Imaginary Numbers Research Articles

Proving Physical Reality and Explaining the Physical Essence of Imaginary Numbers

Proving Physical Reality and Explaining the Physical Essence of Imaginary Numbers

Supersymmetry Via EDM (Electric Dipole Moment)

The article begins by mentioning the accepted correlation between Albert Einstein’s relativistic equations, as well as James Clerk Maxwell’s electromagnetic waves, with the Prandtl-Glauert equations for fluid flow. This equation-free mention nevertheless ends with 180° ΔO = + => - (180 Degree Change in Orientation Equals Positive Becomes Negative). Of course, it also means negative can become positive: 180° ΔO = - => +. Using this article’s new equations, the Electric Dipole Moment is introduced and the charges of the quarks composing a neutron are transformed. EDM’s undetectability is explained from the perspective of a hypothetical someone who doesn’t accept the Big Bang but believes the universe is literally infinite and eternal. Then quantum spin of matter particles is discussed, and extended to astronomy’s black holes, in terms of photons and gravitons not being elementary force-carrying particles but being in possession of EDM. The photon-graviton interaction producing quantum spin is proposed as electromagnetic and gravitational vectors in a figure that might be called “Vector-Tensor-Scalar Geometry”, as well as being related to quaternions plus the Higgs boson and field. Then the article returns to black holes, showing how the inability of light to escape them leads to a 4-dimensional space-time: via binary digits or base 2 mathematics, Mobius bands, figure-8 Klein bottles, and Wick rotation. The electric potential of photons and gravitons is then interpreted not strictly as a neutron-identical Electric Dipole Moment but as a vast array of pulses sharing similarities with modern computers and electronics – the binary digits of the previous sentence. As a consequence of the electric force-carriers bringing them into existence, particles of matter and antimatter are symmetrical with bosons in the sense that every boson or fermion is, at its most fundamental level, composed of binary digits. Imaginary numbers are essential to quantum mechanics, and this article connects the imaginary numbers of Wick rotation to the nature of time. Therefore, the words here are not painting a classical picture of space-time. The 1’s and 0’s of binary digits are compatible with quantum mechanics and may be referred to as the Hidden Variables which Einstein advocated to complete quantum physics, and to give its calculations an exactness which would bring a hidden order to its chaotic randomness and superficial uncertainty. If the universe can be quantized and viewed as comprised of infinitesimal ones and zeros, how could it not obey quantum physics? And if those ones and zeros are all ultimately connected by Quantum Gravity, waves and particles could never be separated but wave-particle duality would rule. To finish, many thanks to Albert Einstein for laying the foundations of this article 105 years ago when he published a paper titled “Do gravitational fields play an essential role in the structure of elementary particles?” Immediately before submission for publication, a few topics were added to this article - dark matter, precession, the Hodge conjecture, and the Riemann hypothesis.

The imaginary unit described as an algebraic ambiguity

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

There are at most finitely many singular moduli that are S-units

We show that for every finite set of prime numbers $S$ , there are at most finitely many singular moduli that are $S$ -units. The key new ingredient is that for every prime number $p$ , singular moduli are $p$ -adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$ -invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.

A NOVEL CONCEPT OF THE THEORY OF DYNAMICS OF NUMBERS AND ITS APPLICATION IN THE QUADRATIC EQUATION

Considering the basic role of numbers in Mathematics, Science, and Technology the author developed a new structure of numbers named as ‘Theory of Dynamics of Numbers.’ According to the Theory of Dynamics of Numbers, the author defined 0 (zero) is the starting point of any number and also defined 0 (zero) as a neutral number. The numbers can move in infinite directions from the starting point 0 (zero) and back to 0 (zero). The author has defined the three types of numbers: 1) Neutral Numbers, 2) Count Up Numbers, and 3) Count Down Numbers. These three types of numbers cover the entire numbers in the number system where there is no necessity for the concept of imaginary numbers. Introducing this new concept the author solved the quadratic equation in one unknown (say x) in the form ax2 + bx + c = 0, even if the numerical value of the discriminant b2 – 4ac < 0 in real numbers without using the concept of imaginary numbers. Already the author solved the quadratic equation x2 + 1 = 0 and proved that √ -1 = -1 by using the Theory of Dynamics of Numbers. The Theory of Dynamics of Numbers is a more powerful tool than that of the real and imaginary number system to explain the truth of nature.

Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges

We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert space followed by an imposition of—what effectively amounts to—a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach–Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlarges the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.Quanta 2023; 12: 164–170.

Quaternion Quantum Mechanics II: Resolving the Problems of Gravity and Imaginary Numbers

We present a quaternion representation of quantum mechanics that allows its ontological interpretation. The correspondence between classical and quaternion quantum equations permits one to consider the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase energy density, and as a result, generate gravity forcing and affect the wave speed. Consequently, gravity could be described by an index of refraction.

HOW MULTIPLE BRANCHES OF MATHEMATICS UNITE THE MOBIUS STRIP WITH MULTIPLE PHENOMENA IN PHYSICS AND COSMOLOGY

Question – How does the Riemann Hypothesis Support Topological Propulsion and Faster-than-light Travel? Answer – 
 
 a) Using the axiom that there indeed are infinitely many nontrivial zeros on the critical line (calculations have confirmed the hypothesis to be true to over 13 trillion places), the critical line is identified as the y-axis of Wick rotation. This suggests the y-axis is literally infinite and that infinity equals zero. In this case, it is zero distance in time and space. Travelling zero distance is done instantly and is therefore faster-than-light travel. 
 b) Wick rotation is essential to this article’s description of a topological (mathematical) universe and the Riemann hypothesis’ identification with Wick means the hypothesis doesn’t just apply to the distribution of prime numbers but also applies to the fundamental structure of the mathematical universe’s space-time. 
 
 An absolutely essential ingredient of time travel by applied maths is being able to journey into the remote past and future. For this reason, a paragraph about the coexistence of past, present, and future (without such coexistence, time travel could never be a possibility) is included in a section titled “Riemann Hypothesis and Wick Rotation Support Time Travel into the Past and Future”.
 This article provides insights into travelling at significant fractions of – as well as faster than - light, the Higgs boson and field, electroweak interaction, dark matter, dark energy, other dimensions, space-time (eg the expanding-universe question and time travel), quantum mechanics, quantum computers, the Riemann hypothesis, and Unidentified Flying Objects. 
 The article is also a brief summary discussing how the Matrix, the Riemann hypothesis, quaternions, Wick rotation, and imaginary numbers unite topology’s Mobius strip with known and unknown dimensions as well as the Higgs boson, Higgs field, dark matter, dark energy, antigravitons, and the static universe. The article also explains Unidentified Flying Objects as the result of future human technology.

Spatial Semantics: Tafereel, Einder and El Lissitzky

Abstract In visual awareness the expanse of pictorial space fills a slab of finite depth between a frontal plane and a backdrop at infinity. It paradoxically coexists with the flatness of a picture. The constructivist El Lissitzky (1890–1941) suggested that a formal account would therefore require paradoxical notions such as non-Euclidean geometry and imaginary numbers. Perhaps so, for pictorial and physical spaces have distinct ontological roots so they are incommensurable. Alberti’s perspective discards the third dimension. Depth is a mental quality that cannot be represented. Since the picture plane is flat, ‘depth’ has to be imaginary. Imaginary dimensions capture the ‘enchantment’ of pictorial vision. Pictures and Alberti’s perspective address the disenchanted (flat), objective aspect. Only arcane geometries with imaginary dimensions might possibly account for the phenomenology — as they indeed are found to do. Linguistic terms that refer to such imaginary objects — such as ‘tafereel’ and ‘einder’ in Dutch — are scarce.

COMPUTATION OF DISPERSION CURVES OF RAYLEIGH-LAMB WAVES BY NEWTON-RAPHSON METHOD

The paper treats the numerical construction of dispersion curves which belong to the propagation of Rayleigh-Lamb waves in isotropic elastic media. The Newton-Raphson method.is applied for computation of real, imaginary and complex wavenumbers in dependence on the frequency of the wave. Due to multivaluedness, unexpected jumps between modes of dispersion curves occur in branch tracing, which prevents successful direct application of the method. This shortcoming is eliminated in the article by introducing the arc-length continuation method. The efficiency of this supplement to the Newton-Raphson method is shown on the example of an aluminum plate with constructions of dispersion curves for real, imaginary and complex wave numbers.

Математика и физическая реальность: тема отсутствия «трансцендентального означаемого» в языке современной фундаментальной науки

Introduction. It is argued that today the expansion of mathematics into the field of physical reality is not being performed in line with the classical paradigm, but in line with the linguistic turn of non-classical philosophy. Theoretical analysis. Such aspect of nature’s “non-fittability” into the conceptual network of mathematics as the problem of asymmetry is explored. It is stated that the world, constructed according to the logic of ideal mathematical symmetries, passes into a state called unmanifested in Hinduism. In the same direction the concept of J. Baudrillard is considered, where the structures of ideal reversibility are supposed to be the most radical way to destroy what exists and the symbol of nothingness. Accordingly, ideal mathematical symmetry describes states that cannot exist as a physical reality. The settlement of “dense” reality in the mathematical language of theoretical physics is seen in irrational, transcendent and imaginary numbers, in which the ideal reversibility of mathematical operations is violated. Conclusion. The search for “supersymmetry” is regarded as a variation of general language problems. Ideal symmetry in mathematics, “supersymmetry” in fundamental physics, like any “transcendental signified”, can only exist “under the sign of strikeout”. Variations on the theme of differance are revealed here. A generating source of difference, according to J. Derrida, is always on the other side of the language.

Confirm that the imaginary number i is a closed field

In the History of mathematics of mankind, some strange symbols appeared when dealing with some mathematical problems, which were defined as imaginary numbers by mankind. The imaginary number has been idle for a long time since it was discovered. Later, mathematicians such as Gauss moved the imaginary number to the mathematical plane (Complex plane). Humans have also learned the difference between imaginary and real numbers, and have obtained the difference between the two types of numbers on the square root. My contribution is to discover the inconsistency between real and imaginary numbers. I have discovered a new method of calculating imaginary number logic that is deeply hidden. 2020 Mathematics Subject Classification: 03G27, 03F07, 03D45, 11U09, 08A05, 08A40.

Graphical representation of a real valued function in the interval where it gives imaginary value

We are introduced to the graph of a real valued function . But when values of are imaginary numbers at some points of real axis, we have no graphical representation of the function at those points. This article demonstrates a model where we construct a 3D graph of the function at those points where values of the function are imaginary numbers.

Complex Analysis and Residue Theorem

The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a functions Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

The Broccoli Curriculum [President’s Message

If you were to ask a controls engineer why one should care about eigenvalues, the answer would be obvious and immediate. Eigenvalues are absolutely critical to anyone designing controllers or wanting to understand how a system behaves. Eigenvalues can literally tell us whether or not an aircraft will crash, or if an infectious disease will turn into a global pandemic. Now, if you were to ask the same question to a college student early on in their educational journey who has just completed their first course in linear algebra, the answer would more than likely be a blank stare or some semiconfused story about how eigenvalues are these mysterious numbers that somehow connect vectors and matrices (which are probably also somewhat mysterious, by the way). Similar statements can be made about Laplace transforms, imaginary numbers, differential equations, random processes, or a whole slew of similarly important yet initially unclear objects. So why in the world would someone keep taking courses covering these topics, with no obvious endgame in sight, to one day in the distant future become a controls engineer?

Good Habitats: Strangely real imaginary numbers, an unsung founder of quantum physics and an endless AI conversation.

Good Habitats: Strangely real imaginary numbers, an unsung founder of quantum physics and an endless AI conversation.

Sum of Three Squares is Equal to a Single Square

Sum of three squares is equal to a single square is a new mathematical formula which is true for real integers,fractional numbers and for imaginary numbers also. The formula is built with two consecutive numbers and their product.

Why is the Incorrect Version of the Special Theory of Relativity Still Being Studied in Physics Textbooks, Despite all its Experimental Refutations

The article states that the existing version of the special theory of relativity (STR) is incorrect, since relativistic formulas obtained therein are incorrect; they have been incorrectly explained by using the incorrect principle of speed of light non-exceedance and entailed wrong conclusions about physical unreality of imaginary numbers and existence of only our visible universe. It mentions experimental evidence of the foregoing, obtained by the author within study of transient and resonant processes in linear electric circuits. It is shown that the existing version of the STR implies conclusions on nonexistence of tsunami and bell ringing, piano music and swinging children’s swings, as well as many other real processes. It is also shown that the existing version of the STR denies even the possibility of existence of radio- and electrical engineering. Therefore, it is concluded that existing university physics textbooks should be corrected.

Existence and stability of the periodic orbits induced by grazing bifurcation in a cantilever beam system with single rigid impacting constraint

Grazing, which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by adopting the discontinuous mapping method. Then, by means of shooting method, the periodic orbits are determined, followed by the acquisition of Jacobian matrix in the case of non-impact is obtained subsequently. In addition, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing, with the stability criterion of the grazing-induced periodic orbits given. Numerical results verify the effectiveness of theoretical analysis. What is more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.

The discovery and development of Imaginary number

The discovery of imaginary numbers and the establishment of complex systems took a long time. These nonexistent numbers have been studied in great detail by mathematicians thousands of years ago, including in Tartaglia's arithmetic competition. The general solution of Girolamo's cubic equation with one variable. All reflect the wisdom of the ancients. In modern times, as more and more mathematicians use more sophisticated methods to explore imaginary numbers, which are rarely studied, theories and systems are becoming more and more robust. Some famous mathematicians such as Euler, Gauss, Newton, Leibniz, and Descartes have made indelible contributions. These research areas also laid the foundation for the future development of science and technology physics. This review was focused on the mathematicians’ outstanding discovery of complex system.