Summary
We already have a sound mathematical basis for believing that complex numbers can act like vectors. I want to show that they obey ALL the rules of complex mathematics. In addition because I have introduced a unique geometry I want to show how the rules of logarithms can be obeyed and how forces that have square roots values originate.
The Consistency of Complex Vectors
Forces that have square roots values
There are two situations with vectors that need clarification because we are not dealing with an entirely mathematical system any more but a clearly defined complex space.
The first is the operation of taking square roots.
In my Analytic Complex Number Theory I define waveforms acting into 3D from 4D as x^2 type forces and I define waveforms acting into 4D from 3D
as x type forces. So I believe it is important to clarify this point. In the figure "3D to 4D Vectors" I treat both the 3D and 4D planes as
being linear so the addition by Pythagoras' Triangle works in both of these planes, It is only when we take a translation from 3D to 4D that
the new rule applies and only then in the specific case of an area being translated upon a distance. This could prove useful with my new
interpretation of eigenvalues.
We get this result if we apply Pythagoras' Triangle to sides with square root values we get (sq rt a)^2 + (sq rt b)^2 = (sq rt c)^2
or a + b = c. In this case it is not that we have a parabolic geometry but that there is a translation by taking square roots.
This operation ignores the complex plane and in fact we do not need to bother about it. Any operation on the 3D side of the complex plane os
reversed in 4D to restore linear distances.
Now although I have defined x^2 and x type forces as I have. I believe that this is arbitrary due to the fact that we can keep multiply
i^2 = -1 by factors of i to get rotations. If we take i^4 = -i^2 then this should result in obtaining the reverse direction of these vectors AND
the possibility of generating forces in 3D that have square root values.
I have shown elsewhere that if we sum the elliptical diameters for the planets in our solar system they form a square root of 2 to a square
root of 3 combination ratio.
Logarithmic rules and forces on the Complex Plane
The second problem is that the logarithmic rules of addition and subtraction should work upon the complex plane I have defined. After trying various alternatives
this is satisfied when all the vectors lie upon the complex plane and to satisfy BOTH rules the surface must not only not be glat but must fold depending upon the vectors used. This is shown in the two diagrams for logarithmic addition and subtraction.
With the logarithmic addition it means that the resultants tend to increase much faster than for subtraction. This may be due in part to the subtraction
representing tension and subtraction representing compression. However I believe that this folding can work because if it does measure
movement perpendicular to the complex plane. A large folding factor therefore represents a perpendicular approaching a parallel condition
to the Y axis (or 4D) while folding factor approaching zero represents the complex plane approaching a parallel condition to the X axis (or 3D).
This gives another possibilty for defining the folding factors as being connected to the inclination of the complex plane.
