The Unified Number

[conway]

Numerus “Numerans Numeratus”



The Unified Number





It is the inherent nature of all things, that they are a compilation of two different, and distinct things. It is axiomatic that these two things are space, and value. The value of any given thing being what it is, while the space is what it occupies.


It is true that abstract, or otherwise, numbers are a thing. Therefore, they must also contain a compilation of space, and value. It is an axiomatic truth, that space is the labeling of quantities of dimensions. It is an axiomatic truth, that value is the labeling of quantities of existence, other than dimensions.



Let all abstract numbers be defined exactly as concrete numbers.


Concrete number: A numerical quantity with a corresponding unit.


Let the corresponding unit of all abstract numbers, exist as an abstract dimension notated with the use of (_).


Let the length, and width of all dimensional units remain abstract, and undeclared.


Let the dimensional unit be equal in quantity, to the numerical quantity it corresponds to.


Let all numerical quantities inhabit their corresponding abstract dimensional units.


Let zero be assigned a single dimensional unit.


Classic Novel

0 = (0) = (0,_) = (0,0_)

1 = (1) = (1,_) = (1,1_)

2 = (2) = (2,_,_) = (2,2_)

3 = (3) = (3,_,_,_) = (3,3_)

(-1) = (-1) = (-1,-(_)) = (-1,-(1_))

(-2) = (-2) = (-2,-(_,_)) = (-2,-(2_))

(-3) = (-3) = (-3,-(_,_,_)) = (-3,-(3_))


Therefore:


Any classic number (n) = novel number (n) = (n,n_)


Where (_) is defined as a dimensional unit, separate from the numerical quantity.


Where (n) is defined as the numerical quantity separate from the dimensional unit.


Where (n_) is defined as the dimensional unit, separate from the numerical quantity, and equal in quantity to the numerical quantity it corresponds to, or (n).


Where (n) is defined as the numerical quantity, not separate from, but inhabiting the dimensional unit quantity.


Let addition, and subtraction exist without change. Except regarding notation: (a+b = c: a+0 = a: a-0 = a: 0+0 = 0: 0-0 = 0).


In any binary expression of multiplication, let one number (n) represent only a numerical quantity or (n). Let the other number (n) represent only a quantity of dimensional unit, equal in quantity to the number it corresponds to, or (n_). Therefore, in all cases of binary multiplication (n*n): (n) is notated as (n*n_)=(n_*n).


In any binary expression of division, let the numerator (n), always exist as a numerical quantity or (n). Let the denominator (n) always exist as a dimensional unit quantity, equal in quantity to the number it corresponds to, or (n_). Therefore, in all cases of binary division (n/n): (n) is notated as (n/n_).


Let multiplication be defined as the placing of a given numerical quantity, with addition, equally into each given quantity of dimensional unit. Then all numerical quantities, in all dimensional units are added.


Let division be defined as the placing of a given numerical quantity, with subtraction, equally into each given quantity of dimensional unit. Then all numerical quantities, in all dimensional units are subtracted except one.


In all binary operations of multiplication containing the number (0), and a non-zero number (n). The notation of the number (0), as (0), or as (0_), will dictate the notations of the binary non-zero number (n) in the operation.


In all cases of a binary expression where the notation is not given for the number (0), the numerical quantity (0) is notated for (0), and the dimensional quantity (n_) is notated for (n).


Therefore: (n*0 = n_*0 = 0*n_ = 0).


Let exponents, and logarithms reflect these rules of multiplication, and division.


Assertion:


All binary operations of multiplication, and division remain unchanged. Except binary operations involving the number (0). As well as defining division by the number (0), as an operation of a given numerical quantity (n), into the dimensional unit quantity of (0_). Therefore, this novel system is isomorphic to the classical system. Except for regarding multiplication, and division by zero.



Multiplication


Classic


2*3 = 6


Isomorphic


2*(_,_,_) = 6 = 2*3_


Where:


Classic (2): Is the numerical quantity.


Classic (3): Is the dimensional unit quantity.


(_,_,_): The dimensional unit quantity of the number (3).


(2,2,2): The numerical quantity (2) added equally into all dimensional unit quantities.


(2+2+2 = 6): The numerical quantity (2) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


2*(_,_,_) = 6


Or,


3*(_,_) = 6 = 3*2_


Where:


Classic (2): Is the dimensional unit quantity.


Classic (3): Is the numerical quantity.


(_,_): The dimensional unit quantity of the number (2).


(3,3): The numerical quantity (3) added equally into all dimensional unit quantities.


(3+3 = 6): The numerical quantity (3) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


3*(_,_) = 6


Classic


2*0 = 0


Novel


2*(_) = 2 = 2*0_


Where:


Classic (2): Is the numerical quantity.


Classic (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(2): The numerical quantity (2) added equally into all dimensional unit quantities.


(2): The numerical quantity (2) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


2*(_) = 2


Or,


0*(_,_) = 0 = 0*2_


Where:


Classic (0): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(0,0): The numerical quantity (0) added equally into all dimensional unit quantities.


(0+0 = 0): The numerical quantity (0) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


0*(_,_) = 0


Classic


0*0 = 0


Isomorphic


0*(_) = 0 = 0*0_


(_): The dimensional unit quantity of the number (0).


(0): The numerical quantity of (0) added equally into all dimensional unit quantities.


(0): The numerical quantity of (0) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


0*(_) = 0


Therefore, the product of binary multiplication by the number (0), with a non-zero number. Is relative to the number (0), declared as a numerical quantity, or as a dimensional unit quantity. In the binary expression.


Isomorphic expressions containing variables.


Where: (n) =/= 0


n*(0_) = n = (0_)*n


n*(_) = n = (_)*n


n_*0 = 0 = 0*n_



Division


Classic


6/2 = 3


Isomorphic


6/(_,_) = 3 = 6/2_


Where:


Classic (6): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(3,3): The numerical quantity (6) subtracted equally into all dimensional unit quantities.


(3): The numerical quantity (6) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


6/(_,_) = 3


Classic


1/4 = .25


Isomorphic


1/(_,_,_,_) = .25 = 1/4_


Where:


Classic (1): Is the numerical quantity.


Classic (4): Is the dimensional unit quantity.


(_,_,_,_): The dimensional unit quantity of the number (4).


(.25,.25,.25,.25): The numerical quantity (1) subtracted equally into all dimensional unit quantities.


(.25): The numerical quantity (1) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Relatively,


(1/4_,1/4_,1/4_,1/4_): The numerical quantity (1) subtracted equally into the dimensional unit quantity of (4_).


(1/4_): The numerical quantity (1), subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


1/(_,_,_,_) = .25 = 1/4_



Therefore, the interchange of fractions, and decimals, remains unchanged.


*All fractions exist as a binary operation of division. Therefore, all decimals exist as a binary operation of division. Therefore, (.n), or (.n_) does not exist. Only (.n), or (n/n_) may exist. *


Classic


0/2 = 0


Isomorphic


0/(_,_) = 0 = 0/2_


Where:


Classic (0): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(0,0): The numerical quantity (0) subtracted equally into all dimensional unit quantities.


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


0/(_,_) = 0


Classic


2/0 = undefined


Novel


2/(_) = 2 = 2/0_


Where:


Classic (2): Is the numerical quantity.


Classic (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(2): The numerical quantity (2) is subtracted equally into all dimensional unit quantities.


(2): The numerical quantity (2) is subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


2/(_) = 2


Classic


0/0 = undefined


Novel


0/(_) = 0 = 0/0_


Where:


Classic numerator (0): Is the numerical quantity.


Classic denominator (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities.


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


0/(_) = 0


Isomorphic, and novel expressions containing variables.


Where (n) =/= 0


n/(0_)= n


n/(_) = n


0/(n_) = 0


Therefore, division by zero is expressible as a quotient. By definition of division, the numerical quantity (0) can never exist as a divisor. Only the dimensional unit quantity of the number (0): or (_), or (0_) may exist as a divisor.


Therefore, all division is defined as a specific operation of a given numerical quantity. Into a given dimensional unit quantity. So that division by zero, is defined as a given numerical quantity operated into the dimensional unit quantity of the number (0).


In keeping with the assertion that the entirety of Numerus "Numerans Numeratus" is isomorphic, except regarding multiplication, and division by zero. The current definition of the negative symbol is adhered to.


Negative is: the opposite of.


Therefore, the negative symbol represents a characteristic of the following given quantity.


Let the given characteristic of the negative symbol, for the numerical quantity exist without change.


Let the numerical quantity of the number 0: exist without an opposite numerical quantity.


Let the dimensional unit quantity of the number (0), or (0_): exist without an opposite.


Therefore:


-1: the opposite of the numerical quantity 1, remains unchanged.


Allow the characteristic of the negative symbol for the dimensional unit quantity to exist as: “place the opposite of”.


Therefore:


-(_): Is defined as “place the opposite of” a given numerical quantity into the dimensional unit quantity of (_).


Classic


-2 * 3 = -6


Isomorphic


-(_,_) * 3 = -6 = -2_ * 3


3: The numerical quantity of the number 3.


-3: The opposite of the numerical quantity given.


-(_,_): The dimensional unit quantity of the number 2. Indicating that the opposite of the given numerical quantity is to be placed into it.


(-3,-3): The opposite of the given numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2.


(-3+-3 = -6): The opposite of the given numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2. Then all numerical quantities, in all dimensional unit quantities are added.


Classic


-2 * -3 = 6


Isomorphic


-(_,_) * -3 = 6 = -2_ * -3


Or relatively:


-(_,_,_) * -2 = 6 = -3_ * -2


-(_,_): The dimensional unit quantity of the number 2. Indicating that the opposite of the given numerical quantity is to be placed into it.


-3: The numerical quantity given.


3: The opposite of the numerical quantity given.


(3+3 = 6): The opposite of the opposite, of the numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2. Then all numerical quantities, in all dimensional unit quantities are added.


Or relatively:


-(_,_,_): The dimensional unit quantity of the number 3. Indicating that the opposite of the given numerical quantity is to be placed into it.


-2: The numerical quantity given.


2: The opposite of the numerical quantity given.


(2+2+2 = 6): The opposite of the opposite, of the numerical quantity of the number 2. Placed into the dimensional unit quantity of the number 3. Then all numerical quantities, in all dimensional unit quantities are added.


Assertion:


The defining of abstract numbers, and the operations of multiplication, and division as given above. Will allow for a mathematical construct, in which it is possible to define division by zero. It will also do so in such a manner, as to not contradict any given field axiom.


*As all operations of addition, and subtraction exist without change. Only the field axioms regarding multiplication will be addressed. *



Field Axioms


Associative: (ab)c = a(bc)

Commutative: ab = ba

Distributive: (a+b)c = ac+bc

Identity: a*1 = a = 1*a

Inverses: a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


For the field axioms to hold, the defining of special operations, for binary multiplication of the number (0), on the number (n) must be considered. In these special cases alone, binary expressions of multiplication may exist without a unique numerical quantity, and a unique dimensional unit quantity.


Allow that: (0*0 = 0)


As the numerical quantity of the number (0), can be added to the numerical quantity of the number (0): But cannot yield a product containing a dimensional unit quantity.


Allow that: (0_*0_ = 0_)


As the dimensional unit quantity of the number (0), can be added to the dimensional unit quantity of the number (0): But cannot yield a product containing a numerical quantity.


Where any number (0) exists as undefined in a binary expression of multiplication:


(0*0=0): (0*0_=0): (0*0=0)


Where (n) =/= (0): and (0) exists as undefined in a binary expression of multiplication:


(n*0)=(n*0)=(n_*0)=0


Associative


(ab)c = a(bc)


Isomorphic/Novel equations.


(a*b)c = a(b*c)


Let: a = 1, b = 2, c = 0: 0 (is a numerical quantity for use in all binary expressions)

(1_*2)0 = 1(2_*0)

2_*0 = 1*0

0 = 1_*0

0 = 0


Let: a = 1, b = 2, c = 0: 0_ (is a dimensional quantity for use in all binary expressions)

(1_*2)0 = 1(2*0_)

2*0_ = 1*2_

2 = 2


Continued isomorphic, and novel examples of the associative axiom.


Let: a = 1, b = 0: 0, c = 0: 0

(1_*0)0 = 1(0*0)

0*0 = 1_*0

0 = 1_*0

0 = 0


Let: a = 1, b = 0: 0_, c = 0: 0_

(1*0_)0 = 1(0_*0_)

1*0_ = 1*0_

1 = 1


Let: a = 1, b = 0: 0, c = 0: 0_

(1_*0)0 = 1(0*0_)

0*0_ = 1*0

0 = 1_*0

0 = 0


Let: a = 1, b = 0: 0_, c = 0: 0

(1*0_)0 = 1(0_*0)

1_*0 = 1*0

0 = 1_*0

0 = 0


Therefore, the associative axiom still holds as true.


Commutative


a*b = b*a


Isomorphic equations.


a*b = b*a


Let: a = 2: 2, b = 3: 3_

2*(_,_,_) = (_,_,_)*2

2*3_ = 3_*2

6 = 6


Let: a = 2: 2_, b = 3: 3

3*(_,_) = (_,_)*3

3*2_ = 2_*3

6 = 6


Continued isomorphic, and novel examples of the commutative axiom.


If (a) = 0: 0

0*b_ = b_*0

0 = 0


If (a) = 0: 0_

0_*b = b*0_

b = b


If (b) = 0: 0

a_ *0 = 0*a_

0 = 0


If (b) = 0: 0_

a*0_ = 0_*a

a = a


Therefore, the commutative axiom still holds true.


Distributive


(a+b)c = a*c+b*c


Isomorphic/Novel equations.


*No classic number (n), in any binary expression of addition, or subtraction may exist as anything other than (n). *

*In all binary expressions containing the number (0): (0) dictated as (0_), or as (0) dictates the other binary number (n): as (n), or (n_) in the binary operation. *

*The “order of operations” is strictly adhered to. Therefore, in all cases of binary parenthesis containing addition, or subtraction. The number (0), cannot be used as (0), or (0_). *


(a+b)c = a*c+b*c


Let: a = 1, b = 2, c = 0: 0

(1+2)0 = 1*0+2*0

3*0 = 1_*0+2_*0

3_*0 = 0+0

0 = 0


Let: a = 1, b = 2, c = 0: 0_

(1+2)0_ = 1*0_+2*0_

3*0_ = 1*0_+2*0_

3*0_ = 1+2

3 = 3


Continued isomorphic, and novel examples of the distributive axiom.


Let: a = n, b = 0, c = 0: 0

(n+0)0 = n*0+0*0

n*0 = n_*0+0

n_*0 = 0+0

0 = 0


Let: a = n, b = 0, c = 0: 0_

(n+0)0_ = n*0_+0*0_

n*0_ = n+0

n*0_ = n

n = n


Let a = n, b = 0, c = 0

(n+0)0 = n*0+0*0

n*0 = n_*0+0

n_*0 = 0+0



0 = 0


Therefore, the distributive axiom still holds as true.


Identity


a*1 = a = 1*a


Isomorphic


a*1 = a = 1*a


For the identity axiom to hold: (a) =/= (0)


Where (a) = 0: the operations of (0) by the multiplicative identity (1) is given previously in the text.


Where (a) =/= 0: All binary expressions not involving zero exist without change.


Therefore, except regarding the number (0), the identity axiom still holds as true.


Inverses


a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


Isomorphic


a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


As all binary expressions not involving zero exist without change, the inverse axiom holds as true.

Where (a) = 0: the number (0) remains without a multiplicative inverse.


The dimensional unit quantity of the number (0): (_), or (0_), cannot be considered the multiplicative inverse of the number (1). By definition the multiplicative inverse of the number (1) must be a numerical quantity. Therefore, the numerical quantity (1) remains the only multiplicative inverse for the number (1).


Therefore, all field axioms continue to exist as true.


Examples as to the validity for the necessity of Numerus “Numerans Numeratus”.


1. Provides for a mathematical construct in which it is possible to define division by zero.


2. As division by zero is defined, any slope formula expressing division by zero is definable. Therefore, the slope of a formula expressing division by zero can be expressed as “vertical”.


3. Allows for division by zero in a field, without contradicting the field axioms.


4. Allows dimensional analysis to define division by zero with “actual concrete numbers”, within the confines of its own system. The possibility of which was previously unexplored, the application of which is applicable to physics.


5. Therefore physics, semantics, philosophy, and mathematics, can be considered to be unified to an extent. As all abstract numbers have been shown to exist, and function, exactly as concrete numbers. Therefore, the unification of abstract, and concrete principles. Both in mathematics, and in physics.

 

[conway]

If it helps to garner interest in this post...

I have inputted this thesis into chat.gpt (A.I.). I specifically asked two questions.

1. Are there any mathematical flaws, or errors in this mathematical construct? It answered NO.
2. Are there any contradictions in the field axioms in this mathematical construct? It answered NO.

This is repeatable for anyone who wishes to copy, and paste it into the chat forum.

*note that while I changed the properties of numbers, including zero. I did not add to, or change the field axioms.

 

[conway]

https://www.physicsforums.com/threads/numerus-numerans-numeratus-part-2.1039833/

There are some interesting points made my other people at the bottom of this thread.

 

[e.jane.aran]

If it helps to garner interest in this post... I have inputted this thesis into chat.gpt (A.I.).

ChatGPT is notoriously bad at even simple arithmetical exercises. Using this to "prove" something is actually fairly well guaranteed to reduce interest.

There are some interesting points made my other people at the bottom of this thread.

Not actually:

I don't see much point in keeping this thread open.

Additionally I have deleted all comments in this thread since they do not add any value.

 

[conway]

I was not aware about gpt, being so horrible at basic math. Thank you. If you had read past the closed thread post, you would see 5 points. I know you have no interest in this. I accept that.

I thank you for your time anyways.

 

[conway]

https://mathhelpforum.com/t/numerus-“numerans-numeratus”.280536/

 

[TheBox]

Numerus “Numerans Numeratus”



The Unified Number





It is the inherent nature of all things, that they are a compilation of two different, and distinct things. It is axiomatic that these two things are space, and value. The value of any given thing being what it is, while the space is what it occupies.


It is true that abstract, or otherwise, numbers are a thing. Therefore, they must also contain a compilation of space, and value. It is an axiomatic truth, that space is the labeling of quantities of dimensions. It is an axiomatic truth, that value is the labeling of quantities of existence, other than dimensions.



Let all abstract numbers be defined exactly as concrete numbers.


Concrete number: A numerical quantity with a corresponding unit.


Let the corresponding unit of all abstract numbers, exist as an abstract dimension notated with the use of (_).


Let the length, and width of all dimensional units remain abstract, and undeclared.


Let the dimensional unit be equal in quantity, to the numerical quantity it corresponds to.


Let all numerical quantities inhabit their corresponding abstract dimensional units.


Let zero be assigned a single dimensional unit.


Classic Novel

0 = (0) = (0,_) = (0,0_)

1 = (1) = (1,_) = (1,1_)

2 = (2) = (2,_,_) = (2,2_)

3 = (3) = (3,_,_,_) = (3,3_)

(-1) = (-1) = (-1,-(_)) = (-1,-(1_))

(-2) = (-2) = (-2,-(_,_)) = (-2,-(2_))

(-3) = (-3) = (-3,-(_,_,_)) = (-3,-(3_))


Therefore:


Any classic number (n) = novel number (n) = (n,n_)


Where (_) is defined as a dimensional unit, separate from the numerical quantity.


Where (n) is defined as the numerical quantity separate from the dimensional unit.


Where (n_) is defined as the dimensional unit, separate from the numerical quantity, and equal in quantity to the numerical quantity it corresponds to, or (n).


Where (n) is defined as the numerical quantity, not separate from, but inhabiting the dimensional unit quantity.


Let addition, and subtraction exist without change. Except regarding notation: (a+b = c: a+0 = a: a-0 = a: 0+0 = 0: 0-0 = 0).


In any binary expression of multiplication, let one number (n) represent only a numerical quantity or (n). Let the other number (n) represent only a quantity of dimensional unit, equal in quantity to the number it corresponds to, or (n_). Therefore, in all cases of binary multiplication (n*n): (n) is notated as (n*n_)=(n_*n).


In any binary expression of division, let the numerator (n), always exist as a numerical quantity or (n). Let the denominator (n) always exist as a dimensional unit quantity, equal in quantity to the number it corresponds to, or (n_). Therefore, in all cases of binary division (n/n): (n) is notated as (n/n_).


Let multiplication be defined as the placing of a given numerical quantity, with addition, equally into each given quantity of dimensional unit. Then all numerical quantities, in all dimensional units are added.


Let division be defined as the placing of a given numerical quantity, with subtraction, equally into each given quantity of dimensional unit. Then all numerical quantities, in all dimensional units are subtracted except one.


In all binary operations of multiplication containing the number (0), and a non-zero number (n). The notation of the number (0), as (0), or as (0_), will dictate the notations of the binary non-zero number (n) in the operation.


In all cases of a binary expression where the notation is not given for the number (0), the numerical quantity (0) is notated for (0), and the dimensional quantity (n_) is notated for (n).


Therefore: (n*0 = n_*0 = 0*n_ = 0).


Let exponents, and logarithms reflect these rules of multiplication, and division.


Assertion:


All binary operations of multiplication, and division remain unchanged. Except binary operations involving the number (0). As well as defining division by the number (0), as an operation of a given numerical quantity (n), into the dimensional unit quantity of (0_). Therefore, this novel system is isomorphic to the classical system. Except for regarding multiplication, and division by zero.



Multiplication


Classic


2*3 = 6


Isomorphic


2*(_,_,_) = 6 = 2*3_


Where:


Classic (2): Is the numerical quantity.


Classic (3): Is the dimensional unit quantity.


(_,_,_): The dimensional unit quantity of the number (3).


(2,2,2): The numerical quantity (2) added equally into all dimensional unit quantities.


(2+2+2 = 6): The numerical quantity (2) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


2*(_,_,_) = 6


Or,


3*(_,_) = 6 = 3*2_


Where:


Classic (2): Is the dimensional unit quantity.


Classic (3): Is the numerical quantity.


(_,_): The dimensional unit quantity of the number (2).


(3,3): The numerical quantity (3) added equally into all dimensional unit quantities.


(3+3 = 6): The numerical quantity (3) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


3*(_,_) = 6


Classic


2*0 = 0


Novel


2*(_) = 2 = 2*0_


Where:


Classic (2): Is the numerical quantity.


Classic (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(2): The numerical quantity (2) added equally into all dimensional unit quantities.


(2): The numerical quantity (2) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


2*(_) = 2


Or,


0*(_,_) = 0 = 0*2_


Where:


Classic (0): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(0,0): The numerical quantity (0) added equally into all dimensional unit quantities.


(0+0 = 0): The numerical quantity (0) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


0*(_,_) = 0


Classic


0*0 = 0


Isomorphic


0*(_) = 0 = 0*0_


(_): The dimensional unit quantity of the number (0).


(0): The numerical quantity of (0) added equally into all dimensional unit quantities.


(0): The numerical quantity of (0) added equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are added.


Therefore:


0*(_) = 0


Therefore, the product of binary multiplication by the number (0), with a non-zero number. Is relative to the number (0), declared as a numerical quantity, or as a dimensional unit quantity. In the binary expression.


Isomorphic expressions containing variables.


Where: (n) =/= 0


n*(0_) = n = (0_)*n


n*(_) = n = (_)*n


n_*0 = 0 = 0*n_



Division


Classic


6/2 = 3


Isomorphic


6/(_,_) = 3 = 6/2_


Where:


Classic (6): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(3,3): The numerical quantity (6) subtracted equally into all dimensional unit quantities.


(3): The numerical quantity (6) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


6/(_,_) = 3


Classic


1/4 = .25


Isomorphic


1/(_,_,_,_) = .25 = 1/4_


Where:


Classic (1): Is the numerical quantity.


Classic (4): Is the dimensional unit quantity.


(_,_,_,_): The dimensional unit quantity of the number (4).


(.25,.25,.25,.25): The numerical quantity (1) subtracted equally into all dimensional unit quantities.


(.25): The numerical quantity (1) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Relatively,


(1/4_,1/4_,1/4_,1/4_): The numerical quantity (1) subtracted equally into the dimensional unit quantity of (4_).


(1/4_): The numerical quantity (1), subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


1/(_,_,_,_) = .25 = 1/4_



Therefore, the interchange of fractions, and decimals, remains unchanged.


*All fractions exist as a binary operation of division. Therefore, all decimals exist as a binary operation of division. Therefore, (.n), or (.n_) does not exist. Only (.n), or (n/n_) may exist. *


Classic


0/2 = 0


Isomorphic


0/(_,_) = 0 = 0/2_


Where:


Classic (0): Is the numerical quantity.


Classic (2): Is the dimensional unit quantity.


(_,_): The dimensional unit quantity of the number (2).


(0,0): The numerical quantity (0) subtracted equally into all dimensional unit quantities.


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


0/(_,_) = 0


Classic


2/0 = undefined


Novel


2/(_) = 2 = 2/0_


Where:


Classic (2): Is the numerical quantity.


Classic (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(2): The numerical quantity (2) is subtracted equally into all dimensional unit quantities.


(2): The numerical quantity (2) is subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


2/(_) = 2


Classic


0/0 = undefined


Novel


0/(_) = 0 = 0/0_


Where:


Classic numerator (0): Is the numerical quantity.


Classic denominator (0): Is the dimensional unit quantity.


(_): The dimensional unit quantity of the number (0).


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities.


(0): The numerical quantity (0) subtracted equally into all dimensional unit quantities. Then all numerical quantities, in all dimensional unit quantities are subtracted except one.


Therefore:


0/(_) = 0


Isomorphic, and novel expressions containing variables.


Where (n) =/= 0


n/(0_)= n


n/(_) = n


0/(n_) = 0


Therefore, division by zero is expressible as a quotient. By definition of division, the numerical quantity (0) can never exist as a divisor. Only the dimensional unit quantity of the number (0): or (_), or (0_) may exist as a divisor.


Therefore, all division is defined as a specific operation of a given numerical quantity. Into a given dimensional unit quantity. So that division by zero, is defined as a given numerical quantity operated into the dimensional unit quantity of the number (0).


In keeping with the assertion that the entirety of Numerus "Numerans Numeratus" is isomorphic, except regarding multiplication, and division by zero. The current definition of the negative symbol is adhered to.


Negative is: the opposite of.


Therefore, the negative symbol represents a characteristic of the following given quantity.


Let the given characteristic of the negative symbol, for the numerical quantity exist without change.


Let the numerical quantity of the number 0: exist without an opposite numerical quantity.


Let the dimensional unit quantity of the number (0), or (0_): exist without an opposite.


Therefore:


-1: the opposite of the numerical quantity 1, remains unchanged.


Allow the characteristic of the negative symbol for the dimensional unit quantity to exist as: “place the opposite of”.


Therefore:


-(_): Is defined as “place the opposite of” a given numerical quantity into the dimensional unit quantity of (_).


Classic


-2 * 3 = -6


Isomorphic


-(_,_) * 3 = -6 = -2_ * 3


3: The numerical quantity of the number 3.


-3: The opposite of the numerical quantity given.


-(_,_): The dimensional unit quantity of the number 2. Indicating that the opposite of the given numerical quantity is to be placed into it.


(-3,-3): The opposite of the given numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2.


(-3+-3 = -6): The opposite of the given numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2. Then all numerical quantities, in all dimensional unit quantities are added.


Classic


-2 * -3 = 6


Isomorphic


-(_,_) * -3 = 6 = -2_ * -3


Or relatively:


-(_,_,_) * -2 = 6 = -3_ * -2


-(_,_): The dimensional unit quantity of the number 2. Indicating that the opposite of the given numerical quantity is to be placed into it.


-3: The numerical quantity given.


3: The opposite of the numerical quantity given.


(3+3 = 6): The opposite of the opposite, of the numerical quantity of the number 3. Placed into the dimensional unit quantity of the number 2. Then all numerical quantities, in all dimensional unit quantities are added.


Or relatively:


-(_,_,_): The dimensional unit quantity of the number 3. Indicating that the opposite of the given numerical quantity is to be placed into it.


-2: The numerical quantity given.


2: The opposite of the numerical quantity given.


(2+2+2 = 6): The opposite of the opposite, of the numerical quantity of the number 2. Placed into the dimensional unit quantity of the number 3. Then all numerical quantities, in all dimensional unit quantities are added.


Assertion:


The defining of abstract numbers, and the operations of multiplication, and division as given above. Will allow for a mathematical construct, in which it is possible to define division by zero. It will also do so in such a manner, as to not contradict any given field axiom.


*As all operations of addition, and subtraction exist without change. Only the field axioms regarding multiplication will be addressed. *



Field Axioms


Associative: (ab)c = a(bc)

Commutative: ab = ba

Distributive: (a+b)c = ac+bc

Identity: a*1 = a = 1*a

Inverses: a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


For the field axioms to hold, the defining of special operations, for binary multiplication of the number (0), on the number (n) must be considered. In these special cases alone, binary expressions of multiplication may exist without a unique numerical quantity, and a unique dimensional unit quantity.


Allow that: (0*0 = 0)


As the numerical quantity of the number (0), can be added to the numerical quantity of the number (0): But cannot yield a product containing a dimensional unit quantity.


Allow that: (0_*0_ = 0_)


As the dimensional unit quantity of the number (0), can be added to the dimensional unit quantity of the number (0): But cannot yield a product containing a numerical quantity.


Where any number (0) exists as undefined in a binary expression of multiplication:


(0*0=0): (0*0_=0): (0*0=0)


Where (n) =/= (0): and (0) exists as undefined in a binary expression of multiplication:


(n*0)=(n*0)=(n_*0)=0


Associative


(ab)c = a(bc)


Isomorphic/Novel equations.


(a*b)c = a(b*c)


Let: a = 1, b = 2, c = 0: 0 (is a numerical quantity for use in all binary expressions)

(1_*2)0 = 1(2_*0)

2_*0 = 1*0

0 = 1_*0

0 = 0


Let: a = 1, b = 2, c = 0: 0_ (is a dimensional quantity for use in all binary expressions)

(1_*2)0 = 1(2*0_)

2*0_ = 1*2_

2 = 2


Continued isomorphic, and novel examples of the associative axiom.


Let: a = 1, b = 0: 0, c = 0: 0

(1_*0)0 = 1(0*0)

0*0 = 1_*0

0 = 1_*0

0 = 0


Let: a = 1, b = 0: 0_, c = 0: 0_

(1*0_)0 = 1(0_*0_)

1*0_ = 1*0_

1 = 1


Let: a = 1, b = 0: 0, c = 0: 0_

(1_*0)0 = 1(0*0_)

0*0_ = 1*0

0 = 1_*0

0 = 0


Let: a = 1, b = 0: 0_, c = 0: 0

(1*0_)0 = 1(0_*0)

1_*0 = 1*0

0 = 1_*0

0 = 0


Therefore, the associative axiom still holds as true.


Commutative


a*b = b*a


Isomorphic equations.


a*b = b*a


Let: a = 2: 2, b = 3: 3_

2*(_,_,_) = (_,_,_)*2

2*3_ = 3_*2

6 = 6


Let: a = 2: 2_, b = 3: 3

3*(_,_) = (_,_)*3

3*2_ = 2_*3

6 = 6


Continued isomorphic, and novel examples of the commutative axiom.


If (a) = 0: 0

0*b_ = b_*0

0 = 0


If (a) = 0: 0_

0_*b = b*0_

b = b


If (b) = 0: 0

a_ *0 = 0*a_

0 = 0


If (b) = 0: 0_

a*0_ = 0_*a

a = a


Therefore, the commutative axiom still holds true.


Distributive


(a+b)c = a*c+b*c


Isomorphic/Novel equations.


*No classic number (n), in any binary expression of addition, or subtraction may exist as anything other than (n). *

*In all binary expressions containing the number (0): (0) dictated as (0_), or as (0) dictates the other binary number (n): as (n), or (n_) in the binary operation. *

*The “order of operations” is strictly adhered to. Therefore, in all cases of binary parenthesis containing addition, or subtraction. The number (0), cannot be used as (0), or (0_). *


(a+b)c = a*c+b*c


Let: a = 1, b = 2, c = 0: 0

(1+2)0 = 1*0+2*0

3*0 = 1_*0+2_*0

3_*0 = 0+0

0 = 0


Let: a = 1, b = 2, c = 0: 0_

(1+2)0_ = 1*0_+2*0_

3*0_ = 1*0_+2*0_

3*0_ = 1+2

3 = 3


Continued isomorphic, and novel examples of the distributive axiom.


Let: a = n, b = 0, c = 0: 0

(n+0)0 = n*0+0*0

n*0 = n_*0+0

n_*0 = 0+0

0 = 0


Let: a = n, b = 0, c = 0: 0_

(n+0)0_ = n*0_+0*0_

n*0_ = n+0

n*0_ = n

n = n


Let a = n, b = 0, c = 0

(n+0)0 = n*0+0*0

n*0 = n_*0+0

n_*0 = 0+0



0 = 0


Therefore, the distributive axiom still holds as true.


Identity


a*1 = a = 1*a


Isomorphic


a*1 = a = 1*a


For the identity axiom to hold: (a) =/= (0)


Where (a) = 0: the operations of (0) by the multiplicative identity (1) is given previously in the text.


Where (a) =/= 0: All binary expressions not involving zero exist without change.


Therefore, except regarding the number (0), the identity axiom still holds as true.


Inverses


a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


Isomorphic


a*a^(-1) = 1 = a^(-1) * a: if a =/= 0


As all binary expressions not involving zero exist without change, the inverse axiom holds as true.

Where (a) = 0: the number (0) remains without a multiplicative inverse.


The dimensional unit quantity of the number (0): (_), or (0_), cannot be considered the multiplicative inverse of the number (1). By definition the multiplicative inverse of the number (1) must be a numerical quantity. Therefore, the numerical quantity (1) remains the only multiplicative inverse for the number (1).


Therefore, all field axioms continue to exist as true.


Examples as to the validity for the necessity of Numerus “Numerans Numeratus”.


1. Provides for a mathematical construct in which it is possible to define division by zero.


2. As division by zero is defined, any slope formula expressing division by zero is definable. Therefore, the slope of a formula expressing division by zero can be expressed as “vertical”.


3. Allows for division by zero in a field, without contradicting the field axioms.


4. Allows dimensional analysis to define division by zero with “actual concrete numbers”, within the confines of its own system. The possibility of which was previously unexplored, the application of which is applicable to physics.


5. Therefore physics, semantics, philosophy, and mathematics, can be considered to be unified to an extent. As all abstract numbers have been shown to exist, and function, exactly as concrete numbers. Therefore, the unification of abstract, and concrete principles. Both in mathematics, and in physics.

An interesting thread that I will have to read over several times to comprehend fully . I am more a physicist than a mathematician !

I can explain to you a few things that may help you in your exploration of math .

Max Planck conceived the smallest possible measure known as the Planck length , 1.616255(18)×10−35 m . However , Planck did this incorrectly because he reduced a measure rather than extending 0 .

Planck could of simply took a point of space that has zero dimensions (x0,y0,z0,) and then displaced this point by another point creating two adjoined points of (x0,y0,z0,)+(x0,y0,z0,)=(x1,y0,z0,) which is the real smallest possible scalar and vector measure available .

The smallest possible area consists of four points , (x1,y1,z0,) and this measure demonstrates that Einstein's famous thought experiment of a light clock is incorrect because (x1,)=(y1,)=(xy1,) . Demonstrating there is no time dilation on the angular ascent or descent of the lights traversing path, being no spacing between points .

The smallest possible three dimensional volume consists of eight points , (x1,y1,z1,) but in this instant 1*1*1 does not equal 1 , it equals eight adjoined points forming the smallest conceivable 3 dimensional measure . This is important to consider when discussing zero point energy because (x1,y1,z1,)=E8 where E represents energy and 8 represents the amount of zero point energy of the volume .


'' It is the inherent nature of all things, that they are a compilation of two different, and distinct things. It is axiomatic that these two things are space, and value. The value of any given thing being what it is, while the space is what it occupies.''
I totally agree with your axiom statement based on some of my own understandings. It is axiomatic that for any thing of value to exist , it requires a space to exist in . For any event to occur , it requires a space to occur in .

Space does not require any thing to exist , it is axiomatic that space cannot be created nor destroyed . We can remove any thing of substance from space but cannot remove the space because space has no physicality .

Finally you mention n which normally represents n-dimensional . I see space in being (x0,y0,y0,............n,) which is n-dimensional space , meaning that in topology any dimensions can fit within and exist within a n-dimensional space . (x0,y0,y0,............n,)=infinite in any direction !

Hope this helps you in your thoughts !

 

[conway]

An interesting thread that I will have to read over several times to comprehend fully . I am more a physicist than a mathematician !

I can explain to you a few things that may help you in your exploration of math .

Max Planck conceived the smallest possible measure known as the Planck length , 1.616255(18)×10−35 m . However , Planck did this incorrectly because he reduced a measure rather than extending 0 .

Planck could of simply took a point of space that has zero dimensions (x0,y0,z0,) and then displaced this point by another point creating two adjoined points of (x0,y0,z0,)+(x0,y0,z0,)=(x1,y0,z0,) which is the real smallest possible scalar and vector measure available .

The smallest possible area consists of four points , (x1,y1,z0,) and this measure demonstrates that Einstein's famous thought experiment of a light clock is incorrect because (x1,)=(y1,)=(xy1,) . Demonstrating there is no time dilation on the angular ascent or descent of the lights traversing path, being no spacing between points .

The smallest possible three dimensional volume consists of eight points , (x1,y1,z1,) but in this instant 1*1*1 does not equal 1 , it equals eight adjoined points forming the smallest conceivable 3 dimensional measure . This is important to consider when discussing zero point energy because (x1,y1,z1,)=E8 where E represents energy and 8 represents the amount of zero point energy of the volume .


'' It is the inherent nature of all things, that they are a compilation of two different, and distinct things. It is axiomatic that these two things are space, and value. The value of any given thing being what it is, while the space is what it occupies.''
I totally agree with your axiom statement based on some of my own understandings. It is axiomatic that for any thing of value to exist , it requires a space to exist in . For any event to occur , it requires a space to occur in .

Space does not require any thing to exist , it is axiomatic that space cannot be created nor destroyed . We can remove any thing of substance from space but cannot remove the space because space has no physicality .

Finally you mention n which normally represents n-dimensional . I see space in being (x0,y0,y0,............n,) which is n-dimensional space , meaning that in topology any dimensions can fit within and exist within a n-dimensional space . (x0,y0,y0,............n,)=infinite in any direction !

Hope this helps you in your thoughts !

Very helpful. There is a lot for me to unpack here. But first I need your thoughts here before I delve into your ideas any further. I myself am a philosopher not a mathematician or physicist. But I am under the belief, and as a consequence of the axioms of this thesis. That No space, or zero space does not exist. That a point on a number line is not dimensionless. It is only that it is to small to define from our perspective. (unless we change that perspective). Consider a black hole. The math leads us to division by zero. But the mass the hole has absorbed, and the space still exist. It is only that they are both to small to define, again only from our perspective. Your thoughts here on the Planck scale, and the non-existence of zero space is immensely desired.

Waring! If you bother to go any further with this thesis you will be considered a "flat-earther" mathematician. You will be chastised, and criticised. It may not be worth your time, relative to your goals.

Thanks again for your time and thoughts!

 

[TheBox]

That No space, or zero space does not exist.

Zero space as you call it may well exist in regions of infinite space where light may not as yet reached those regions . Emptiness is often a term we don't associate with space but this is based on our localised understanding of relativity .
Black holes should rightful so involve spherical math as black holes aren't exactly a hole , they are isotropic in nature affecting the surrounding space in an isotropic manner . You mention division by zero but I don't see it that way , I observe spherical volume times mass or energy in some mathematical way.
The thing with energy is that an infinite amount of energy can occupy a single point of space but it is not inherent that this energy can be conserved indefinitely . Although perhaps a singular point of energy could be conserved indefinitely creating a fabric of space via an inherent conservation property of space that allows a 1:1 ratio of conservation creating zero point energy .
Mass is different , often associated with things that have more solidity . This is where density plays a part , there is only a certain amount of density that can occupy a point of space before it essentially expands in measure into other points of space in growth .

Now when discussing the gravitation properties of a BH , we have to consider that the properties of the BH does not behave in a typical manner like a body of mass attracted to another body of mass . The BH behaves in such a way that it attracts light and mass . Light of course being massless in a conventual sense . This information tells me that a BH may have two attractive forces in use at anytime , one that attracts mass and one that attracts light .
However , only the mass can expand the measure of the BH because light has no limits of occupancy of the same space .

I think any sort of BH math must be very complex and perhaps something we do not or cannot understand at this time . Maybe something we will never understand .

 

[conway]

So my understanding was wrong. Yet it is the case then that "zero space" is not the absence of space. So as we have agreed previously...there is no such thing as "no space". Either "zero space" out in the universe where light hasn't reached. Or the space inside a BH. My understanding of BH math is that it "blows up". That is to say it arrives at division by zero. Now if I may reach way way out of a limb here. If we introduce varying types of zeros (sizes). As zero is our perspective point on a number line. We can introduce a mathematical structure where division by zero yields a quotient that is on a number line that has a different perspective, or a zero that is smaller in size than the original zero. Abstractly the same thing as a BH yielding an entire universe on the inside of it, but only infinitely smaller than ours.

 

[TheBox]

So my understanding was wrong. Yet it is the case then that "zero space" is not the absence of space. So as we have agreed previously...there is no such thing as "no space". Either "zero space" out in the universe where light hasn't reached. Or the space inside a BH. My understanding of BH math is that it "blows up". That is to say it arrives at division by zero. Now if I may reach way way out of a limb here. If we introduce varying types of zeros (sizes). As zero is our perspective point on a number line. We can introduce a mathematical structure where division by zero yields a quotient that is on a number line that has a different perspective, or a zero that is smaller in size than the original zero. Abstractly the same thing as a BH yielding an entire universe on the inside of it, but only infinitely smaller than ours.

When discussing space , we can consider 0 is the absence of dimensions (x0,y0,z0,) . We can also consider that 0 is the absence of energy and/or mass . We can never consider that 0 is the absence of space because space has no physicality , nothingness cannot be absence from itself .

When considering 0 is the absence of energy and/or mass , I like to consider in regards to Eigen states and Eigen values . For example a region of space that was absent of energy and/or mass , I would suggest that had an Eigen state of 0 and an Eigen value of zero but this is not to say that region is without dimensions .

The big bang suggests that in the beginning there was nothing but how do we determine what is really meant by nothing . Georges Lemaître in his thoughts removed all energy and mass from the picture but did he really ever mean there was actually no space ?
Nothing could describe an infinite space that had (x0,y0,z0,........n) dimension but with an Eigen state and Eigen value of 0 which in essence is nothing or as I like to put it , space with the absence of things !

Georges Lemaître explains that the Universe started from a hot dense state but surely before a hot dense state could form , there is a must for a low energy sparse state .

The problem is that math does not and cannot cover ever single thing or event that there is . We can attempt this math but most of the time the answers won't even make sense or produce a result value .

For example let us say that there is a point of space (x0,y0,z0,) that has an Eigen state of 1 and an Eigen value of 1 . Let us then say that this single point energy is torn apart and divided by surrounding space . We can write the action has

E1÷ (x0,y0,z0,........n) =? but because we don't have any specific volume , we can never give the miniscule answer .

This is similar to any sort of BH math .

 

[conway]

I think we have to work on our ontology and epistemology, or what we actually mean by the words space, dimensions etc. I do not think there is such a thing as no dimension. It is only that the dimension in question is to small to be measured. Such as a number line. A point is not dimensionless. If it was then it cannot give rise to a dimension, as every point on it is dimensionless. A number line by definition is a dimension. When I use the word space, I just mean dimension. I am saying that there is no such thing as "nothingness". There is only dimension and value. Numerical quantities, and dimensional quantities. It is even understood that a BH is not dimensionless, it is only that the dimension is to infinitely small to make any sense to us. (From our perspective). So I am posing to you that 0 is empty space/dimension. And that therefore if we varied our sizes of these empty dimensions. We would be looking at varying amounts of zero. Zero on a number line is our starting point, or perspective of that number line. So in cases of Bh math and this Eigen math you speak of. We could devise a number system that allows different 0's to change our perspective of the infinitely small numbers, there by hopefully making sense out of them. In any case I am in way over my head here, and really am just blabbing wildly.

 

[TheBox]

I do not think there is such a thing as no dimension.

Have you never seen 0 on a ruler or tape measure ? Of course there is such a thing as no dimension . A spatial point (x0,y0,z0,) is worldly accepted by science etc .

Anyway I can see this conversation isn't really going anywhere so I will wish you good day and thank you for the chat !

 

[conway]

Yes I have. 0 is on every caliper known to man. So to mathematicians claim any given point on a number line is dimensionless. You claim 0 is the absence of dimension.

When discussing space , we can consider 0 is the absence of dimensions (x0,y0,z0,) .

Which is it? The "absence of dimension" exists or it does not. In any case, sorry I frustrated you. Thank you for the conversation. You have been helpful to me.