## Mathematics

### Real and Natural Numbers

Mathematics has been very helpful in evaluating and understanding the world around us. Therefore, we frequently look at mathematics to build physical models. We have two different mathematical choices we can adopt to build a model for space-time.

Real numbers with infinite amount of decimals (2.567854332234…) stand for continuity in the field. If we take real numbers to represent space, we may see the world as a continuum. The problem is as far as we know the universe is finite. therefore, real number system can not represent the universe. In contrast, natural numbers (1, 2, 3, 4, 5…), with their discrete nature, symbolize discontinuity of the field. If we adopt the natural number system as the model, then our space-time at a fundamental level has to be grainy and discrete.Then the question arises, what is the nature of the void in between the units?

### Imaginary Numbers

In mathematics, imaginary numbers are the square root of negative numbers. In geometry, the square root means the sides of a square with area a. Imaginary numbers (square root of minus numbers) are frequently used as an integral part of quantum mechanical mathematics.

By definition, the square root of a (√-a) is the side of a square with area equal to -a. What kind of a square would have negative sides? Such a square cannot have a meaningful, real (tangible) area. It cannot have a meaningful dimension either. Dimensions of such a square are not similar to the four dimensions of the real world. They are therefore imaginary dimensions.

In theoretical physics, imaginary numbers represent components of hyper-space (a space beyond our four familiar dimensions). The assumption that hyper-space exists is necessary to build theories about nature of the universe. So, it seems that, contrary to the assumptions of string theory, hyper-space cannot have a real dimension, or area. In this model, I call hyper-space the singularity, which does not possess any real dimension. However, it can accommodate the abstract quantities.

## Complex Number Mathematics

Let us return now to the complex number system (a two-dimensional system that consists of real and imaginary coordinates). The system is explained in detail in the article “Complex Numbers”

Assertion C3 postulates that the discontinuity of real numbers always occurs around zero point. One of the strange characteristics of imaginary numbers is the fact that in unit circle, the real value of any parameter when coupled with (multiplied by) any imaginary number will be reduced to zero.

The algebraic formula can be written as follows:

(X+0*i*) *i* = X*i* + 0(*ii*) = X*i*

X in the unit circle is one. Therefore,

X*i* =* i*

As we go counter-clock wise around the circle the real value gradually disappears and the imaginary value fully expresses itself at 90^{0} degrees.This is analogous to our attention span. The more we focus on quality of an object the more we lose focus on its quantity or measurable features.

In trigonometry we can show this fact as follows:

X (the real Value) = r Cos a, since we took a = 90 and, Cos a = 0

then, X = 0

The complex number equation Z = R [cos a + i sin a] indicates that these numbers also have a periodic nature. Therefore, they lose their real number value and hit zero twice in each period, which indicates discontinuity in the real number field.

Therefore, any space-time element (space, time, matter) shown by the X coordinate, as it couples (multiplies) by imaginary number, loses and regains its real value periodically. For example, if X indicates distance in space, the space has to disappear and reappear during each period. This is the basis for our assumption that space and time are discrete and not continuous.

Erwin Schroedinger was one of the first physicists to suggest a discrete space. Einstein, in his last published paper, having quantum theory in mind, also proposed that a theory based on discrete space might be the way forward for physics.

As mentioned, in this model we take the zero point on the complex number plane to represent singularity. The imaginary number (*i*) represents the singularity effect on different space-time phenomenon.

Paul Davis writes, “Undecidable propositions run through mathematics like threads of gristle that criss-cross a steak in such a dense way that they cannot be cut out without the entire steak being destroyed" ^{[1]}

What can we make out of the above statement? The undecidable propositions to me represent physical findings which are not directly related to the pre-determined or expected outcome of calculations. They represent outcomes that are ambivalent and therefore cannot point to a single definite answer related to space-time elements.

Assertion C4 indicates that any point in the plane can be considered zero point. Therefore zero point is interwoven with any minuscule portion of the space-time fabric. Or we can further assume that zero is present in between each atom of space-time.

In mathematics considering every point of a domain as zero is called “blowing up the origin.” This is done with a mind-state that localizes the origin somewhere inside space-time. Spreading zero all over the domain, on the other hand, is considered “exploding the origin.” However, in this model zero point is a separate entity, which is accessible at every point of space-time. Therefore, the origin remains intact. Assumption C_{6} (in the section "Complex Numbers") speculates that the singularity and our universe are two separate domains.

"There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition; and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call "the Twilight Zone"^{[2]}

Using common sense mathematics, imagine that we have a bunch of pebbles and we remove them from the scene one by one. At the end, there will not be any pebbles left. The reality is that there are no pebbles in the scene, but zero is still there. The scene still exists.

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