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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, including 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. In short, this book will teach you to engage with mathematics. A Programmer's Introduction to Mathematics is written by Jeremy Kun, who has been writing about math and programming for 8 years on his blog "Math Intersect Programming." As of 2018, he works in datacenter optimization at Google.
Our Goal
Like Programming, Mathematics has a Culture
Polynomials, Java, and Definitions
A Little More Notation
Existence &amp Uniqueness
Realizing it in Code
Application: Sharing Secrets
Cultural Review
Exercises
Chapter Notes
On Pace and Patience
Sets
Sets, Functions, and Their -Jections
Clever Bijections and Counting
Proof by Induction and Contradiction
Application: Stable Marriages
Cultural Review
Exercises
Chapter Notes
Variable Names, Overloading, and Your Brain
The Definition of a Graph
Graph Coloring
Register Allocation and Hardness
Planarity and the Euler Characteristic
Application: the Five Color Theorem
Approximate Coloring
Cultural Review
Exercises
Chapter Notes
The Many Subcultures of Mathematics
Calculus with One Variable
Lines and Curves
Limits
The Derivative
Taylor Series
Remainders
Application: Finding Roots
Exercises
On Types and Tail Calls
Linear Algebra
Linear Maps and Vector Spaces
Linear Maps, Formally This Time
The Basis and Linear Combinations
Dimension
Matrices
Conjugations and Computations
One Vector Space to Rule Them All
Geometry of Vector Spaces
Application: Singular Value Decomposition
Exercises
Chapter Notes
Live and Learn Linear Algebra (Again)
Eigenvectors and Eigenvalues
Eigenvalues of Graphs
Limiting the Scope: Symmetric Matrices
Inner Products
Orthonormal Bases
Computing Eigenvalues
The Spectral Theorem
Application: Waves
Cultural Review
Exercises
Chapter Notes
Rigor and Formality
Generalizing the Derivative
Linear Approximations
Multivariable Functions and the Chain Rule
Computing the Total Derivative
The Geometry of the Gradient
Optimizing Multivariable Functions
The Chain Rule: a Reprise and a Proof
Gradient Descent: an Optimization Hammer
Gradients of Computation Graphs
Application: Automatic Differentiation and a Simple Neural Network
Exercises
Chapter Notes
The Argument for Big-O Notation
Groups
The Geometric Perspective
The Interface Perspective
Homomorphisms: Structure Preserving Functions
Building Blocks of Groups
Geometry as the Study of Groups
The Symmetry Group of the Poincaré Disk
The Hyperbolic Isometry Group as a Group of Matrices
Application: Drawing Hyperbolic Tessellations
Exercises
Chapter Notes
A New Interface
About the Author and Cover
Index