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Irrationality and Transcendence
Irrationalität und Transzendenz
Last Updated: 2026-02-05 15:24:28
Abstract
This lecture is for a better understanding of irrational numbers, namely we discuss irrationality proofs, division into algebraic and transcendental irrational numbers, theorems of Liouville and Roth, theorem of Lindemann. Especially, we will prove the irrationality and transcendence of e and pi.
Objective
Irrationality proofs, algebraic and transcendental real numbers, theorems of Liouville and Roth, theorem of Lindemann and the transcendence of e and pi.
Content
At the beginning of each mathematical education the real numbers are introduced and discussed; however, the larger part of these numbers, namely the irrational numbers, remain mysterious. In this course we start with easy irrationality proofs, e.g. for sqrt{2},e,e^2,e^sqrt{2}, and then discuss more general ones, which for example imply the irrationality of pi. For a more detailed discussion we divide the real numbers into algebraic and transcendental numbers and prove the algebraic structure of algebraic numbers and the existence of transcendental ones. Moreover, we will prove the theorem of Liouville, which allows the construction of transcendental numbers explicitly, and the famous theorem of Roth. Afterwards we prove the transcendence of e and pi; more generally the theorem of Lindemann will be proved. The course ends with a list of further results and open problems.
Resources
Lecture Notes
Lecture notes will be provided during the semester.
General Information
- Language
- German
- Levels
- BSC
Examination
- Type
- end-of-semester examination
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Irrationalität und Transzendenz |
|
2 h weekly |