*** I have attempted to fix the ambiguous definition and sloppy notations ***

*** My apologies for being slow on the up take on figuring out the document transfer methods ***

*Revised* Numerus “Numerans-Numeratus”

Let all abstract numbers be defined exactly as concrete numbers.

Concrete number: A numerical quantity with a corresponding unit.

Let the corresponding unit 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 Isomorphic

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) = isomorphic (__n__) = (n,n_).

Where (_) is defined as a dimensional unit, the quantity of which corresponds to a given 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.

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_).

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 a 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__).

Let exponents and logarithms exist without change. Except regarding notation: (__a__^__b__ = __c__).

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 (0_).

**Multiplication**

Classic

2*3 = 6

Isomorphic

2*(_,_,_) = __6__

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__

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

Isomorphic

2*(_) = __2__

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__

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__

(_): 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 unity 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__

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__

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.

Therefore:

1/(_,_,_,_) = .__25__

Classic

0/2 = 0

Isomorphic

0/(_,_) = __0__

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

Isomorphic

2/(_) = __2__

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

Isomorphic

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 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__).

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__)

Therefore:

(__n__+0 = __n__): (__n__+0_ = __n__): (__n__+__0__ = __n__)

Where (__n__) =/= (__0__): and (__0__) exists as undefined in a binary expression:

(__n__*__0__) = (n*__0__) = (n_*0) = __0__

**Associative**

(ab)c = a(bc)

Isomorphic 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 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 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 equations.

(__a__+__b__)__c__ = __a__*__c__+__b__*__c__

Let: __a__ = __1__, __b__ = __2__, __c__ = __0__: 0

(__1__+__2__)__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__+__2__

__3__ = __3__

Continued isomorphic examples of the distributive axiom.

Let: __a__ = __n__, __b__ = __0__: 0, __c__ = __0__: 0

(__n__+__0__)__0__ = n_*0+0*0

n_*0 = __0__+0

__0__ = __0__

Let: __a__ = __n__, __b__ = __0__: 0_, __c__ = __0__: 0_

(__n__+__0__)__0__ = n*0_+0_*0_

n*0_ = __n__+0_

__n__ = __n__

Let: __a__ = __n__, __b__ = __0__: 0, __c__ = __0__: 0_

(__n__+__0__)__0__ = n*0_+0*0_

n*0_ = __n__+__0__

__n__ = __n__

Let: __a__ = __n__, __b__ = __0__: 0_, __c__ = __0__: 0

(__n__+__0__)__0__ = n_*0+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.