To revise complex mathematics we need to deal with 2 situations that have baffled mathematicians for centuries to the point that they do
not accept that complex space is a real or valid entity.
Without explaining how I can interpret i^2 = -1 and show how complex numbers are ordered no valid revision of complex mathematics is
going to have a sound basis. That is why we must persist in explaining these two theories!
Summary
The explanation below seems longer than I would like. I have tried to answer any possible criticism.
Briefly, what I am saying is that the complex numbers can be interpreted as a translation
between a circular (spherical) geometry and a cubic geometry.
This works in both directions which explains why multiplying by a factor of i can be and is traditionally translated as a rotation.
This avoids the need for quadrature (changing a circle into a square).
classical quantum mechanics mathematicians have always used complex numbers and in so doing have unwittingly made these translations without being aware of it!
A Translation Between two Geometries
This equation can be interpreted as a translation between circular and square geometries with a rotation included. This is very beneficial if we consider 3D forces act spherically while I suggest that 4D forces act cubically. No difficult mathemathics need now be attempted for quadrature. We just use complex mathematics.
This is why Classical Quantum Mechanics works mathematically. It uses complex mathematics and makes these translations without any awareness that they exist!
My revison of complex mathematics includes a new interpretation (in fact the only interpretation) of this equation.
1> I interpret this equation as a translation, that can be considered in vector terms, between a spherical or circular geometry that we see
in our normal existence into a cubic or square geometry that we visualise in complex space.
2> We compare and translate the area of a circle onto a square.
The equations for which differ by the factor pi. This factor can be eliminated by noting that a translation from a spherical to a flat
geometry automatically eliminates the factor pi.
3> One important point to note is that the equation has an equals sign that really operates in both directions from left to right.
That is any interpretation of this equation has to take into account that by multiplying by a factor of i changes real components into
imaginary ones and vice versa and so reverses the direction of the equation!
4> We can get an area for the circle or the square by letting the radius of the circle/length of the side of the square equal +1 or i
imaginary units. This creates an area of +1 or -1 which we can interpret as being in real or complex space respectively and indicating a directionality.
See image below.
The Complex Translation
Let us take the situation in spherical or circular space and geometry and interpret how the factors of i can apply:
The Unit Circle
This can exist into ^2 basic forms. That is with a radius of 1 real unit (case 1) or with a radius of 1 imaginary unit (case 2).
In case 2 we have a circular force acting towards complex space.
This situation can occur, for example, in a strut that is loaded at both ends (one dimensionally) yet bends into the second dimension as
it buckles. I interpret this has being the result of a very small complex stress being created at the point of bending. If you look at
the geometry of my ztar on the Yahoo Group site you will see how the ceometry of complex space can allow this tiny stress to develop but that it
then increases exponentially.
The Unit Square
Case 3 is a square on the complex plane with a x axis measurement of 1 real unit and a y axis measurement of 1 imaginary unit giving an
area of i.
Case 4 is a square on the complex plane with a x axis measurement of 1 real unit and a y axis measurement of i^2 or -1 unit giving an
area of -1 to provide a reverse translation possibility.
Case 5 is a square on the complex plane with a x axis measurement of 1 imaginary unit i and a y axis measurement of i units giving an
area of -1 to provide a second reverse translation possibility.
Case 3 is a square (in the 4th dimension and on the exterior side of the complex plane) that occupies that space with no forces acting
into or out of complex space.
Case 4 is a square (in the 4th dimension and on the exterior side of the complex plane) that occupies that space with a REAL
force/vector of 1 unit acting out of complex space and into real space. If we think in a three dimensional framework then it can be interpretted as a rotation of the y axis onto the x axis but on the negative half of the Argand Diagram.
Perhaps a better explanation is to suggest that the z axis can be real or imaginary depending upon the vector acting along it.
In this case the y axis simply rotates towards the z axis but it has a directionality towards real space.
Case 5 is a square (in the 4th dimension and on the exterior side of the complex plane) that occupies that space with a COMPLEX
force/vector of i units acting out of complex space and into real
space. I accept that this situation in case 5 is not presently accepted. I can give you the example of the recent nuclear accident
in Japan in which technicians were thrown to the ground by some phenomena.
This interpretation of i^2 = -1 therefore allows for real and complex forces to act in or out of real space and a similar
situation to be possible in complex space.
We have therefore succeeded in showing how a vector mechanism can exist between the complex and real planes.
The translation given by i^2 = -1 has been shown in the image below to work in two directions. This obeys complex mathematics with i^3 = -i and i^4 = +1 representing further rotations. In my interpretation I prefer to think of these rotations as being of one axis into the Z axis.
The Complex Plane can now be considered as a plane halfway between 3D and 4D.
