The Theoretical Significance and Reality of Imaginary Number

It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.

Killing Imaginary Numbers? From Today’s Asymmetric Number System to a Symmetric System

In this paper, we point out an interesting asymmetry in the rules of fundamental mathematics between positive and negative numbers. Further, we show an alternative numerical system identical to today’s system, but where positive numbers dominate over negative numbers. This is like a mirror symmetry of the existing number system. The asymmetry in both systems leads to imaginary and complex numbers. We also suggest an alternative number system with perfectly symmetrical rules—that is, where there is no dominance of negative numbers over positive numbers or vice versa, and where imaginary and complex numbers are no longer needed. This number system seems to be superior to other numerical systems, as it brings simplicity and logic back to areas that complex rules have dominated for much of the history of mathematics. Finally, we also briefly discuss how the Riemann hypothesis may be linked to the asymmetry in the current number system. The foundation rules of a number system can, in general, not be proven incorrect or correct inside the number system itself. However, the ultimate goal of a number system is, in our view, to describe nature accurately. The optimal number system should therefore be developed with feedback from nature. If nature, at a very fundamental level, is ruled by symmetry, then a symmetric number system should make it easier to understand nature than an asymmetric number system would. We hypothesize that a symmetric number system may thus be better suited to describing nature. Further, such a number system should eliminate imaginary numbers in space-time and quantum mechanics, for example, two areas of physics that are clouded in mystery to this day.

Imaginary Numbers

Mathematicians knew that the multiplication of two negative numbers yields a positive product Thus, not only does+1×+1=+1,but-1×-1=+1. What number,then,multiplied by itself yields -1? To put it another way,what is the square root of -1? Mathematicians can invent the necessary number, call it an imaginary number,and symbolize it as i

Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy

Analysis of WMAP and Planck spacecraft data has proved that we live in an invisible Multiverse, referred to as hidden, that has a quaternion structure. It explains the reason for the mutual invisibility of parallel universes contained in the hidden Multiverse. It is shown that the hidden Multiverse includes most likely twenty parallel universes from different dimensions, six of which are adjacent to our universe. Besides, edges of the hidden Multiverse are connected to other (from one to four) Multiverses, which are observable neither by electromagnetic nor by gravitational manifestations. The Multiverse described contains four matter-antimatter pairs, annihilation of which is prevented by relative spatial position of the universes. The experimental proof of existence of the hidden Multiverse is explained to be the phenomenon of dark matter and dark energy that correspond to other invisible parallel universes, except ours, included in the hidden Multiverse. General scientific principle of physical reality of imaginary numbers, refuting some of the statements of the existing version of the special theory of relativity, is a physical and mathematical foundation of the outlined conception of the hidden Multiverse. The article presents relativistic formulas of the theory of special relativity adjusted in accordance with the principle. It also offers appropriate interpretation of multidimensional space of the hidden Multiverse.

Ohm’s Law Refutes Current Version of the Special Theory of Relativity

It is shown that Ohm’s law is not only the main electrical engineering law, but also a generally scientific worldview law of the extreme significance, as in the interpretation of Steinmetz it proves physical reality of imaginary numbers theoretically and experimentally in the most indisputable way. Thus, it refutes the principle of light speed non-exceedance, which is fundamental in the special theory of relativity. Moreover, unlike the MINOS and OPERA experiments recognized by physical com-munity as not enough reliable, which were conducted for the same purpose, alternative experiments were performed during the research of oscillation processes in linear electric circuits. Therefore, they are absolutely reliable and conclusive as can be repeated and verified in any electrical engineering laboratory. The principle of phy-sical reality of imaginary numbers proven by the electrical engineering experiments is generally scientific, since mathematics is the universal language of the exact sciences. Therefore, all scientific theories and hypotheses in quantum mechanics, relativity theory, geo-physics, cosmology, optics, radio electronics and other sciences should be adjusted accordingly, given the principle of physical reality of imaginary numbers. There is an example of how this can be done in the special theory of relativity and astrophysics. This approach allowed explanation of dark matter and dark energy, which correspond to the invisible parallel universes existing in extra dimensions.

What Physical World Do We Live in?

It is shown that the hypothesis of Monoverse relevant to the existing version of the special relativity theory is incorrect. Incorrect is also the existing version of the special relativity theory itself. Its revised version, the relativistic formula of which allows to propose the hypothesis of hidden Multiverse is presented. The hypothesis of hidden Multiverse is based on a proof by the author general scientific principle of physical reality of imaginary numbers. It is shown that this hypothesis, unlike many other hypotheses of Multiverse and the hypothesis of Monoverse, is already verifiable. And the experimental proof of the reality of the hidden Multiverse existence is a phenomenon of dark matter and dark energy. Other possible experimental proof of that is proposed as well. As a result, the data processing is obtained by WMAP and Planck spacecrafts, and the structure of hidden Multiverse is determined, which has proved to be a quaternion, and its parameters are defined. According to these calculations, the hidden Multiverse contains from twenty to twenty-two parallel universes, five-six neighboring universes of which constitute dark matter, and the rest parallel universes constitute dark energy. Moreover, it is shown that our hidden Multiverse neighbors with other hidden Multiverses, inaccessible to our observations not only by electromagnetic but also by gravitational manifestations, which together form a Supermultiverse.

Universes Being Invisible on Earth outside the Portals Are Visible in Portals

The authors of the existing version of the special theory of relativity had to use the princi-ple of light speed non-exceedance to explain relativistic formulas obtained therein within the space of real numbers. The principle implied existence of only our visible universe and absence of physical content in imaginary numbers. However, since the principle is just a postulate, i.e. an unproven assumption, it has always carried little credibility. The paper refutes the principle by the experimentally proven principle of physical reality of imagi-nary numbers. As follows from the principle of physical reality of imaginary numbers, relativistic formulas of the existing version of the STR are wrong and incorrectly ex-plained, and conclusions drawn from them are misguided. In other words, this version is incorrect1 and, thus, the STR was not actually created in the 20th century. Moreover, it could not be created in the 20th century, as its creation required experimental data ob-tained in the 21st century. The paper provides an alternative version of the STR contain-ing relativistic formulas corrected given experimental data obtained in the 21st century. These formulas imply that instead of the Monoverse, whose existence is supposed in the existing version of the STR, there is a Multiverse, whose universes are mutually invisible (and the invisibility is explainable) and therefore it has been referred to as hidden. The paper explains that existence of invisible universes of the hidden Multiverse gives rise to the phenomenon of dark matter and dark energy that is actually a sort of optical effect (however, not electromagnetic, but gravitational), a shadow, rather than some physical substance2. It also explains that existence of other invisible universes outside the hidden Multiverse gives rise to the phenomenon of dark space. Invisible universes are claimed to really exist, which can be experimentally proved by astronomical observations in portals, where universes not visible outside portals become partially visible. Therefore, constella-tions observed in the starry sky inside the portals are different from those observed in our visible universe. The alternative version of the special theory of relativity can also suc-cessfully solve other issues of astrophysics. In particular, it can explain where antimatter is located and why it does not annihilate with matter, as well as where tachyons are locat-ed and why they don’t violate the principle of causality, etc.

Antimatter, Anti-Space, Anti-Time

The article shows that the special theory of relativity (STR) created in the last century was based on postulates due to the lack of the required experimental information and turned out to be incorrect, as its principle of light speed non-exceedance was refuted by studies of special processes in linear electric circuits in the 21st century. And thus, it made obsolete the unsuccessful OPERA and ICARUS experiments carried out at the Large Hadron Collider. Therefore, an alternative version of the STR has been proposed. Its relativistic formulas imply the existence of numerous mutually invisible parallel universes and antiverses. It is explained how they can be seen. There is anti-matter, as well as anti-space and anti-time in antiverses in the same quantities as matter, space and time in universes.

Gauss Sum of Index 4:(2)Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.