Numerus “Numerans-numeratus”

Let all abstract numbers be defined exactly as concrete numbers.

Concrete number: A numerical quantity with a corresponding dimensional unit.

Let the corresponding dimensional unit be equal in quantity to the numerical quantity it is assigned to.

Let the length and width of each dimensional unit remain abstract and undeclared.

Let zero be assigned a single dimensional unit.

0 = (0,_)

1= (1,_)

2= (2,_,_)

3= (3,_,_,_)

Etc…

Therefore:

Any abstract number (n) = (n,n_).

Where (n_) is defined as a dimensional unit quantity equal in quantity to (n).

Therefore:

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

Let addition and subtraction exist without change.

In any binary expression of multiplication let one number represent only a numerical quantity, let the other number represent only a quantity of dimensional unit equal in quantity to the number it represents.

In any binary expression of division let the numerator always exist as a numerical quantity, let the denominator always exist a dimensional unit equal in quantity to the number it represents.

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 any binary expression involving one or more zero’s, each zero must be declared as a numerical quantity or as the dimensional unit quantity.

In all equations with varying amounts of binary expressions involving zero, the declarations for zero as a numerical quantity or as a dimensional unit (with-in each set of binary expressions), must hold the same in all given expressions in the given equation.

Let exponents and logarithms exist without change.

Assertion:

All binary operations of multiplication and division remain unchanged except with respect to zero.

Multiplication

Classic

2 * 3 = 6

Isomorphic

2 * (_,_,_) = 6

Where:

2 = numerical quantity

3 = dimensional unit quantity

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

(2,2,2): the numerical quantity 2 placed additionally and equally into all dimensional unit quantities.

(2+2+2= 6 ): the numerical quantity 2 placed additionally and equally into all dimensional unit quantities then added.

Therefore:

2 * (_,_,_) = 6

Or,

3 * (_,_) = 6

Where:

2 = dimensional unit quantity

3 = numerical quantity

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

(3,3): the numerical quantity 3 placed additionally and equally into all dimensional unit quantities.

(3+3= 6 ): the numerical quantity 3 placed additionally and 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:

2 = numerical quantity

0 = dimensional unit quantity

(_): the dimensional unit quantity of zero.

(2): the numerical quantity of 2 placed additionally and equally into all dimensional unit quantities.

(2): the numerical quantity of 2 placed additionally and equally into all dimensional unit quantities, then all numerical quantities in all dimensional unit quantities are added.

Therefore:

2 * (_) = 2

Or,

0 * (_,_) = 0

Where:

0 = numerical quantity

2 = dimensional unit quantity

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

(0,0): the numerical quantity of 0 placed additionally and equally into all dimensional unit quantities.

(0+0= 0 ): The numerical quantity of 0 placed additionally and 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 0.

(0): the numerical quantity of 0 placed additionally and equally into all dimensional unit quantities.

(0): the numerical quantity of 0 placed additionally and 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 zero is relative to zero declared as a numerical quantity or as a dimensional unity quantity.

Isomorphic expressions containing variables.

Where: (n) =/= 0

n * (_) = n

n_ * 0 = 0

0 * (_) = 0

Division

Classic

6/2 = 3

Isomorphic

6/(_,_) = 3

Where:

6 = numerical quantity

2 = dimensional unit quantity

(_,_): dimensional unit quantity of 2.

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

(3): all numerical quantities in all dimensional unit quantities are subtracted except one.

Therefore:

6/(_,_) = 3

Classic

Ľ = .25

Isomorphic

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

Where:

1 = numerical quantity

4 = dimensional unit quantity

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

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

(.25): the numerical quantity of 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

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

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

(0): the numerical quantity of 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

(_): the dimensional unit quantity of 0.

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

(2): the numerical quantity of 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

(_): the dimensional unit quantity of 0.

(0): the numerical quantity of 0 is subtracted equally into all dimensional unit quantities.

(0): the numerical quantity of 0 subtracted equally into all dimensional unit quantities, then all numerical quantities in all dimensional unit quantities subtracted except one.

Therefore:

0/(_) = 0

Isomorphic expressions containing variables.

Where (n) =/= 0

n/(_) = n

0/(n_) = 0

0/(_) = 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 zero, (_) may exist as a divisor.

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

Assertion:

The defining of abstract numbers and the operations of multiplication and division as given above will allow for the defining of division by zero. It will also do so in such a manner as to not infringe upon the validity of 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 continue to hold as true, the use of any given zero as a numerical quantity or as a dimensional unit quantity most remain uniform throughout the equation.

Associative

(ab)c = a(bc)

Isomorphic equations.

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

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

Or,

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

Continued isomorphic examples of the associative axiom.

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

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

n_ * 0 = 0 * 0_

0 = 0

Or,

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

n_ * 0 = n_ * 0

0 = 0

Therefore, the associative axiom still holds as true.

Commutative

a * b = b * a

Isomorphic equations.

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

6 = 6

Or,

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

6 = 6

Continued isomorphic examples of the commutative axiom.

If a = 0

0 * b = b * 0

0 = 0

Or,

0_ * b = b * 0_

b = b

If b = 0

a * 0 = 0 * a

0 = 0

Or,

a * 0_ = 0_ * a

a = a

Therefore, the commutativity axiom still holds true.

Distributive

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

Isomorphic equations.

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

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

0 = 0

Or,

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

3 = 3

Continued isomorphic examples of the distributive axiom.

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

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

0 = 0

Or,

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

n = n

Therefore, the distributive axiom still holds as true.

Identity

a * 1 = a = 1 * a

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

Proof for the validity of such a statement found in its identical use in the inverse axiom property.

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

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

Therefore, except regarding zero, the identity axiom of multiplication still exists as true.

Inverses

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

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

Where (a) = 0: the numerical quantity of 0 remains without a multiplicative inverse.

As well as the dimensional unit quantity of 0: (_), or (0_), cannot be considered the multiplicative inverse of the number 1. By definition the multiplicative inverse of 1 must be a numerical quantity. Thus, 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 of division by zero.

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

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

4. Allows dimensional analysis to define division by zero with “actual concrete numbers”, within the confines of its own system. The details of which have been 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.