Math for Computer Scientists 2 (MA2)

Deepen your mathematical foundations: You will learn about number theory, linear algebra, and the fundamentals of cryptography. Using mathematical proof techniques, you will analyze and solve problems in computer science.

Contents

• The Mathematics 2 lecture introduces fundamental mathematical concepts from number theory and linear algebra that are of central importance for numerous applications in computer science, particularly in cryptography, computer graphics, and data processing.The first part of the lecture covers fundamental concepts of number theory. These include prime numbers and their factorization, Euler’s phi function, and various positional numeral systems for representing numbers. In addition to the familiar decimal system, numeral systems with arbitrary bases as well as the binary system are considered, providing an understanding of how numbers are represented internally by computers.Building on this foundation, the second part develops the mathematical principles underlying cryptography. The main topics include the extended Euclidean algorithm, modular arithmetic, and residue class rings. The lecture examines when elements in a residue class ring are invertible with respect to multiplication and introduces the Fermat–Euler theorem as one of the key results underlying many modern cryptographic methods, such as RSA.The third part of the lecture introduces the fundamentals of linear algebra and demonstrates their wide range of applications in computer science. Topics include systems of linear equations and their solution using the Gaussian elimination algorithm, R hoch n as a vector space, and basic operations such as the dot product and cross product. Building on these concepts, geometric objects such as lines and planes, as well as matrices and their applications, are covered, including rotation matrices, reflections, and symmetric matrices. The section concludes with inverse matrices, determinants, and eigenvalues, which are of particular practical relevance in computer graphics, data analysis, and numerous other areas of computer science.Overall, the lecture provides a solid mathematical foundation, covering both the theoretical principles and their practical applications in key areas of computer science.

Competencies

Students are able to

• understand mathematical reasoning and proof techniques.
• apply mathematical problem-solving strategies to computer science problems.
• understand and analyze fundamental cryptographic problems.
• classify and analyze data-related problems.
• apply methods of linear algebra.

Literature

• M. Plaue and M. Scherfner, Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis, 2nd ed. Berlin, Germany: Springer Spektrum, 2019. doi: 10.1007/978-3-662-58352-4.
• J. Liesen and V. Mehrmann, Lineare Algebra: Ein Lehrbuch über die Theorie mit Blick auf die Praxis, 4th ed. Berlin, Germany: Springer Spektrum, 2024. doi: 10.1007/978-3-662-67944-9.
• M. Hien, Lineare Algebra I: Geeignet zum Selbststudium oder für Inverted-Classroom-Vorlesungen. Berlin, Germany: Springer Spektrum, 2024. doi: 10.1007/978-3-662-69211-0.
• G. Fischer and B. Springborn, Lineare Algebra: Eine Einführung für Studienanfänger, 20th ed. Berlin, Germany: Springer Spektrum, 2025. doi: 10.1007/978-3-662-71261-0.
• G. Stroth and R. Waldecker, Elementare Algebra und Zahlentheorie, 3rd ed. Cham, Switzerland: Birkhäuser, 2024. doi: 10.1007/978-3-031-39771-4.
• C. Karpfinger, Algebra: Gruppen – Ringe – Körper, 6th ed. Berlin, Germany: Springer Spektrum, 2024. doi: 10.1007/978-3-662-68656-0.
• J. Frischauf and T. Netzer, Zahlen und Algebra: Ein Kurs im Lehramtsstudium. Berlin, Germany: Springer Spektrum, 2024. doi: 10.1007/978-3-662-69776-4.
• M. Aigner, Zahlentheorie: Eine Einführung mit Übungen, Hinweisen und Lösungen. Wiesbaden, Germany: Springer Spektrum, 2012. doi: 10.1007/978-3-8348-8344-5.
• N. Oswald and J. Steuding, Elementare Zahlentheorie: Ein sanfter Einstieg in die höhere Mathematik. Berlin, Germany: Springer, 2015. doi: 10.1007/978-3-662-44248-7.
• K. Reiss and G. Schmieder, Basiswissen Zahlentheorie: Eine Einführung in Zahlen und Zahlbereiche. Berlin, Germany: Springer Spektrum.
• F. Lemmermeyer, 4000 Jahre Zahlentheorie: Geschichte – Kulturen – Menschen I. Von Babel bis Abel. Berlin, Germany: Springer Spektrum, 2023. doi: 10.1007/978-3-662-68110-7.
• S. Müller-Stach and J. Piontkowski, Elementare und algebraische Zahlentheorie: Ein moderner Zugang zu klassischen Themen. Berlin, Germany: Springer, 2008. doi: 10.1007/978-3-8348-9064-1.

Lecturer

• Prof. Dr. Elena Fimmel
• Prof. Dr. Miriam Föller-Nord
• Prof. Dr. Lutz Strüngmann
• Prof. Dr. Ivo Wolf

Recommended Previous Knowledge 

• Math for Computer Scientists 1 (MA1)

Module Details

Semester	2
Lecture Language	German
Frequency	Every Term
Credit Points (ECTS)	5
Course Coordinator	Prof. Dr. Lutz Strüngmann
Duration	1 Semester
Course Achievement	None
Prerequisite for exam	Compulsory Assignment (PU)
Exam	Written Exam 90 Minutes (K90)

Weekly Hours (SWS) 

Lecture	2 SWS
Exercises	2 SWS
Sum	4 SWS

Work Load 

Lecture	45 h
Self Study	60 h
Assignments	15 h
Exam Preparation	30 h
Sum	150 h