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"Complex number"
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Fantastic numbers and where to find them : a journey to the edge of physics
For particularly brilliant theoretical physicists like James Clerk Maxwell, Paul Dirac or Albert Einstein, the search for mathematical truths - via ever more mind-boggling numbers - led to strange new understandings of reality. But what are these mysterious numbers that explain the universe? In this book, leading theoretical physicist and YouTube star Antonio Padilla takes us on an irreverent cosmic tour of nine of the most extraordinary numbers in physics.
BOOK
Generalization of Dempster–Shafer theory: A complex mass function
Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster’s combination rule is exploited. In contrast to the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between the evidences [inline-graphic not available: see fulltext] is released in the generalized Dempster’s combination rule. Hence, it is more general and applicable than the classical Dempster’s combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences [inline-graphic not available: see fulltext] is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm.
Journal Article
Fuzzy stationary Schrödinger equation with correlated fuzzy boundaries
This article introduces the space of A-linearly correlated fuzzy complex numbers. Using this space, we study the stationary Schrödinger equation with boundary conditions are given by fuzzy complex numbers. This equation plays an special role in Quantum Mechanics describing the state of the system. We apply the formalism to the step potential, generating quantum results consistent with traditional quantum results.
Journal Article
On the Lucas-Leonardo numbers in complex and dual-complex number systems
This study aimed to introduce the Lucas-Leonardo numbers in 2-dimensional real algebra and 4-dimensional real Clifford algebra, namely, complex and dual-complex Lucas-Leonardo numbers, respectively. In this sense, basic algebraic properties of these numbers were presented as well as some Fibonacci-type identities such as Cassini, Catalan, and d'Ocagne. The generating function and Binet formula were constructed for the complex and dual-complex forms of Lucas-Leonardo numbers. Some relations between these numbers and other well-known integer sequences were proven. Moreover, some formulas related to the sums of the terms of these sequences were established.
Journal Article
Self-evident Automated Geometric Theorem Proving Based on Complex Number Identity
How to automatically generate short and easy-to-understand proofs for geometric theorems has long been an issue of concern in mathematics education. A novel automated geometric theorem proving method based on complex number identities is proposed in this paper, which acts as a bridge between geometry and algebra. According to the proposed method, the geometric relations in the given proposition are first transformed into a complex number expression, then the complex number identity is generated by the elimination method; finally, the closure property under all four operations of real numbers is employed to prove the proposition. A test on more than 300 geometric problems shows that the proposed method is highly effective, and the corresponding proofs are short, with obvious geometric meaning.
Journal Article
Methods for solving LR-bipolar fuzzy linear systems
In this paper, we propose a technique to solve
LR
-bipolar fuzzy linear system(BFLS),
LR
-complex bipolar fuzzy linear (CBFL) system with real coefficients and
LR
-complex bipolar fuzzy linear (CBFL) system with complex coefficients of equations. Initially, we solve the
LR
-BFLS of equations using a pair of positive
(
∗
)
and negative
(
∙
)
of two
n
×
n
LR
-real linear systems by using mean values and left-right spread systems. We also provide the necessary and sufficient conditions for the solution of
LR
-BFLS of equations. We illustrate the method by using some numerical examples of symmetric and asymmetric
LR
-BFLS equations and obtain the strong and weak solutions to the systems. Further, we solve the
LR
-CBFL system of equations with real coefficients and
LR
-CBFL system of equations with complex coefficients by pair of positive
(
∗
)
and negative
(
∙
)
two
n
×
n
real and complex
LR
-bipolar fuzzy linear systems by using mean values and left-right spread systems. Finally, we show the usage of technique to solve the current flow circuit which is represented by
LR
-CBFL system with complex coefficients and obtain the unknown current in term of
LR
-bipolar fuzzy complex number.
Journal Article
Spectral Properties of Dual Unit Gain Graphs
In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.
Journal Article
On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication
Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π -periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.
Journal Article
The Exponential Versus the Complex Power ez Function Revisited
The complex exponential function exp is a well-known entire function. In this paper, we recall its relation with the definition of the complex power of a complex number, which emanates that the complex power ez may coincide with it at some complex values. Still, on most occasions, the power represents a much broader spectrum of complex values. We also outsight how the software Mathematica may become a valuable tool for evaluating and visualizing complex power functions, in some cases by introducing some specific commands that have not been implemented in the software.
Journal Article
Multidual Complex Numbers and the Hyperholomorphicity of Multidual Complex-Valued Functions
We develop a rigorous algebraic–analytic framework for multidual complex numbers DCn within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to DCn, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development.
Journal Article