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Imaginary Mathematics for Computer Science

  • John Vince

Table of contents

  1. Front Matter
    Pages i-xvii
  2. John Vince
    Pages 1-10
  3. John Vince
    Pages 11-53
  4. John Vince
    Pages 55-109
  5. John Vince
    Pages 111-140
  6. John Vince
    Pages 141-151
  7. John Vince
    Pages 153-188
  8. John Vince
    Pages 195-219
  9. John Vince
    Pages 221-227
  10. John Vince
    Pages 229-236
  11. John Vince
    Pages 237-278
  12. John Vince
    Pages 287-291
  13. John Vince
    Pages 293-296
  14. Back Matter
    Pages 297-301

About this book

Introduction

The imaginary unit i = √-1 has been used by mathematicians for nearly five-hundred years, during which time its physical meaning has been a constant challenge. Unfortunately, René Descartes referred to it as “imaginary”, and the use of the term “complex number” compounded the unnecessary mystery associated with this amazing object. Today, i = √-1 has found its way into virtually every branch of mathematics, and is widely employed in physics and science, from solving problems in electrical engineering to quantum field theory.

John Vince describes the evolution of the imaginary unit from the roots of quadratic and cubic equations, Hamilton’s quaternions, Cayley’s octonions, to Grassmann’s geometric algebra. In spite of the aura of mystery that surrounds the subject, John Vince makes the subject accessible and very readable. 

The first two chapters cover the imaginary unit and its integration with real numbers. Chapter 3 describes how complex numbers work with matrices, and shows how to compute complex eigenvalues and eigenvectors. Chapters 4 and 5 cover Hamilton’s invention of quaternions, and Cayley’s development of octonions, respectively. Chapter 6 provides a brief introduction to geometric algebra, which possesses many of the imaginary qualities of quaternions, but works in space of any dimension. The second half of the book is devoted to applications of complex numbers, quaternions and geometric algebra. John Vince explains how complex numbers simplify trigonometric identities, wave combinations and phase differences in circuit analysis, and how geometric algebra resolves geometric problems, and quaternions rotate 3D vectors. There are two short chapters on the Riemann hypothesis and the Mandelbrot set, both of which use complex numbers. The last chapter references the role of complex numbers in quantum mechanics, and ends with Schrödinger’s famous wave equation. 

Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to imaginary mathematics for computer science.

Keywords

Imaginary Algebra Complex Algebra Quaternion Algebra Geometric Algebra Applications for Imaginary Algebra

Authors and affiliations

  • John Vince
    • 1
  1. 1.Bournemouth UniversityPooleUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-94637-5
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Computer Science
  • Print ISBN 978-3-319-94636-8
  • Online ISBN 978-3-319-94637-5
  • Buy this book on publisher's site
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