New Foundations in Mathematics

The Geometric Concept of Number

  • Garret Sobczyk

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Garret Sobczyk
    Pages 1-21
  3. Garret Sobczyk
    Pages 23-42
  4. Garret Sobczyk
    Pages 43-66
  5. Garret Sobczyk
    Pages 67-83
  6. Garret Sobczyk
    Pages 85-93
  7. Garret Sobczyk
    Pages 95-105
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    Pages 107-116
  9. Garret Sobczyk
    Pages 117-136
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    Pages 137-151
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    Pages 153-179
  12. Garret Sobczyk
    Pages 181-199
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    Pages 201-222
  14. Garret Sobczyk
    Pages 223-241
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    Pages 243-251
  16. Garret Sobczyk
    Pages 253-274
  17. Garret Sobczyk
    Pages 275-295
  18. Garret Sobczyk
    Pages 297-327
  19. Garret Sobczyk
    Pages 329-351
  20. Back Matter
    Pages 353-370

About this book

Introduction

The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.

The book begins with a discussion of modular numbers (clock arithmetic) and modular polynomials. This leads to the idea of a spectral basis, the complex and hyperbolic numbers, and finally to geometric algebra, which lays the groundwork for the remainder of the text. Many topics are presented in a new
light, including:

* vector spaces and matrices;
* structure of linear operators and quadratic forms;
* Hermitian inner product spaces;
* geometry of moving planes;
* spacetime of special relativity;
* classical integration theorems;
* differential geometry of curves and smooth surfaces;
* projective geometry;
* Lie groups and Lie algebras.

Exercises with selected solutions are provided, and chapter summaries are included to reinforce concepts as they are covered. Links to relevant websites are often given, and supplementary material is available on the author’s website.
 
New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

Keywords

Clifford algebra Lie algebras geometric algebra interpolation linear and multilinear algebra special relativity spectral basis

Authors and affiliations

  • Garret Sobczyk
    • 1
  1. 1., Departamento de Físico-MatemáticasUniversitad de Las AméricasCholula, PueblaMexico

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-8385-6
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Birkhäuser, Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-8384-9
  • Online ISBN 978-0-8176-8385-6
  • About this book