Laczkovich Miklós. Conjecture and Proof

Mathematical Association of America, 2001 — 140 p. — ( Classroom resource materials) — ISBN 9780883857229, 0883857227The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.Proofs of Impossibility, Proofs of Nonexistence
Proofs of Irrationality
The Elements of the Theory of Geometric Constructions
Constructible Regular Polygons
Some Basic Facts About Linear Spaces and Fields
Algebraic and Transcendental Numbers
Cauchy’s Functional Equation
Geometric Decompositions
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 MethodHints