The Enjoyment of Math

Front Cover
Princeton University Press, 1966 - Mathematics - 205 pages
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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.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.
  

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Contents

The Sequence of Prime Numbers
9
Traversing Nets of Curves
13
Some Maximum Problems
17
Incommensurable Segments and Irrational Numbers
22
A Minimum Property of the Pedal Triangle
27
A Second Proof of the Same Minimum Property
30
The Theory of Sets
34
Some Combinatorial Problems
43
The Spanning Circle of a Finite Set of Points
103
Approximating Irrational Numbers by Means of Rational Numbers
111
Producing Rectilinear Motion by Means of Linkages
119
Perfect Numbers
129
Eulers Proof of the Infinitude of the Prime Numbers
135
Fundamental Principles of Maximum Problems
139
The Figure of Greatest Area with a Given Perimeter
142
Periodic Decimal Fractions
147

On Warings Problem
52
On Closed SelfIntersecting Curves
61
Is the Factorization of a Number into Prime Factors Unique?
66
The FourColor Problem
73
The Regular Polyhedrons
82
Pythagorean Numbers and Formats Theorem
88
The Theorem of the Arithmetic and Geometric Means
95
A Characteristic Property of the Circle
160
Curves of Constant Breadth
163
The Indispensability of the Compass for the Constructions of Elementary Geometry
177
A Property of the Number 30
187
An Improved Inequality
192
Notes and Remarks
197
Copyright

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About the author (1966)

Rademacher was an Affiliate of Rockefeller University and Professor Emeritus of Mathematics at the University of Pennsylvania.

Otto Toeplitz (1881-1940) was a leading scholar in linear algebra and functional analysis. He was the author of many scholarly articles and coauthor of "The Enjoyment of Mathematics,"

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