A Programmer’s Introduction to Mathematics

Description

A Programmer's Introduction to Mathematics uses your familiarity with ideas from programming and software to teach mathematics. You'll learn about the central objects and theorems of mathematics, covering graphs, calculus, linear algebra, eigenvalues, optimization, and more. You'll also be immersed in the often unspoken cultural attitudes of mathematics, learning both how to read and write proofs while understanding why mathematics is the way it is. Between each technical chapter is an essay describing a different aspect of mathematical culture, and discussions of the insights and meta-insights that constitute mathematical intuition. As you learn, we'll use new mathematical ideas to create wondrous programs, from cryptographic schemes to neural networks to hyperbolic tessellations. Each chapter also contains a set of exercises that have you actively explore mathematical topics on your own. By the end of the book, you will be able to learn mathematics on your own. In short, this book will teach you to engage with mathematics. Contents Our Goal i Chapter 1. Like Programming, Mathematics has a Culture Chapter 2. Polynomials 2.1 Polynomials, Java, and Definitions 2.2 A Little More Notation 2.3 Existence & Uniqueness 2.4 Realizing it in Code 2.5 Application: Sharing Secrets 2.6 Cultural Review 2.7 Exercises 2.8 Chapter Notes Chapter 3. On Pace and Patience Chapter 4. Sets 4.1 Sets, Functions, and Their -Jections 4.2 Clever Bijections and Counting 4.3 Proof by Induction and Contradiction 4.4 Application: Stable Marriages 4.5 Cultural Review 4.6 Exercises 4.7 Chapter Notes Chapter 5. Variable Names, Overloading, and Your Brain Chapter 6. Graphs 6.1 The Definition of a Graph 6.2 Graph Coloring 6.3 Register Allocation and Hardness 6.4 Planarity and the Euler Characteristic 6.5 Application: the Five Color Theorem 6.6 Approximate Coloring 6.7 Cultural Review 6.8 Exercises 6.9 Chapter Notes Chapter 7. The Many Subcultures of Mathematics Chapter 8. Calculus with One Variable 8.1 Lines and Curves 8.2 Limits 8.3 The Derivative 8.4 Taylor Series 8.5 Remainders 8.6 Application: Finding Roots 8.7 Cultural Review 8.8 Exercises Chapter 9. On Types and Tail Calls Chapter 10. Linear Algebra 10.1 Linear Maps and Vector Spaces 10.2 Linear Maps, Formally This Time 10.3 The Basis and Linear Combinations 10.4 Dimension 10.5 Matrices 10.6 Conjugations and Computations 10.7 One Vector Space to Rule Them All 10.8 Geometry of Vector Spaces 10.9 Application: Singular Value Decomposition 10.10 Cultural Review 10.11 Exercises 10.12 Chapter Notes Chapter 11. Live and Learn Linear Algebra (Again) Chapter 12. Eigenvectors and Eigenvalues 12.1 Eigenvalues of Graphs 12.2 Limiting the Scope: Symmetric Matrices 12.3 Inner Products 12.4 Orthonormal Bases 12.5 Computing Eigenvalues 12.6 The Spectral Theorem 12.7 Application: Waves 12.8 Cultural Review 12.9 Exercises 12.10 Chapter Notes Chapter 13. Rigor and Formality Chapter 14. Multivariable Calculus and Optimization 14.1 Generalizing the Derivative 14.2 Linear Approximations 14.3 Vector-valued Functions and the Chain Rule 14.4 Computing the Total Derivative 14.5 The Geometry of the Gradient 14.6 Optimizing Multivariable Functions 14.7 Gradient Descent: an Optimization Hammer 14.8 Gradients of Computation Graphs 14.9 Application: Automatic Differentiation and a Simple Neural Network 14.10 Cultural Review 14.11 Exercises 14.12 Chapter Notes Chapter 15. The Argument for Big-O Notation Chapter 16. Groups 16.1 The Geometric Perspective 16.2 The Interface Perspective 16.3 Homomorphisms: Structure Preserving Functions 16.4 Building Blocks of Groups 16.5 Geometry as the Study of Groups 16.6 The Symmetry Group of the Poincaré Disk 16.7 Application: Drawing Hyperbolic Tessellations 16.8 Cultural Review 16.9 Exercises 16.10 Chapter Notes Chapter 17. A New Interface Appendix A. Notation Appendix B. A Summary of Proofs B.1 Propositional and first-order logic B.2 Methods of proof B.3 How does one actually prove things? Appendix C. Annotated Resources C.1 Fundamentals and Foundations C.2 Polynomials C.3 Graph Theory and Combinatorics C.4 Calculus and Analysis C.5 Linear Algebra C.6 Optimization C.7 Abstract Algebra (Groups, etc.) C.8 Topology C.9 Computer Science, Theory, and Algorithms C.10 Fun and Recreation About the Author and Cover Index