Imaginary Numbers Research Articles

On Theoretical Incomprehensibility

This contribution tentatively outlines the presumed conceptual duality between the issues of incompleteness and incomprehensibility—The first being more formal in nature and able to be declined in various ways until specified in the literature as theoretical incompleteness. This is theoretical and not temporary, which is admissible and the completion prosecutable. As considered in the literature, theoretical incompleteness refers to uncertainty principles in physics, incompleteness in mathematics, oracles for the Turing Machine, logical openness as the multiplicity of models focusing on coherence more than the optimum selections, fuzziness, quasiness, e.g., quasi-crystals, quasi-systems, and quasi-periodicity, which are intended as the space of equivalences that allow for coherent processes of emergence. The issue of incomprehensibility cannot be considered without reference to an agent endowed with cognitive abilities. In this article, we consider incomprehensibility as understood here as not generally scientifically explicable, i.e., with the available knowledge, as such incomprehensibility may be temporary, pending theoretical and technological advances, or deemed to be absolute as coincident with eventual definitive, theoretical non-explicability, and incomprehensibility. We considered the theoretically incomprehensibility mostly in three main ways: as the inexhaustibility of the multiplicity of constructivist reality as given by the theoretically incomprehensible endless loop of incomprehensible–comprehensible, and by existential questions. Moreover, theoretical incomprehensibility is intended as evidence of the logical openness of both the world and of understanding itself. The role of theoretical incomprehensibility is intended as a source of theoretical research issues such as paradoxes and paradigm shifts, where it is a matter of having cognitive strategies and approaches to look for, cohabit, combine, and use comprehensibility and (theoretical) incomprehensibility. The usefulness of imaginary numbers comes to mind. Can we support such research for local, temporary, and theoretical incomprehensibility with suitable approaches such as software tools, for instance, that simulate the logical frameworks of incomprehensibility? Is this a step toward a kind of artificial creativity leading to paradigm shifts? The most significant novelty of the article lies in the focus on the concept of theoretical incomprehensibility and distinguishing it from incomprehensibility and considering different forms of understanding. It is a matter of identifying strategies to act and coexist with the theoretically incomprehensible, to represent and use it, for example when dealing with imaginary numbers and quantum contexts where classical comprehensibility is theoretically impossible. Can we think of forms of non-classical understanding? In this article, these topics are developed in conceptual and philosophical ways.

Imagination captured

Imaginary numbers have a chequered history, and a sparse — if devoted — following. Abigail Klopper looks at why a concept as beautiful as i gets such a bad rap.

Imaginary Numbers of Climate Change Migrants?

Within the extensive scientific and policy discussions about climate change migrants, detailed analyses continue to highlight the lack of evidence thus far for climate change directly causing migration. To understand better how climate change might or might not lead to migration, this paper explores possibilities for developing a robust, repeatable, and verifiable method to count or calculate the number of people migrating or not migrating due to climate change. The discussion starts by examining definitions of “climate change” and “migration”, then looking at how to determine numbers of climate change migrants based on those definitions. These points lead to descriptions of the subjectivity and arbitrariness of the decisions needed for counting or calculating climate change migrants and non-migrants. While the scientific study of working out numbers of climate change migrants and non-migrants is challenging and interesting, especially due to its complexity, changing baselines alongside legitimate concerns about necessary assumptions lead to questions regarding the usefulness of the calculations for policy and action. Ultimately, labelling, counting, and calculating climate change migrants and non-migrants depend on political choices, so any numbers reached might not be scientifically robust. Improved understanding of people’s motivations for migrating and not migrating under different circumstances, including under climate change and perceptions thereof, would be preferable to a starting point assuming that climate change inevitably causes migration.

The Way They Did Theory Then

ABSTRACTIn this article, I take sixteenth-century mathematics as a test case for considering at its most general level what doing theory entails. Ideally, doing theory aims to change rather than merely describe reality, and in order to do so, it represents that reality differently. Such transformations, I argue, need special techniques of representation. In early modern mathematics, two techniques in particular enable new theoretical content: symbolic notation, and the real continuum. Together they create a language that can redescribe mathematical objects, for example, so that a single general formula can solve for all specific instances, and simple counting numbers can be reidentified in the real continuum as limits. I argue for the formative contribution of the imagination in such developments, that is, for an ability to reassemble givens of a problem in such a way as to solve it; and to concoct new objects or possibilities that do not exist in reality. In the sixteenth century, imaginary numbers (roots of negative values) most explicitly demonstrate the importance of speculating beyond what is real. As we consider what doing theory in the twenty-first century entails, it is worth remembering, in light of the article’s analysis, how form or technique itself has agency, and the ability to transform theoretical content.

The Mote in God’si: Fourier, Fractals, and Imaginary Numbers in Tom Stoppard’sArcadia

Scholars of Tom Stoppard’s Arcadia have long recognized its deep interest in science and mathematics as well as its complex use of scientific and mathematical concepts to illustrate its thematic concerns. But Stoppard criticism has often overlooked several important concepts, including the imaginary and complex numbers of fractal geometry, Brownian motion and chaos, and the Fourier transform. This article fills in these historical and contextual gaps and argues that critics of Arcadia should take a more integrative approach by examining these ideas as they relate to and reinforce each other rather than in isolation.

A Short History of Imaginary Numbers

This paper is discussing how and where imaginary numbers came to be and how their extension to our classic number line helped mathematics to grow even faster. We talk about the beginning of imaginary numbers and the set of rules that come with them. We show how an error that occurred in an equation started the discovery of these. These numbers also help us achieve a better perspective towards the parabolas we see every day. At the end, you can see how these new numbers found the perfect place on the number line and fit in well with different categories we all know.

Explanation of Dark Matter, Dark Energy and Dark Space: Discovery of Invisible Universes

Theoretical and experimental studies of special processes in linear electric circuits have proved the principle of physical reality of imaginary numbers discovered 500 years ago. This principle of physical reality of imaginary numbers has allowed astrophysics to prove the existence of invisible parallel universes and explain the phenomena of dark matter, dark energy and dark space associated therewith. Physical entities corresponding to imaginary, complex and hypercomplex numbers in the other exact sciences together form an invisible and still unknown world that is to be learned by science of the future.

Bent Lyre

SANDRA LIM | 127 Bent Lyre Sandra Lim poetry I would bring out the co≠ee, bread and butter, salt, and a piece of fruit. The pigeons ran around in berserk patterns outside. Our warm, clandestine complicity had the force of a new actuality. Many mornings, we drank our co≠ee without any pleasantries. At the table, along an axis of real and imaginary numbers, you made your notes. I loved how true the imaginary numbers could be; it was much more interesting than life—you could have company and you could have loneliness in it. Your hardness and your terrifying honesty could be bitter, but I forgave the bitterness. Next to all that, my words to you must have the ring of false contrition; but like a lack of medicine or money, they are sincere. ...

The paradigm of complex probability and Monte Carlo methods

In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers and this by adding a supplementary three original axioms. Therefore, any experiment can be performed in the set of complex probabilities which is the summation of the set of real probabilities and the set of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the ‘real’ laboratory in and hence to evaluate all the probabilities. Consequently, the probability in the entire set = + is permanently equal to one no matter what the stochastic distribution of the input random variable in is, therefore the outcome of the probabilistic experiment in can be determined perfectly. This is due to the fact that the probability in is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. This novel complex probability paradigm will be applied to the classical probabilistic Monte Carlo numerical methods and to prove as well the convergence of these stochastic procedures in an original way.

Quantifying the imaginarity of quantum mechanics

The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating (RNG) operations, and stochastically RNG operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.

Discovery of Dark Space

It is currently a generally accepted opinion in astrophysics that the phenomenon of dark matter and dark energy can be explained only by existence of microcosm contents unknown to us. This assumption seems to be quite reasonable, since the Monoverse hypothesis following from the postulate about non-exceedance of the speed of light in the special theory of relativity, leaves no place for dark matter and dark energy in macrocosm. However, the assumption is erroneous, as dark matter and dark energy are actually invisible parallel universes in six-dimensional space, as shown in the article. Therefore, they can be detected only during research of macrocosm. Research of the phenomenon of dark matter and dark energy is presented in the article. Moreover, it is performed using previously unknown experiments that have allowed proof of general scientific principle of physical reality of concrete imaginary numbers. In the special theory of relativity, this principle has refuted the postulate of light speed non-exceedance. It has also allowed adjustment of relativistic formulas. It follows from the adjusted relativistic formulas that we live in a hidden Multiverse. The article explains the nature of mutual invisibility of parallel universes of the hidden Multiverse and proves verifiability of the hypothesis of the hidden Multiverse. The hypothesis has explained the nature of dark matter and dark energy as well as given an idea where antimatter is and why it does not annihilate with matter. It has also explained where tachyons are and why they do not violate the principle of causality. According to the hypothesis, the hidden Multiverse is connected to other Multiverses through portals, forming dark space. Multiverses of dark space together with the hidden Multiverse form a Hyperverse.

Physical Reality and Essence of Imaginary Numbers in Astrophysics: Dark Matter, Dark Energy, Dark Space

Dark matter and dark energy phenomenon which has been totally incomprehensible until very recently is explained by existence, besides our Universe, other invisible parallel universes in the hidden Multiverse. Such explanation of dark matter and dark energy phenomenon in astrophysics has become possible only after proving of the principle of physical reality of imaginary numbers discovered in mathematics 500 years ago. And the principle of physical reality of imaginary numbers has made the postulate of light speed non-exceedance in the special theory of relativity unnecessary. Therefore, all relativistic formulas in this theory should be adjusted. As follows from the adjusted relativistic formulas, we live in Multiverse rather than in Monoverse. All parallel universes of the Multiverse are mutually invisible, therefore Multiverse is called hidden. WMAP and Planck data analysis has showed that the hidden Multiverse has a quaternion structure and contains 20 - 22 parallel universes, of which 5 - 6 are adjacent to our universe, and others are further. And these parallel universes are linked by unidirectional and bidirectional portals. Some portals link the hidden Multiverse with other Multiverses, which form a dark space. The multiverses of dark space together with the hidden Multiverse form Hyperverse.

A Psychological View of Complex Numbers through Classical and Quantum Computing

This article explores a psychological view of zero and complex numbers under the embrace of classical with quantum computing. At first, the author broaches classical computing, after-the-fact learning, and the autistic finiteness of the first attention spared in high-functioning autistics. Secondly, he deals with quantum computing, before-the-fact learning, and the less-than-perfect infinity of the second attention impaired in the autistic spectrum before the age of three years. The author emphasizes that quantum computing in the second step agrees with quantum coherence, the angle Greek Pi (p) in radians, Euler’s Identity (eip + 1 = 0), and the approach of daring explorers to the schizophrenia attached to algebraic zero. Thirdly, this article shows that the alliance of classical with quantum computing, or of the first with the second attention, allows the atoned self to reach transcendental zero, quantum decoherence, and a renovated home in the arms of the Third Attention. In turn, the renewal brought by the Third Attention can be linked to the angle twice Greek Pi (t, or Tau), the “Tau” identity (eit -1 = 0), the union of real with imaginary numbers in complex numbers, and the proposition by Pythagoras that “all is number.”

Stable optical soliton in the ring-cavity fiber system with carbon nanotube as saturable absorber

Main attention focuses on the theoretical study of the ring-cavity fiber laser system with carbon nanotubes (CNT) as saturable absorber (SA). The system is modelled as a non-standard Schrödinger equation with the coefficients blended real and imaginary numbers. New stable exact soliton solution is constructed by the bilinear transformation method for the system. The influences of the key parameters related to CNTs and SA on the optical pulse soliton are discussed in simulation. The soliton amplitude and phase can be tuned by choosing suitable parameters.

A systemized parameter set applicable to microwave absorption for ferrite based materials

When electromagnetic waves are absorbed by materials, the most significant factor is the resistance to the induced current and the induced oscillations of electric and magnetic dipoles in the material. In this paper, a set of related parameters has been integrated into a theoretical system to characterize electromagnetic wave absorption from materials and to reveal the connection of the different physical processes for wave absorption. The parameters used are all associated with imaginary numbers. The system is general and can be applied to the important ferrite based materials. The system characterizes absorption that is common to a variety of physical processes. A problem found in the literature concerning the measurement of reflection loss has been corrected.

Beam driven temporal growth and spatial amplification of electrostatic waves in a partially spin polarized degenerate magnetized plasma

The temporal growth and spatial amplification of electrostatic waves due to the interaction of an electron beam with a partially spin polarized degenerate magnetized plasma are studied. Using Fourier analysis, a generalized linear dispersion relation is derived for parallel propagating electrostatic waves in the presence of electron spin polarization. The electron spin polarization gives birth to a new spin-dependent wave (spin electron acoustic wave) in the real wave spectrum. Furthermore, increasing spin polarization reduces the temporal growth rate of the unstable mode. Regarding spatial amplification (magnitude of the imaginary wave number), it is found that spin polarization significantly affects the magnitude of both the real and the imaginary wave numbers. Additionally, how the various other parameters such as electron beam density, beam thermal speed, and streaming speed of the beam electron influence the temporal growth and spatial rates is also discussed. These effects would have a strong bearing on instability phenomena in the degenerate metallic and semiconductor plasmas and in astrophysical plasmas.

Generalized Trigonometric Functions and Matrix Parameterization

The generalized trigonometric functions (GTF) have been introduced using an appropriate redefinition of Euler type identities involving non-standard forms of imaginary numbers, realized by different types of matrices. In this paper we use the GTF to get parameterization of practical interest for non-singular matrices. The possibility of using this procedure to deal with applications in electron transport is also touched on.

I for Imaginary numbers

This paper surveys the history of complex numbers from their initial discovery to their complete acceptance and their use in contemporary physics and mathematics. The initial rejection of these new numbers by figures as important as Leibniz and Cauchy is what strikes us most and makes us reflect on how distrust, by even competent people, sometimes delays the adoption of important and valuable content.

Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy

There are currently a large number of Multiverse hypotheses, which are, however, non-verifiable, i.e. they can be neither confirmed nor refuted experimentally even in the distant future. In contrast, the hypothesis of the hidden Multiverse considered in the article is verifiable and therefore has a right to be called a theory. The theory uses the principle of physical reality of imaginary numbers discovered 500 years ago, including complex and hypercomplex numbers, as fundamental and proved by the author theoretically and experimentally. This principle has allowed revealing a number of serious mistakes in the special theory of relativity. An adjusted version of the special theory of relativity has been proposed and the theory of the hidden Multiverse has been developed on its basis. The Multiverse has been referred to as hidden, because parallel universes it contains are mutually invisible. The nature of their invisibility is explained in the article. It is shown that dark matter and dark energy are other universes of the hidden Multiverse apart from ours. Analysis of data from WMAP and Planck spacecrafts has shown that the hidden Multiverse has quaternion structure comprising four pairs of universes and antiverses (i.e., four pairs of matter and antimatter).

The Ghost Operator and Its Applications to Reveal the Physical Meaning of Reactive Power for Electrical and Mechanical Systems and Others

In science and engineering, many concepts are introduced without a clear physical meaning, e.g., imaginary numbers in mathematics and reactive power in electrical engineering. In this paper, a new operator, coined the ghost operator $g$ , is introduced to physically construct the ghost of a system. It satisfies $g^{2}=-1$ but is different from the imaginary operator. With the help of the port-Hamiltonian systems theory, it is proved that the ghost of a system behaves exactly in the opposite way as the original system. This brings the ghost of a system into reality and paves the way to reveal the physical meaning of some imaginary concepts. Two applications are given as an example. One is to reveal the physical meaning of reactive power in electrical systems: it is the (real) power of the ghost system, which leads to a significantly simplified instantaneous power theory called the ghost power theory. The other is to define the reactive power for mechanical systems to complete the electrical–mechanical analogy. As a matter of fact, the resulting instantaneous power theory is generic and applicable to any dynamic system that can be described by the port-Hamiltonian model.