Hallucinary Number Theory

A few years ago some friends and I invented the "HALLUCINARY NUMBERS". In our view the numbers, as we know them up to now, are too limited. The introduction of the hallucinary numbers brings us a complete new way of thinking towards the Unification Theory, the Theory of everything ...and more. Hallucinary numbers form together with the ordinary real and imaginary numbers the hyper-complex numbers. All physical laws written in the context of this hyper-complex number space have such an elegant form, that GOD must have known about these numbers much before we even thought (hallucinate) about it. Before I will explain about these new numbers, I will give a short introduction in the world of numbers.

• Real Numbers
  These are also the basic numbers ordinary people use to express money in valuta, weights in kilograms etc. We can divide this group of real number into many sub groups. So we have positive and negative numbers, integer and non-integer numbers, rational and irrational numbers.

• Imaginary Numbers
  The scientific world uses also a different type of numbers, which is not used in daily life, the imaginary numbers. It is not possible to buy a imaginary number of kilograms at the grocery or to have a imaginary number of valuta on your account. However for scientific problems they are very practical. The basic unit of the imaginary numbers is the number $i$. This number is defined as the square root of minus one

  $$i=\sqrt{-1}$$ (1)

  
  

  Most people have learned at school that there is no solution for the square root of a negative number. However at the university you get taught that there is one, this one is not real , but still you can 'imagine' that there is a solution, which we call $i$. This may sound strange and useless, however introducing this number $i$ gives enormous advantages for mathematics. Physics deals with quantities, which can be measured and are consequently real. However, in theoretical formulas or in theoretical calculations, we often use imaginary numbers.

  Real numbers and imaginary numbers form the group of complex numbers. Mathematicians have derived a whole set of rules how to calculate with complex numbers. Some simple examples of calculations with complex numbers are the following:

  
  

  $$\sqrt{-4} = \sqrt{4} \times \sqrt{-1}=2 i$$

  $$(1+3i) \times (2+i) = 1\times 2 + 1 \times i + 3i \times 2 + 3i \times i$$

  $$= 2+i+6i-3 =-1+7i$$ (2)

  
  

  
  or a bit more complicated
  
  

  $$e^{i \pi}=-1$$ (3)

  
  

  The proof of last expression requires a bit more advanced mathematics. The reason to put it here is just to show the beauty of some mathematical formulas with complex numbers. It is quite surprising that the expression with all those complicated numbers $e, \pi$ and $i$ has such a simple result.
  More information about imaginary numbers, real numbers, rational, irrational numbers, etc., can be found
  here or here.
  A slightly different interpretation of imaginary numbers can also be found in literature:
  Waterson Calvin and Hobbes .

• hallucinary numbers
  Still we think this is too limited. Besides real numbers and numbers you only can imagine, we want to introduce 'hallucinary' numbers. Numbers beyond imagination. The unit for these kind of numbers is $h$ (Don't get confused with the Planck-constant! ). This number $h$ is defined as:
  
  
  
  
  

  $$h \times h^* =-1.$$ (4)

  
  
  
  
  
  where $^*$ denotes the complex conjugate. This implies that: $\vert h \vert=i$.
  
  
  
  
  So the hallucinary numbers have an imaginary absolute value!!!.
  This is something which is impossible in ordinary complex number theory. The hallucinary numbers extend the complex numbers to the hyper-complex numbers. Each hyper-complex number $z$ can now be written as the sum of a real, an imaginary and a hallucinary part:
  
  $z=a+bi+ch$ (with $a,b,c$ real).

Hallucinary Numbers in Physics

The hyper-complex numbers are extremely useful in Einstein's theory of Relativity. Here I will show you a single result to prove this. Einstein improved the classical mechanics theory, by a reformulation of the basic physical laws, known from Newton's Philosophiae naturalis principia mathematica. The rules formulated by Einstein are independent on the inertial system of the observer. Basically this means that the laws of physics noticed by you should be always the same, no matter if you are standing still, are in a plane, space shuttle, on Earth or on Mars. The spatial and time observations of a certain event in space (e.g. explosion of a neutron star, collision between two black holes etc.) can differ for two observers, who are at different places and at different times somewhere in the universe.

However there are quantities which they can both observe and which are exactly the same for both. The event in space can be described with a so called position four-vector $x^\mu$, in which time and space are combined into a four dimensional superspace: the Minkowski-space

$$x^\mu\equiv(x^0,x^1,x^2,x^3)\equiv (ct,x,y,z)$$ (5)

with $c$ the speed of light in vacuum.



This formalism also brought a new way of thinking to wards space and time. In Einstein's philosophy time is actually nothing else than a fourth dimension (or the zero-th dimension, if you like).
Ofcourse the space location and time location of the event are different for the two observers. We can however relate the difference in time and space of two events (say event A and event B). In Newton's theory the two observers will always measure the same time difference

$\Delta t=t_A-t_B$

and the same spatial difference

$dr=\sqrt{ (x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2}$,

but in reality this is not true. It is even possible the observer one thinks event A happens before event B, while observer two thinks event B happens first.
The quantity $dS$ which is the same for both observers is a combination of the time and spatial difference.

$$dS\equiv\sqrt{dx^\mu dx_\mu} \equiv \sqrt{(t_A-t_B)^2-(x_A-x_B)^2-(y_A-y_B)^2-(z_A-z_B)^2 }$$ (6)

The first part deals with contravariant and covariant representation of the position four-vector, but I will not bother you about this here. What is important, is that $dS$ is approximately the length of the four-vector $dx^\mu$, but not completely, as you need to put the plus and minus signs on its right position. However it would be much easier if we really could think of $dS$ as the length of a four-vector. That is possible if we would have used the correct position four-vector!. With the help of hallucinary numbers, we can now define a new position four-vector:

$$\chi \equiv(x^0,x^1,x^2,x^3)\equiv (cth,x,y,z)$$ (7)

and now we can write

$$dS=\sqrt{d\chi d\chi^*}=\vert d\chi\vert$$ (8)

This is just one example, but all physical formulas become much more simple and more elegant when we use hallucinary numbers. No need for covariant and contravariant representations anymore. The elegance of the new formulas proof that this is the only and true way how we should think.
There is no other way, then that the four-vector postulated in eq. (7) is the only and true representation of space and time and is thus favourable to the old position four-vector of eq. (5). One striking and inevitable conclusion from eq. (7) is:

Time is the hallucinary dimension of space!!


I think, subconsciously, you were all aware of that, but finally there is scientific proof!!