Technical Summary
Animated numbers versus complex numbers
Technically, there is zero difference between these two labels.
Complex numbers have been understood as having the capacity to represent numbers in the plane since the seventeen hundreds. In such a context, the difference between the real and imaginary numbers is an arbitrarily assigned axis. Why the real axis involves expansion and contraction, while the imaginary one utilizes rotation in the plane is just the way it has to be.
Time is commonly involved in systems that use complex numbers. This is a common accident. I believe that time, and all time-like measurements in physics, map to the real numbers. Time has a number of unique qualities. Zero time is now. Negative time, the past, can only be remembered. Positive time, the future, can only be predicted. Space and space-like quantities are not like that. Mirrors can be used to flip positions.
Animated numbers can only be faithfully represented using computers not books printed on paper. The future of animated numbers will be visually interesting.
The equations on the complex manifold
The same equation is used throughout these pages:
3 + 3 = 6
This may strike some as wrong for the animated or complex numbers surely involve three different equations:
- Time: (3, 0) + (3, 0) = (6, 0)
- Space: (0, 3) + (0, 3) = (0, 6)
- Motion: (3, 3) + (3, 3) = (6, 6)
Those are the quations for the complex numbers over the manifold R2. For this site however, I only work over the complex manifold C1. The way to write pure real and pure imaginary number is then:
- Real: (z + z* )/2
- Imaginary: (z - z*)/2
- Diagonal: z z* + z z* i
The equation stays the same, but the manifold changes. I considered the subject of manifolds beyond the reach of "Numbers 101".
The same pattern happened for the space-time numbers or quaternions.
Space-time numbers versus quaternions
Technically, there is zero difference between these two labels.
Quaternions have a long, twisted history. Even today, people can have strong opinions about quaternions. The name in isolation of that history says "quat", four. Other than that, one needs to read up on the subject.
Space-time numbers, as a label, says a bit more. It is not simply that there are four equal partners, but there are the familiar three dimensions of space in a tense marriage with scalar time. Space has properties that are deeply different from time, yet with space-time numbers, they never part. The label can tie directly into the real world by pointing to events there-then, the "there" being space while "then" is the time.
Full disclosure: I own quaternions.com so the label quaternions is important to me. I now use both words interchangeably. Space-time numbers feels more like a real world term. Quaternions as a label feels more abstract. I find it helpful to have both terms in my efforts to understand the accounting system used by the Universe.
Space-time numbers as the new King
Mathematical Physics is written using tensor calculus often over the field of complex numbers.
My own effort is to replace all such work using quaternions, a division algebra born to work with space-time.
It has been my experience that when someone in the literature has published that real-valued quaternions could not be used for a certain problem, I was able to find an approach that worked. This happened for representing the Lorentz group, for deriving the Maxwell equations, and now I have an invariance principle that may do the work of gravity.
In the future, I hope that an area of study will not be considered complete until the researchers can provide animations.
The source of probability in quantum mechanics
Once there is some probability in quantum mechanics, so much follows. The riddle is why there should be any? My idea is that in high energy physics, the normal fundamental operations of math - addition, subtraction, multiplication, and division - become indistinguishable. There is either zero, the vacuum, or unity made up of all possible histories. The quaternion group Q8 describes the properties of space-time numbers.
It is my belief that the standard model of physics will need to be torn down and reconstructed using only the normalized quaternion group. I don't know how to start such a quest, so it will remain a personal belief.