Mathematical Structures

This book is chapter 1 of a series of book. The entire series is an introduction to combinatory, number theory, topology, groups, rings, fields, modules, algebraic and integral extensions, noncommutative algebra, algebraic number theory, algebraic geometry, algebraic topology, and even more. The title of the series is Abstract Algebra and Discrete Mathematics.The focus is on breadth, rather than depth. Excellent books already exist for any one of these topics in detail, and I don't want to reinvent that wheel. Instead, this book knits them all together, providing a foundation for each topic in turn. By analogy, you might point your backyard telescope to every corner of the galaxy, in an effort to comprehend its scope, beauty, and diversity. You might not understand the Crab Nebula in all its detail, this is not the Hubble Space Telescope, but you should walk away with an appreciation for the vastness and the wonder of the galaxy, in this case, the galaxy of modern mathematics. If, from time to time, you find yourself saying, "How did anybody ever think of that?", then I have succeeded.Each chapter builds on the information that has gone before, and forward references are rare, though they do happen from time to time. I hope this series of books is more accessible than a sea of disconnected web pages, which is the hallmark of most math websites.This book is Chapter 1, Data Structures, By Karl Dahlke and Kermit Rose, Copyright (c) 2015Table of ContentsPage 041.01 Properties of ArithmeticPage 051.02 Modular MathPage 051.03 HomomorphismPage 081.04 Casting out NinesPage 091.05 Monomorphism, Epimorphism, IsomorphismPage 111.06 Discrete LogsPage 131.07 Binary SearchPage 141.08 Function CompositionPage 141.09 PermutationsPage 171.10 Even and Odd PermutationsPage 191.11 MatricesPage 231.12 Linear FunctionsPage 251.13 Matrix as FunctionPage 271.14 DeterminantPage 331.15 Gaussian EliminationPage 361.16 Elementary Row OperationsPage 371.17 The Determinant of the ProductPage 381.18 Matrix Identities and Inverses Page 431.19 OrthogonalPage 441.20 Determinant equals VolumePage 461.21 The Gram Schmidt ProcessPage 471.22 The Shoelace FormulaPage 501.23 Orthonormal and Rotations Page 551.24 Cross Product Page 571.25 Vandermonde MatrixPage 581.26 A Matrix of Matrices Page 591.27 Complex Numbers and the Gaussian IntegersPage 611.28 Eisenstein IntegersPage 631.29 QuaternionsPage 641.30 Half Integer QuaternionsPage 651.31 Projective Space and an Interesting HomomorphismPage 681.32 The Square Root of a Rotation Page 691.33 The Hairy Ball Theorem Page 701.34 Generalized Euclidean SpacePage 711.35 PolynomialsPage 721.36 Synthetic Division, Roots, and GCDPage 741.37 Formal DerivativePage 751.38 Power and Laurent SeriesPage 761.39 p-adic Numbers