Math, branch by branch¶
Each book follows one branch of mathematics from its first idea to the questions mathematicians are still working on. Read a book in order: every chapter needs only the ones before it.
Books or Topics? The Topics shelf is a ladder of short lessons by age, for children and teens. A book is one continuous course for a reader who wants the whole road. Its first chapters move faster and point to the matching Topics lessons for anyone who wants them slower; its later chapters go well past high school, through university mathematics to the frontier.
How a chapter works¶
- It opens with a box saying what you will learn, which chapters you need first, and which sections are beginner, intermediate, advanced or expert. Skip the harder sections on a first read and come back later.
- Every idea comes with its reason. Each theorem is proved, at the level of the chapter; where a proof is too long for the page, the chapter says so, gives the idea, and names where the full proof is.
- Worked examples show every step.
- Check yourself ends each chapter: questions from easy to hard, each with a full worked answer folded under it. Try first, then open it.
The books¶
| Book | From | To | |
|---|---|---|---|
| Algebra | Letters for numbers | Galois theory, representations and open problems | 22 chapters |
| Geometry | Points, lines and angles | Curved space, general relativity and open problems | 21 chapters |
| Number theory | Which numbers divide which | The Riemann hypothesis, elliptic curves and Fermat's last theorem | 21 chapters |
| Combinatorics | How many ways? | Ramsey theory, the probabilistic method and open problems | 20 chapters |
| Probability | Coin flips and dice | Brownian motion, measure theory, entropy and open problems | 20 chapters |
| Statistics | Reading data honestly | Causal inference, high-dimensional statistics and the replication crisis | 20 chapters |
| Calculus and analysis | Speed and area | Lebesgue integration, Fourier series and the Navier-Stokes problem | 22 chapters |
| Linear algebra | Arrows and equations | The spectral theorem, the SVD, tensors and the matrix multiplication exponent | 20 chapters |
| Logic and foundations | True and false | Gödel's incompleteness theorems, undecidability and the busy beaver | 20 chapters |
| Topology | Rubber sheets and bridges | Homology, knots and the Poincaré conjecture | 20 chapters |
All ten branches are on the shelf. Each book can grow: new chapters and sections will be added over time.