The Nature and Power of Mathematics

Popular mathematics can be done at a level that is also suitable for "advanced" survey courses in mathematics. This book is an existence proof of that. Davis does a superb job of tracing the development of five major areas of mathematics, some of which are more historical than others. The areas are: *) Greek mathematics of irrational numbers, Euclidean geometry, conic sections and Kepler's discovery of the motion of the planets. *) The development of non-Euclidean geometry, and examples of non-Euclidean geometries such as hyperbolic and spherical geometries. This includes a brief discussion of Einstein's theories of relativity. *) Number theory, factorization, prime numbers, the Euclidean algorithm, congruence arithmetic, and the work of Pierre de Fermat. *) Cryptography, in particular that number theory is the theoretical foundation of modern cryptographic methods. *) Fractal geometry and how iteration is used to construct the Mandelbrot and Julia sets. Davis does an excellent job in explaining how the math was developed and is used and makes no attempt to "spoil the reader by sparing the equation", although equations are not multiplied beyond their necessity. A large number of exercises are included at the end of each section, which makes it possible to use it as a textbook.