The numbers π, φ and e (Bachelor thesis)

Λαμπίδη, Κυριακούλα/ Lampidi, Kyriakoula

This master’s dissertation researches into the irrational numbers π, φ and e. It attempts to answer questions such as: what these numbers are, how they occurred, who studied them and when. The first chapter concerns the sets of numbers. At the beginning, it presents the meaning of the set and relevant definitions, for example which sets are called equal, what a subtotal is and how they are represented. Next, it analyses the sets of numbers emphasizing the irrational numbers. The first chapter finishes with the algebraic and the transcendental numbers. The second chapter is about the number π. Firstly, it provides a historical overview and approximations of the number by mathematicians. Then, it proves the irrationality and transcendence of π. The second chapter closes with the relation between the number π and the problem of squaring the circle. The third chapter deals with golden section φ. After a historical overview, approximations of the number are mentioned, followed by a reference to its relevance to algebra. In other words, its definition and structure are given. Subsequently, the connection between geometry and the number φ is shown. This chapter ends with the presence of the number φ in art and more specifically in painting, sculpture, architecture and even music. Finally, the fourth chapter presents the number e, by emphasizing the historical overview and evidence of irrationality and transcendence. The final chapter concludes with Euler’s identity, which not only combines three of the most important arithmetic operations (addition, multiplication and exponentiation) but also links the most fundamental mathematical constants, the number 0, the number 1, the number π and the number e.
Institution and School/Department of submitter: Δημοκρίτειο Πανεπιστήμιο Θράκης. Σχολή Επιστημών Αγωγής. Τμήμα Επιστημών της Εκπαίδευσης στην Προσχολική Ηλικία
Subject classification: Number Theory
Keywords: Αριθμός π,Aριθμός φ,Aριθμός e,Αρρητότητα,Υπερβατικότητα,Νumber of π,Νumber of φ,Νumber of e,Irrationality,Transcendence

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