Softcover ISBN:  9781470472412 
Product Code:  CLRM/15.S 
List Price:  $37.00 
MAA Member Price:  $27.75 
AMS Member Price:  $27.75 
Electronic ISBN:  9781470458324 
Product Code:  CLRM/15.E 
List Price:  $37.00 
MAA Member Price:  $27.75 
AMS Member Price:  $27.75 

Book DetailsClassroom Resource MaterialsVolume: 15; 2001; 118 pp
The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to Englishspeaking 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 BanachTarski 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. 
Table of Contents

Chapters

Part I. Proofs of Impossibility, Proofs of Nonexistence

1. Proofs of Irrationality

2. The Elements of the Theory of Geometric Constructions

3. Constructible Regular Polygons

4. Some Basic Facts About 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 $\mathbf {R}^n$

12. The Problem of Invariant Measures

13. The Banach–Tarski Paradox

14. Open and Closed Sets in $\mathbf {R}$. The Cantor Set

15. The Peano Curve

16. Borel Sets

17. The Diagonal Method


Additional Material

Reviews

I love this book! It's a very well designed problem driven course done in the splendid Hungarian tradition.
Michael Bert, MAA Online


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 Book Details
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The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to Englishspeaking 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 BanachTarski 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.

Chapters

Part I. Proofs of Impossibility, Proofs of Nonexistence

1. Proofs of Irrationality

2. The Elements of the Theory of Geometric Constructions

3. Constructible Regular Polygons

4. Some Basic Facts About 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 $\mathbf {R}^n$

12. The Problem of Invariant Measures

13. The Banach–Tarski Paradox

14. Open and Closed Sets in $\mathbf {R}$. The Cantor Set

15. The Peano Curve

16. Borel Sets

17. The Diagonal Method

I love this book! It's a very well designed problem driven course done in the splendid Hungarian tradition.
Michael Bert, MAA Online