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Evokation
  
 
Index
  

 

ADVENT 2167 ADVENT

 

I, TIME, EMIT, MITE, MINUTE, MIN, HOURS, HORUS, MINUTE,
SECOND, HOUR
SIXTY, SECONDS
ONE, HOUR
NAMES OF GOD, MOHAMMED, PEACE, BE, UPON, HIM
EQUATE, EQUATION, EQUATIONS, COUNT, COUNTED, COUNTING,
NUMBER, SEQUENCE
NUMBERS, SEQUENCES
COUNT, NUMBER, SEQUENCE
COMPLETE, COMPLETES, COMPLETED
AND-SO-ON, ELLIPSIS
CHAIN, CHAINS
NUMBER, CHAIN
ARITHMETIC
THOUGHT, ZERO, OUGHT, NOUGHT
THOUGHT, NOUGHT
THOUGHTS, NOUGHTS
PEACE, LOVE, LIGHT
THE, ABSENT, SELF
ZERO, ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE
SOLO, DUEL, TRIO, QUARTET, QUINTET
MYRIAD, MYRIADS
ATTIC, ACRIPHONIC
ALGEBRA, ALGEBRAIC
TENNESSEE
IRRISISTABLE
SPATIAL, MAGNITUDES, MOTION, TIME
SPATIAL MAGNITUDE
TIME, T, I, ME
SINE QUA NON, SINE, QUA, NON
THE, PHAEDO
PLATO
PHAEDRUS
ABSOLUTE
SOURCE, WHOLE
WHOLE NUMBER, WHOLE, NUMBER

 

 

-
-
-
-
-
LIGHT SOUND
-
-
-
L
=
3
-
5
LIGHT
56
29
2
S
=
1
-
5
SOUND
73
28
1
-
-
4
-
10
LIGHT SOUND
129
57
3
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
-
-
-
-
-
1+2
1+2
-
-
-
4
-
1
LIGHT SOUND
3
3
3

 

 

-
-
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
29
-
-
-
56
29
29
-
-
-
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
28
-
-
-
73
28
19
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

 

-
-
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

THE

HEART THE R HEAT

 

-
-
-
-
-
LIGHT HEAT
-
-
-
L
=
3
-
5
LIGHT
56
29
2
H
=
8
-
4
HEAT
34
16
7
-
-
11
-
9
LIGHT HEAT
90
45
9
-
-
1+1
-
-
-
9+0
4+5
-
-
-
2
-
9
LIGHT HEAT
9
9
9
-
-
-
-
-
-
1+2
1+2
-
-
-
2
-
9
LIGHT HEAT
3
3
9

 

 

-
-
-
-
-
LIGHT HEAT
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
29
-
-
-
56
29
29
-
-
-
-
-
-
-
-
-
-
H
=
8
6
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
A
=
1
8
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
16
-
-
-
34
16
16
-
1
2
3
4
5
6
7
16
9
-
-
-
-
-
LIGHT HEAT
-
-
-
-
-
-
-
-
-
-
-
1+6
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
7
9
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
6
4
5
6
7
7
9
-
-
1+1
-
-
-
9+0
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
9
LIGHT HEAT
9
9
9
-
1
2
6
4
5
6
7
7
9

 

 

-
-
-
-
-
LIGHT HEAT
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
4
-
6
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
4
-
6
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
4
-
6
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
H
=
8
6
1
H
8
8
8
-
-
-
-
4
-
6
-
8
-
E
=
5
7
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
A
=
1
8
1
A
1
1
1
-
1
-
-
4
-
6
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
16
9
-
-
-
-
-
LIGHT HEAT
-
-
-
-
-
-
-
-
-
-
-
1+6
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
7
9
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
6
4
5
6
7
7
9
-
-
1+1
-
-
-
9+0
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
9
LIGHT HEAT
9
9
9
-
1
2
6
4
5
6
7
7
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
LIGHT HEAT
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
8
1
A
1
1
1
-
1
-
-
4
-
6
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
4
-
6
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L
=
3
1
1
L
12
3
3
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=
5
7
1
E
5
5
5
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5
6
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G
=
7
3
1
G
7
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8
4
1
H
8
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8
6
1
H
8
8
8
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2
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I
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9
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16
9
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LIGHT HEAT
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1+6
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L
=
3
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5
LIGHT
56
29
2
-
1
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6
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9
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=
8
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4
HEAT
34
16
7
-
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-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
6
4
5
6
7
7
9
-
-
1+1
-
-
-
9+0
4+5
-
-
-
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-
-
-
-
-
-
-
-
-
2
-
9
LIGHT HEAT
9
9
9
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1
2
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6
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9

 

 

-
-
-
-
-
LIGHT HEAT
-
-
-
-
1
2
3
5
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
3
5
7
8
9
-
-
-
-
-
-
-
-
-
-
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-
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A
=
1
8
1
A
1
1
1
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1
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-
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-
T
=
2
9
1
T
20
2
2
-
-
2
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
E
=
5
7
1
E
5
5
5
-
-
-
-
5
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
8
-
H
=
8
6
1
H
8
8
8
-
-
-
-
-
-
8
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
16
9
-
-
-
-
-
LIGHT HEAT
-
-
-
-
-
-
-
-
-
1+6
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
5
7
7
9
H
=
8
-
4
HEAT
34
16
7
-
-
-
-
-
-
-
-
-
-
11
-
9
LIGHT HEAT
90
45
9
-
1
2
6
5
7
7
9
-
-
1+1
-
-
-
9+0
4+5
-
-
-
-
-
-
-
-
-
-
-
2
-
9
LIGHT HEAT
9
9
9
-
1
2
6
5
7
7
9

 

 

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 351 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 126 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 9 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

 

 

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 351 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 126 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 9 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

 

 

www.merriam-webster.com/dictionary/algorithm

a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an ...

algorithm [ˈælgəˌrɪðəm]
n
1. (Mathematics) a logical arithmetical or computational procedure that if correctly applied ensures the solution of a problem Compare heuristic
2. (Mathematics) Logic Maths a recursive procedure whereby an infinite sequence of terms can be generated Also called algorism
[changed from algorism, through influence of Greek arithmos number]
algorithmic adj
aal·go·rithm (lg-rm)
n.
A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.
algorithmically adv

algorithm (lg-rthm)
A finite set of unambiguous instructions performed in a prescribed sequence to achieve a goal, especially a mathematical rule or procedure used to compute a desired result. Algorithms are the basis for most computer programming.

Noun 1. algorithm - a precise rule (or set of rules) specifying how to solve some problem
algorithmic program, algorithmic rule
formula, rule - (mathematics) a standard procedure for solving a class of mathematical problems; "he determined the upper bound with Descartes' rule of signs"; "he gave us a general formula for attacking polynomials"
sorting algorithm - an algorithm for sorting a list
stemming algorithm, stemmer - an algorithm for removing inflectional and derivational endings in order to reduce word forms to a common stem algorithm
any methodology for solving a certain kind of problem.
See also: Mathematics

 

Algorithm
From Wikipedia, the free encyclopedia

Flow chart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
In mathematics and computer science, an algorithm (i/ˈælɡərɪðəm/) is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.

More precisely, an algorithm is an effective method expressed as a finite list[1] of well-defined instructions[2] for calculating a function.[3] Starting from an initial state and initial input (perhaps empty),[4] the instructions describe a computation that, when executed, will proceed through a finite [5] number of well-defined successive states, eventually producing "output"[6] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[7]

Though al-Khwārizmī's algorism referred to the rules of performing arithmetic using Hindu-Arabic numerals and the systematic solution of linear and quadratic equations, a partial formalization of what would become the modern algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"[8] or "effective method";[9] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.[10]

Informal definition
For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.
While there is no generally accepted formal definition of "algorithm," an informal definition could be "a set of rules that precisely defines a sequence of operations."[11] For some people, a program is only an algorithm if it stops eventually; for others, a program is only an algorithm if it stops before a given number of calculation steps.[12]

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.

Boolos & Jeffrey (1974, 1999) offer an informal meaning of the word in the following quotation:

No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus an algorithm can be an algebraic equation such as y = m + n—two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in language understood by "the computer")[14] for a fast, efficient, "good"[15] process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities)[16] to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively"[17] produce, in a "reasonable" time,[18] output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.

[edit] Formalization

Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):

Minsky: "But we will also maintain, with Turing . . . that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. Although this may seem extreme, the arguments . . . in its favor are hard to refute".[19]

Gurevich: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".[20]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.

[edit] Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite state machine, state transition table and control table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Representations of algorithms can be classed into three accepted levels of Turing machine description:[21]
1 High-level description:
"...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head." 2 Implementation description:
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function." 3 Formal description:
Most detailed, "lowest level", gives the Turing machine's "state table". For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.
[edit] Implementation

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

[edit] Computer algorithms

Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test=true THEN GOTO step xxx) (a diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf Tausworthe 1977:100,114).
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the target machines to produce output from given input (perhaps null).

"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: ". . .we want good algorithms in some loosely defined aesthetic sense. One criterion . . . is the length of time taken to perform the algorithm . . .. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"[22] Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"[23]
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid).

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".[24]

Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example of using Euclid's algorithm will be shown below.

Computers (and computors), models of computation: A computer (or human "computor"[25]) is a restricted type of machine, a "discrete deterministic mechanical device"[26] that blindly follows its instructions.[27] Melzak's and Lambek's primitive models[28] reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters[29] (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.[30]

Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".[31] Minsky's machine proceeds sequentially through its five (or six depending on how one counts) instructions unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution)[32] operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1).[33] Rarely will a programmer have to write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.[34]

Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[35] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.[36]

This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[37] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" (division) instruction available rather than just subtraction (or worse: just Minsky's "decrement").

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code" a programmer can write structured programs using these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".[38] Tausworthe augments the three Böhm-Jacopini canonical structures:[39] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.[40] An additional benefit of a structured program will be one that lends itself to proofs of correctness using mathematical induction.[41]

Canonical flowchart symbols[42]: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only 4: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm-Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.

 

 

EVOLVE LOVE EVOLVE

LOVES SOLVE LOVES

EVOLVE LOVE EVOLVE

 

LOGARITHM


Logarithm - Wikipedia
en.wikipedia.org › wiki › Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed ...
Logarithmic scale · ?Natural logarithm · ?Common logarithm · ?Binary logarithm

 

Introduction to Logarithms - Math is Fun
www.mathsisfun.com › algebra › logarithms
Introduction to Logarithms. In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number?

?Working with Exponents · ?Logarithms Can Have Decimals · ?Exponents, Roots

 

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ALGORITHM
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LOGARITHM
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L
=
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L
12
3
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O
=
6
4
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O
15
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6
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G
=
7
3
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G
7
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7
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A
=
1
1
1
A
1
1
1
-
1
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5
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R
=
9
5
1
R
18
9
9
-
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5
-
-
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9
I
=
9
6
1
I
9
9
9
-
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5
-
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T
=
2
7
1
T
20
2
2
-
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5
-
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H
=
8
8
1
H
8
8
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M
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LOGARITHM
103
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4+9
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1+0
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1+0+3
4+9
4+9
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LOGARITHM
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LOGARITHM
4
4
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9

 

 

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ALGORITHM
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LOGARITHM
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A
=
1
1
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A
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=
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20
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=
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12
3
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M
=
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9
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M
13
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=
6
4
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O
15
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6
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G
=
7
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H
=
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8
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H
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R
=
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18
9
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I
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I
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49
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LOGARITHM
103
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7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

LOGARITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL MORDER

 

-
-
-
-
10
ALGORITHM
103
49
49
-
1
2
3
4
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
LOGARITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

MINUS 5

 

LOGARITHM

ALGORITHM

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
4
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
4
4
4

 

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

ALGORITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL MORDER

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

MINUS THE 5

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

Algorithm - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Algorithm

In mathematics and computer science, an algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and ...

 

A
=
1
-
9
ALGORITHM
103
49
4
A
=
1
-
10
ALGORITHMS
122
59
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
-
ALGORITHMS
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
L
12
3
3
-
-
-
-
1
G
7
7
7
-
-
-
-
1
O
15
6
6
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
H
8
8
8
-
-
-
-
1
M+S
32
14
5
A
=
1
-
10
ALGORITHMS
122
59
50
-
-
-
-
1+0
-
1+2+2
5+9
5+0
A
=
1
-
1
ALGORITHMS
5
14
5
-
-
-
-
-
-
-
1+4
-
A
=
1
-
1
ALGORITHMS
5
5
5

 

 

-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
9
-
8
-
1
+
=
24
2+4
=
6
=
6
-
-
-
-
-
15
-
9
-
8
-
19
+
=
51
5+1
=
6
=
6
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
3
7
-
9
-
2
-
4
-
+
=
26
2+6
=
8
=
8
-
-
1
12
7
-
18
-
20
-
13
-
+
=
71
7+1
=
8
=
8
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
12
7
15
18
9
20
8
13
19
+
=
122
1+2+2
=
5
1+0
5
-
-
1
3
7
6
9
9
2
8
4
1
+
=
50
5+0
=
5
1+0
5
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
9
-
-
-
-
-
-
9
occurs
x
1
=
9
5
10
A
L
G
O
R
I
T
H
M
S
-
-
27
-
-
10
-
41
-
1+0
-
-
-
-
9
9
-
-
-
-
-
-
2+7
-
-
1+0
-
4+1
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5
-
-
1
3
7
6
9
9
2
8
4
1
-
-
-
-
-
-
-
-
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
T
=
2
-
1
T
20
2
2
L
=
3
-
1
L
12
3
3
M
=
4
-
1
M
13
4
4
5
-
5
-
-
5
-
-
5
O
=
6
-
1
O
15
6
6
G
=
7
-
1
G
7
7
7
H
=
8
-
1
H
8
8
8
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Preface to the New Edition

Page 5
Twelve years ago a little boy entered my imagination as he hopped across the centuries and played with numbers. I began to see how the simple architecture of our decimal system could be constructed in secret ways — not a building project this time but an abstract one. On the surface of our arithmetic countless combinations of numbers take part in tedious and exacting calculations but underneath it all there is pattern, governed by a repeating code of integers. The Sigma Code reduces numbers to a single digit and the illusion of the many is seen to be but the reflection of a few. This is not a book on maths: this is a book for anyone who can carry out simple sums in their heads, and who won't be short-changed knowingly.
When Number 9 first came out I received mail from many who played with numbers. They chased patterns; some had special numbers and even mystical systems. I was tempted to write about numerology but resisted. I wanted to write about the intricacy of what the.. numbers actually do and leave the reader to wonder about the larger irrational that seems to hover around such constructions.
If I were writing this book today the numbers would have featured in a wider context of structuring nature's patterns, and also playing the role of animator in algorithms that create unique architectural forms and shapes. I would also include my previous research into other base systems. But this book was a first step which came from a child-like urge, like playing with building blocks, to build out of our numbers — just the simple 1, 2, 3, up to number 9.

 

RESEARCH R E SEARCH ER RESEARCH

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

NUMBER = 534259 = 1 = 534259 NUMBER

NUMBER = 234559 NUMBER

NUMBER = 534259 = 1 = 534259 NUMBER

 

 

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

SBUMNER = 1234559 = SBUMNER

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

THE

BALANCING

ONE TWO THREE FOUR

FIVE

NINE EIGHT SEVEN SIX

4 FIVE 42 24 6

1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1

15 ONE TWO THREE FOUR 208 82 1
4 FIVE 42 24 6
17 NINE EIGHT SEVEN SIX 208 91 1

1234 5 6789

 

 

prime number - Whatis Techtarget
https://whatis.techtarget.com/definition/prime-number
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

 


Prime number - Wikipedia
https://en.wikipedia.org/wiki/Prime_number

The first 25 prime numbers (all the prime numbers less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in the OEIS).
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several proofs of the theorem.

 

Prime numbers
www-groups.dcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html
Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to ...
Prime numbers

Number theory index History Topics Index

Version for printing

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

 

PRIME NUMBERS

 

P
=
2
-
5
PRIME
61
34
7
N
=
2
-
7
NUMBERS
92
38
2
-
-
17
-
12
Add to Reduce
153
72
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
8
-
3
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
34
-
11
-
61
34
34
-
-
-
-
-
-
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
10
-
92
38
29
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

6 +8 = 14 1+4 = 5

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
7
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
7
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
7
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
7
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
7
9

 

 

I = 9 9 = I
ME = 9 9 = ME
BRAIN + BODY = 9 9 = BODY + BRAIN
LIGHT + DARK = 9 9 = DARK + LIGHT
ENERGY + MASS = 9 9 = MASS +ENERGY
MIND + MATTER = 9 9 = MATTER + MIND
MAGNETIC + FIELD = 9 9 = FIELD + MAGNETIC
POSITIVE + NEGATIVE = 9 9 = NEGATIVE + POSITIVE
973 OM AZAZAZAZAZAZAZAZAZZAZAZAZAZAZAZAZAZAOM 973

 


thermodynamic properties of hydrogen-helium plasmas
ntrs.nasa.gov

Thermodynamic properties of hydrogen-helium plasmas The thermodynamic properties of an atomic hydrogen-helium plasma are calculated and tabulated for ...

 

Thermodynamic properties of hydrogen-helium plasmas. - AIAA
arc.aiaa.org

The shock layer plasma consists of electrons, protons, atomic hydrogen, atomic helium, singly ionized helium, and doubly ionized helium. The thermodynamic ...
by HF NELSON · ?1972

 


Atomic processes in a low-density hydrogen-helium plasma ...
www.sciencedirect.com › science › article › pii

For a partially ionized hydrogen-helium plasma at 104–106 °K the processes bremsstrahlung, radiative and dielectronic recombination, excitation of discrete ...
by RJ Gould · ?1970 · ?Cited by 53 · ?Related articles

 
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
H
=
8
=
8
HYDROGEN
96
51
6
H
=
8
=
6
HELIUM
68
32
5
P
=
7
=
6
PLASMA
62
26
8
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
-
2+3
-
2+0
-
2+2+6
1+0+9
1+9
-
-
5
-
20
HYDROGEN HELIUM PLASMA
10
10
10
-
-
-
-
2+0
-
1+0
1+0
1+0
-
-
5
-
20
HYDROGEN HELIUM PLASMA
1
1
1

 

 
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
1
2
3
4
5
6
7
8
9
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
-
-
-
-
-
-
H
=
8
=
6
HELIUM
68
32
5
-
-
-
-
-
-
-
-
-
-
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
H
=
8
1
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
Y
=
7
2
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
51
-
-
-
96
51
51
-
-
-
-
-
-
-
-
-
-
H
=
8
9
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
L
=
3
11
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
12
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
U
=
3
13
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
32
-
-
-
68
32
32
-
-
-
-
-
-
-
-
-
-
P
=
7
15
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
L
=
3
16
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
17
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
S
=
1
18
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
M
=
4
19
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
A
=
1
20
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
26
-
-
-
62
26
17
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
3
2
9
12
15
6
21
16
18
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
1+2
1+5
-
2+1
1+6
1+8
H
=
8
=
6
HELIUM
68
32
5
-
3
2
9
3
6
6
3
7
9
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
2+2+6
1+0+9
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
10
10
10
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
1+0
1+0
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
HYDROGEN HELIUM PLASMA
1
1
1
-
3
2
9
3
6
6
3
7
9

 

 
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
1
2
3
4
5
6
7
8
9
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
-
-
-
-
-
-
H
=
8
=
6
HELIUM
68
32
5
-
-
-
-
-
-
-
-
-
-
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
H
=
8
1
1
H
8
8
8
-
-
2
-
-
-
-
-
8
-
Y
=
7
2
1
Y
25
7
7
-
-
2
-
-
-
-
7
-
-
D
=
4
3
1
D
4
4
4
-
-
2
-
4
-
-
-
-
-
R
=
9
4
1
R
18
9
9
-
-
2
-
-
-
-
-
-
9
O
=
6
5
1
O
15
6
6
-
-
2
-
-
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
2
-
-
-
-
7
-
-
E
=
5
7
1
E
5
5
5
-
-
2
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
2
-
-
5
-
-
-
-
H
=
8
9
1
H
8
8
8
-
-
2
-
-
-
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
2
-
-
5
-
-
-
-
L
=
3
11
1
L
12
3
3
-
-
2
3
-
-
-
-
-
-
I
=
9
12
1
I
9
9
9
-
-
2
-
-
-
-
-
-
9
U
=
3
13
1
U
21
3
3
-
-
2
3
-
-
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
2
-
4
-
-
-
-
-
P
=
7
15
1
P
16
7
7
-
-
2
-
-
-
-
7
-
-
L
=
3
16
1
L
12
3
3
-
-
2
3
-
-
-
-
-
-
A
=
1
17
1
A
1
1
1
-
1
2
-
-
-
-
-
-
-
S
=
1
18
1
S
19
10
1
-
1
2
-
-
-
-
-
-
-
M
=
4
19
1
M
13
4
4
-
-
2
-
4
-
-
-
-
-
A
=
1
20
1
A
1
1
1
-
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
3
2
9
12
15
6
21
16
18
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
1+2
1+5
-
2+1
1+6
1+8
H
=
8
=
6
HELIUM
68
32
5
-
3
2
9
3
6
6
3
7
9
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
2+2+6
1+0+9
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
10
10
10
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
1+0
1+0
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
HYDROGEN HELIUM PLASMA
1
1
1
-
3
2
9
3
6
6
3
7
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED INTO NUMERICAL ORDER

 
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
1
2
3
4
5
6
7
8
9
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
-
-
-
-
-
-
H
=
8
=
6
HELIUM
68
32
5
-
-
-
-
-
-
-
-
-
-
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
17
1
A
1
1
1
-
1
2
-
-
-
-
-
-
-
S
=
1
18
1
S
19
10
1
-
1
2
-
-
-
-
-
-
-
A
=
1
20
1
A
1
1
1
-
1
2
-
-
-
-
-
-
-
L
=
3
11
1
L
12
3
3
-
-
2
3
-
-
-
-
-
-
L
=
3
16
1
L
12
3
3
-
-
2
3
-
-
-
-
-
-
U
=
3
13
1
U
21
3
3
-
-
2
3
-
-
-
-
-
-
D
=
4
3
1
D
4
4
4
-
-
2
-
4
-
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
2
-
4
-
-
-
-
-
M
=
4
19
1
M
13
4
4
-
-
2
-
4
-
-
-
-
-
E
=
5
7
1
E
5
5
5
-
-
2
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
2
-
-
5
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
2
-
-
5
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
2
-
-
-
6
-
-
-
Y
=
7
2
1
Y
25
7
7
-
-
2
-
-
-
-
7
-
-
G
=
7
6
1
G
7
7
7
-
-
2
-
-
-
-
7
-
-
P
=
7
15
1
P
16
7
7
-
-
2
-
-
-
-
7
-
-
H
=
8
1
1
H
8
8
8
-
-
2
-
-
-
-
-
8
-
H
=
8
9
1
H
8
8
8
-
-
2
-
-
-
-
-
8
-
R
=
9
4
1
R
18
9
9
-
-
2
-
-
-
-
-
-
9
I
=
9
12
1
I
9
9
9
-
-
2
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
3
2
9
12
15
6
21
16
18
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
1+2
1+5
-
2+1
1+6
1+8
H
=
8
=
6
HELIUM
68
32
5
-
3
2
9
3
6
6
3
7
9
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
2+2+6
1+0+9
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
10
10
10
-
3
2
9
3
6
6
3
7
9
-
-
-
-
2+0
-
1+0
1+0
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
HYDROGEN HELIUM PLASMA
1
1
1
-
3
2
9
3
6
6
3
7
9

 

 
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
1
3
4
5
6
7
8
9
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
-
-
-
-
-
-
H
=
8
=
6
HELIUM
68
32
5
-
-
-
-
-
-
-
-
-
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
1
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
17
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
18
1
S
19
10
1
-
1
-
-
-
-
-
-
-
A
=
1
20
1
A
1
1
1
-
1
-
-
-
-
-
-
-
L
=
3
11
1
L
12
3
3
-
-
3
-
-
-
-
-
-
L
=
3
16
1
L
12
3
3
-
-
3
-
-
-
-
-
-
U
=
3
13
1
U
21
3
3
-
-
3
-
-
-
-
-
-
D
=
4
3
1
D
4
4
4
-
-
-
4
-
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
4
-
-
-
-
-
M
=
4
19
1
M
13
4
4
-
-
-
4
-
-
-
-
-
E
=
5
7
1
E
5
5
5
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
5
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
5
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
6
-
-
-
Y
=
7
2
1
Y
25
7
7
-
-
-
-
-
-
7
-
-
G
=
7
6
1
G
7
7
7
-
-
-
-
-
-
7
-
-
P
=
7
15
1
P
16
7
7
-
-
-
-
-
-
7
-
-
H
=
8
1
1
H
8
8
8
-
-
-
-
-
-
-
8
-
H
=
8
9
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
12
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
-
-
-
-
3
9
12
15
6
21
16
18
H
=
8
=
8
HYDROGEN
96
51
6
-
-
-
1+2
1+5
-
2+1
1+6
1+8
H
=
8
=
6
HELIUM
68
32
5
-
3
9
3
6
6
3
7
9
P
=
7
=
6
PLASMA
62
26
8
-
-
-
-
-
-
-
-
-
-
-
23
-
20
HYDROGEN HELIUM PLASMA
226
109
19
-
3
9
3
6
6
3
7
9
-
-
-
-
2+0
-
2+2+6
1+0+9
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
20
HYDROGEN HELIUM PLASMA
10
10
10
-
3
9
3
6
6
3
7
9
-
-
-
-
2+0
-
1+0
1+0
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
2
HYDROGEN HELIUM PLASMA
1
1
1
-
3
9
3
6
6
3
7
9

 

 

1
I
9
9
9
4
TIME
47
20
2
4
EMIT
47
20
2
4
MITE
47
20
2
6
MINUTE
82
28
1
3
MIN
36
18
9
5
HOURS
81
27
9
5
HORUS
81
27
9
6
MINUTE
82
28
1
6
SECOND
60
24
6
4
HOUR
62
26
8

 

 

5
SIXTY
97
25
7
7
SECONDS
79
25
7
-
-
-
-
-
6
SECOND
60
24
6

 

 

3
ONE
34
16
7
4
HOUR
62
26
8

 

 

10
NAMES OF GOD
99
45
9
8
MOHAMMED
72
36
9
-
-
-
-
-
-
-
-
-
-
5
PEACE
30
21
3
2
BE
7
7
7
4
UPON
66
21
3
3
HIM
30
21
3
14
Add to Reduce
133
70
16
1+4
Reduce to Deduce
1+3+3
7+0
1+6
5
Essence of Number
7
7
7

 

 

6
EQUATE
69
24
6
8
EQUATION
102
39
3
9
EQUATIONS
121
40
4
5
COUNT
73
19
1
7
COUNTED
82
28
1
8
COUNTING
103
40
4
6
NUMBER
73
28
1
8
SEQUENCE
89
35
8

 

 

7
NUMBERS
92
29
2
9
SEQUENCES
108
36
9

 

 

5
COUNT
73
19
1
6
NUMBER
73
28
1
8
SEQUENCE
89
35
8
19
First Total
235
82
10
1+9
Add to Reduce
2+3+5
8+2
1+0
10
Second Total
10
10
1
1+0
Reduce to Deduce
1+0
1+0
-
1
Essence of Number
1
1
1

 

ESSENES = 5+9 = ESSENES

E + SSENES = 5+9 = E +SSENES

9 = S+S+E+N+E+S = 9

E = 5 = E

 

E
=
5
-
-
ESSENES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
6
S+S+E+N+E+S
81
45
9
E
=
5
-
7
ESSENES
86
50
14
-
-
-
-
-
-
8+6
5+0
1+4
E
=
5
-
7
ESSENES
14
5
5
-
-
-
-
-
-
1+4
-
-
E
=
5
-
7
ESSENES
5
5
5

 

S+S+E+N+E+S = 9 = S+S+E+N+E+S

E SSENES = E SSENES = E SSENES

5 SSENES = 5 SSENES = 5 SSENES

E SSENES = E SSENES = E SSENES

S+S+E+N+E+S = 9 = S+S+E+N+E+S

 

E
=
5
-
-
ESSENES
-
-
-
-
-
-
-
4
E+E+N+E
29
20
2
-
-
-
-
3
S+S+S
57
30
3
E
=
5
-
7
ESSENES
86
50
5
-
-
-
-
-
-
8+6
5+0
-
E
=
5
-
7
ESSENES
14
5
5
-
-
-
-
-
-
1+4
-
-
E
=
5
-
7
ESSENES
5
5
5

 

 

E
=
5
-
-
ESSENES
-
-
-
-
-
-
-
1
5
5
5
5
-
-
-
-
1
S
19
10
1
-
-
-
-
1
S
19
10
1
-
-
-
-
1
5
5
5
5
-
-
-
-
1
5
14
5
5
-
-
-
-
1
5
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
-
7
ESSENES
86
50
23
-
-
-
-
-
-
8+6
5+0
2+3
E
=
5
-
7
ESSENES
14
5
5
-
-
-
-
-
-
1+4
-
-
E
=
5
-
7
ESSENES
5
5
5

 


The followers of the first of which are the Pharisees; of the second, the Sadducees; and the third sect, which pretends to a severer discipline, are called Essenes. These last are Jews by birth, and seem to have a greater affection for each other than other sects have.
Essenes - Wikipedia

 

Essene | ancient Jewish sect | Britannica
www.britannica.com › ... › Religious Beliefs
3 Apr 2008 — Like the Pharisees, the Essenes meticulously observed the Law of Moses, the sabbath, and ritual purity. They also professed belief in immortality and divine punishment for sin. But, unlike the Pharisees, the Essenes denied the resurrection of the body and refused to immerse themselves in public life.

 

ESSENES 5115551 ESSENES

 

E
=
5
-
-
ESSENES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
-
-
-
-
1
S
19
10
1
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
-
7
ESSENES
86
50
23
-
-
-
-
-
-
8+6
5+0
2+3
E
=
5
-
7
ESSENES
14
5
5
-
-
-
-
-
-
1+4
-
-
E
=
5
-
7
ESSENES
5
5
5

 

 

-
-
-
-
-
ESSENES
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
86
50
23
-
3
2
3
4
20
6
7
8
9
-
-
-
-
-
-
8+6
5+0
2+3
-
-
-
-
-
2+0
-
-
-
-
E
=
5
-
7
ESSENES
14
5
5
-
3
2
3
4
2
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
5
5
5
-
3
2
3
4
2
6
7
8
9

 

 

-
-
-
-
-
ESSENES
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
N
=
5
-
1
N
14
5
5
-
-
2
3
4
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
86
50
23
-
3
2
3
4
20
6
7
8
9
-
-
-
-
-
-
8+6
5+0
2+3
-
-
-
-
-
2+0
-
-
-
-
E
=
5
-
7
ESSENES
14
5
5
-
3
2
3
4
2
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
5
5
5
-
3
2
3
4
2
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
ESSENES
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
3
4
-
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
N
=
5
-
1
N
14
5
5
-
-
2
3
4
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
86
50
23
-
3
2
3
4
20
6
7
8
9
-
-
-
-
-
-
8+6
5+0
2+3
-
-
-
-
-
2+0
-
-
-
-
E
=
5
-
7
ESSENES
14
5
5
-
3
2
3
4
2
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
5
5
5
-
3
2
3
4
2
6
7
8
9

 

 

-
-
-
-
-
ESSENES
-
-
-
-
1
5
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
S
=
1
-
1
S
19
10
1
-
1
-
S
=
1
-
1
S
19
10
1
-
1
-
E
=
5
-
1
E
5
5
5
-
-
5
E
=
5
-
1
E
5
5
5
-
-
5
N
=
5
-
1
N
14
5
5
-
-
5
E
=
5
-
1
E
5
5
5
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
7
ESSENES
86
50
23
-
3
20
-
-
-
-
-
-
8+6
5+0
2+3
-
-
2+0
E
=
5
-
7
ESSENES
14
5
5
-
3
2
-
-
-
-
-
-
1+4
-
-
-
-
-
E
=
5
-
7
ESSENES
5
5
5
-
3
2

 

 

PHARISEES 781991551 PHARISEES

 

P
=
7
-
-
PHARISEES
-
-
-
-
-
-
-
1
P
16
7
7
-
-
-
-
2
H
8
8
8
-
-
-
-
2
A
1
1
1
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
4
S
19
10
1
-
-
-
-
4
E
5
5
5
-
-
-
-
4
E
5
5
5
-
-
-
-
4
S
19
10
1
P
=
7
-
9
PHARISEES
100
64
37
-
-
-
-
-
-
1+0+0
6+4
3+7
P
=
7
-
9
PHARISEES
1
10
10
-
-
-
-
-
-
-
1+0
1+0
P
=
7
-
9
PHARISEES
1
1
1

 

 

P
=
7
-
-
PHARISEES
-
-
-
-
-
-
-
1
P
16
7
7
-
-
-
-
2
H+A
9
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
4
S+E+E+S
48
12
3
P
=
7
-
9
PHARISEES
100
46
37
-
-
-
-
-
-
1+0+0
4+6
3+7
P
=
7
-
9
PHARISEES
1
10
10
-
-
-
-
-
-
-
1+0
1+0
P
=
7
-
9
PHARISEES
1
1
1

 

 

SADDUCEES 114433551 SADDUCEES

 

S
=
1
-
-
SADDUCEES
-
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
2
A
1
1
1
-
-
-
-
1
D
4
4
4
-
-
-
-
1
D
4
4
4
-
-
-
-
1
U
21
3
3
-
-
-
-
1
C
3
3
3
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
S
=
1
-
9
SADDUCEES
81
45
27
-
-
-
-
-
-
8+1
4+5
2+7
S
=
1
-
9
SADDUCEES
9
9
9
-
-
-
-
-
-
-
-
-
S
=
1
-
9
SADDUCEES
9
1
1

 

 

S
=
1
-
-
SADDUCEES
-
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
2
A
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
D
4
4
4
-
-
-
-
1
D
4
4
4
-
-
-
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
-
1
C
3
3
3
-
-
-
-
-
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
-
-
-
-
-
-
-
-
-
1
S
19
10
1
S
=
1
-
9
SADDUCEES
81
45
27
-
-
-
-
-
-
8+1
4+5
2+7
S
=
1
-
9
SADDUCEES
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
SADDUCEES
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
D
=
4
-
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
D
=
4
-
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
-
9
SADDUCEES
81
45
27
-
3
2
6
8
10
6
7
8
9
-
-
-
-
-
-
8+1
4+5
2+7
-
-
-
-
-
1+0
-
-
-
-
S
=
1
-
9
SADDUCEES
9
9
9
-
3
2
6
8
1
6
7
8
9

 

 

-
-
-
-
-
SADDUCEES
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
-
-
-
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
2
-
-
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
2
-
4
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
2
-
4
-
6
7
8
9
U
=
3
-
1
U
21
3
3
-
-
2
3
-
-
6
7
8
9
C
=
3
-
1
C
3
3
3
-
-
2
3
-
-
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
-
-
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
-
-
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
-
-
-
6
7
8
9
S
=
1
-
9
SADDUCEES
81
45
27
-
3
2
6
8
10
6
7
8
9
-
-
-
-
-
-
8+1
4+5
2+7
-
-
-
-
-
1+0
-
-
-
-
S
=
1
-
9
SADDUCEES
9
9
9
-
3
2
6
8
1
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED INTO NUMERICAL ORDER

 

-
-
-
-
-
SADDUCEES
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
-
-
-
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
2
-
-
-
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
2
-
-
-
6
7
8
9
U
=
3
-
1
U
21
3
3
-
-
2
3
-
-
6
7
8
9
C
=
3
-
1
C
3
3
3
-
-
2
3
-
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
2
-
4
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
2
-
4
-
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
-
-
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
2
-
-
5
6
7
8
9
S
=
1
-
9
SADDUCEES
81
45
27
-
3
2
6
8
10
6
7
8
9
-
-
-
-
-
-
8+1
4+5
2+7
-
-
-
-
-
1+0
-
-
-
-
S
=
1
-
9
SADDUCEES
9
9
9
-
3
2
6
8
1
6
7
8
9

 

 

-
-
-
-
-
SADDUCEES
-
-
-
-
1
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
-
-
-
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
-
-
-
6
7
8
9
U
=
3
-
1
U
21
3
3
-
-
3
-
-
6
7
8
9
C
=
3
-
1
C
3
3
3
-
-
3
-
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
-
4
-
6
7
8
9
D
=
4
-
1
D
4
4
4
-
-
-
4
-
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
-
-
5
6
7
8
9
E
=
5
-
1
E
5
5
5
-
-
-
-
5
6
7
8
9
S
=
1
-
9
SADDUCEES
81
45
27
-
3
6
8
10
6
7
8
9
-
-
-
-
-
-
8+1
4+5
2+7
-
-
-
-
1+0
-
-
-
-
S
=
1
-
9
SADDUCEES
9
9
9
-
3
6
8
1
6
7
8
9

 

 

-
-
-
-
-
SADDUCEES
-
-
-
-
1
3
4
5
S
=
1
-
1
S
19
10
1
-
1
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
U
=
3
-
1
U
21
3
3
-
-
3
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
D
=
4
-
1
D
4
4
4
-
-
-
4
-
D
=
4
-
1
D
4
4
4
-
-
-
4
-
E
=
5
-
1
E
5
5
5
-
-
-
-
5
E
=
5
-
1
E
5
5
5
-
-
-
-
5
S
=
1
-
9
SADDUCEES
81
45
27
-
3
6
8
10
-
-
-
-
-
-
8+1
4+5
2+7
-
-
-
-
1+0
S
=
1
-
9
SADDUCEES
9
9
9
-
3
6
8
1

 

 

Mishnah - Wikipedia

en.wikipedia.org › wiki › Mishnah
The Mishnah or Mishna is the first major written collection of the Jewish oral traditions known as ... In this last context, the word mishnah means a single paragraph of the work, i.e. the smallest unit of structure, ... on Kelim and Ohalot (the commentary on the rest of Tehorot and on Eduyot is lost) by Gershon Henoch Leiner, the ...
?Mishnah Yomis · ?Category:Mishnah · ?Judah ha-Nasi · ?List of Talmudic tractates

 

MISHNAH - JewishEncyclopedia.com

www.jewishencyclopedia.com › articles › 10879-mishnah
"Mishnah," the derivative of the verb "shanah," means therefore: (1) ... there are many "'eduyot" which are expressly said to have modified the earlier Mishnah; ...

 

What Is the Talmud? Definition and Comprehensive Guide ...
www.chabad.org › library › article_cdo › aid › jewish

The Talmud is a collection of writings that covers the full gamut of Jewish law and tradition… ... With it we can understand what the Torah means, and determine the details of the various commandments. Furthermore ... Mishnah, Eduyot 1:6. 15.

 

-
-
13
MISHNAH EDUYOT
-
-
-
M
=
4
=
7
MISHNAH
72
45
9
E
=
5
=
6
EDUYOT
90
27
9
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
-
-
-
1+3
-
1+6+2
4+5
3+6
-
-
-
-
4
MISHNAH EDUYOT
9
9
9

 

 
-
-
13
MISHNAH EDUYOT
-
-
-
-
1
2
3
4
5
6
7
8
9
M
=
4
=
7
MISHNAH
72
45
9
-
-
-
-
-
-
-
-
-
-
E
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
M
=
4
1
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
3
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
H
=
8
7
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
36
-
-
-
72
45
36
-
-
-
-
-
-
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
U
=
3
10
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
Y
=
7
11
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
T
=
2
13
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
27
-
-
-
90
27
27
-
2
2
3
8
10
6
7
16
9
-
-
13
MISHNAH EDUYOT
-
-
-
-
-
-
-
-
1+0
-
-
1+6
-
M
=
4
=
7
MISHNAH
72
45
9
-
2
2
3
8
1
6
7
7
9
N
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
2
2
3
8
1
6
7
7
9
-
-
-
-
1+3
-
1+6+2
4+5
3+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
MISHNAH EDUYOT
9
9
9
-
2
2
3
8
1
6
7
7
9

 

 
-
-
13
MISHNAH EDUYOT
-
-
-
-
1
2
3
4
5
6
7
8
9
M
=
4
=
7
MISHNAH
72
45
9
-
-
-
-
-
-
-
-
-
-
E
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
M
=
4
1
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
3
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
H
=
8
7
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
U
=
3
10
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
Y
=
7
11
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
T
=
2
13
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
3
8
10
6
7
16
9
-
-
13
MISHNAH EDUYOT
-
-
-
-
-
-
-
-
1+0
-
-
1+6
-
M
=
4
=
7
MISHNAH
72
45
9
-
2
2
3
8
1
6
7
7
9
N
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
2
2
3
8
1
6
7
7
9
-
-
-
-
1+3
-
1+6+2
4+5
3+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
MISHNAH EDUYOT
9
9
9
-
2
2
3
8
1
6
7
7
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 
-
-
13
MISHNAH EDUYOT
-
-
-
-
1
2
3
4
5
6
7
8
9
M
=
4
=
7
MISHNAH
72
45
9
-
-
-
-
-
-
-
-
-
-
E
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
3
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
13
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
10
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
1
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
Y
=
7
11
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
H
=
8
7
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
2
2
3
8
10
6
7
16
9
-
-
13
MISHNAH EDUYOT
-
-
-
-
-
-
-
-
1+0
-
-
1+6
-
M
=
4
=
7
MISHNAH
72
45
9
-
2
2
3
8
1
6
7
7
9
N
=
5
=
6
EDUYOT
90
27
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
13
MISHNAH EDUYOT
162
72
18
-
2
2
3
8
1
6
7
7
9
-
-
-
-
1+3
-
1+6+2
4+5
3+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
MISHNAH EDUYOT
9
9
9
-
2
2
3
8
1
6
7
7
9

 

 

8
COMPLETE
89
35
8
9
COMPLETES
108
36
9
9
COMPLETED
93
40
4

 

 

7
AND-SO-ON
82
28
1
8
ELLIPSIS
101
38
2

 

 

5
CHAIN
35
26
8
6
CHAINS
54
27
9

 

 

6
NUMBER
73
28
1
5
CHAIN
35
26
8
11
Add to Reduce
108
54
9
1+1
Reduce to Deduce
1+0+8
5+4
-
2
Essence of Number
9
9
9

 

 

10
ARITHMETIC
106
52
7

 

 

7
THOUGHT
99
36
9
4
ZERO
64
28
1
5
OUGHT
71
26
8
6
NOUGHT
85
31
4

 

 

7
THOUGHT
99
36
9
6
NOUGHT
85
31
4

 

 

8
THOUGHTS
118
37
1
7
NOUGHTS
104
32
5

 

 

5
PEACE
30
21
3
4
LOVE
54
18
9
5
LIGHT
56
29
2
14
First Total
140
68
14
1+4
Add to Reduce
1+4+0
6+8
1+4
5
Second Total
5
14
5
-
Reduce to Deduce
-
1+4
-
5
Essence of Number
5
5
5

 

 

3
THE
33
15
6
6
ABSENT
61
16
7
4
SELF
42
15
6
13
First Total
136
46
19
1+3
Add to Reduce
1+3+6
4+6
1+9
4
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
4
Essence of Number
1
1
1

 

 

0
8
=
Z
ZERO
O
=
6
1
6
=
O
ONE
E
=
5
2
2
=
T
TWO
O
=
6
3
2
=
T
THREE
E
=
5
4
6
=
F
FOUR
R
=
9
5
6
=
F
FIVE
E
=
5
6
1
=
S
SIX
X
=
6
7
1
=
S
SEVEN
N
=
5
8
5
=
E
EIGHT
T
=
2
9
5
=
N
NINE
E
=
5
45
42
-
42
-
-
-
54
4+5
4+2
-
4+2
-
-
-
5+4
9
6
-
6
-
-
-
9

 

 

S
=
1
1
4
SOLO
61
16
7
D
=
4
2
4
DUEL
42
15
6
T
=
2
3
4
TRIO
62
26
8
Q
=
8
4
7
QUARTET
102
30
3
Q
=
8
5
7
QUINTET
106
34
7

 

 

M
=
13
=
4
6
MYRAID
70
34
7
M
=
13
=
4
7
MYRIADS
89
35
8

 

 

5
ATTIC
53
17
8
10
ACROPHONIC
102
57
3

 

 

7
ALGEBRA
46
28
1
9
ALGEBRAIC
58
40
4

 

 

9
T
E
N
N
E
S
S
E
E
-
-
-
-
-
--
-
-
-
-
-
-
-
5
5
-
1
1
-
-
+
=
12
1+2
=
3
=
3
=
3
-
-
-
14
14
-
19
19
-
-
+
=
66
6+6
=
12
1+2
3
=
3
9
T
E
N
N
E
S
S
E
E
-
-
-
-
-
--
-
-
-
-
-
2
5
-
-
5
-
-
5
5
+
=
22
2+2
=
4
=
4
=
4
-
20
5
-
-
5
-
-
5
5
+
=
40
4+0
=
4
=
4
=
4
9
T
E
N
N
E
S
S
E
E
-
-
-
-
-
--
-
-
-
-
-
20
5
14
14
5
19
19
5
5
+
=
106
1+0+6
=
7
=
7
=
7
-
2
5
5
5
5
1
1
5
5
+
=
34
3+4
=
7
=
7
=
7
9
T
E
N
N
E
S
S
E
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
1
occurs
x
2
=
2
=
2
-
2
-
-
-
-
-
-
-
-
-
-`
2
occurs
x
1
=
2
=
2
-
-
5
5
5
5
-
-
5
5
-
-
5
occurs
x
6
=
30
3+0
3
9
T
E
N
N
E
S
S
E
E
-
-
8
-
1
9
-
34
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+4
-
-
9
T
E
N
N
E
S
S
E
E
-
-
8
-
-
9
-
7
-
7

 

 

-
12
I
R
R
E
S
I
S
T
I
B
L
E
-
-
-
-
-
--
-
-
-
-
-
-
9
-
-
-
1
9
1
-
9
-
-
-
+
=
29
2+9
=
11
1+1
2
=
2
-
-
9
-
-
-
19
9
19
-
9
-
-
-
+
=
65
6+5
=
11
1+1
2
=
2
-
12
I
R
R
E
S
I
S
T
I
B
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
9
5
-
-
-
2
-
2
3
5
+
=
35
3+5
=
8
=
8
=
8
-
-
-
18
18
5
-
-
-
20
-
2
12
5
+
=
80
8+0
=
8
=
8
=
8
-
12
I
R
R
E
S
I
S
T
I
B
L
E
-
-
-
-
-
--
-
-
-
-
-
-
9
18
18
5
19
9
19
20
9
2
12
5
+
=
145
1+4+5
=
10
1+0
1
=
1
-
-
9
9
9
5
1
9
1
2
9
2
3
5
+
=
64
6+4
=
10
1+0
1
=
1
-
12
I
R
R
E
S
I
S
T
I
B
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
2
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
9
9
-
-
9
-
-
9
-
-
-
-
-
9
occurs
x
5
=
45
3+6
9
25
12
I
R
R
E
S
I
S
T
I
B
L
E
-
-
20
-
-
12
-
64
-
19
2+5
1+2
9
9
9
-
-
9
-
-
9
-
-
-
-
-
2+0
-
-
1+2
-
6+4
-
1+9
7
3
I
R
R
E
S
I
S
T
A
B
L
E
-
-
2
-
-
3
-
10
-
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+0
-
1+2
7
3
I
R
R
E
S
I
S
T
A
B
L
E
-
-
2
-
-
3
-
1
-
1

 

 

7
SPATIAL
78
24
6
10
MAGNITUDES
113
59
4
6
MOTION
86
32
5
4
TIME
47
20
2

 

 

16
SPATIAL MAGNITUDE
172
64
1

 

 

4
TIME
-
-
-
-
T
20
2
2
-
I
9
9
9
-
M+E
18
9
9
4
TIME
47
20
20
-
-
4+7
2+0
2+0
4
TIME
11
2
2
-
-
1+1
-
-
4
TIME
2
2
2

 

 

-
SINE QUA NON
-
-
-
4
SINE
47
20
2
3
QUA
39
12
3
3
NON
43
16
7
10
SINE QUA NON
129
48
12
1+0
-
1+2+9
4+8
1+2
1
SINE QUA NON
12
12
3
-
-
1+2
1+2
-
1
SINE QUA NON
3
3
3

 

 

HOMER

H

ROME

 

5
HOMER
-
-
-
-
H
23
5
5
-
ROME
18
9
9
5
HOMER
59
23
23
-
-
5+9
2+3
2+3
5
HOMER
14
5
5
-
-
1+4
-
-
5
HOMER
5
5
5

 

ROME

H

HOMER

 

5
HOMER
-
-
-
-
H+O
23
5
5
-
M+E
18
9
9
-
R
18
9
9
5
HOMER
59
23
23
-
-
5+9
2+3
2+3
5
HOMER
14
5
5
-
-
1+4
-
-
5
HOMER
5
5
5

 

 

5
HOMER
-
-
-
-
H
8
8
8
-
O
15
6
6
-
M
13
4
4
-
E
5
5
5
-
R
18
9
9
5
HOMER
59
23
23
-
-
5+9
2+3
2+3
5
HOMER
14
5
5
-
-
1+4
-
-
5
HOMER
5
5
5

 

 

5
HOMER
-
-
-
-
H+O+M
36
18
5
-
E
18
9
9
-
R
18
9
9
5
HOMER
59
23
23
-
-
5+9
2+3
2+3
5
HOMER
14
5
5
-
-
1+4
-
-
5
HOMER
5
5
5

 

 

3
THE
33
15
6
6
PHAEDO
49
31
4
9
-
82
46
10
-
-
8+2
4+6
1+0
9
-
10
10
1
-
-
1+0
1+0
-
9
-
1
1
1

 

 

5
PLATO
64
19
1

 

 

8
PHAEDRUS
92
38
2

 

 

8
ABSOLUTE
95
23
5

 

 

6
SOURCE
81
18
9
5
WHOLE
63
27
9

 

 

-
WHOLE NUMBER
-
-
-
5
WHOLE
63
27
9
6
NUMBER
73
28
1
11
WHOLE NUMBER
136
55
10
1+1
-
1+3+6
5+5
1+0
2
WHOLE NUMBER
10
10
1
-
-
1+0
1+0
-
2
WHOLE NUMBER
1
1
1

 

I

THE

MESSAGE

unless integral to quoted work.

all arithmetical machinations, emphasis,

comment, insertions subterfuge and insinuations

are those of the Zed Aliz Zed as recorded by the far yonder scribe.

 

 

THE MAGIC MOUNTAIN

Thomas Mann 1875-1955

Page 466

"Had not the normal, since time was, lived on the achievements of the abnormal? Men consciously and
voluntarily descended into disease and madness, in search of knowledge which, acquired by fanaticism, would lead back to health; after the possession and use of it had ceased to be conditioned by that heroic and abnormal act of sacrifice.

That was the true death on the cross, the true Atonement."

 

 

HOLY BIBLE

Scofield References

Page 1117 A.D. 30.

Jesus answered and said unto him, Verily, verily,
I say unto thee, Except a man be born again,
He cannot see the kingdom of God.
St  John  Chapter   3  verse  3
3     +     3     3     x     3
6        x        9
54
5 + 4
9

 

 

IN SEARCH OF THE MIRACULOUS

Fragments of an Unknown Teaching

P.D.Oupensky 1878- 1947

Page 217

" 'A man may be born ,but in order to be born he must first die, and in order to die he must first awake.' "
" 'When a man awakes he can die; when he dies he can be born' " Thus spake the prophet Gurdjieff.

 

 

THE MAGIC MOUNTAIN

Thomas Mann 1875-1955

Page 496

" There is both rhyme and reason in what I say, I have made a dream poem of humanity.

I will cling to it. I will be good. I will let death have no mastery over my thoughts.

For therein lies goodness and love of humankind, and in nothing else."

 

"Love stands opposed to death. It is love, not reason, that is stronger than death . Only love, not reason, gives sweet thoughts. And from love and sweetness alone can form come: form and civilisation, friendly and enlightened , beautiful human intercourse-always in silent recognition of the blood-sacrifice. Ah, yes, it is it is well and truly dreamed. I have taken stock I will keep faith with death in my heart, yet well remember that faith with death and the dead is evil, is hostile to mankind, so soon as we give it power over thought and action.

For the sake of goodness and love, man shall let death have no sovereignty over his thoughts.
- And with this -I awake. For I have dreamed it out to the end, I have come to my goal."

Page 496 / 497

After a short meeting with their good and trusted friend Thomas. Alizzed and the scribe thanked him most genuinely for the benefit of his wisdom, in the matter of their quest, and in saying their good byes, wished the other well, a not unusual seven times, and of course, promised, not to leave it quite so long in the future.

 

GOD WITH US AND US WITH GOD

 

3
GOD
26
17
8
4
WITH
60
24
6
2
US
40
4
4
9
Add to Reduce
126
45
18
-
Reduce to Deduce
1+2+6
4+5
1+8
9
Essence of Number
9
9
9

 

 

"The virgin will conceive and give birth to a son, and they will call him Immanuel" (which means "God with us"). “Behold, the virgin shall conceive and bear a son, and they shall call his name Immanuel” (which means, God with us).

Matthew 1:23 "The virgin will conceive and give birth to a ...

biblehub.com/matthew/1-23.htm

 

The Meaning of Immanuel, God with Us

www.orlutheran.com/html/immanuel.html

And this very special Christmas name, as Matthew tells us, means "God with us." Jesus Christ is Immanuel, "God with us," and I'd like to share why this is so ...

Matthew 1:23 "The virgin will conceive and give birth to a ...

matthew/1-23.
“Behold, the virgin shall conceive and bear a son, and they shall call his name Immanuel” (which means, God with us). New American Standard Bible "BEHOLD ...

 

Christ Emmanuel or God with Us - Grace Gems!

www.gracegems.org/W/e1.htm

"They shall call His name Emmanuel, which being interpreted is, God with us. ... give birth to a son, and they will call him Immanuel– which means, 'God with us.

 

Isaiah 7:14 Explained - Immanuel God With Us

www.bibleanswerstand.org/immanuel.htm

This study is aimed at finding the true meaning of Immanuel in Isaiah 7:14. ... texts for the deity of Jesus Christ because of the words, “Immanuel,” (God with us).

 

Why wasn't Jesus named Immanuel? - GotQuestions.org

www.gotquestions.org/Immanuel-Jesus.html

by S. Michael Houdmann - Jesus was God making His dwelling among us (John 1:1,14). No, Jesus' name was not Immanuel, but Jesus was the meaning of Immanuel, "God with us.

 

Words Around "Emmanuel" in the English Dictionary

"The word Immanuel/Emmanuel means, "God with us." It conveys the idea of God come down in the flesh, mingling alongside mankind, subject to their brutality, while extending his love in bringing their redemption."

 

GOD WITH US AND US WITH GOD

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
8
-
W
=
5
-
4
WITH
60
24
6
-
-
-
-
-
-
6
-
-
-
U
=
3
-
2
US
40
4
4
-
-
-
-
4
-
-
-
-
-
-
-
1+5
-
9
Add to Reduce
126
45
18
-
1
2
3
4
5
6
7
8
9
-
-
6
-
-
Reduce to Deduce
1+2+6
4+5
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
GOD WITH US
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
-
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
45
-
9
GOD WITH US
126
54
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
1+2+6
5+4
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
GOD WITH US
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
GOD WITH US
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
5
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
7
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
D
=
7
-
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
3
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
5
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
5
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
3
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
45
-
9
GOD WITH US
126
54
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
1+2+6
5+4
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
GOD WITH US
9
9
9
-
1
2
3
4
5
6
7
8
9

 

GOD WITH US 123456789 987654321 US WITH GOD

 

 

BLESSED ARE THE PEACEMAKERS FOR THEY SHALL BE CALLED THE CHILDREN

OF

GOD

 

Jean-Jacques Rousseau
(1712-1788)

"Man is born free, and everywhere he is in chains. One man thinks himself the master of others, but remains more of a slave than they are."

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
M
=
4
-
2
MAN
28
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
2
IS
28
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
-
10
BORN
49
22
4
-
-
-
-
4
-
-
-
-
-
K
=
2
-
4
FREE
34
25
7
-
-
-
-
-
-
7
-
-
-
O
=
6
-
2
AND
19
10
1
-
1
-
-
-
-
-
-
-
-
K
=
2
-
5
EVERYWHERE
134
62
8
-
-
-
-
-
-
-
-
8
-
K
=
2
-
4
HE
13
13
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IS
28
10
1
-
1
-
-
-
-
-
-
-
-
K
=
2
-
4
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
D
=
4
-
7
CHAINS
54
27
9
-
-
-
-
-
-
-
-
-
9
-
-
56
-
38
First Total
410
203
41
-
4
2
3
8
5
7
7
8
9
-
-
5+6
-
3+8
Add to Reduce
4+1+0
2+0+3
4+1
-
-
-
-
-
-
-
-
-
-
-
-
11
-
11
Second Total
5
5
5
-
4
2
3
8
5
7
7
8
9
-
-
1+1
-
1+1
Add to Reduce
-
-
 1+0
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
Essence of Number
5
5
5
-
4
2
3
8
5
7
7
8
9

 

 

7
MESSIAH
-
-
-
-
M+E
18
9
9
-
S+S
38
20
2
-
I
9
9
9
-
A+H
9
9
9
7
MESSIAH
74
47
29
-
-
7+4
4+7
2+9
7
MESSIAH
11
11
11
-
-
1+1
1+1
1+1
7
MESSIAH
2
2
2

 

THE MESSIAH AT THE GATES OF ROME
Post by Black_Rose » 03 Sep 2020 20:30

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6
-
-
-
M
=
4
-
7
MESSIAH
74
29
2
-
-
2
-
-
-
-
-
-
-
A
=
1
-
2
AT
21
3
3
-
-
-
3
-
-
-
-
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6
-
-
-
G
=
7
-
5
GATES
52
25
7
-
-
-
-
-
-
-
7
-
-
O
=
3
-
2
OF
21
12
3
-
-
-
3
-
-
-
-
-
-
R
=
9
-
4
ROME
51
24
6
-
-
-
-
-
-
6
-
-
-
-
-
28
-
26
First Total
285
123
33
-
1
2
6
4
5
18
7
8
9
-
-
2+8
-
2+6
Add to Reduce
4+1+0
1+2+3
3+3
-
-
-
-
-
-
1+8
-
-
-
-
-
10
-
11
Second Total
6
6
6
-
1
2
6
4
5
9
7
8
9
-
-
1+0
-
1+1
Add to Reduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
2
Essence of Number
6
6
6
-
1
2
6
4
5
9
7
8
9

 

THE MESSIAH AT THE GATES OF ROME

"The Messiah at the Gates of Rome" is a traditional story, Mashal or parable in the Jewish tradition, from the Babylonian Talmud, Sanhedrin 98a.
The Wind in the Willows is a children's novel by Scottish novelist Kenneth Grahame.

 

 

THE PIPER AT THE GATES OF DAWN

"The Piper at the Gates of Dawn" tells how Mole and Rat search for Otter's missing son Portly, whom they find in the care of the god Pan.

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6
-
-
-
P
=
7
-
5
PIPER
64
37
1
-
1
-
-
-
-
-
-
-
-
A
=
1
-
2
AT
21
3
3
-
-
-
3
-
-
-
-
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6
-
-
-
G
=
7
-
5
GATES
52
25
7
-
-
-
-
-
-
-
7
-
-
O
=
6
-
2
OF
21
12
3
-
-
-
3
-
-
-
-
-
-
D
=
4
-
4
DAWN
42
15
6
-
-
-
-
-
-
6
-
-
-
-
-
29
-
24
First Total
266
113
32
-
1
2
6
4
5
18
7
8
9
-
-
2+9
-
2+4
Add to Reduce
2+6+6
1+1+3
3+2
-
-
-
-
-
-
1+8
-
-
-
-
-
11
-
6
Second Total
14
5
5
-
1
2
6
4
5
9
7
8
9
-
-
1+1
-
-
Add to Reduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
6
Essence of Number
5
5
5
-
1
2
6
4
5
9
7
8
9

 

 

Homer is the presumed author of the Iliad and the Odyssey, two epic poems that are the central works of ancient Greek literature
9 Sep 2019 - The Greek poet Homer was born sometime between the 12th and 8th centuries BC, possibly somewhere on the coast of Asia Minor.

He is famous for the epic poems
The Iliad and The Odyssey, which have had an enormous effect on Western culture, but very little is known about their alleged author.

 

-
THE UNKNOWN GOD
-
-
-
3
THE
33
15
6
7
UNKNOWN
112
31
4
3
GOD
26
17
8
13
THE UNKNOWN GOD
171
63
18
1+3
-
1+7+1
6+3
1+8
4
THE UNKNOWN GOD
9
9
9

 

 
 
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