Mathematics Is Not a Spectator Sport

It is often said that mathematics and music go together, and that people with a special aptitude for mathematics often have similar gifts in music. Some music is very profound, and listeners find that there is far more in it than they appreciated at a first hearing. A similar point can be made about an understanding of mathematics.
This book introduces the reader to various topics in mathematics and is intended for precocious high school students and college students just beginning their study of mathematics. The topics discussed in this book include a variety of results in number theory involving squares, and also complex numbers, early algebraic ideas such as the Euclidean algorithm, geometrical constructions created by the Greeks, and more recent topics such as group theory.

This book introduces the reader to various topics in mathematics and is intended for precocious high school students and college students just beginning their study of mathematics. Topics include the Euclidean algorithm, geometrical constructions created by the Greeks, and ancient Babylonian and Chinese proofs of the Pythagorean theorem.

Compared to other popular math books, there is more algebraic manipulation, and more applications of algebra in number theory and geometry Presents an exciting variety of topics to motivate beginning students May be used as an introductory course or as background reading Includes supplementary material: sn.pub/extras

Auteur
George Phillips is Professor of Mathematics at St. Andrews University, Scotland. He is the author of two previous books also published by Springer, Two Millenia of Mathematics (2000), and Interpolation and Approximation by Polynomials (2003).

Texte du rabat

Mathematics Is Not a Spectator Sport challenges the reader to become an active mathematician. Beginning at a gentle pace, the author encourages the reader to get involved, with discussions of an exciting variety of topics, each placed in its historical context, including:

• The surprising achievements of early Babylonian mathematics;

• The fascinating arithmetic of continued fractions;

• Geometric origins of the Euclidean algorithm;

• Infinite sets and the pioneering work of Georg Cantor;

• The sieve of Eratosthenes, which is used for finding primes;

• Gauss's conjecture about the density of primes;

• Special methods for finding really large primes, and a discussion of the famous Riemann hypothesis;

• A combinatorial interpretation of the Fibonacci numbers;

• A study of properties of the triangle, including one named after Napoleon;

• The application of algebraic methods to solve geometrical problems;

• The study of symmetries using algebraic methods;

• The foundations of group theory;

• An algebraic interpretation of the Platonic solids.

The chapters are largely self-contained and each topic can be understood independently. However, the author draws many connections between the various topics to demonstrate their interplay and role within the context of mathematics as a whole. Lots of carefully chosen problems are included at the end of each section to stimulate the reader's development as a mathematician.

This book is intended for those beginning their study of mathematics at the university level, as well as the general reader who would like to learn more about what it means to "do" mathematics.

Contenu
Squares.- Numbers, Numbers Everywhere.- Fibonacci Numbers.- Prime Numbers.- Choice and Chance.- Geometrical Constructions.- The Algebra of Group.