Polysign Numbers

Description

graphic
Polysign numbers are a family of number systems having a natural number of signs.
It includes the real numbers as the reals have two signs.
Two-signed numbers are consistent with the real numbers by design.
However to understand higher sign numbers one must abandon tradition.
This will feel harmful to your perception which is based in real numbers.
Upon reorienting yourself to polysigned numbers perhaps you will look at the real numbers differently.
Three-signed numbers are equivalent to complex numbers.
The polysigned family has natural correspondence to spacetime.
Polysigned numbers are intimately tied to dimensionality.
This comes as a result of superposition and the identity relation.

The Identity Law (Cancellation)

In general a zero sum always has identical component magnitudes in every sign yielding proper cancellation.
For example, in two-signed math (the reals) for any value x:
    - x + x = 0.
In three-signed math (P3) for any value x:
    - x + x * x = 0. (where "*" is a new sign pronounced 'star')
In four-signed math (P4) for any value x:
    - x + x * x # x = 0. (where "#" is a new sign pronounced 'sharp')
The general and elegant form is:
Sum from s = 1 to n ( s x ) = 0 where s is sign and x is either a magnitude or an n-signed value.

Rudimentray numbers can be graphed on a series of rays emanating from an origin, one ray for each sign.
Simply choosing a point on the structure yields a rudimentary value.

   Three-Signed Number Graph
Upon choosing a point on the above structure you will have chosen one sign and one magnitude. Thus the form sx is elemental and it is combinations of such values that we will consider. It is important to understand that when we speak of the sign plus or '+' we do so in one domain only. For instance in P3(the three-signed numbers) the plus sign means something different than in P2. This is also challenging since traditionally the plus sign means superposition. In P3 and beyond the sign mechanics are modified. Every instance of usage of the plus sign as superposition will be noted carefully to avoid such confusion. In general when you see a plus sign it means sign two unless noted otherwise.

The Sum

Summation or superposition is similar to the real numbers.
Quantities of similar sign preserve their sign but increase in magnitude so that:
   - 1 - 1 = - 2 .
   + 4.2 + 5.1 = + 9.3 .
   * 2 * 3 = * 5 .
 
This behavior can be generally stated as

    s1 x1 + s1 x2 = s1 ( x1 + x2 )

Where s is sign and x is magnitude and the operator '+' means superposition.
General sums in three-signed math are more dynamic since the identity law applies:

- 2 * 3 =  + 2 - 4 * 5 .
Above the reduced form is to the left, but the expressions are equivalent.
The right hand side may be rewritten as
( + 2 - 2 * 2 ) - 2 * 3
where the quantity in parenthesis evaluates to zero by the identity law.
This is the same process that occurs in the real numbers where one might write:
+ 2 = - 3 + 5 = ( -3 + 3 ) + 2 .

Polysigned numbers form a nonorthogonal coordinate system.
The three-signed numbers map to the plane:
Three-Signed Geometry

Dimensionality

Any n-signed value can always be reduced to at most n - 1 terms.
This is a direct consequence of the identity law.
In the three-signed example above + 2.3 - 0.5 cannot be further simplified.
Sum from s = 1 to n ( s x ) = 0 As the three-signed numbers map to the plane so the four-signed numbers map to a three-dimensional space.
General sums in two signs are one-dimensional.
Geometrically they form a line.
General sums in three signs are two-dimensional.
Geometrically they form a plane via the triangular coordinate system in the diagram above.
General sums in four signs are three-dimensional.
Geometrically they form a tetrahedral coordinate system.
General sums in n signs are (n-1) dimensional.
Geometrically they form a symmetrical n-1 dimensional n-hedron (simplex) coordinate system.
A Cartesian transform exists based on the symmetry of the system.
A native distance function exists.
Polysign lattices behave differently than the Cartesian lattice.

The Product

Product rules exist and work much like the real numbers.
The sign symbols are useful mnemonics.
Each sign is the number of pen strokes that it takes to draw it.
The smallest sign is - (one), then + (two), then * (three), then # (four), and so on.
When two signs are combined via a product the resulting sign is the wrapped sum of the two source signs.
The count wraps at the highest sign of the system.
The identity sign is always the highest sign of the system.
This rule is logical and coexists with the traditional real (two-signed) numbers.
However the special properties of + do not extend beyond the two-signed domain.
Here are the sign product rules for three-signed math:

( - 1 ) ( - 1 ) = + 1
( - 1 ) ( + 1 ) = * 1
( - 1 ) ( * 1 ) = - 1
( + 1 ) ( +1 ) = - 1
( + 1 ) ( * 1 ) = + 1
( * 1 ) ( * 1 ) = * 1

The resultant sign is simply the modulo 3 sum of the source signs with zero representing the highest sign.
The product is commutative as well so that (+1)(*1) = (*1)(+1) = +1 .
For example in three-signed:
   - (-1) = + 1 .
   + ( + 2 * 3 ) = - 2 + 3 .
   * ( + 2 * 3 )  = + 2 * 3 .
   ( - 3)( + 4 ) = * 12 .
   ( + 1.2 )( * 2.2 ) = + 2.64 .
General products conform to the distributive law.
Because the * sign is the identity sign in three-signed it is used for summation math as + is used in two-signed:
   ( - 2 + 3 )( + 1 * 4 ) = (- 2)(+ 1) * (- 2)(* 4) * (+ 3)(+ 1) * (+ 3)(* 4)
    = * 2 - 8 - 3 + 12
    = - 9 + 10 .
The general three-signed product
( - a + b * c )( - d + e * f )
where a,b,c,d,e, and f are arbitrary magnitudes is equivalent to
+ ad * ae - af
* bd - be + bf
- cd + ce * cf .
This is just the expansion using the distributive property and using the sign rules layed out above.
It is proven that three-signed numbers are equivalent to the complex numbers.
The elegant form of the product rule is
   ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2
where s is sign and x is magnitude. The sum ( s1 + s2 ) is a modulo sum in the signature of the components.

The Zero Sign

The need for a zero sign equivalent to the identity sign is exposed in the product logic. The symbol '@' is the zero sign though its use is thus far avoided. Rather than needing to focus on which sign is the identity sign we can always treat
   z1 @ z2
as the superposition of z1 and z2 and the product rules then conform to straight modulo math. The strict usage of the zero sign is not necessary as evidenced by the consistency with the real numbers (P2) where
   @ 1 = + 1 .
To force the usage of the zero sign will break congruence with existing math by deleting the plus sign for the real numbers. Arguably it is good habit to use the zero sign though the extension might be difficult for beginners.
Using the @ symbol the basic operators of the polysign can be rewritten as:

  s1 x1 @ s1 x2 = s1( x1 @ x2 )

  (s1 x1)(s2 x2) = (s1 @ s2) x1 x2    (modulo n sum)
Sum from s = 1 to n ( s x ) = 0

Magnitude Property

One property of products on real numbers and complex numbers is:
| A B | = | A | | B | .
This property only holds on polysigned numbers up to three signs.
Beyond three signs this property breaks.
A random analysis of magnitude shows an interesting pattern.
In four-signed numbers the product behavior can be depicted graphically.
The product and sum rules still work for the higher signs as evidenced in the Mandelbrot study of  polysigned numbers.
Magnitude can also be leveraged as a basis for analysis and generating dimensional data with magnitude sweep functions.

So What?

Physicists generally accept space-time because it is observed.
Few trouble over why.
String theorists have added curled dimensions but maintain three 'extended' dimensions manually.
If a theoretical basis can be created that naturally generates space-time then it is likely that such a basis will apply to physics.
The polysigned numbers form such a basis.
They might be the root for why we observe three spacial
dimensions and time as the physical basis.
The topolgy would be:
P1 x P2 x P3 ...
where P1 is one-signed numbers, P2 two-signed and P3 three-signed.
P1 matches time.
P2 is a one-dimensional line.
P3 is a two-dimensional plane.
Altogether they are a sufficient representation of spacetime.
Because the magnitude behavior of the product is variant beyond P3 there is a natural break in the progression.
Even if the progression continues the support for spacetime remains.
This suggests that the product plays an important role in physics.
Currently this discussion ends with only philosophical value.
The polysigned numbers await practical application.
Mathematically they are primitive yet have interesting behaviors.
Thank you for spending some time on the polysign numbers.