Complex Numbers

Researchers are often perplexed when their machine learning algorithms are required to deal with complex number. Various strategies are commonly employed to project complex number into real number, although it is frequently sacrificing the information contained in the complex number. This paper proposes a new method and four techniques to represent complex number as real number, without having to sacrifice the information contained. The proposed techniques are also capable of retrieving the original complex number from the representing real number, with little to none of information loss. The promising applicability of the proposed techniques has been demonstrated and worth to receive further exploration in representing the complex number.

Any cyclic quadrilateral whose sides are not parallel can define a triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the continuations of the quadrilateral’s pairs of opposite sides. Using a cyclic quadrilateral and this triangle, the following four circles may be defined: the circumcircle of the quadrilateral, two circles with diameters that are the sides of the triangle that issue from the point of intersection of the quadrilaterals diagonals, and the Euler circle of the triangle. In the current paper, we shall prove four properties that relate the quadrilateral, the triangle and these circles. We shall show that the two circles whose diameters are the sides of the triangle are perpendicular to the circumcircle of the quadrilateral. We shall prove equalities that relate the angle between the midlines in the quadrilateral with other angles. We shall show that the point of intersection of th...

These are my original research of elementary complex numbers applied on the triangle. The vertices of the triangle are given as the initial elements with which in the complex plane are defined sides, area, center of gravity, orthocenter and circumcenter of the triangle along with some other features that have been discovered along the way.

Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.

Abstract: We find the applications of complex numbers in almost all fields of modern science and engineering. Due to its much application a lot of research work is underway to reduce the computation requirements for the Complex binary numbers. Recent research indicates ...

The aim of this study was to examine the abstraction process of tenth grade students' complex number knowledge. A semi-structured interview was held with two students who were successful at mathematics and who voluntarily participated in the study. In this interview, the students faced three different research problems prepared by the researcher in order to reveal the students' knowledge formation processes. The students studied these research problems together. In the meantime, their cognitive processes related to recognizing, building-with, and constructing of discourses was analysed. The analyses indicated that both of the students recognized and built-with the preliminary knowledge necessary for them to construct the knowledge of complex numbers. Moreover, it was understood that one of these students recognized and build-with their previous knowledge of linear functions, coordinate systems, and parallel displacement. In conclusion, it could be said that this student cons...

This paper, through the exploration of complex numbers, attempts to conjecture that "1/2 does not equal 2/4".

In 〖(-b)〗^(p/q)  = 〖(-b)〗^(2kp/2kq)  , p/q = 2kp/2kq is true when the numerator is even. For an odd numerator, it only yields true when the absolute value of the result is taken. This indicates that odd/odd fractions does not always equal even/even  fractions. An example of this is that 1/3 does not always equal 2/6. This indicates that odd/even fractions does not always equal even/even  fractions. An example of this is that 1/2 does not always equal 2/4.

It is considered the elements of complex numbers. In particular, rotation in standard complex plane, the real product (dot product), with some applications in geometry. The generalizations to complex matrices and quaternions are included. You will notice that the known definition of dot product in standard theory is not valid more, so I suggested to call it " compound sum " .

Abstract: We find the applications of complex numbers in almost all fields of modern science and engineering. Due to its much application a lot of research work is underway to reduce the computation requirements for the Complex binary numbers. Recent research indicates ...

In this paper we will propose a direct method which is accurate and fast to solve the equations of the second degree in the general case (which means with complex coefficients). We will also propose an algorithm that will make our calculations more easy. Finaly we found also the general formula that gives us the second roots (square root) of a complex number.

(e^iPi-1=0) Dokaz (Proof)
Identitet sadrži pet značajnih konstanti: o 0, neutral sabiranja, (The additive identity) o 1, neutral množenja, (The multiplicative identity) o e, Ojlerov broj (Kalkulus) o π, (Geometrija) o i, (Imaginarni, kompleksni brojevi a+bi √) Broj e, poznat kao Ojlerov broj ili Neperova konstanta, osnova je prirodnog logaritma. Osim što je iracionalan i realan, ovaj broj je još i transcedentan (nije rešenje nijedne algebarske jednačine sa racionalnim koeficijentima. Konstantu je otkrio švajcarski matematičar Jacob Bernoulli tokom proučavanja složenih kamata. e ≈ 2,71828 18284 59045 23536 02874 71352... Pi ili π je matematička konstanta, danas široko primenjivana u matematici i fizici. Njena približna vrednost je 3,14159, a definiše se kao odnos obima i prečnika (d=2r) kruga ili kao odnos površina kruga i kvadrata nad njegovim poluprečnikom. Pi je takođe poznato i kao Arhimedova konstanta. Uređeni par realnih brojeva a i b, označen sa (a,b), pri čemu su a i b realni brojevi naziva se kompleksan (složen) broj. Skup svih ovakvih parova, odnosno svih kompleksnih brojeva, označavamo sa C i on je suštinski Dekartov proizvod C=RXR. Uređeni par (a,b), kao kompleksan broj, zapisuje se još kao a+bi, element i naziva se imaginarna jedinica, i ima svojstvo da je Imaginarni broj se u fizici često obeležava i latiničnim slovom j.

Les nombres complexes sont des objets qui se situent à la transition entre lycée et université. Mais ce sont également des objets interdisciplinaires : construits par le mathématicien, ils sont utilisés par le physicien et discutés par le philosophe. Nous examinons au cours de cet atelier des exemples de scénarios pédagogiques visant à croiser les regards sur les nombres complexes. Nous montrons également, en nous appuyant sur cet exemple, comment la réflexion épistémologique peut-être utile afin de faciliter la transition lycée-université.

Today's methods for computing orientation are quaternion and rotation matrix. However, their efficiencies are tarnished by the complexity of the rotation matrix and the counterintuitivity of quaternion. A better method is presented here. It uses complex multiplication for rotating vectors in 3D space and can compute orientation without angle and trigonometric functions, which is simple, intuitive and fast.

This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

This paper is informed by Charles Sanders Peirce’s philosophy as semiotics or the
doctrine of signs. The paper’s purpose is to explore Peirce’s category of abduction as not being
limited to the inference to the best explanation. In the context of the logic of discovery, abduction
is posited as a necessary although not sufficient condition for the production of meanings. The
structure of a genuine sign is triadic and represents a synthesis between precognitive ideas and
conceptual representations. The novel model of reasoning is offered, based on the mathematical
formalism borrowed from Gauss’ interpretation of the complex number. It is suggested that
this model in a form of a diagram not only represents a semiotic process-structure but also
overcomes the long-standing paradox of new knowledge. For Peirce, it is a diagram as a visual
representation that may yield solutions to the otherwise unsolvable logical problems. What
appears to us as a paradox is the very presence of abductive, or hypothetical, inference, as
Peircean generic category of Firstness within the Thirdness of the total thought-process. Firstness
(feeling), Secondness (action), and Thirdness (reason) together constitute a dynamic structure of
experience.

The paper presents Mirrored Vedic Vertically and Crosswise Multiplication Technique (MVVCMT) which is an algorithm based on Vedic Vertically and Crosswise Multiplication Technique. Vedic Vertically and Crosswise Multiplication Technique is an ancient Indian technique used to shorten the process of mental multiplication especially for big numbers. In India, the multiplication technique is still taught to kids to enhance their skills in mental multiplication. The proposed algorithm in this research was inspired by this ancient yet practical, easy to understand and apply multiplication technique. Vedic Vertically and Crosswise Multiplication Technique was rediscovered in 1965 by Swami Bharati Krishna Tirthaji in his book Vedic Mathematics. The proposed algorithm runs with only 100 multiplications rather than n2 based number of multiplications. In this paper, the first section explains the Vedic Vertically and Crosswise Multiplication Technique with and example and algorithm. Then the p...

The topic has its source from complex analysis, univalent function theory and geometric function theory. Authors have studied the Bernoulli Lemniscate in different ways and their interesting results are inherent everywhere. But the class of (j,i) -symmetric functions in this direction are not so common in literature. This research work shall introduce a new class of functions defined using the concept of (j,i) – symmetric functions that are related to the Bernoulli lemniscate. It would be used to provide further results in this direction. The initial coefficients were obtained to obtain the second Hankel determinant.

La idea de construcción matemática está determinada por la concepción lógica de nuestro pensamiento. Existe una tendencia natural en esta concepción por la generalización de los conceptos que ha servido como una metodología para alcanzar un entendimiento más claro de dichos conceptos. Algo similar sucede en la construcción de sistemas numéricos que ha dado lugar al desarrollo del álgebra abstracta. Por la conquista de la universalidad, en la lucha del pensamiento por alcanzar dicha generalización, nos hemos visto en la irremediable situación de generar un sistema que permita que cada paso dado sea sólido y definitivo. Al definir ciertas propiedades matemáticas, un sistema algebraico, es puesto en alguna categoría algebraica bien definida que delimita los alcances de dicha estructura algebraica. El uso o aplicaciones que tenga algún sistema algebraico servirá de herramienta y abrirá camino en el entendimiento de la naturaleza de los fenómenos que conciernen al pensamiento humano. Se ha dilucidado de manera muy breve el camino que se ha seguido en la historia de la extensión del campo de los complejos; un enigma interesante y elusivo, sin duda. Algunas propuestas a este problema han emergido de grandes mentes y se les ha visto fracasar por su alto grado de complejidad o su falta de solidez en su estructura matemática. Algunas otras no convencen del todo a la comunidad científica que se dispone a hacer uso de estas herramientas. Tal es el caso de los cuaternios que funcionan de manera sobresaliente en cuanto a que este sistema conserva la mayoría de las propiedades de campo excepto que no es un álgebra conmutativa y se limita a 1+3 dimensiones.

This work shows how the Riemann zeta function can be transformed into a class of equations that are found in continuous optimization. The Diophantine attribute of the Riemann Zeta function is also discussed for the integral values of. Riemann Zeta function is also written as a reminiscence of its non-trivial zeros. Matrix representations of the Riemann zeta function are also obtained through which the general form of the point spectrums and the trace of the Riemann zeta function can be generated.

The paper presents an analysis of imaginary quantity before Gauss based on the notion of continuity and symbolic representation. Its aim is to uncover subtle roots of the "impossible", "sophisticated" or "absurd" entities that, as we claim, stem from the Renaissance notion of nature and from the "symbolic turn" which occurred in that period. In order to grant impossible quantities a reasonable (operational) meaning, it is necessary to establish an equation (formal continuity) between real and imaginary. It is possible only if the real is in a sense subsumed within the symbolic which holds paradigmatically for the notions of number and magnitude. For, once number and magnitude become symbolic representations of the same universal intellectual matter of quantity, an operational analogy and continuity between them may be established. Three "continuities" shall be distinguished on the path to such "universal mathematics" at the end of which the imaginary entities may acquire the citizenship in the Republic of numbers.

The analytic continuation formula of (), () is presented as the power of , in this paper. The Riemann Zeta functions (Riemann 1859), is also presented as a function with a real and imaginary parts; thus being able to write () , which is the similitude of its zeros [1].

The second part of the presentation of complex numbers illustrate its representation on a plane known as the Argend diagram as well as further properties of complex numbers such as its principal argument. Finally, the presentation also illustrates the conversion of complex numbers from Cartesian form to polar and exponential forms.

The paper presents Mirrored Vedic Vertically and Crosswise Multiplication Technique (MVVCMT) which is an algorithm based on Vedic Vertically and Crosswise Multiplication Technique. Vedic Vertically and Crosswise Multiplication Technique is an ancient Indian technique used to shorten the process of mental multiplication especially for big numbers. In India, the multiplication technique is still taught to kids to enhance their skills in mental multiplication. The proposed algorithm in this research was inspired by this ancient yet practical, easy to understand and apply multiplication technique. Vedic Vertically and Crosswise Multiplication Technique was rediscovered in 1965 by Swami Bharati Krishna Tirthaji in his book Vedic Mathematics. The proposed algorithm runs with only 100 multiplications rather than n 2 based number of multiplications. In this paper, the first section explains the Vedic Vertically and Crosswise Multiplication Technique with and example and algorithm. Then the paper presents a hands-on example to show the simplicity of the original technique. Next, the paper presents the proposed algorithm name "Mirrored Vedic Vertically and Crosswise Multiplication Technique" which is dubbed (MVVCMT).

The imaginary number “i” is an amphibian as it links two real numbers and renders the ordered pair of real numbers complex: z=x+iy.  Its discovery has made rapid stride in mathematics and its application. The equation e^iπ+1=0 is one of the most profound equations. The solution to quadratic equation  x^2+1=0 are ±i.  Geometric algebra, pioneered by Hestenes, Baylis and others, has given a better place for i. It can be considered as the analogue of scalar I. We discuss briefly  about polar vectors and axial (pseudo) vectors, scalars and pseudo scalars.

It is considered the elements of complex numbers. In particular, rotation in standard complex plane, the real product (dot product), with some applications in geometry. The generalizations to complex matrices and quaternions are included. You will notice that the known definition of dot product in standard theory is not valid more, so I suggested to call it " compound sum ". To the second edition were added quaternions with complex coefficients and the Pauli matrices.

La teoría cuántica moderna se basa en el supuesto de que los estados cuánticos están representados por elementos de un espacio de Hilbert complejo. Se espera que en la futura teoría cuántica el campo numérico no se postule sino que se derive de principios más generales. Consideramos la elección del campo numérico en la Teoría Cuántica basada en un Campo de Galois (GFQT) discutido en nuestras publicaciones anteriores.

Dado que cualquier campo de Galois no está algebraicamente cerrado, en el caso general no hay garantía de que incluso un operador hermitiano tenga necesariamente eigenvalores.

Suponemos que el Álgebra de Simetría es el análogo de Campo de Galois del álgebra de De Sitter so(1,4) y consideramos representaciones irreductibles sin espín de este álgebra. Se muestra que el análogo de campo de Galois de números complejos es la extensión mínima del módulo de campo de residuo 'p' para lo cual las representaciones son completamente descomponibles.

The paper revisits the learning paradox first posited by Socrates in his famous dialogue with Meno. The paradox of new knowledge has been steadily attracting attention of educational researchers (see, e.g., Bereiter 1985; Petrie 1991; Prawat 1999). My paper briefly examines Kierkegaard's classical solution in terms of the necessity of the decisive moment arguing that, contrary to Kierkegaard, there is no miraculous knowledge. The paper justifies this assertion by suggesting a two-fold approach towards re-solving the Socratic paradox: first, using Charles Sanders Peirce's triadic scheme and his semiotics that incorporates the logical category of abduction; and second, by extending his notion of diagrammatic reasoning and suggesting a model for a cognitive structure constructed on the complex (Gauss) plane. Peirce pointed out that his categories of Firstness, Secondness and Thirdness are the "conceptions of complexity" (Peirce CP 1. 526). The paper concludes by asserting that the problem of new knowledge debated in antiquity by Socrates and Meno can be solved by adopting the geometry of complexity as a representation of the logic of the included middle, constituting the core of the triadic semiotics.