Excursions in the World of Mathematics

Entry requirements

General entry requirements and Mathematics 3b/3c or Mathematics C

Learning outcomes

On completion of the course, the student should be able to:

• use the most basic concepts and terms of number theory;
• describe the principles of Peano's axiomatic system for positive integers, be able to outline the main construction features of the number system from positive integers to octonions, and be able to carry out basic calculations with complex numbers and quarternions;
• describe with some types of integers of special historical interest and prove some important theorems related to them;
• solve number theoretical problems by using methods dealt with in the course;
• explain the history about Fermat's last theorem;
• explain the concepts denumerability and superdenumerability and the most important calculation rules for transfinite numbers;
• use the fundamental combinatorial concepts;
• explain the foundation of Euclidean geometry and its axiomatic structure, be able to prove some central theorems, know and be able to carry out basic geometric constructions with ruler and compass, and know the classical impossibility results;
• explain Euler's polyeder theorem.

Content

• Basic set theory.
• The construction of the number system from positive integers to octonions.
• The basis of number theory; figurative numbers, perfect numbers, divisibility and prime numbers.
• Basic knowledge of transfinite numbers (infinitely large numbers) and calculation with such numbers.
• Problem solving by number theory and congruence arithmetic.
• The basic concepts of combinatorics: the multiplication principle, permutations and combinations, the binomial theorem.
• The axiomatic construction of Euclidean geometry.
• Geometric constructions with ruler and compass.
• Euler's polyeder theorem. Platonic bodies.

Instruction

Lectures, lessons and exercises.

Assessment

A paper on a historical theme (3 credit points) and a final test (4.5 credit points).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.