
± 1,1,2,3,5,8,13,21,34,55
1/7 ± 1,4,4,10,10,12,12,15,15,21,21,23,23,26,26,32,34,34,34
1/13 ± 7,7,10,10,11,11,18,18,21,21,22,22,29
Phi
± 1,6,6,13,13,16,16,22
74
CHAPTER FIFTEEN
MULTIPLICATIVE ASSESSMENT
THE CIRCULARITY OF “ONENESS”
In previous chapters we added zeroes to numbers to produce balanced binaries.
In actuality, any sequence that begins and ends with the same number creates a
perfectly balanced binary. This is most likely due to The Law of Conservation as
identified by the great mathematician Emmy Noether. In the foregoing, we have
started and ended with zeroes to produce perfectly balanced binaries. This is the
only number that can be added to a sequence to create balanced binaries
without changing the “value” of the sequence.
In this chapter, we will explore the binaries created using repunits (repeating
ones) and repdigits (repeating digits) multiplied by each other. It will be noted
that series of ones, when multiplied by series of ones, always result in the
number line from one through 9 and back again. Sometimes, there are even
cycles within the cycles.
1. 1 x 1 = 1
2. 11 x 11 = 121
3. 111 x 111 = 12321
4. 1111 x 1111 =1234321
5. 11111 x 11111 = 123454321
6. 111111 x 111111 = 12345654321
7. 1111111 x 1111111 = 1234567654321
8. 11111111 x 11111111 = 123456787654321
9. 111111111 x 111111111 = 12345678987654321
10. 1111111111 x 1111111111 = 1234567900987654321
11. 11111111111 x 11111111111 = 123456790120987654321
12. 111111111111 x 111111111111 = 12345679012320987654321
13. 1111111111111 x 1111111111111 = 1234567901234320987654321
14. 11111111111111 x 11111111111111 = 123456790123454320987654321
15. 111111111111111 x 111111111111111 = 12345679012345654320987654321
16. 1111111111111111 x 1111111111111111 = 12345679012345677654320987654321
17. 11111111111111111 x 11111111111111111 = 1.2345679012345678765432098765432e+32
18. 111111111111111111 x 111111111111111111 = 1.2345679012345678987654320987654e+34
19. 1111111111111111111 x 1111111111111111111 = 1.2345679012345679009876543209877e+36
20. 11111111111111111111 x 11111111111111111111 = 1.2345679012345679012098765432099e+38
21. 111111111111111111111 x 111111111111111111111 = 1.2345679012345679012320987654321e+40
22. 1111111111111111111111 x 111111111111111111111=1.2345679012345679012343209876543e+42
75
Natural patterns within the lines numbered 1 – 22, when multiplying ones by each
other, resulted in balanced binaries. Starting with # 10, there developed
segments within the segments, creating binaries within binaries.
Line Patterns 1. 1 2. +1 -1 3. +2 -2 4. +3 -3 5. +4 -4 6. +5 -5 7. +6 -6 8. +7 -7 9. +8 -8 10. Change point, beginning of segments in segments, starting with 00; from this point on, only the segments within segments underlined will be assessed. Zeroes did not need to be added – they were already there). 11. 0120 +2 -2 12. 012320 +3 -3 13. 01234320 +4 -4 14. 0123454320 +5 -5 15. 012345654320 +6 -6 16. 012345677654320 +7 -7 17. 0123456787654320 +8 -8 18. 012345678987654320 +9 -9 19. Change point, two segments within the line segment:
0123456790 0987654320 +9 -9 +9 -9 20. 0123456790120 +11 -11 21. 012345679012320 +12 -12 22. 01234567901234320 +13 -13
76
In the foregoing sections, even groupings of ones are multiplied by the same
number of ones. In the following examples, uneven groupings of ones are
multiplied. The results are still the same, however, balanced binaries.
1111 x 11111 = 12344321
1111111 x 11111 = 12345554321
111 x 111111111 = 12333333321
11 x 11111 = 122221
THE CIRCULARITY OF “TWONESS” When groupings of twos are multiplied by other groupings of twos, the result
always begins and ends with the number four, creating balanced binaries in the
process.
2 x 2 = 4 4 22 x 22 = 484 +4 -4 222 x 222 = 49284 +11 -11 2222 x 2222 = 4937284 +15 -15 22222 x 22222 = 493817284 +22 -22 222222 x 222222 = 49382617284 +26 -26 2222222 x 2222222 = 4938270617284 +33 -33 22222222 x 22222222 = 493827150617284 +37 -37 222222222 x 222222222 = 49382715950617284 +41 -41 2222222222 x 2222222222 = 4938271603950617284 +47 -47 22222222222 x 22222222222 = 493827160483950617284 +52 -52
THREENESS
Groupings of three multiplied by each other, unlike ones and twos, do not
produce balanced binaries. They always result in +8, and when zeroes are
added, +11 -11.
333 x 33333 = 11099889
77
“FOURNESS” THROUGH “EIGHTNESS:”
These numbers form predictable patterns when multiplied by themselves, but are
not naturally circular and do not form balanced binaries unless zeroes are added.
THE NUMBER NINE:
The number nine is the only number that generates a negative number,
which is always –8. (It is the converse of #3 which always generates +8). When
zeroes are added, it always generates +11 -11, just as # 3 does.
78
CHAPTER SIXTEEN
FACTORIALS
A factorial is the product of a given series of whole numbers. The numbers 1! –
10! yield the following patterns:
Assessment (Phantom) Values
1! = 1 010 = +1 -1
2! = 2 020 = +2 -2
3! = 6 060 = +6 -6
4! = 24 0240 = +4 -4
5! = 120 0120 = +2 -2
6! = 720 0720 = +7 -7
7! = 5040 05040 = +9 -9
8! = 40320 040320 = +7 -7
9! = 362880 0362880 = +12 -12
10! = 3628800 03628800 = +12 -12
Note that the factorial values are always even numbers with the exception of #1,
yet the phantom (assessment) values alternate.
79
CHAPTER SEVENTEEN
CONCLUSIONS AND IMPLICATIONS
The End of Randomness
Kurt Gödel (1906 – 1978) turned the world of mathematics upside-down with his
famous proof that showed that if a system is consistent, then you can show that it
is incomplete. Mathematician Gregory J. Chaitin of the IBM Thomas J. Watson
Research Center, author of The Limits of Mathematics suggests that the
structure of arithmetic is random. He states, “Although almost all numbers are
random, there is no formal axiomatic system that will allow us to prove this fact.”
Increment Analysis and The Law of Digit Balance point to an underlying order in
mathematics that defies randomness, and 50 + element mirror image increments
lends credence to this notion. Moreover, zero seems to be the background
“noise” of all numbers. Increment analysis opens up a whole new set of options
for analyzing number sequences. New unsuspected relationships and hidden
patterns become apparent that are beautifully symmetrical. As a result of the
evidence presented, the following implications become apparent:
• There is a Law of Digit Balance wherein the increments
between integers of any number sequence sum to zero. This
includes sequences comprised of positive and negative
numbers.
♦ The increments between digits of circular numbers manifest mirror image
patterns that sum to zero in all cases.
♦ The first 218 digits of pi have increments that converge to zero when a
quaternary skip sequence is utilized to isolate digits.
80
♦ The number seven is prominent in many numbers that produce mirror image
groupings that cancel to zero
♦ The prime numbers are organized around the digits of the reciprocal of seven
♦ Transcendental numbers have increments that sum to zero when extracting a
sequence of any size, hence, transcendental numbers sum to zero.
♦ Zero does not merely represent the absence of something, it also represents
a fullness and a balance.
♦ Fibonacci and Lucas numbers have increments that are perfect mirrors with
anomalies that add to nine, and demonstrate 24 – step repetition cycles.
♦ All numbers begin and end in zero.
Practical Applications?
When considering the practical applications of The Law of Digit Balance I am
reminded of a popular television commercial for an insurance company. In it, a
whimsical cartoon character is handed a black dot. The character, puzzling,
doesn’t know what to do with it until, suddenly, something falling from the sky
appears which threatens to squash him. He drops the dot, which becomes a hole
in the ground offering a means of escape. The point is that he had a tool that
was useless until more information was forthcoming. Further research may open
broader vistas in the world of numbers and yield surprising practical applications.
Chief among these might be information that can lead to a better understanding
of “trinary logic” and its application to trinary computers.
Actually, the idea of trinary computers is not a remote dream. One was
developed (SETUN) in 1958 by Nikolai Brusentsov and his team at the Moscow
State University and was based on ternary (trinary) logic (-1, 0, 1), which
distinguishes it completely from the usually binary operating computers of the
81
present and the past. While the binary logic just allows two states yes (1) | no (0),
the ternary logic has three different logical states: yes (1) | no (-1) | both or
maybe (0). In this book, Increment Analysis always reveals balanced positive
and negative numbers resulting in zero underlying all number sequences.
The sheer beauty of pattern revealed by The Law of Digit Balance, however,
provides esthetic satisfaction in itself.
82
83
NATURAL NUMBERS 1 – 100
Number Sum of increments
1 010 = +1 -1 = 0
2 020 = +2 –2 = 0
3 030 = +3 –3 = 0
4 040 = +4 –4 + 0
5 050 = +5 –5 = 0
6 060 = +6 –6 = 0
7 070 = +7 –7 = 0
8 080 = +8 – 8 = 0
9 090 = +9 –9 = 0
10 010 = +1 -1 = 0
11 0110 = +1 0 –1 = 0
84
12 0120 = +1 +1 – 2 =
+2 -2 = 0
13 0130 = +1 +2 –3 =
+3 –3 = 0
14 0140 = +1 +3 –4 =
+4 -4 = 0
15 0150 = +1+4 -5 =
+5 -5 = 0
16 0160 = +1 +5 –6 =
+6 –6 = 0
17 0170 = +1+6 –7 =
+7 –7 = 0
85
18
0180 = +1 +7 –8 =
+8 - 8 = 0
19 0190 = +1 +8 –9 =
+9 – 9 = 0
20 020 = +2 – 2 = 0
21 0210 = +2 –1 –1 =
+2 -2 = 0
22 0220 = +2 0 -2 = 0
23 0230 = +2 +1 –3
+3 -3 = 0
24 0240 = +2 +2 –4 =
+4 –4 = 0
86
25
0250 = +2 +3 –5 =
+5 -5 = 0
26 0260 = +2 +4 –6 =
+6 –6 = 0
27 0270 = +2 +5 –7 =
+7 -7 = 0
28 0280 = +2 +6 –8=
+8 –8 = 0
29 0290 = +2 +7 –9 =
+9 -9 = 0
30 030 = +3 -3 = 0
87
31 0310 = +3 –2 –1 =
+3 -3 = 0
32 0320 = +3 –1 –2 =
+3 -3 = 0
33 0330 = +3 0 -3 = 0
34 0340 = +3 +1 –4 =
+4 -4 = 0
35 0350 = +3 +2 –5 =
+5 -5 = 0
36 0360 = +3 +3 -6 =
+6 -6 = 0
88
37 0370 = +3 +4 –7 =
+7 -7 = 0
38 0380 = +3 +5 –8 =
+8 –8 = 0
39 0390 = +3 +6 –9 =
+9 -9 = 0
40 040 = +4 –4 = 0
41 0410 = +4 –3 –1 =
+4 -4 = 0
42 0420 = +4 –2 –2 =
+4 -4 = 0
43 0430 = +4 –1 –3 =
+4 -4 = 0
89
44 0440 = +4 0 -4 = 0
45
0450 = +4+1 –5 =
+5 –5 = 0
46
0460 = +4+2 –6 =
+6 -6 = 0
47 0470 = +4+3 – 7 =
+7 -7 = 0
48 0480 = +4+4 –8 =
+8 –8 = 0
49 0490 = +4+5 –9 =
+9 -9 = 0
90
50 050 = +5 –5 = 0
51
0510 = +5 –4 –1 =
+5 -5 = 0
52 0520 = +5 –3 –2 =
+5 -5 = 0
53 0530 = +5 –2 –3 =
+5 -5 = 0
54 0540 = +5 –1 –4 =
+5 -5 = 0
55 0550 = +5 0 -5 = 0
91
56 0560 = +5 +1 -6 =
+6 -6 = 0
57 0570 = +5 +2 –7 =
+7 -7 = 0
58 0580 = +5+3 -8 =
+8 -8 = 0
59 0590 = +5+4 -9 =
+9 -9 = 0
60 060 = +6 –6 = 0
61 0610 = +6 -5 –1 =
+6 -6 = 0
92
62 0620 = +6 -4 –2 =
+6 -6 = 0
63 0630 = +6 -3 –3 =
+6 -6 = 0
64 0640 = +6 -2 –4 =
+6 -6 = 0
65 0650 = +6 -1 –5 =
+6 -6 = 0
66 0660 = +6 0 -6 = 0
67
0670 = +6 +1 –7 =
+7 -7 = 0
93
68 0680 = +6+2 – 8 =
+8 -8 = 0
69 0690 = +6+3 –9 =
+9 -9 = 0
70 070 = +7 –7 = 0
71 0710 = +7 –6 –1 =
+7 -7
72 0720 = +7 –5-2 =
+7 -7 = 0
73 0730 = +7 –4 –3 =
+7 -7 = 0
74 0740 = +7 –3 –4 =
+7 -7 = 0
94
75 0750 = +7 -2 –5 =
+7 -7 = 0
76 0760 = +7 -1 -6 =
+7 -7 = 0
77 0770 = =7 0 -7 = 0
78 0780 = +7 +1 –8 =
+8 -8 = 0
79 0790 = +7 +2 –9 =
+9 -9 = 0
80 080 = +8 -8 = 0
81 0810 = +8 –7 –1 =
+8 -8 = 0
95
82 0820 = +8 –6 –2 =
+8 -8 = 0
83 0830 = +8 -5 –3 =
+8 -8 = 0
84 0840 = +8 -4 –4 =
+8 -8 = 0
85
0850 = +8 -3 –5 =
+8 -8 = 0
86 0860 = +8 -2 -6 =
+8 -8 = 0
96
87
0870 = +8 -1 -7 =
+8 -8 = 0
88 0880 = +8 0 -8 = 0
89 0890 = +8 +1 -9 =
+9 -9 = 0
90 090 = +9 -9 = 0
91 0910 = +9 -8 -1 =
+9 -9 = 0
92 0920 = +9 -7 –2 =
+9 -9 = 0
93 0930 = +9 -6 –3 =
+9 -9 = 0
97
94
0940 = +9 –5 –4 =
+9 -9 = 0
95 0950 = +9 -4 -5 =
+9 -9 = 0
96 0960 = +9 -3 –6 =
+9 -9 = 0
97 0970 = +9 -2 -7 =
+9 -9 = 0
98 0980 = +9 -1 -8 =
+9 -9 = 0
99 0990 = +9 0 -9 = 0
100 0100 = +1 -1 0 = 0
98
RECIPROCALS of #1 – 100
(If zeros are added before and after the number, it always sums to zero).
NUMBER
PATTERN MODE
1/1 = 1.0
+1 -1 Oscillating 1s
½ = 0.50
+5 -5 Oscillating 5s
1/3 = 0.333333333
( +3) 0000000000
0
¼ = 0.250
+2 +3 –5 second order: (+5 -5)
Oscillating 5s
1/5 = 0.20
+2 -2
Oscillating 2s
1/6 = 0.166666666666…
(+1 +5) 00000000
0
1/7 = 142857142857…
+3 -2 +6 -3 +2 -6
Six-element mirror
1/8 = 0.125
+1 +1 +3 second order: +5
Mixed (Ps = +5)
1/9 =0.1111111111…
(+1) 000000000
0
1/10 = 0.10
+1 -1
Oscillating 1s
1/11 = 0.0909090909090…
+9 -9 +9 -9…
Oscillating 9s
99
1/12 = 0.0833333333333…
(+8 -5) 00000000000
0
1/13 = 0.0769230769230…
+7 -1 +3 -7 +1 -3
Six-element mirror
1/14 = 0.07142857142857…
(+7) –6 +3 -2 +6 -3 +2 …
Six-element mirror
1/15 = 0.06666666666….
(+6) 000000000000…
0
1/16 = 0.0625
+6 -4 +3
Mixed (Ps = +5)
1/17 = 0.05882352941176470
+5 +3 0 -6 +1 +2 -3 +7 -5 -3 0 +6 -1 -2 +3 -7
16-element mirror
1/18 = 0.0555555555…
(+5) 00000000000…
0
1/19 = 0.0526315789473684210…
+5 -3 +4 -3 -2 +4 +2 +1 +1 -5 +3 -4 +3 +2 -4 -2 -1 -1
18-element mirror
1/20 = 0.05
+5 -5
Oscillating 5s
1/21 = 0.047619047619…
+4 +3 -1 -5 +8
Mixed (Ps = 9)
1/22 = 0.04545454545…
+1 -1 +1 -1…
Oscillating 1s
100
1/23 = 0.04347826086956521739130…
+4 -1 +1 +3 +1 -6 +4 –6 +8 -2 +3 -4 +1 -1 -3 -1 +6 -4 +6 -8 +2 -3
22-element mirror
1/24 = 0.041666666666666…
(+4 -3 +5) 0000000000000
0
1/25 = 0.04
+4 -4
Oscillating 4s
1/26 = 0.03846153846153846153846…
( +3 ) +5 -4 +2 -5 +4 -2 +3 –4 +2
6-element mirror
1/27 = 0.037037037037…
+3 +4 -7 +3 +4 -7 Second order: +7 -7
Oscillating 7s
1/28 = 0.0357142857142857…
(+3 +2) +2 -6 +3 -2 +6 -3
6-element mirror
1/29 = 0.03448275862068965517241379310…
+3 +1 0 +4 -6 +5 -2 +3 -2 -4 -2 +6 +2 +1 -3 -1 0 -4 +6 -5 +2 -3 +2 +4 +2 -6 -2 -1
28-element mirror
1/30 = 0.033333333333333333… 1/31 = 0.0322580645161290…
(+3) 00000000000000000
0
101
1/32 = 0.03125
+3 -2 +1 +3
Mixed (Ps = +5)
1/33 = 0.030303030303…
+3 -3 +3 -3 +3 -3…
Oscillating 3s
1/34 = 0.02941176470588235…
(+2 +7) -5 -3 0 +6 -1 -2 +3 -7 +5 +3 0 -6 +1 +2 -3 +7 -5 -3 0
16-element mirror; Oscillating –5 -3 0, +5 +3 0
1/35 = 0.02857142857142857…
(+2) +6 -3 + 2 -6 +3 -2 +6 -3…
6-element mirror
1/36 = 0.027777777777…
(+2 +5) 000000000000
0 (Ps = +7)
1/37 = 0.027027027027…
+2 +5 -7 +2 +5 -7 +2 +5 -7… Second order: +7 -7 +7 -7 +7 -7…
Oscillating 7s
1/38 = 0.0263157894736842105263…
(+2) +4 -3 -2 +4 +2 +1 +1 -5 +3 -4 +3 +2 -4 -2 -1 -1 +5 -3 (+4 -3)
18-element mirror
1/39 = 0.025641025641…
+2 +3 +1 -2 -3 -1…
6-element mirror Ps = 0
102
1/40 = 0.025
+2 +3
Mixed (Ps = +5)
1/41 = 0.02439 02439…
+2 +2 -1 +6…
Mixed (Ps = +9)
1/42 = 0.0238095 238095…
(+2) +1 +5 -8 +9 -4 -3…
Mixed (Ps = 0)
1/43 = 0.0232558139534883720930 232558139535 1/44 = 0.02272727272727…
+2 +1 -1 +3 0 +3 -7 +2 +6 -4 -2 +1 +4 0 -5 +4 -5 -2 +9 -6 -3…
Mixed; (Ps = 0)
1/45 = 0.02222222222222…
(+2) 000000000000
0
1/46 = 0.021739130434782608695652…
-1 +6 -4 +6 -8 +2 -3 +4 -1 +1 +3 +1 -6 +4 -6 +8 -2 +3 -4 +1 -1 -3
22-element mirror Ps = 0
103
1/47 = 0.0212765957446808510638297872340425…
+2 -1 +1 +5 -1 -1 +4 -4 +2 –3 0 +2 +2 -8 +8 -3 -4 -1 +6 -3 +5 -6 +7 -2 +1 -1 -5 +1 +1 -4 +4 -2 +3 0 -2 -2 +8 -8 +3 +4 +1 -6 +3 –5 +6 -7***
46-element mirror Ps = 0
1/48 = 0.02083333333333333333…
(+2 -2 +8 -5 ) 000000000000000000
0
1/49 = 0.0204081632653061224489795918367346…
+2 -2 +4 -4 +8 -7+5 -3 -1 +4 –1 -2 -3 +6 -5 +1 0 +2 0 +4 +1 -2 +2 -4 +4 -8 +7 -5 +3 +1 –4 +1 +2 +3 -6 +5 -1 0 -2 0 -4 -1 ***
42-element mirror Ps = 0
1/50 = 0.02
+2 (-2)
+2 ( -2)
1/51 = 0.01960784313725490…
+1 +8 -3 -6 +7 +1 -4 -1 0 +4 -5 +3 -1 +5 -9
Mixed (Ps = 0)
1/52 = 0.019230769230769230…
( +1 +8) -7 +1 -3 +7 -1 +3…
6-element mirror
104
1/53 = 0.01886792452830 1886792452830 1886792…
+1 +7 0 -2 +1 +2 -7 +2 +1 -3 +6 -5 -3
Mixed (Ps = 0)
1/54 = 0.0185185185185185…
(+1) +7 -3 -4 +7 -3 -4… Second order: (+1) +7 -7 +7 -7
Oscillating sevens (Ps = 0)***
1/55 = 0.018181818181818…
(+1) +7 -7 +7 -7 +7 -7…
Oscillating sevens
1/56 = 0.017857 142857 142857…
(+1 +6 +1) -3 +2 -6 +3 -2 +6…
Six-element mirror
1/57 = 0.0175438596491228070 175438596491228…
+1 +6 -2 -1 -1 +5 -3 +4 -3 -2 +5 -8 +1 0 +6 -8 +7 -7…
Mixed (Ps = 0)
1/58 = 0.01724137931034482758620689655 17241…
(+1) +6 -5 +2 -3 +2 +4 +2 -6 -2 -1 +3 +1 0 +4 -6 +5 -2 +3 -2 -4 -2 +6 +2 +1 -3 -1 0 -4…
28-element mirror
105
1/59 = 0.0169491525423728813559322033898305…
(+1) +5 +3 -5 +5 -8 +4 -3 +3 -1 -2 +1 +4 -5 +6 0 -7 +2 +2 0 +4 -6 -1 0 -2 +3 0 +5 +1 -1 -5 -3 +5 -5 +8 -4 +3 -3 +1 +2 -1 -4 +5 -6 0 +7 -2 -2 0 -4 +6 +1 0 +2 -3 0 -5 -1 +1
58-element mirror
1/60 = 0.016666666666666666…
(+1 +5) 0000000000000000000
0
1/61 = 0.0163934426229508196721311475409836…
+1 +5 -3 +6 -6 +1 0 -2 +4 -4 0 +7 -4 -5 +8 -7 +8 -3 +1 -5 -1 +2 -2 0 +3 +3 -2 -1 -4 +9 -1 -5 +3……
60-element mirror?
1/62 = 0.0161290322580645 161290322580645 161…
(+1) +5 -5 +1 +7 -9 +3 -1 0 +3 +3 -8 +6 -2 +1 -4 +5 -5…
Mixed (Ps = 0 )
106
1/63 = 0.015873 015873 015873 015873 015873 0158…
+1 +4 +3 -1 - 4 -3
Six-element mirror
1/64 = 0.015625
+1 +4 +1 -4 +3
Mixed (Ps = 5)
1/65 = 0.0153846 153846 153846153846 153846 153…
(+1) +4 -2 +5 -4 +2 -5
Six-element mirror
1/66 = 0.0151515151515….
(+1) +4 -4 +4 -4 +4 -4 +4 -4…
Oscillating 4s (Ps = 0)
1/67 = 0.014925373134328358208955223880597…
(+1) +3 +5 -7 +3 -2 +4 -4 -2 +2 +1 -1 -1 +6 -5 +2 +3 -6 -2 +8 +1 -4 0 -3 0 +1 +5 0 -8 +5 +4 -2 …
62-element mirror???
1/68 = 0.014705882352941176 4705882352941176…
(+1 +3) +3 -7 +5 +3 0 -6 +1 (+2) -3 +7 -5 -3 0 +6 -1
14-element mirror
107
1/69 = 0.01449275362318840579710 14492753623…
+1 +3 0 +5 -7 +5 -2 -2 +3 -4 +1 -2 +7 0 -4 -4 +5 +2 +2 -2 -6 -1….
Mixed (Ps = 0 )
1/70 = 0.0142857 142857142857…
(+1) +3 -2 +6 -3 +2 -6
6-element mirror
1/71 = 0.0140845070422535211267605633802817…
+3-4+8-4+1-5+7-7+4-2 0+3-2+2-3-10+1+4+1-1-6+5+1-3 0+5-8+2+6-7+6
Mixed
1/72 = 0.0138888888888888888…
(+1) +2 +5 000000000000
0
1/73 = 0.013698630 13698630 13698630 136986301…
+1 +2 +3 +3 -1 -2 -3 -3
8-element mirror
1/74 = 0.0135135135135135135…
(+1) +2 +2 -4 +2 +2 -4 Second order: +4 -4 +4 -4…..
Oscillating 4s
1/75 = 0.01333333333333…
(+1) +2 00000000000000
0
108
1/76 = 0.01315789473684210526 31578947368421…
(+1) +2 -2 +4 +2 +1 +1 -5 +3 -4 -3 +2 -4 -2 -1 -1 +5 -3 +4 -3
18-element mirror
1/77 = 0.0129870 129870 129870 129870 129870…
+1 +1 +7 -1 -1 -7
6-element mirror
1/80 = 0.0125
+1 +1 +3
Mixed (Ps = 5)
1/81 = 0.012345679 012345679 012345679…
+1 +1 +1 +1 +1 +1 + 1 +2
+1
1/82 = 0.012195 12195 12195 1219512195 12195…
(+1) +1 -1 +8 -4
Mixed (Ps = +5 )
1/83 = 0.0120481927710843373493975903614458
+1+1-2+4+4-7+8-7+5 0 –6-1+8-4-1 0 +4-4+1+5-6+6-2-2+4-9+3+6-5+3 0 +1+3…
Mixed
1/84 = 0.01190476 190476 190476 190476 19047619
(+1 0) +8 -9 +4 +3 -1
Mixed (Ps = 6)
109
1/85 = 0.01176470588235294 1176470588235294…
(+1) 0 +6 -1 -2 +3 -7 +5 +3 0 -6 +1 +2 -3 +7 -5 -3
16-element mirror
1/86 = 0.0116279069767441860465 116279069767…
+1 0 +5 –4+5+2-9+6+3-2-1+1-3 0 –3+7-2-6+4+2-1=4 0 +5-4+5+2-9+6+3-2-1+1
Mixed
1/87 = 0.01149425287356321839080459770 11494…
+1 = +3 +5-5-2+3-3+6-1-4+2+1-3-1-1+7-5+6-9+8-8+4+1+4-2 0 -– +1 0 +3 +5-5
Mixed
1/88 = 0.0113636363636363636…
(+1) 0 -3 +3 -3 +3 -3 +3….
Oscillating 3s
110
1/89 = 0.0112359550561797752808988764044944
+1 0 +1 +1 +2 +4-4 0 –5+5+1-5+6+2-2 0 –2 –3 +6 –8+8 +1-1 0 –1-1-2-4+4 0 +5 –5 0
44-element mirror
1/90 = 0.0111111111111111111111111111111111…
(+1) 000000000000000
0
1/91 = 0.010989 010989 010989 010989 010989…
+1 -1 +9 -1 +1 -9
6-element mirror
1/92 = 0.0108695652173913043478260869565217
(+1-1)+8-2+3-4 +1-1-3-1+6-4+6-8+2-3 +4- 1+1+3+1-6+4-6 +8 -2+3-4+1-1-3-1+6
22-element mirror
1/93 = 0.010752688172043 010752688172043…
+1 -1 +7 -2 -3 +4 +2 0 +6 -5 -2 +4 -1 (-3)
Mixed
1/94 = 0.0106382978723404255319148936170213
(+1-1)+6-3+5-6+7-2+1-1-5+1+1-4+4-2+3 0 –2-2+8-8+3+4+1 -6+3-5+6-7+2-1+2
14-element mirror
111
1/95 = 0.0105263157894736842105263157894737
(+1-1)+5-3+4-3-2+4+2+1+1-5+3-4+3+2-4-2-1-1+5-3+4-3-2+4+2+1+1 -5+3-4+4…
+5-3+4-3-2+4+2+1+1 +5-3+4-3-2+4+2 +1-1
1/96 = 0.010416666666666666666666666666666…
(+1-1)+4-3+5 0 0 0…
Ps = 6
1/97 = 0.0103092783505154639175257731958763
(+1-1)+3-3+9-7+5+1-5+2-5+5-4+4-1+2-3+6-8+6-2 -3+3+2 0 –4-2+8-4+3-1-1-3
Mixed
1/98 = 0.0102040816326530612244897959183673
(+1-1)+2-2+4-4+8-7+5-3 -1+4-1-2-3+6-5+1 0 +2 0+4+1-2+2 -4+4-8+7 -5+3+1-4…
46-element mirror
1/99 = 0.0101010101010101010101010101010101
+1-1+1-1….
Oscillating 1s
1/100 = 0.01
+1-1
0
112
MIRROR SEQUENCES
(For reciprocals of numbers between 1 – 100)
1/7 = 142857142857…
+3 -2 +6 -3 +2 -6
Six-element mirror
1/13 = 0.0769230769230…
+7 -1 +3 -7 +1 -3
Six-element mirror
1/14 = 0.07142857142857…
(+7) –6 +3 -2 +6 -3 +2 …
Six-element mirror
1/17 = 0.05882352941176470
+5 +3 0 -6 +1 +2 -3 +7 -5 -3 0 +6 -1 -2 +3 -7
16-element mirror
1/19 = 0.0526315789473684210…
+5 -3 +4 -3 -2 +4 +2 +1 +1 -5 +3 -4 +3 +2 -4 -2 -1 -1
18-element mirror
1/23 = 0.04347826086956521739130…
+4 -1 +1 +3 +1 -6 +4 –6 +8 -2 +3 -4 +1 -1 -3 -1 +6 -4 +6 -8 +2 -3
22-element mirror
1/26 = 0.03846153846153846153846…
( +3 ) +5 -4 +2 -5 +4 -2 +3 –4 +2
6-element mirror
1/28 = 0.0357142857142857…
(+3 +2) +2 -6 +3 -2 +6 -3
6-element mirror
1/29 = 0.03448275862068965517241379310…
+3 +1 0 +4 -6 +5 -2 +3 -2 -4 -2 +6 +2 +1 -3 -1 0 -4 +6 -5 +2 -3 +2 +4 +2 -6 -2 -1
28-element mirror
113
1/34 = 0.02941176470588235…
(+2 +7) -5 -3 0 +6 -1 -2 +3 -7 +5 +3 0 -6 +1 +2 -3 +7 -5 -3 0
16-element mirror; Oscillating –5 -3 0, +5 +3 0
1/35 = 0.02857142857142857…
(+2) +6 -3 + 2 -6 +3 -2 +6 -3…
6-element mirror
1/38 = 0.0263157894736842105263…
(+2) +4 -3 -2 +4 +2 +1 +1 -5 +3 -4 +3 +2 -4 -2 -1 -1 +5 -3 (+4 -3)
18-element mirror
1/39 = 0.025641025641…
+2 +3 +1 -2 -3 -1…
6-element mirror Ps = 0
1/46 = 0.021739130434782608695652…
-1 +6 -4 +6 -8 +2 -3 +4 -1 +1 +3 +1 -6 +4 -6 +8 -2 +3 -4 +1 -1 -3
22-element mirror Ps = 0
1/47 = 0.0212765957446808510638297872340425…
+2 -1 +1 +5 -1 -1 +4 -4 +2 –3 0 +2 +2 -8 +8 -3 -4 -1 +6 -3 +5 -6 +7 -2 +1 -1 -5 +1 +1 -4 +4 -2 +3 0 -2 -2 +8 -8 +3 +4 +1 -6 +3 –5 +6 -7
46-element mirror Ps = 0
1/49 = 0.0204081632653061224489795918367346…
+2 -2 +4 -4 +8 -7+5 -3 -1 +4 –1 -2 -3 +6 -5 +1 0 +2 0 +4 +1 -2 +2 -4 +4 -8 +7 -5 +3 +1 –4 +1 +2 +3 -6 +5 -1 0 -2 0 -4 -1
42-element mirror Ps = 0
114
1/52 = 0.019230769230769230…
( +1 +8) -7 +1 -3 +7 -1 +3…
6-element mirror
1/56 = 0.017857 142857 142857…
(+1 +6 +1) -3 +2 -6 +3 -2 +6…
Six-element mirror
1/58 = 0.01724137931034482758620689655 17241…
(+1) +6 -5 +2 -3 +2 +4 +2 -6 -2 -1 +3 +1 0 +4 -6 +5 -2 +3 -2 -4 -2 +6 +2 +1 -3 -1 0 -4…
28-element mirror
1/59 = 0.0169491525423728813559322033898305…
(+1) +5 +3 -5 +5 -8 +4 -3 +3 -1 -2 +1 +4 -5 +6 0 -7 +2 +2 0 +4 -6 -1 0 -2 +3 0 +5 +1 -1 -5 -3 +5 -5 +8 -4 +3 -3 +1 +2 -1 -4 +5 -6 0 +7 -2 -2 0 -4 +6 +1 0 +2 -3 0 -5 -1 +1
58-element mirror
1/61 = 0.0163934426229508196721311475409836…
+1 +5 -3 +6 -6 +1 0 -2 +4 -4 0 +7 -4 -5 +8 -7 +8 -3 +1 -5 -1 +2 -2 0 +3 +3 -2 -1 -4 +9 -1 -5 +3……
60-element mirror?
115
1/63 = 0.015873 015873 015873 015873 015873 0158…
+1 +4 +3 -1 - 4 -3
Six-element mirror
1/65 = 0.0153846 153846 153846153846 153846 153…
(+1) +4 -2 +5 -4 +2 -5
Six-element mirror
1/67 = 0.014925373134328358208955223880597…
(+1) +3 +5 -7 +3 -2 +4 -4 -2 +2 +1 -1 -1 +6 -5 +2 +3 -6 -2 +8 +1 -4 0 -3 0 +1 +5 0 -8 +5 +4 -2 …
? 66-element mirror???
1/68 = 0.014705882352941176 4705882352941176…
(+1 +3) +3 -7 +5 +3 0 -6 +1 (+2) -3 +7 -5 -3 0 +6 -1
14-element mirror
1/70 = 0.0142857 142857142857…
(+1) +3 -2 +6 -3 +2 -6
6-element mirror
1/73 = 0.013698630 13698630 13698630 136986301…
+1 +2 +3 +3 -1 -2 -3 -3
8-element mirror
1/76 = 0.01315789473684210526 31578947368421…
(+1) +2 -2 +4 +2 +1 +1 -5 +3 -4 -3 +2 -4 -2 -1 -1 +5 -3 +4 -3
18-element mirror
116
1/77 = 0.0129870 129870 129870 129870 129870…
+1 +1 +7 -1 -1 -7
6-element mirror
1/85 = 0.01176470588235294 1176470588235294…
(+1) 0 +6 -1 -2 +3 -7 +5 +3 0 -6 +1 +2 -3 +7 -5 -3
16-element mirror
1/89 = 0.0112359550561797752808988764044944…
+1 0 +1 +1 +2 +4 -4 0 -5 +5 +1 -5 +6 +2 -2 0 -2 –3 +6 -8 +8 +1 -1 0 -1 -1 -2 -4 +4 0 +5 -5 -1 +5 -6 -2 +2 0 +2 +3 -6 +8 -8 -1**
44-element mirror
1/91 = 0.010989 010989 010989 010989 010989…
+1 -1 +9 -1 +1 -9
6-element mirror
1/92 = 0.0108695652173913043478260869565217
(+1 -1) +8 -2 +3 -4 +1 -1 -3 -1 +6 -4 +6 -8 +2 -3 +4 -1 +1 +3 +1 -6 +4 -6
22-element mirror
1/95 = 0.0105263157894736842105263157894737
(+1 -1) +5 -3 +4 -3 -2 +4 +2 +1 +1 -5 +3 -4 +3 +2 -4 -2 -1 -1
18-element mirror
117
1/98 = 0.0102040816326530612244897959183673…
(+1 -1) +2 -2 +4 -4 +8 -7 +5 -3 -1 +4 -1 -2 -3 +6 -5 +1 0 +2 0 +4 +1 -2 +2 -4 +4 -8 +7 -5 +3 +1 -4…
42-element mirror
118
OTHER NUMBER SEQUENCES
(When zeros are added before and after segment, the mode is always zero)
NUMBER
INCREMENT ANALYSIS PATTERN
MODE
Mass, Gravity, Communications harmonic (Bruce Cathie) 16944
+5 +3 -5 0 0
repeaters
23 = 2.828428
+6 -6 +6 -6 +6 (2nd order sequence)
Oscillating 6s
Synergetics conversion constant (R. Buckminster Fuller) 1.06066
-1 +6 -6 +6 0
Oscillating 6s
Scheherazade Number (first 8 primes factorial) R. B. Fuller 30, 030 x 17 = 510,510!
-5 +5 -5
Oscillating 5s
Scheherazade Number (first 7 primes factorial) R. B. Fuller 30, 030
-3 0 +3 -3
Oscillating 3s
Primes through 29 (R.B. Fuller) 6,469,693,230
-2 +2 +3 -3 +3 -3 -1 +1 -3
Oscillating binary pairs
Seven-illion Scheherazade Number (R.B. Fuller) 24,421,743,243,121,524,300,000
+1 -1 +1 (5th order sequence)
Oscillating 1s
119
Matter plus anti-matter Earth Field Harmonic (Bruce Cathie) 4114846801
-3 0 +3 +4 -4 +4 -8 +1 (2nd order sequence)
Oscillating 3s and 4s
e = 2.718281828459045…
+5 -6 +7 -6 +6 -7 +7 -6 +6 -4 +1 +4
4-element mirrors (-6 +7, +6 –7) binary pairs
Planck’s Constant h = 6.626069 x 10-34
0 -4 +4 -6 +6 +3
Binary pairs
Speed of light in a vacuum 2.99792458 x 108 m/s
+7 0 -2 +2 -7 +2 +1 +3
Binary pairs
Atomic mass unit 1.6605655
+5 0 -6 +6 -1 0 (2nd order sequence)
Binary pair (-6 +6)
C (Euler-Mascheroni Constant) 0.5772156649
+2 0 -5 -1 +4 +1 0 -2 +5
Pseudo-mirrors
√√√√2= 1.414213…
+3 -3 +3 -3 (2nd order sequence)
Oscillating 3s
Schrodinger Constant 1.644934067
+1 -1 +1 +9 (2nd order sequence)
Oscillating 1s
120
Electric permittivity constant 8.85418781...x10
-7 +7 -1 +1 -7 (2nd order sequence)
Binary pairs, oscillating 7s
Euler’s constant reciprocal = 1.732454714600636056606841…
-3 +3 -6 +6 0 -6 +6 +2 -4 -3 ( 2nd order sequence)
Oscillating 3s, 6s
Reciprocal of #137 (a rough approximation of the fine structure constant) 0.007299270072...
0 +7 -5 +7 0 -7 +5 -7 0 +7
6-element mirrors
Reciprocal of #49 0.020408163265306…
+2 -2 +4 -4 +8 -7 +5 -4 +4 -6 +6 (2nd order sequence)
Binary pairs and 4-element mirrors
7/9 + 9/7 = 1.396825396825… up to 13 digits
+2 +6 -3 +2 -6 +3 -2 +6 -3 +2 -6 +3
6-element mirrors
7/9 + 9/7 reciprocal = 0.715909090909… up to 12 digits
-6 +4 +4 -9 +9 -9 +9 -9 +9 -9 +9
Oscillating 9s
137/13 reciprocal = 0.094890510…
+9 -5 +5 -9 +5 -5 6-element mirrors
121
137/13 = 10.538461538461…
-1 +5 -2 +5 -4 +2 -5 +4 -2 +5 -4 +2 -5 +4
6-element mirrors
137/7 reciprocal = 0.051094890510…
+5 -4 -1 +9 -5 -4 +1 -9
4-element mirrors (-1 +9, +1 –9)
137/7 = 19.5714285714…
+8 -4 +2 -6 +3 -2 +6 -3 +2 -6 +3
6-element mirrors
Coupling constant = 0.08542455
+8 -3 -3 +3 0 (2nd order sequence)
Oscillating 3s
13/81 = 6.2307692307
-4 +1 -3 +7 -1 +3 -7 +1 -3 +7
6-element mirrors
13/81 reciprocal = 0.160493827…
+5 -6 +4 +5 -6 +5 -6 +5
Repeaters (+5 –6)
13/49 = 0.265306122…
+4 -6 +6 -4 0 (2nd order sequence)
4-element mirrors
122
√26/81 = 0.566557723
+1 0 -1 0 +2 0 -5 +1
4-element mirrors
9/123 = 0.073170731…
+7 -6 +6 -7 +7 -6 (2nd order sequence)
Binary pairs
28/274 =0.102189781
-1 +2 -1 +7 +1 -2 +1 -7
8-element mirrors
Connected with electron spin (Harold Aspden) 1.001159652193
-1 0 +1 0 +8 -8 +8 -6
Oscillating 8s, 1s
Absolute zero; average period of human gestation: 273: reciprocal = 1/273 = 0.003663003 (Peter Plichta: God’s Secret Code)
+3 +3 0 -3 0 0 +3
“threeness”
Sidereal month: 27.32 days
+5 -5 (2nd order sequence)
Binary pair
Synodic month reciprocal, 1/29.53 days = 0.033863867
+3 0 +5 -5 +5 -1
Oscillating 5s;
123
Euler’s Constant = 0.577215664901532
+2 0 -5 -2 +5 0 -2 +5 -9 +5 -2 -1 (2nd order sequence)
Mirrors; repeaters
Golden ratio = 1.618033988749894…
+5 -5 +7 -8 +3 0 +6 -1 0 -1 +3 +5 -1 +1 -5
Oscillating fives
Magnetic permeability constant = 1.2566370614…x1?
+4 -4 +4 -2 -4 0 -1 (3rd order sequence)
Oscillating 4s
Nuclear magneton = 5.050783 x 10 –22 J/T
-5 +5 -5 +7 +1 -5
Oscillating 5s
#0.094890510
+9 -5 +5 -9 +5 -5
6-element mirrors
#7787537
+1 -1 -4 +4 (2nd order sequence)
Binary pairs
#7078760
-7 +7 +1 -1 -7
Oscillating 7s, binary pairs
124
0.213471897639213471897639… (Cyclic divisibility, mathpages.com)
-1 +2 +1 +3 -6 +7 +1 -2 -1 -3 +6 -7…
12-element mirrors
#134718976392 (Cyclic divisibility, mathpages.com)
+2 +1 +3 -6 +7 +1 -2 -1 -3 +6 -7
10-element mirrors
#706320182568
-7 +6 -3 -1 -2 +1 +7 -6 +3 +1 +2
10-element mirrors
#865281023607 (cyclic integer)
-2 -1 -3 +6 -7 -1 +2 +1 +3 -6 +7
10-element mirrors
Auric Key reciprocal; 1/2720 = 3.968253968253… (Syndex Synergetics Synopsis, Iona Miller)
+6 -3 +2 -6 +3 -2
6-element mirrors
125
BIBLIOGRAPHY
Dunham, William. The Mathematical Universe. John Wiley & Sons, Inc. New York. 1994. King, Jerry P. The Art of Mathematics. Fawcett Columbine. New York. 1992. Schwaller de Lubicz. R.A. A Study of Numbers. Inner Traditions International. Rochester, Vermont. 1986. Fuller, R. Buckminster. Applewhite, E.J. Synergetics: Explorations in the Geometry of Thinking. Macmillan Publishing Co. New York. 1975. Balmond, Cecil. Number 9: The Search for the Sigma Code. Prestel. Munich-New York. 1998. Blatner, David. The Joy of Pi. Walker and Company. New York. 1997. Georges, Ifrah. The Universal History of Numbers. John Wiley & Sons, Inc. New York. 2000. Merrill, Helen A. Mathematical Excursions. Dover Publications, Inc. U.S.A. 1933, 1957. Clapham, Christopher. The Concise Oxford Dictionary of Mathematics, 2nd Edition. Oxford University Press. Oxford-New York. 1996. Boyer, Carl B. Merzbach, Uta C. A History of Mathematics 2nd Edition. John Wiley & Sons, Inc. New York. 1991. Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books. London, England. 1986. Ogilvy, Stanley C. Anderson, John T. Excursions in Number Theory. Dover Publications, Inc. New York. 1988.
126
Plichta, Peter. God’s Secret Formula: Deciphering the Riddle of the Universe and the Prime Number Code. Element Books. Rockport, MA. 1997. Bennett, Deborah J. Randomness. Harvard University Press. Cambridge, MA. 1998. Gilles, William F. The Magic and Oddities of Numbers. Vantage Press, Inc. New York. 1953. Camm, F. J. Mathematical Tables and Formulae. Philosophical Library. New York. 1958. Watkins, Matthew. Useful Mathematical & Physical Formulae. Walker & Company. New York. 2000. Dwight, Herbert Bristol. Mathematical Tables of elementary and some higher mathematical functions. Dover Publications, Inc. New York. 1961. Saradar, Ziauddin. Ravetz, Jerry. Van Loon, Borin. Introducing Mathematics. Totem Books. New York. 1999. Salem. Testard. Salem. The Most Beautiful Mathematical Formulas. John Wiley and Sons, Inc. 1992. Dunham, William. Journey Through Genius (The Great Theorems of Mathematics). Penguin Books, John Wiley & Sons. 1990.
Sardar, Ziauddin. Ravetz, Jerry. Van Loon, Borin. Introducing Mathematics. Totem Books. U.S. 1999.
127
INTERNET RESOURCES
Gamboa, Ana. Alpha – The Fine Structure Constant
http://www.physicspost.com/articles.php?articleId=11
Can Negative Numbers be Prime? Prime Pages FAQ
http://www.utm.edu/research/primes/notes/faq/negative_primes.html
Michael Conrad Tilstra (Tadpol) Basic Infinite Math.
http://tadpol.org/theory/infinitemath.html
Mathworld. Transcendal Number.
http://mathworld.wolfram.com/TranscendentalNumber.html
Pascal’s Triangle. http://www.csam.montclair.edu/~kazimir/patterns.html
Ancient Light and Uncertain Theories.
http://www.geocities.com/syzygywjp/AncientLight.html
Sirag, Saul-Paul. The World as Cryptogram. International Space Sciences
Organization. November 7, 2000.
The First 100 Fibonacci Numbers.
http://math.holycross.edu/~davis/fibonacci/fib0-99.html
128
Riemann Zeta Function – MathWorld
http://mathworld.wolfram.com/RiemannZetaFunction.html
The Riemann Hypothesis. Prime Pages.
http://www.utm.edu/research/primes/notes/rh.html
When constants are not constant. Physics Web. October, 2001.
http://physicsweb.org/article/world/14/10/4
Rice, Aaron. Infinity, The Truth About. 1997/02/10
http://www.galactic-guide.com/articles/8R69.html
The First 1000 Primes.
http://www.utm.edu/research/primes/lists/small/1000.txt
The First 200 Lucas Numbers and their factors.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucas200.html
Devlin, Keith. How Euler discovered the zeta function.
http://www.maa.org/devlin/Zeta.PDF
129
The 23 Paris Problems. http://www.math.umn.edu/~wuttnab/problems2.html
Latin Squares. http://lwww.cut-the-knot.org/arithmetic/latin.shtml
Ivars Peterson’s MathTrek. The Limits of Mathematics.
http://www.sciencenews.org/sn_arc98/2_21_98/mathland.htm
Magic Square – from MathWorld.
http://mathworld.wolfram.com/MagicSquare.html
The Math Forum. Why are Operations of Zero so Strange?
http://mathforum.org/library/drmath/view/55764.html
Re: Infinity. http://www.math.tulane.edu/bulletin/messages/100.html
Michael Conrad Tilstra (Tadpol) Zero over Zero.
http://tadpol.org/theory/zero.html
Complex Infinity. http://www.mupad.com/doc/eng/stdlib/cinfty.shtml
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“I have examined Dr. Bowen’s abstract and found it an innovative contribution to mathematics.”
Emmanuel Gugwor, Mathematics Instructor
Prologue H.S. and Truman College, Chicago
“Looking at the patterns of these numbers you might discover the deeper secrets of many lives.”
Sow Aboubacar Sidy, Ph.D.
Adjunct Professor of Mathematics Loyola University of Chicago
Professor of Mathematics, Columbia College, Chicago
ISBN 0-9615454-2-9