A New Number Theory

α β + = ⋅ + ⋅ + ⋅ ≡    j k s u x v y w z re Behaves as a complex number with three dimensions provided that: To do the sums it must always be used the definition (1) [Cartesian notation]. To do the products it must always be used the definition (2) [polar notation]. The definition can be extended to 4 dimensions. The relations between P= ( , , ) x y z and ( , , r α β ) are given by the following formulas:


Definitions
Let us consider the 3d space that can be represented by the tern , , u v w    as shown in the Figure 1 The point P can be written as: P= ( , , ) x y z = u x v y w z ⋅ + ⋅ + ⋅    But also, using the polar notation the point P can be written as P= The operator k e β ⋅ raises the lying vectors in the plane , u v   of β radians along w  ; The operator j e α ⋅ rotates the lying vectors in the plane , u v   of α radians.

Definition of the sum:
1) 1  Behaves as a complex number with three dimensions provided that: To do the sums it must always be used the definition (1) [Cartesian notation]. To do the products it must always be used the definition (2) [polar notation]. The definition can be extended to 4 dimensions.
The relations between P= ( , , ) x y z and ( , , r α β ) are given by the following formulas: For 0 α = , the vector is lying in the plane ,   u w , and the polar notation coincides with a vector in the vertical rotation. The y values for 0 α = are 0 by default. The transition from one format to another is always possible, because the tern ( , , ) x y z always and uniquely identifies the tern ( , , ) r α β through the formulas (3). The methods of symbolic computation are identical to those of the standard complex numbers.
The operations, of calculation, must always take into account the two rules (1) and (2) above for the sums and products. To assess the calculator expressions you can use the Reverse Polish Notation (RPN).

Calculus
The question is: does the calculus work? Fixing: The definition above, can't coincides at the infinitesimal to the differential of ( ) Practically the polar notation is useful only to define the concept of the commutative product, necessary for the symbolic operations and for the real calculation of the product between the numbers.
As a last consideration, we can also define

Dimensions
Let us consider s as a 4 dimension number as defined above: Is it still true the analysis above? The answer is yes, but we have to introduce the operator h e λ ⋅ which raises the vectors in the cube , , The unity vector q  is orthogonal to u  , v  and w  , but it can't be represented graphically; the formulas between the two representations, one Cartesian, the other polar, can be detected from an idea given by the Figure 2.
The formulas: Because ' r can't be negative, it is clear that γ must be reduced to

Examples: Calculating the volume of a sphere
We need a little core of the new algebra, see below.
Here a couple of routines (written in Visual Basic) to estimates the volume of a sphere with the algebra above. The more Kloop is high, the more accurate is the estimation.

Dim S As Complex3d
Dim dAlfa As Double, dBeta As Double, dr As Double Dim I As Long, J As Long, K As Long, KLoop As Long Dim Vol As Double, dv As Double