The Nature and Power of Mathematics

by Donald M. Davis

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The publisher's blurb reads as follows:

In this engaging book, Donald Davis explains some of the most fascinating ideas in mathematics to the nonspecialist, highlighting their philosophical and historical interest, their often surprising applicability, and their beauty. The three main topics discussed are non-Euclidean geometry, with its application to the theory of relativity; number theory, with its application to cryptography; and fractals, which are an application in art, among other areas, of early mathematical work on iteration. Other topics include the influence of Greek mathematics on Kepler's laws of planetary motion, and the theoretical work that led to the development of computers.

Assuming the reader has some background in basic algebra and geometry, Davis relies on exercises to develop some of the important concepts. These exercises are designed to improve the reader's ability in logic, and enable him or her actually to experience mathematics at increasingly advanced levels.

Table of Contents

• 1. Some Greek mathematics
  • 1.1 pi and irrational numbers: an introduction to Greek mathematics
  • 1.2 Euclidean geometry
  • 1.3 Greek mathematics and Kepler
• 2. Non-Euclidean geometry
  • 2.1 Formal axiom systems
  • 2.2 Precursors to non-Euclidean geometry
  • 2.3 Hyperbolic geometry
  • 2.4 Spherical geometry
  • 2.5 Models of hyperbolic geometry
  • 2.6 The geometry of the universe
• 3. Number theory
  • 3.1 Prime numbers
  • 3.2 The Euclidean algorithm
  • 3.3 Congruence arithmetic
  • 3.4 The Little Fermat Theorem
• 4. Cryptography
  • 4.1 Some basic methods of cryptography
  • 4.2 Public key cryptography
• 5. Fractals
  • 5.1 Fractal dimension
  • 5.2 Iteration and computers
  • 5.3 Mandelbrot and Julia sets