Synopses & Reviews
What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through.
Review
"Each chapter is a gem of mathematical exposition.... [The book] will not only stretch the imagination of the amateur, but it will also give pleasure to the sophisticated mathematician."--American Mathematical Monthly
Review
"A thoroughly enjoyable sampler of fascinating mathematical problems and their solutions."--Science
Review
A thoroughly enjoyable sampler of fascinating mathematical problems and their solutions. Science
Review
Each chapter is a gem of mathematical exposition.... [The book] will not only stretch the imagination of the amateur, but it will also give pleasure to the sophisticated mathematician. American Mathematical Monthly
Synopsis
Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting.
Synopsis
What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through.
Table of Contents
Preface v
Introduction 5
1. The Sequence of Prime Numbers 9
2. Traversing Nets of Curves 13
3. Some Maximum Problems 17
4. Incommensurable Segments and Irrational Numbers 22
5. A Minimum Property of the Pedal Triangle 27
6. A Second Proof of the Same Minimum Property 30
7. The Theory of Sets 34
8. Some Combinatorial Problems 43
9. On Waring's Problem 52
10. On Closed Self-Intersecting Curves 61
11. Is the Factorization of a Number into Prime Factors Unique?66
12. The Four-Color Problem 73
13. The Regular Polyhedrons 82
14. Pythagorean Numbers and Fermat's Theorem 88
15. The Theorem of the Arithmetic and Geometric Means 95
16. The Spanning Circle of a Finite Set of Points 103
17. Approximating Irrational Numbers by Means of Rational Numbers ill
18. Producing Rectilinear Motion by Means of Linkages 119
19. Perfect Numbers 129
20. Euler's Proof of the Infinitude of the Prime Numbers 135
21. Fundamental Principles of Maximum Problems 139
22. The Figure of Greatest Area with a Given Perimeter 142
23. Periodic Decimal Fractions 147
24. A Characteristic Property of the Circle 160
25. Curves of Constant Breadth 163
26. The Indispensability of the Compass for the Constructions of Elementary Geometry 177
27. A Property of the Number 30 187
28. An Improved Inequality 192
Notes and Remarks 197