Concept Of Complex Numbers Research Articles

A NOVEL CONCEPT: THE PRODUCT OF TWO NEGATIVELY DIRECTED NUMBERS IS A NEGATIVELY DIRECTED NUMBER THOUGH THE NEGATIVE OF A NEGATIVE NUMBER IS A POSITIVE NUMBER

In this paper, the author proved that the product of two negatively directed numbers is a negatively directed number. This article is the outcome of previously published (2021-2024) ten (10) articles of this author. It is true that the negative of a negative number is a positive number. It has been done by applying the inversion process to a negative number. The process of inversion does not satisfy the basic concept of multiplication. Multiplication is defined as the adding of a number concerning another number repeatedly. So, the process of inversion does not comply with the fundamental concept of multiplication. According to the Theory of Dynamics of Numbers there exist three types of numbers: (1) Neutral or discrete numbers (2) Positively directed numbers (3) Negatively directed numbers. In general, we use four types of operators: addition (+), subtraction (-), multiplication (x), and division (÷) in mathematical calculations. Besides these, we use the negative sign (-) as an inversion operator. The positive sign (+) and negative sign (-) also represent the direction of neutral or discrete numbers. In this paper, the author introduced Fermat's Last Theorem: xn + yn = zn for n = 2 in Bhattacharyya's Theorem to prove that the product of two negatively directed numbers is a negatively directed number using the concept of the Theory of Dynamics of Numbers. In this paper, the author framed new laws of multiplication and inversion. Also, the author has given a comparative study between the conventional method of multiplication and the present concept of multiplication citing some practical examples. The author has become successful in finding the root of a quadratic equation in real numbers even if the discriminant, b2 – 4ac < 0 without using the concept of the imaginary number and also in determining the radius of a circle even if g2 + f2 < c, in real number without using the concept of complex numbers. With one example the author proved that this theorem is applicable in ‘Calculus’ also.

A NOVEL CONCEPT OF THE BHATTACHARYYA’S THEOREM: √{-(x2+ y2)}= – √( x2+ y2 ) TO FIND THE SQUARE ROOT OF ANY NEGATIVE NUMBER INTRODUCING FERMAT’S LAST THEOREM IN REAL NUMBERS WITHOUT USING THE CONCEPT OF COMPLEX NUMBERS

In this paper, the author stated and proved Bhattacharyya’s Theorem: √{-(x2 + y2 )} = -√(x2 + y2). With the help of this theorem, the author finds the square root of any negative number introducing Fermat’s last theorem without using the concept of complex numbers. The author has introduced Fermat’s Last Theorem in Bhattacharyya’s Theorem to find the square root of any negative number in real numbers in a very simple way. Indeed it is a new invention in mathematics in this era.

복소수 개념에 관한 학생 오류 해석에 필요한 교사지식

In this study, the teacher’s knowledge and its features necessary to analyze student errors about the concept of complex numbers are revealed. The instruments used for the diagnosis of and correcting student’s responses were administered to 36 pre-service teachers. The results and analysis of this study showed that. First, a teacher’s knowledge in the step of error judgement is the key to determine the direction and accuracy of diagnosis of and correcting student’s errors. Secondly, it was confirmed that there is the need for the precise knowledge about the representation of complex numbers and the knowledge about the essential prerequisites for the harmony of the complex number system and real number system. Thirdly, pre-service teachers need to have a critical approach to the definition and operations of complex numbers. Lastly, the knowledge about the internal and external value and the usefulness of complex numbers was proposed as the teacher's knowledge necessary to analyze student's errors about the concept of complex numbers.

Revisiting Burmester theory with complex forms of Bottema’s instantaneous invariants

The objective of this study is to take advantage of using the concept of complex numbers for instantaneous geometric properties of planar motion of rigid bodies. We consider that both the fixed and the moving planes are Gaussian. Then we give Bottema’s instantaneous invariants in complex forms and we study the kinematic geometry of infinitesimally separated positions of a moving Gaussian plane regarding this formulation. The followed analytical method provides a straightforward way to describe order properties of motion. We obtain the complex forms of the inflection circle, cubic of stationary curvature and cubic of twice stationary curvature. Moreover, we give the existence conditions of Ball, Burmester and Ball–Burmester points.

수학적 지식의 발달에서 연속성 원리의 역할

When imaginary numbers were first encountered in the 16th century, mathematicians were able to calculate the imaginary numbers the same as they are today. However, it required 200 years to mathematically acknowledge the existence of imaginary numbers. The new mathematical situation that arose with a development in mathematics required a harmony of real numbers and imaginary numbers. As a result, the concept of complex number became clear. A history behind the development of complex numbers involved a process of determining a comprehensive perspective that ties real numbers and imaginary numbers in a single category, complex numbers. This came after a resolution of conflict between real numbers and imaginary numbers. This study identified the new perspective and way of mathematical thinking emerging from resolving the conflicts. Also educational implications of the analysis were discussed.

Steinmetz and the Concept of Phasor: A Forgotten Story

The history of the concept of phasor is often neglected when it is introduced in textbooks on circuits. The presentation does not emphasize the historical aspects, which is natural. This paper intends to recover the original and creative way the concept of complex numbers, then almost unknown to engineers, was applied to electric circuits in sinusoidal steady-state. As usual in physics and engineering, the theory of phasor could have been anticipated by earlier researchers, if they had followed their original reasonings. Maxwell and Heaviside had proved the meal, but could not, or were not interested in writing the recipe.

교과서에 표현된 복소수와 이에 대한 학생들의 이해 실태 분석

In this study, contents of `the 2007 revised curriculum handbook` and 16 kinds of mathematics textbooks were analyzed first. The purpose of this study is to examine the understanding state of students at general high schools by making questionnaires to survey the understanding state on contents of chapter of complex number based on above analysis. Results of research can be summarized as follows. First, the content of chapter of complex number in textbook was not logically organized. In the introduction of imaginary number unit, two kinds of marks were presented without any reason and it has led to two kinds of notation of negative square root. There was no explanation of difference between delimiter symbol and operator symbol at all. The concepts were presented as definition without logical explanations. Second, students who learned with textbook in which problems were pointed out above did not have concept of complex number for granted, and recognized it as expansion of operation of set of real numbers. It meant that they were confused of operation of complex numbers and did not form the image about number system itself of complex number. Implications from this study can be obtained as follows. First, as we came over to the 7th curriculum, the contents of chapter of complex number were too abbreviated to have the logical configuration of chapter in order to remove the burden for learning. Therefore, the quantitative expansion and logical configuration fit to the level for high school students corresponding to the formal operating stage are required for correct configuration of contents of chapter. Second, teachers realize the importance of chapter of complex number and reconstruct the contents of chapter to let students think conceptually and logically.

Geometric and algebraic approaches in the concept of complex numbers

This study explores pupils’ performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from high schools in Greece (17–18 years old). Results shed light on pupils’ use of two distinct approaches to solve complex number tasks: the geometric and the algebraic approach. The geometric approach was used more frequently, while the pupils used the algebraic approach more consistently and in a more persistent way. The phenomenon of compartmentalization indicating a fragmental understanding of complex numbers was revealed among pupils who implemented the geometric approach. A common phenomenon was pupils’ difficulty in complex number problem solving, irrespective of their preferred type of approach.