Conjecture and proof

Includes index

Includes bibliographical references

1. Proofs of irrationality 2. The elements of the theory of geometric constructions 3. Constructible regular polygons 4. Some basic facts on linear spaces and fields 5. Algebraic and transcendental numbers 6. Cauchy's functional equation 7. Geometric decompositions Part II. Constructions, Proofs of Existence: 8. The pigeonhole principle 9. Liouville numbers 10. Countable and uncountable sets 11. Isometries of Rn 12. The problem of invariant measures 13. The Banach-Tarski paradox 14. Open and closed sets in R. The Cantor set 15. The Peano curve 16. Borel sets 17. The diagonal method.

The 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.