A New Number Theory-Algebra Analysis II

The basis of this quaternions algebra. The problem of the j k i² i² ⋅ product. 3d (and 4d) product and division in algebraic form; also, the algebraic forms of the product and of the division are differentiable. Questions about the possibility of extend this algebra to more dimensions.

Paper [1] implicitly gave the definitions of the norm (or modulus) of a 3d number, of the inverse of a 3d number, and of the conjugate. x j y k z the scalar product and the vector product are also well defined (see code 3d -2.4g in appendix of paper [2]).
3d scalar product: s s x x y y z z ∧ = ⋅ + ⋅ + ⋅ 3d vector product: in algebraic form: s s y z z y j z x x z k x y y x × = ⋅ − ⋅ + ⋅ ⋅ − ⋅ + ⋅ ⋅ − ⋅   So, we have the same symbolic of the standard 2d complex numbers.
Paper [2] analyzed some aspects of this algebra, we have seen that this algebra is not distributive, and that this produces some limitations in derivatives and integrals, also we have seen the extended definitions of functions such as sin(s) and cos(s) may be meaningless.
The problem is because this 3d space is a curved space, the transformations that permit to define the product as a commutative product, are not linear.
Someone could object that the algebraic definition of the j k ⋅   product, in paper [1], is undefined (in polar notation is defined and it is differentiable); in 1843 William Rowan Hamilton has defined the j k ⋅ product in an algebraic form but, with that definition, Hamilton created a non-commutative algebra.
I try now to give an answer about the generic algebraic definition of the product between two 3d numbers as defined in paper [1].
Given: For a generic s (3d) number we can write: now, we need to analyze 4 cases: c z z c r r by substituting: so in this case: The generic case (1) of the algebraic product of a b s s ⋅ is differentiable. The other cases are limit case and are differentiable too.
The algebraic form of s 1 is quite simple: given: it can be observed that ( ) again we have to analyze 4 cases: in this case it can be observed that The algebraic definition of the k j   ⋅ product can be seen as defined by the algebraic analysis of above and, in particular, it is the limit case (3) (see appendix B for the solved code of the algebraic definition of the 3d product and division).
Another consequence of this analysis is that, now, it is possible to try to analyze a generic 2° order (3d) equation: Because now we have an algebraic differentiable definition of the product and of the division, it is clear that if we have two 3d functions (see paper [2]) such as: y z z z are all differentiable functions and they give real results, the product: and the division: This was another open argument of paper [2].

Conclusion
The above analysis has shown it is possible to give an algebraic definition of the product and of the division for 3d numbers as defined in paper [1] and that these algebraic definitions are differentiable.
Same algebraic analysis can also be done for 4d product and division, even that it is a bit more complex (see Figure 2 and appendix A).
For a 4d number we can write: and so on for the inverse property, the norm (or modulus) etc.
In paper [2] I gave a proposal generic sum definition. The objectionable point was to assign by default the v3space' sign set to 1 in the case that ' ' ; also, it could be questionable the generic 4d scalar product definition. This was a mistake.
The solution is much simpler; the γ angle must be treated in the same way of the β angle; β rotates, but in fact, at the end of calculations is reduced to 2 / β π <= (see Figure 2). The same must be done for γ, at the end of calculations γ must be reduced to 2 / γ π <= .
So the sum in 4d space is the same of the sum in 3d space (see appendix C) and, because the schema for 4d is the same for 3d, it is obvious that this idea can be extended to more dimensions.
There are no problems to extend the scalar product formula to more dimensions; a last consideration can be done for the extended definition of the 4d vector product. A 3d number (or a 4d number) can be seen as a vector (x,y,z). Let us consider C as the 3d resulting vector product between two 3d vectors A and B; C can be seen as an orthogonal vector whose length is the area of the parallelogram identified by the two non-parallel vectors A and B, so the D 4d resulting vector product between tree 4d vectors A, B and C can be defined as an orthogonal vector to A, B and C whose length is the volume of the solid identified by the tree non-coplanar 4d vectors A, B and C (for simplicity, you can think that A, B and C are 4d numbers whose h  component value is 0) .
The D vector as result of 4d vector product of A, B and C, is given by the following operative formula: The formula can be extended to more dimensions. Versus (sign) of D depends on the tern A, B and C, but these are all well-known questions.

Appendix A: 4d numbers analysis
Consider s a 4d number: given: The 4d product between two 4d numbers b a s s s ⋅ = ' is: The result in polar notation is: r t r t r r now we have to analyze 7 cases: The generic case (1) of the 4d algebraic product of b a s s ⋅ is differentiable. The other cases are limit case and are also differentiable. Limit case (4) may be an objectionable limit case, but is the same questionable problem we have seen above for the division in 3d; the differential depends on the z a and z b sign.
The algebraic form of s s 1 '= : given: and it is differentiable.
Also here we have to analyze 7 cases: sign z x y you can observe that in this case The generic case (1a) of the 4d algebraic division s a /s b is differentiable. The other cases are limit case and are also differentiable.
Limit case (4a) may be an objectionable limit case, but, again, is the same questionable problem we have seen above for the division in 3d; the differential depends on the z a and z b sign.
Appendix B: 3d core visual basic source code 'reference to the code 3d-2.4g in appendix of paper [2] 'The algebric product 'Creates ds from a vector S and dAlfa,dBeta and dr Function Differentiate_Vector_4d(S As Complex4d, dAlfa As Double, dBeta As Double, dGamma As Double, dr As Double) As Complex4d Dim dx As Double, dy As Double, dz As Double, dt As Double, ds As Complex4d Dim dr1 As Double, R1 As Double Dim Wx As Complex4d, Wy As Complex4d, Wz As Complex4d, Wt As Complex4d, R As Complex4d, R0 As Double Dim X As Double, Y As Double, Z As Double, T As Double Dim BVx As Double, BVy As Double, BVz As Double, BVt As Double