• Title/Summary/Keyword: Complex Number

### Evolution of Geometric Interpretation of Complex Number : Focused on Descarte, Wallis, Wessel (복소수의 기하적 해석의 발달 : Descarte, Wallis, Wessel를 중심으로)

• Lee, Dong-Hwan
• Journal for History of Mathematics
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• v.20 no.3
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• pp.59-72
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• 2007
• In this paper we find the germ of geometric interpretation of complex number in the Euclid Element and try to show the evolution of geometric interpretation of complex number by through Descarte, Wallis, Vessel. As a result, relations and differences between them are found. They related line with complex number and interpreted complex number geometrically by generalizing the multiplication operation.

### On Teaching Materials by Using the Rotations about the Origin and the Reflections in Lines through the Origin

• Tanaka Masaki;Yamaguti Kiyosi
• Research in Mathematical Education
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• v.9 no.3 s.23
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• pp.257-267
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• 2005
• When notions of numbers are expanded from natural number to complex number, a similar mathematical phenomenon can be observed in each number. As a case study, to complex number, the phenomenon is investigated carefully and teaching materials are created. Then complex number is expressed with matrices and is geometrically treated, so a new number which is an extension of complex number is discovered. Thus, teaching material regarding to complex number and matrices is made for students of ordinary level. Moreover, for talented students, material about an extension of complex number can be added to the previous one.

### The algebraic completion of the rational numbers based on ATD (ATD에 근거한 유리수의 대수학적 completion에 관한 연구)

• Kim, Boo-Yoon;Chung, Gyeong-Mee
• The Mathematical Education
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• v.50 no.2
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• pp.135-148
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• 2011
• We can say that the history of mathematics is the history on the development of the number system. The number starts from Natural number and is constructed to Integer number and Rational number. The Rational number is not the complete number analytically so that Real number is completed by the idea of the nested interval method. Real number is completed analytically, however, is not by algebra, so the algebraically completed type of the rational number, through the way that similar to the process of completing real number, is Complex number. The purpose of this study is to show the most appropriate way for the development of the human being thinking about the teaching and leaning of Complex number. To do this, We have to consider the proof of the existence of Complex number, the background of the introduction of Complex number and the background knowledge that the teachers to teach Complex number should have. Also, this study analyzes the knowledge to be taught of Complex number based on the anthropological theory of didactics and finally presents the teaching method of Complex number based on this theory.

### Complex number on textbooks and Analysis on understanding state of students (교과서에 표현된 복소수와 이에 대한 학생들의 이해 실태 분석)

• Park, Seon-Ho;Pyo, Sung-Soo
• The Mathematical Education
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• v.51 no.1
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• pp.1-19
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• 2012
• 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.

### Analysis of planar optical waveguides using incident angle of complex number (복소수 입사각을 이용한 평판 광도파로 해석)

• 임영준;김창민
• Journal of the Korean Institute of Telematics and Electronics A
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• v.33A no.5
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• pp.149-154
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• 1996
• We propose the concept of incident angle of complex number and analyze planar optical waveguides by applying the concept. The incident angle of complex number is concerned with the modeling of prism-gap-waveguide structures. It is shown that, when optical waveguides are analyzed by use of the transfer matrix method, the proposed concept enable us to find solutions faster and more accurately than ghatak's method which introduces the leaky structure.

### A New Complex-Number Multiplication Algorithm using Radix-4 Booth Recoding and RB Arithmetic, and a 10-bit CMAC Core Design (Radix-4 Booth Recoding과 RB 연산을 이용한 새로운 복소수 승산 알고리듬 및 10-bit CMAC코어 설계)

• 김호하;신경욱
• Journal of the Korean Institute of Telematics and Electronics C
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• v.35C no.9
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• pp.11-20
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• 1998
• High-speed complex-number arithmetic units are essential to baseband signal processing of modern digital communication systems such as channel equalization, timing recovery, modulation and demodulation. In this paper, a new complex-number multiplication algorithm is proposed, which is based on redundant binary (RB) arithmetic combined with radix-4 Booth recoding scheme. The proposed algorithm reduces the number of partial product by one-half as compared with the conventional direct method using real-number multipliers and adders. It also leads to a highly parallel architecture and simplified circuit, resulting in high-speed operation and low power dissipation. To demonstrate the proposed algorithm, a prototype complex-number multiplier-accumulator (CMAC) core with 10-bit operands has been designed using 0.8-$\mu\textrm{m}$ N-Well CMOS technology. The designed CMAC core contains about 18,000 transistors on the area of about 1.60 ${\times}$ 1.93 $\textrm{mm}^2$. The functional and speed test results show that it can operate with 120-MHz clock at V$\sub$DD/=3.3-V, and its power consumption is given to about 63-mW.

### LEVEL-m SCALED CIRCULANT FACTOR MATRICES OVER THE COMPLEX NUMBER FIELD AND THE QUATERNION DIVISION ALGEBRA

• Jiang, Zhao-Lin;Liu, San-Yang
• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.81-96
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• 2004
• The level-m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level-m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given.

### REPRESENTATION OF INTUITIONISTIC FUZZY SOFT SET USING COMPLEX NUMBER

• KHAN, MOHSIN
• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.331-347
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• 2017
• Soft sets are fantastic mathematical tools to handle imprecise and uncertain information in complicated situations. In this paper, we defined the hybrid structure which is the combination of soft set and complex number representation of intuitionistic fuzzy set. We defined basic set theoretic operations such as complement, union, intersection, restricted union, restricted intersection etc. for this hybrid structure. Moreover, we developed this theory to establish some more set theoretic operations like Disjunctive sum, difference, product, conjugate etc.

### Development of the concept of complex number and it's educational implications (복소수 개념의 발달과 교육적 함의)

• Lee, Dong-Hwan
• Journal for History of Mathematics
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• v.25 no.3
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• pp.53-75
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• 2012
• 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.

### A high-speed complex multiplier based on redundant binary arithmetic (Redundant binary 연산을 이용한 고속 복소수 승산기)

• 신경욱
• Journal of the Korean Institute of Telematics and Electronics C
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• v.34C no.2
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• pp.29-37
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• 1997
• A new algorithm and parallel architecture for high-speed complex number multiplication is presented, and a prototype chip based on the proposed approach is designed. By employing redundant binary (RB) arithmetic, an N-bit complex number multiplication is simplified to two RB multiplications (i.e., an addition of N RB partial products), which are responsible for real and imaginary parts, respectively. Also, and efficient RB encoding scheme proposed in this paper enables to generate RB partial products without additional hardware and delay overheads compared with binary partial product generation. The proposed approach leads to a highly parallel architecture with regularity and modularity. As a results, it results in much simpler realization and higher performance than the classical method based on real multipliers and adders. As a test vehicle, a prototype 8-b complex number multiplier core has been fabricated using $0.8\mu\textrm{m}$ CMOS technology. It contains 11,500 transistors on the area of about $1.05 \times 1.34 textrm{mm}^2$. The functional and speed test results show that it can safely operate with 200 MHz clock at $V_{DD}=2.5 V$, and consumes about 90mW.