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Florentin Smarandache - One of the best experts on this subject based on the ideXlab platform.
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finite neutrosophic Complex Numbers
arXiv: General Mathematics, 2011Co-Authors: W Vasantha B Kandasamy, Florentin SmarandacheAbstract:In this book for the first time the authors introduce the notion of real neutrosophic Complex Numbers. Further the new notion of finite Complex modulo integers is defined. For every $C(Z_n)$ the Complex modulo integer $i_F$ is such that $2F_i = n - 1$. Several algebraic structures on $C(Z_n)$ are introduced and studied. Further the notion of Complex neutrosophic modulo integers is introduced. Vector spaces and linear algebras are constructed using these neutrosophic Complex modulo integers.
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Finite Neutrosophic Complex Numbers
2011Co-Authors: W. B. Vasantha Kandasamy, Florentin SmarandacheAbstract:In this book for the first time the authors introduce the notion of real neutrosophic Complex Numbers.
Dorin Andrica - One of the best experts on this subject based on the ideXlab platform.
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Complex Numbers and Geometry
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:This chapter is dealing with the first connections between Complex Numbers and geometry and it is organized into six sections containing a rich material involving the following aspects: simple geometric notions and properties in the language of Complex Numbers, conditions for collinearity, orthogonality and concyclicity in terms of Complex Numbers, similar triangles, equilateral triangles, elements of analytic geometry in the Complex plane, the study of circles. The theoretical material is illustrated by suggestive examples and problems of various level of difficulty.
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Complex Numbers in Trigonometric Form
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:The second chapter is devoted to the study of the trigonometric form of Complex Numbers and it contains two sections dealing with the following aspects: polar representation of Complex Numbers, and the nth roots of unity. Suggestive examples and problems of various level of difficulties are included.
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Complex Numbers in Algebraic Form
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:The first chapter is devoted to the presentation of the basic concepts and results involving the algebraic form of Complex Numbers. The material is organized into two sections realizing a first connection to the plane geometry : algebraic representation of Complex Numbers, and geometric interpretation of the algebraic operations. Suggestive examples and problems are included.
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Complex Numbers from A to ... Z
2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:It is impossible to imagine modern mathematics without Complex Numbers. The second edition of Complex Numbers from A to … Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics. The exposition concentrates on key concepts and then elementary results concerning these Numbers. The reader learns how Complex Numbers can be used to solve algebraic equations and to understand the geometric interpretation of Complex Numbers and the operations involving them. The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. Many new problems and solutions have been added in this second edition. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented. The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics
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Complex Numbers from a to z
2005Co-Authors: Titu Andreescu, Dorin AndricaAbstract:Preface.- Complex Numbers in Algebraic Form.- Complex Numbers in Trigonometric Form.- Complex Numbers and Geometry.- More on Complex Numbers and Geometry.- Olympiad-Caliber Problems.- Answers, Hints and Solutions to Proposed Problems.- Glossary.- References.- Index of Authors.
Stephenie Anderson Dyben - One of the best experts on this subject based on the ideXlab platform.
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Secondary teachers’ conception of various forms of Complex Numbers
Journal of Mathematics Teacher Education, 2015Co-Authors: Gulden Karakok, Hortensia Soto-johnson, Stephenie Anderson DybenAbstract:This study explores in-service high school mathematics teachers’ conception of various forms of Complex Numbers and ways in which they transition between different representations of these forms. One 90-min interview was conducted with three high school mathematics teachers after they completed three professional development sessions, each 4 h, on Complex Numbers. Results indicate that, in general, these teachers did not necessarily have a dual conception of Complex Numbers. However, they demonstrated varying conceptions with different forms of Complex Numbers. Teachers worked at an operational level with the exponential form of Complex Numbers, but there was no evidence to indicate that they had a structural conception of this form. On the other hand, two teachers were very comfortable with the Cartesian form and exhibited a process/object duality by translating between different representations of this form. These results indicate that high school teachers need more opportunities to help them develop a dual conception of each form ( multiple duals ), which in turn can result in developing a dual conception of Complex Numbers. An interesting phenomenon that we found was that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks. This particular observation may suggest that teachers’ teaching assignments coupled with appropriate professional development activities could facilitate their understanding of these concepts.
Titu Andreescu - One of the best experts on this subject based on the ideXlab platform.
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Complex Numbers and Geometry
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:This chapter is dealing with the first connections between Complex Numbers and geometry and it is organized into six sections containing a rich material involving the following aspects: simple geometric notions and properties in the language of Complex Numbers, conditions for collinearity, orthogonality and concyclicity in terms of Complex Numbers, similar triangles, equilateral triangles, elements of analytic geometry in the Complex plane, the study of circles. The theoretical material is illustrated by suggestive examples and problems of various level of difficulty.
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Complex Numbers in Trigonometric Form
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:The second chapter is devoted to the study of the trigonometric form of Complex Numbers and it contains two sections dealing with the following aspects: polar representation of Complex Numbers, and the nth roots of unity. Suggestive examples and problems of various level of difficulties are included.
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Complex Numbers in Algebraic Form
Complex Numbers from A to ... Z, 2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:The first chapter is devoted to the presentation of the basic concepts and results involving the algebraic form of Complex Numbers. The material is organized into two sections realizing a first connection to the plane geometry : algebraic representation of Complex Numbers, and geometric interpretation of the algebraic operations. Suggestive examples and problems are included.
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Complex Numbers from A to ... Z
2014Co-Authors: Titu Andreescu, Dorin AndricaAbstract:It is impossible to imagine modern mathematics without Complex Numbers. The second edition of Complex Numbers from A to … Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics. The exposition concentrates on key concepts and then elementary results concerning these Numbers. The reader learns how Complex Numbers can be used to solve algebraic equations and to understand the geometric interpretation of Complex Numbers and the operations involving them. The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. Many new problems and solutions have been added in this second edition. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented. The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics
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Complex Numbers from a to z
2005Co-Authors: Titu Andreescu, Dorin AndricaAbstract:Preface.- Complex Numbers in Algebraic Form.- Complex Numbers in Trigonometric Form.- Complex Numbers and Geometry.- More on Complex Numbers and Geometry.- Olympiad-Caliber Problems.- Answers, Hints and Solutions to Proposed Problems.- Glossary.- References.- Index of Authors.
Yatsuka Nakamura - One of the best experts on this subject based on the ideXlab platform.
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The Inner Product and Conjugate of Finite Sequences of Complex Numbers
2005Co-Authors: Wenpai Chang, Hiroshi Yamazaki, Yatsuka NakamuraAbstract:Summary. The concept of “the inner product and conjugate of finite sequences of Complex Numbers” is defined here. Addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of “conjugate of finite sequences of field”. Many equations for such operations consist like a case of “conjugate of finite sequences of field”. Some operations on the set of n-tuples of Complex Numbers are introduced as well. Additionally, difference of such n-tuples, complement of a n-tuple and multiplication of these are defined in terms of Complex Numbers.
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The Inner Product and Conjugate of Matrix of Complex Numbers
2005Co-Authors: Wenpai Chang, Hiroshi Yamazaki, Yatsuka NakamuraAbstract:Concepts of the inner product and conjugate of matrix of Complex Numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a Complex field. Many equations for such operations consist like a case of the conjugate of matrix of a field and some operations on the set of sum of Complex Numbers are introduced.