Conjecture and Proof

Classroom Ressource Materials

Miklos Laczkovitch

118 pages, parution le 29/10/2002

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Résumé

This book is a compilation of the lecture notes for a course designed and initiated by Paul Erdös, László Lovász, Vera Sós, and László Babai for a one-semester course given by the Budapest Seminars in Mathematics.
By introducing a variety of advanced topics, the book functions, in part, as a survey of topics from number theory, geometry, measure theory, and set theory. It can be used as a supplement in courses that introduce abstract mathematics to undergraduates. The ideas that are presented are deeper and more sophisticated than those typically encountered in sophomore-level "transition" courses. However, talented students in such courses should find this book to be an exciting excursion into new areas of mathematics--and more importantly, new ways of thinking about mathematical problems. Because of its unusual depth and the fact that some of the sections can stand alone or be combined with a few others to form a unit, this book is ideally suited for upper-level undergraduate seminars or capstone courses.
Although the text discusses questions from various fields including number theory, algebra and geometry, it is centered around the real number system and the problem of measure. Thus, the number theoretic sections are concerned with rational and irrational and with algebraic and transcendental numbers; the problems of geometric constructions clarify the nature of constructible numbers (as a subset of algebraic numbers), and the questions of geometric dissections serve as motivation for general problems of equidecomposability.

Contents

I. Proofs of impossibility, proofs of non-existence:

Proofs of irrationality
The elements of the theory of geometric constructions
Constructible regular polygons
Some basic facts on linear spaces and fields
Algebraic and transcendental numbers
Cauchy's functional equation
Geometric decompositions.

II: Constructions, proofs of existence:

The pigeonhole principle
Liouville numbers
Countable and uncountable sets
Isometries of Rⁿ
The problem of invariant measures
The Banach-Tarski paradox
Open and closed sets in R. The Cantor set
The Peano curve
Borel sets
The diagonal method.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	Miklos Laczkovitch
Parution	29/10/2002
Nb. de pages	118
Format	15 x 23
Couverture	Broché
Poids	180g
Intérieur	Noir et Blanc
EAN13	9780883857229
ISBN13	978-0-88385-722-9