Proofs from the Notebook (MATH 409)

2023 Fall
Faculty of Engineering and Natural Sciences
10/6 ECTS (for students admitted in the 2013-14 Academic Year or following years)
Kağan Kurşungöz,
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Formal lecture,Interactive lecture
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The aim of this course is to introduce a selection of proofs of some important theorems. These proofs require moderate background but high ingenuity. Among the topics are: Division algorithm, prime factorization theorem, some primitive results on the distribution of primes. Greatest common divisor. Euler's totient function. Phytagorean triples. A short survey of metric spaces; continuity, compactness, connectedness. Stone- Weierstrass approximation theorem. Geometry of the sphere. Brouwer fixed point theorem. Borsuk's antipodal mapping theorem.


To introduce creative ideas in different fields of mathematics with as little
mathematical background as possible, to show how different fields of mathematics are
related to each other, to enhance students' analytical skills.


  • Upon completion of the course, student should be able to Perceive relations between different branches of mathematics,
  • Understand advanced creative ideas involved in some of the fundamental results,
  • Develop proofs using ideas similar to those introduced in the course,
  • Have a sense of the history of mathematics,
  • Learn some of the major open problems in mathematics.


1. Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. 1

2. Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 5

3. Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects; have the ability to continue to educate him/herself. 4

4. Communicate effectively in Turkish and English by oral, written, graphical and technological means. 3

5. Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. 1

1. Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 3

2. Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 1

3. Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. 1

4. Have the ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose. 1

5. Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 1

6. Possess knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 1

7. Possess knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; knowledge of behavior according to ethical principles, understanding of professional and ethical responsibility. 1

8. Have the ability to write effective reports and comprehend written reports, prepare design and production reports, make effective presentations, and give and receive clear and intelligible instructions. 1

1. Use mathematics (including derivative and integral calculations, probability and statistics, differential equations, linear algebra, complex variables and discrete mathematics), basic sciences, computer and programming, and electronics engineering knowledge to (a) Design and analyze complex electronic circuits, instruments, software and electronics systems with hardware/software or (b) Design and analyze communication networks and systems, signal processing algorithms or software 1

1. Applying fundamental and advanced knowledge of natural sciences as well as engineering principles to develop and design new materials and establish the relation between internal structure and physical properties using experimental, computational and theoretical tools. 1

2. Merging the existing knowledge on physical properties, design limits and fabrication methods in materials selection for a particular application or to resolve material performance related problems. 1

3. Predicting and understanding the behavior of a material under use in a specific environment knowing the internal structure or vice versa. 1

1. Formulate and analyze problems in complex manufacturing and service systems by comprehending and applying the basic tools of industrial engineering such as modeling and optimization, stochastics, statistics. 1

2. Design and develop appropriate analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy. 1

3. Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 1


  Percentage (%)
Final 25
Midterm 25
Assignment 25
Participation 25



Auxiliary reading is:
Tosun Terzioğlu, "Bir Analizcinin Defterinden Seçtikleri"
(Nesin Yayınları, ikinci basım, 2014)
It is accessible as online resource through IC. However, the auxiliary reading is in Turkish. We will just follow the order of contents there.

Another source for auxiliary reading is:
Martin Aigner, Günter M. Ziegler, ?Proofs from THE BOOK?
(Springer, 2nd ed., 2013)