Preface

Most people meet mathematics, computer science, artificial intelligence, and hardware engineering as separate subjects, in separate courses, from separate books. The connections, which are what an engineer actually uses, are left for later, and later rarely comes. A derivative is taught without the program that computes it; a gradient is taught without the network it trains; a drone is flown without the force balance that keeps it in the air.

I built this book to close those gaps. It started as the study system I designed for myself while running a small software studio and working through university mathematics. It grew into one path that runs from algebra and your first program, through three semesters of calculus, algorithms, systems, linear algebra, probability, and graduate-level computing, to the work engineers are doing right now: training and inspecting AI models, designing and flying drones, proving geometry, and writing fast, correct code under a clock.

Who this book is for#

• Students in high school, college, or university who want calculus, discrete mathematics, and computer science to make sense together, and who want more practice than a course provides.
• Self-taught developers who can ship software and want the mathematics, the theory, and the habits of proof and measurement that let them move into AI, graphics, robotics, or systems work.
• Working engineers moving between fields: software engineers learning machine learning or hardware, mechanical and electrical engineers learning software, and anyone returning to mathematics after years away.
• Researchers and technical leaders who need a dependable reference for the foundations under the tools they use.

You do not need to read it all, or in order. “How to use this book”, next, gives reading paths for each of these goals.

What makes it different#

• One integrated path. Programs appear beside the mathematics they compute, proofs beside the algorithms they justify, and physics beside the code that simulates it. Workshops at the seams connect the subjects explicitly.
• Picture first. Wherever a picture can carry it, a chapter’s central idea comes with one: a tangent line at the instant it measures, a gradient flowing backward through a graph, the thrust and weight on a hovering drone. More than 250 figures were drawn for this book, and each part opens with a map of its route.
• Working code beside the theory. Notes headed “From the engine” show a chapter’s idea at work in software I build: a game engine, a browser game, a tactical shooter, and two aerospace labs, one that tracks satellites and one that lands a rocket. Each quotes a short piece of that code unchanged and says what the mathematics is doing there. This web edition has all twenty-five; the printed book carries eleven of them.
• Practice that makes skill stay. The book has 954 exercises, each with a complete solution, tagged from foundation work to optional stretch, with hints that come in rungs so you take only the help you need. Spaced reviews, faded examples, and closed-book gates are built into every track.
• Checked, and honest about what was checked. Hundreds of answers were verified by script, the tests of the AI, Engineering, and Algorithms labs were themselves tested against deliberately wrong versions, geometry figures are constructed from exact coordinates, and every fact that can go out of date (a software version, a job requirement, a regulation) carries its source and the date it was checked. “How this book was checked”, at the back, states exactly what was verified and what was not.
• Honest boundaries. A “Boundary of the claim” box states what a result does and does not establish. The book tells you what one consumer GPU can and cannot train, what a degree requirement means, and when a simulation is not yet evidence.

How the book is organized#

Part I is about learning itself: how the book’s sessions work, how to use AI tools without losing the skill you are building, a planner for fitting several tracks into one week, and diagnostics that tell you where to start. Parts II–VI are the mathematical and computing core: foundations, Calculus I, algorithms and systems, Calculus II, and Calculus III. Part VII applies linear algebra, probability, statistics, and numerical methods to performance engineering, GPUs, machine learning, robotics, and aerospace computing. Part VIII is five bounded engineering projects, and Part IX is the graduate core. Parts X–XIV are the frontier tracks: an engineering map of the industry with complete chapters on feedback control and performance engineering; the AI Lab; the Engineering Lab, built around a drone you design and fly; Geometry, from angle chasing to olympiad circles; and the Algorithms Lane. Each track opens with a map of the whole route and teaches its first module in full.

Conventions#

Exercises carry an identifier, such as M12.09 or C07.04, that links to its solution, and a tag: F foundation, C standard course work, T transfer to a new setting, S optional stretch, L a runnable lab or project. Boxes mark the book’s teaching moves: Predict first before a calculation or a run, Trap for a mistake learners really make, Say it back after a key idea, Your turn for a faded example, and Mastery gate for the closed-book check that ends a module. Code appears in C++ and Python.

The companion website#

You are reading the web edition, whose home is bezhig.com/book. The same address holds the interactive explorers that accompany the frontier tracks, the starters and tests for every lab, and any corrections to this edition.

A note on dates#

This edition was completed in 2026. Mathematics does not expire, but tools, prices, job postings, and laws do. Each fact of that kind carries the date it was checked; treat it as a snapshot and check the source again before you rely on it.

Caleb Kious

Oklahoma, 2026