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The notion of complex numbers implies that any entity should have an imaginary dimension to its nature. In other words,

Assertion C2: Any being has both a notable objective portion and not-as-obvious subjective qualities

Although the views of classical science, which deal mainly with the observable, are good for simplification purposes, they cannot reveal the whole truth. To get the whole picture, we have to broaden our scope and include non-observable properties of physical elements as well.

In this text, we take the liberty of appointing the point zero in the Argand diagram to represent an assumed version of singularity (explained in the Singularity section) and the imaginary portion of the diagram to symbolize the obscure aspects of reality.

Imaginary numbers are sometimes called magic numbers. One of the strange characteristics of these numbers is that in De Moivre diagram shown below, any pure real number coupled with (multiplied by) them will be reduced to zero.  

complexnumber multiplication

As shown in the diagram above, when we multiply any so-called real quantity by i, its value  is reduced to zero. Mind you that in mathematics,  natural numbers (e.g. 1 2 3….) represent discreteness in the field.

In trigonometry we can show discreteness as follows:

X = r Cos a, if we take a = 90, then Cos a = 0, therefore X = 0.

AssertionC3: Although the real number field may create the illusion of continuity, the more accurate complex number version shows us that the continuity of the real number line breaks down periodically. (see notes to zero and infinity section in the Singularity Chapter)

This introduces a limit for any real value (Zeno effect). Later on, I will deduce that fundamental physical elements (e.g., space, time, and matter) as we know them have to be discrete and not continuous. In calculus we can show this by evaluating the function of (x) in any equation. We take y = x IxI as an example,

yfx 

yfx

The second derivative exhibits discontinuity around point zero. A lack of smoothness and continuity in derivatives of real number functions

yfx

y ̋ = f  ̋ (x) =2 +4θ(x)   For any other function of finite real numbers, we can come to a derivative that shows a lack of smoothness and continuity at finer scales.

   
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