It is the inherent nature of all things thatthey are a compilation of two different and distinct things. It is axiomaticthat these two things are space and value. The value of any given thing beingwhat 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 anaxiomatic truth that space is the labeling of quantities of dimensions. It isan axiomatic truth that value is the labeling of quantities of existence, otherthan dimensions.

It is an axiomatic truth that space and value exist in one oftwo forms. So that any given quantity of space or value is first labeled asdefined or undefined. It is reasonable to say that any given number, that hashad both its quantities of space and value labeled as undefined, requires nofurther question as to its nature. If however a given number, has had both itsquantities of space and value labeled as defined, it is then necessary to furtherdefine the given quantities. That is to say what is the nature of the space andvalue's that are defined.

There are four axiomatic steps in the further defining of adefined quantity of space and value. First it is that, after a given quantityof space and value is labeled as defined, a symbol is given to identify theamount of quantities given. Second it is that the given amounts of definedspace and value are labeled as finite or infinite. Third it is that the givenamounts of defined space and value, that are finite or infinite, are labeled asfractional or whole. Fourth it is that the given amounts of defined space andvalue, that are finite or infinite, fractional or whole, are labeled aspositive or negative.

It is the case that all forms of the defining of quantitiesof space and value, are from the perspective of our humanity. This then showsthat there is a collection of only four kinds of numbers. That is there arenumbers that possess an undefined space and an undefined value. Otherwise representedas a ( Uv + Us ). Such a number not requiring further defining. There arenumbers that possess a defined value and a defined space. Otherwise representedas a ( Dv + Ds ). Such a number requiring further defining. There are numbersthat possess a defined value and an undefined space. Otherwise represented as a( Dv + Us ). There are numbers that possess an undefined value and a definedspace. Otherwise represented as a ( Uv + Ds ).

It is reasonable to say that natural numbers have both theirquantities of space and value labeled as defined. That is that a natural numberis a ( Dv + Ds ). It is then through the process of further defining, that anatural number such as 2 is labeled as having ( 2Dv + 2Ds ). The symbol 2 thenis the symbol identifying the amounts of quantities contained. It is then thatthe given quantities are labeled as finite. Otherwise represented as a ( 2DvF +2DsF ). It is then that the given quantities are labeled as whole. Otherwiserepresented as a ( 2DvFW + 2DsFW ). It is then that a positive is assigned tothe compilation of space and value, and it is so on for any natural number.

It is also the case that fractions are labeled as a ( Dv + Ds). That is any given fraction has both its quantities of space and value labeledas defined. So that such a number as .2 is labeled as ( 2DvFF + 2DsFF ). Then apositive is assigned to the compilation of space and value. Additionally, afractional symbol may replace the decimal symbol.

It is also the case that infinite numbers are labeled as a (Dv + Ds ). So that such a number as 2infinite is defined as a ( 2DvIW + 2DsIW). As well as fractional infinites, such as .2infinite. Which is labeled as (2DvIF + 2DsIF ). Then a positive is assigned to both compilations of space andvalue, and it is so on for any infinite or fractionally infinite number.

Remaining are numbers that are a ( Uv + Ds ) and numbers thatare a ( Dv + Us ). Such numbers do not necessarily require further defining. Asan undefined quantity of space or value composites the given number. So thensuch numbers can only be limitedly defined relative to the given definedquantity. If then a number possess a defined value and an undefined space, thesum is then relative to the defined value. So that such a number as ( Dv + Us )is then a 1 relative. Otherwise represented as a 1r.

If then a number possess an undefined value and a definedspace, the sum is then relative to the defined space. So that such a number asa ( Uv + Ds ) is then a zero. As no quantity of value is defined, and as onequantity of space is defined. The space of zero is clearly defined on anynumber line. The equation ( 1 + (-1) = 0 ) proves this in that, if zero did notoccupy a defined space on the number line, then the equation would equal ( -1), and not zero.

It is the case in multiplication and division, that neithernumber given is an actual number. Not in the fashion that each symbol containsboth space and value. It is that one symbol is representing a value, and thatone symbol is representing a space. It is the case that in multiplication thelabeling of the given symbols as space or value in a specific order is notnecessary. The sum yielded is always the same.

It is the case that in division the labeling of the givensymbols as space or value in a specific order changes the sum that is yielded.So that as an axiom the first given symbol is labeled as value, while thesecond given symbol is labeled as space.

It is then that in multiplication the given value is placedadditionally into the given spaces. Then all values are added in all spaces. Itis then that in division the given value is placed divisionally into all givenspaces. Then all values are subtracted except one.

So that in the equation ( 2 x 0 = X ), there is a givendefined value of ( 2DvFW ), that is placed additionally into the given definedspace of ( Ds ). Then all values are added in all spaces. This process thenyields the number 2.

Whereas the equation ( 0 x 2 = X ), there is a givenundefined value of ( Uv ), that is placed additionally into the defined spaceof ( 2DsFW ). Then all values are added in all spaces. This process then yieldsthe number zero.

So then in the equation ( 2 / 0 = X ), there is a definedvalue of ( 2DvFW ), that is placed divisionally into the defined space of ( Ds). Then all values are subtracted except one. This process then yields thenumber 2.

Whereas the equation ( 0 / 2 = X ), there is an undefinedvalue of ( Uv ), that is placed divisionally into the defined space of ( 2DsFW). Then all values are subtracted except one. This process then yields thenumber zero.

As an addition to all current field axioms these ideas areexpressed as stated.

" For every A in S there exists a Z1 and Z2,constituting A, such that any A in operation of multiplication or division isonly representing Z1 or Z2 in any given equation. Such that Z1 for allA's other than zero equal A. Such that Z2 for all A's other than zeroequal A. Such that Z1 for zero equals zero. Such that Z2 for zeroequals 1. "

It is possible that further defining of the given definedvalue of a relative number, and the given defined space of a zero, isapplicable and necessary. It is possible to either leave the same, oradapt exponents and logarithms. Naturally further axioms will be neededfor adaption. Such as exponents of zero existing as a space representation ofzero (z2). Logarithms of zero existing as a value representation of zero(z1).

It is possible to here-in re-address the idea of thecontinuum theory. If the definitions for numbers and their groups, areadapted as stated, and with further exploration into the defining of ( Dv+ Us ) relative numbers, ( Uv + Ds ) zero numbers, ( Uv + Us ) undefinednumbers, and their placement onto the number line. The idea here being toshow all numbers originating from and returning to ( Uv + Us ) on any givennumber line.