gaffo wrote: ↑Sun Nov 25, 2018 1:19 am
Eodnhoj7 wrote: ↑Fri Nov 16, 2018 10:40 pm
Monadic Numbers
1) All numbers exist through continuums. 1 exists through 2,3,4..., with each number respectively existing through further numbers as a continuum of continuum.
A. This continuous nature of number necessitates not just a number line, but effectively the number as having a directional property.
Example:
1 -> 2 -> 3 -> 4 -> 5....
This nature of all numbers as progressive in nature as defined by thier directional qualities observes all numbers as linear directed movement.
B. The continuous nature of number observes and inherent circulatory nature through the number line where all number cycle to form the number line as progressive oscillation.
Example:
(1 -> 1) -> 2
(1 <- 2) -> 3
(2 -> 2) -> 4
...
The nature of number as 1 directed through itself to maintain itself as 1 through 2, with 2 as an extension of 1 directed towards itself as 4 and 2 directed back to 1 as 3, observes that 1 maintains itself as absolute while progressing through a continuum which observes the same qualities.
C. All numbers, through 1 as extensions of 1, are points of oscillation where this point of oscillation observes the number directed through itself as itself through an approximate number which this approximate number being 1 observed through variations of one. These variations occur as 1 directed mirroring 0 as absent of structure. 1 mirroring 0, with 0 as void of meaning, observes 1 replicating this formless nature through multiple variations with 0 as void being a point of inversion between unity and multiple units.
1 exists as a point of origin existing through itself as itself.
All numbers as an origin point(s) through point(s) as points(s) observe all monastic numbers as infinite Continuums of numbers.
2) All numbers exist through continuums as continuous functions with the number line as a continuum observing all functions as continuums.
a. 1= 1+1-1, 1+2-2, 1+3-3, 1+4-4, 1+5-5...
b. 1= 2-1, 3-2, 4-3, 5-4, 6-5, 7-6...
c. 1= 1*1/1, 1*2/2, 1*3/3, 1*4/4, 1*5/5...
d. 1= 1^1/1, 1^2/2, 1^3/3, 1^4/4, 1^5/5...
f. 1= 2root1, 2root2/2, 2root3/3, 2root4/4...
With infinite variation all showing all number exists as continuums through there inherent functions. While the numbers progressively vary within the continuum, the functions maintain themselves as constants as well as all number being extensions of one.
The number as inseparable from the function, as well as all number existing as an extension of one while being one continuum in itself (ex: 2 -> 4 -> 6...), observes 1 as a continuous function. All numbers are absolute as all numbers are continuums.
3. The number as composed of continuums, with these continuums existing through arithmetic functions as continuums, observes not just the number as a continuum but the function as a continuum as well where the number as form cannot be seperated from it as a function.
4. All numbers are inseperable from all functions. Hence +1 is inseperable from addition, multiplication and powers. -1 is inseperable from subtraction, division and root. One, as point 2 observes, is a continuous function as the standard arithmetic functions are continuous through one.
These arithmetic functions, as inseparable from all numbers through the 1, in themselves oscillate to form eachother in a similar manner to the number line.
Addition oscillate through itself as itself while manifesting multiplication as "the addition of addition".
Multiplication follows the same form and function where it maintains both addition and multiplication manifesting powers as the multiplication of multiplication.
Powers maintains itself as itself through addition and multiplication in turn acts a foundation for addition and multiplication.
Addition, multiplication and powers oscillate through eachother as eachother where 1 exists as an approximation of the other.
Negative functions (subtraction, division and root) follow this same oscillatory pattern.
5. All numbers as form/functions, must exist through further numbers as form/functions, hence all number inherently are directed through themselves as themselves where any replication is strictly an extension of the original mirror "numbers" as a replication of them. The number and the structural extension of that number through further numbers observe all numbers as mirroring as fundamentally oscillcations, progressing to further oscillations as an oscillation.
6. example:
1⊙= the number of oscillation of (2⊙,3⊙)
⧂ = The oscillation of the numbers into further oscillations.
1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)
All numbers as a directive nature observes the directional qualities of the numbers forming new numbers under certain circumstances.
a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27
These mirror number in turn form further relations relative to other mirror numbers where all numbers exist through all numbers an each number is an extension of 1⊙. The monadic number as an oscillation of itself through further monadic numbers as oscillations observes an inherent repitition of monadic numbers, through monadic numbers as simultaneous extensions of the monadic numbers and acting as self referencing in accords to the monadic numbers they exist through.
All monadic calculus is a proof in itself as the repitition of monadic numbers forming an inherent symmetry resulting in both the statement and the numbers as self referencing absolutes as proofs in themselves.
Each monadic number, as a form/function, cycles to further proofs where all calculations are approximations of the Monad and have an inherent element of randomness. This randomness, where the repitition of structural extensions of the Monad are approximations of it, observes all proofs as both random and having self maintained structure reflecting this randomness through multiple monadic numbers as extensions of eachother and 1.
Each monadic number as a set of infinite numbers observe through a continuum, is in itself a continuum where one monadic number directed towards another is a set of infinite fractals in one respect while simultaneously results in a set of monadic numbers.
nice sophistry/word salad.
where do imaginary salads - i mean numbers, fit into your salad?
No different than the rest, but considering the monadic number is primarily both a form and function through its nature as an infinite series of numbers.
The imaginary number i is defined solely by the property that its square is −1:
i 2 = − 1. {\displaystyle i^{2}=-1.} {\displaystyle i^{2}=-1.}
With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.
https://en.wikipedia.org/wiki/Imaginary_unit
1⊙= (i^2= -1)+2, (i^6= -1)+2, (i^10= -1)+2...
(i^2= -1)+2-1+1, (i^6= -1)+2-2+2, (i^10= -1)+2-3+3...
...
i⊙= i^1,i^5,i^9...
The real question, relative to the monadic numbers is considering their inherent nature as arithmetic functions if a positive or negative monadic number has any different function in itself.
The statement:
1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)
Is premised strictly in positive form and function:
a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27
But leads to further questions if each number is a quantitative series and function:
If it is premised in a strictly negative form and function:
1⊙(-2⊙,-3⊙) ⧂ (-6⊙, -5.19615242i⊙, -5⊙,-4.24264069i⊙, -4⊙, -3⊙, -2⊙, -1.73205081i⊙, -1.41421356i⊙,-1 1/2⊙, -1⊙, 2/3⊙, 1⊙, 1 1/2⊙, 1.41421356i⊙, 1.73205081i⊙, 4.24264069i⊙, 5.19615242i⊙)
a. -2-3= -5
b. -3-2= -5
c. -2/3 = -2/3
c1. -2/-3 = 2/3
d. -3/2 = -1 1/2
d1. -3/-2 = 1 1/2
e. 2root-3 = 1.73205081i
e1. -2root-3 = -1.73205081i
f. 3root-2 = 4.24264069i
f1. -3root-2 = -4.24264069i
g. -2-2 = -4
h. -2/2 = -1
h1. -2/-2 = 1
i. 2root-2 = 1.41421356i
i1. -2root-2 = -1.41421356i
j. -3-3 = -6
k. -3/3 = -1
k1. -3/-3 = 1
l. 3root-3 = 5.19615242i
l1. -3root-3 = -5.19615242i
It even extends further considering 1⊙ of 1⊙ (2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙) observes the number of cycles:
a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27
and changes when: 2⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)
(8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)
****Considering all monadic functions are dependent upon the replication of cycles, The oscillation of the numbers into further oscillations always contains the premise within the answer with the premise and answer always containing 1⊙ as the number of cycles replicating into further cycles is always 1 cycle containing further cycles. These oscillations observe an inherent arithmetic function nature involved.
Hence while:
1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)
2⊙(2⊙,3⊙) ⧂ (8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)
a. (2+3= 5 + 2+3= 5) = 10
(2+3= 5 * 2+3= 5) = 25
(2+3= 5 ^ 2+3= 5) = 3125
b. (3+2= 5 + 3+2= 5) = 10
(3+2= 5 * 3+2= 5) = 25
(3+2= 5 ^ 3+2= 5) = 3125
c. (2*3 = 6 + 2*3 = 6) = 12
(2*3 = 6 * 2*3 = 6) = 36
(2*3 = 6 ^ 2*3 = 6) = 46,656
d. (3*2 = 6 + 3*2 = 6) = 12
(3*2 = 6 * 3*2 = 6) = 36
(3*2 = 6 ^ 3*2 = 6) = 46,656
e. (2^3 = 8 + 2^3 =
= 16
(2^3 = 8 * 2^3 =
= 64
(2^3 = 8 ^ 2^3 =
= 16,777,216
f. (3^2 = 9 + 3^2 = 9) = 18
(3^2 = 9 * 3^2 = 9) = 81
(3^2 = 9 ^ 3^2 = 9) = 387,420,489
g. (2+2 = 4 + 2+2=4) = 8
(2+2 = 4 * 2+2=4) = 16
(2+2 = 4 ^ 2+2=4) = 256
h. (2*2 = 4 + 2*2 = 4) = 8
(2*2 = 4 * 2*2 = 4) = 16
(2*2 = 4 ^ 2*2 = 4) = 256
i. (2^2 = 4 + 2^2 = 4) = 8
(2^2 = 4 * 2^2 = 4) = 16
(2^2 = 4 ^ 2^2 = 4) = 256
j. (3+3 = 6 + 3+3 = 6) = 12
(3+3 = 6 * 3+3 = 6) = 36
(3+3 = 6 ^ 3+3 = 6) = 46656
k. (3*3 = 9 + 3*3 = 9) = 18
(3*3 = 9 * 3*3 = 9) = 81
(3*3 = 9 ^ 3*3 = 9) = 387,420,489
l. (3^3 = 27 + 3^3 = 27) = 54
(3^3 = 27 * 3^3 = 27) = 729
(3^3 = 27 ^ 3^3 = 27) = 27^27
This further occurs respectively where the monadic number is a negative as well.
Because the number of cycles contains as series of cycles within it, the Oscillator (2⊙ in this case) always contains the series of cycles prior to it in the answer. So if the oscillator is 33⊙, it would contain as an answer the oscillations of 1⊙, 2⊙, 3⊙...33⊙. However considering (2⊙,3⊙) in the above example is a structural extension, or rather localization of the oscillations inherent within the oscillations of 1⊙ and 2⊙ these are not viewed as series. In these respects it maybe view as:
1⊙(2⊙,3⊙)◬ ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)◬
and
2⊙(2⊙,3⊙)◬ ⧂
(1⊙, 2⊙, 3⊙, 4⊙, 5⊙, 6⊙, 8⊙, 9⊙, 27⊙, 8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)◬
Where each monadic number in itself is true as both an infinite series and function but as a set it is still an approximation. In these respects all monadic calculus through monadic numbers exists dually with a "triadic" number and "triadic function" (approximation of a whole conducive to randomness observed in the localization of number through any standard equation). The Monadic form/function exists dually with the Triadic form/function with this dualistic tension between the Monadic and Triadic observing a third neutral component where the premise and conclusion is the answer itself.
In these respects all monadic "equations" are localizations of self-referencing numbers and from a base level are best limited to 1 cycle. This is considering that because of the complexity, the majority of calculation would require a computer program as 1 simple equation of
33⊙(27⊙,23⊙) ⧂ x may be equivalent to pages upon pages of answers where the proof itself is strictly the book as self-referencing equation.
The real philosophical problem occurs considering all monadic numbers are composed of simultaneous positive and negative functions of arithmetic, as positive and negative number lines, the monadic number in itself cannot be strictly addition/multiplication/powers or subtraction/division/roots
but effectively both.
So the interpretation of 1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙) changes to:
when observing all monadic numbers as either positive or negative having inherent positive and negative values within them considering all of them exist through a constant series of functions.
results in:
*******Unfinished, out of time.
a. 2+3= 5
b. 2-3= -1
c. -2-3= -5
c. 3+2= 5
d. 3-2 = 1
e. -3-2=-5
e. 2*3 = 6
x. -2*3 = -6
x. 2*-3 = -6
x. -2*-3= 6
f. 2/3 = 2/3
g. -2/-3 = 2/3
x. -2/3 = -2/3
x. 2/-3 = -2/3
d. 3*2 = 6
x. -3*2 = -6
x. 3*-2 = -6
x. -3*-2 = 6
x. 3/2 = 3/2
x. -3/2 = -3/2
x. 3/-2 = -3/2
x. -3/-2 = 3/2
e. 2^3 = 8
x. 2root3=
x. 2^-3= 0.125
x. 2root-3=
x. -2^3= -8
x. -2root3=
x. -2^-3= -0.125
x. -2root-3
f. 3^2 = 9
x. 3root2=
g.-3^2 = 9
x.-3root2=
x. 3^-2 = 0.111111111
x. 3root-2=
x. -3^-2 = -0.111111111
x -3root-2=
g. 2+2 = 4
x. -2+2 = 0
x. 2+-2 = 0
x. -2-2 =4
h. 2*2 = 4
x. -2*2=-4
x. 2*-2=-4
x. -2*-2=4
i. 2^2 = 4
x. 2root2=
j. -2^2 = 4
x 2^-2
x. -2^-2
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27
1⊙(-2⊙,-3⊙) ⧂ (-6⊙, -5.19615242i⊙, -5⊙,-4.24264069i⊙, -4⊙, -3⊙, -2⊙, -1.73205081i⊙, -1.41421356i⊙,-1 1/2⊙, -1⊙, 2/3⊙, 1⊙, 1 1/2⊙, 1.41421356i⊙, 1.73205081i⊙, 4.24264069i⊙, 5.19615242i⊙)
a. -2-3= -5
b. -3-2= -5
c. -2/3 = -2/3
c1. -2/-3 = 2/3
d. -3/2 = -1 1/2
d1. -3/-2 = 1 1/2
e. 2root-3 = 1.73205081i
e1. -2root-3 = -1.73205081i
f. 3root-2 = 4.24264069i
f1. -3root-2 = -4.24264069i
g. -2-2 = -4
h. -2/2 = -1
h1. -2/-2 = 1
i. 2root-2 = 1.41421356i
i1. -2root-2 = -1.41421356i
j. -3-3 = -6
k. -3/3 = -1
k1. -3/-3 = 1
l. 3root-3 = 5.19615242i
l1. -3root-3 = -5.19615242i
The next question occurs if both positive and negative monadic numbers exist simultaneously through positive and negative functions is there really a difference between them?
