Introductory complex analysis at two British Columbia universities: The first week–complex numbers

Using the ERFI Function in the Problem of the Shape Optimization of the Compressed Rod

International Journal of Applied Mechanics and Engineering ◽

10.2478/ijame-2020-0021 ◽

2020 ◽

Vol 25 (2) ◽

pp. 75-87

Author(s):

J. Marcinowski ◽

M. Sadowski

Keyword(s):

Shape Optimization ◽

Axial Force ◽

Complex Analysis ◽

Mass Conservation ◽

Structural Mechanics ◽

Complex Numbers ◽

Real Numbers ◽

The Real ◽

Numerical Example ◽

Luca Pacioli

AbstractThe shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible.The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows.There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range.One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]).We can see, that already then, they were definitely important (complex numbers).

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Developing complex analysis textbook to enhance students’ critical thinking

JRAMathEdu (Journal of Research and Advances in Mathematics Education) ◽

10.23917/jramathedu.v5i1.8741 ◽

2020 ◽

Vol 5 (1) ◽

pp. 26-37

Author(s):

Fariz Setyawan ◽

Puguh Wahyu Prasetyo ◽

Burhanudin Arif Nurnugroho

Keyword(s):

Critical Thinking ◽

Complex Analysis ◽

Thinking Skills ◽

Critical Thinking Skills ◽

Complex Numbers ◽

Thinking Skill ◽

Critical Thinking Skill ◽

Open Questions ◽

The Right ◽

D Model

Critical thinking skill was needed since problems in daily life required a solution. To solve the problems, the students collect the data as a consideration in making the right decision. Moreover, the students also need good critical thinking skills. To develop a good critical thinking skill, the researcher develops a textbook of Complex Analysis and Application, especially in complex numbers. In this research, a qualitative investigation was carried out in the development of the textbook. This textbook supports the mathematics education students of Universitas Ahmad Dahlan in learning complex analysis course. Researchers did a walkthrough to the expert in assessing the content and construct to validate the textbook. The development model used by researchers is a 4-D model. A 4-D model used in this study is consists of defining, designing, developing, and disseminating. In defining, the researcher adequate materials related to the complex numbers such as the rational number and real number and its properties in addition and multiplication. In designing, some explanation in the textbook is followed by Why and How questions. Open questions are used to stimulate the students’ critical thinking. Based on the walkthrough result, some notation in the textbook, especially mathematics symbols, must be revised using a proper mathematics equation. In addition, some concept should be clearly defined such as where the value of . As a result of the validity construct, the explanation given in the textbook used a step-by-step exercise. In disseminating, the students can easily access the textbook and answer the questions using their reasoning. The textbook is valid and readable.

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