Volume II — Mathematics

Mathematics is the one subject school teaches backwards. You are handed rules, drilled on them, tested on them, and almost never told what problem the rule was invented to solve or what it lets you see. Then the exam ends and it all evaporates, because nothing was attached to anything.

This volume puts it back the right way round. Every idea arrives with the problem that forced someone to invent it, the plain-language picture of what it does, the real notation written out and read aloud, worked examples with every step shown, and the place it turns up in ordinary life — your loan interest, the map on your phone, the noise cancelling in your headphones, the shape of a bridge cable.

There are no quizzes here, and nothing to memorise for a test. The goal is that in twenty years you can still explain why a negative times a negative is positive, why e^{i\pi} = -1 is not mysticism, and what a p-value actually claims — and explain it to somebody else.

The eleven parts

1. Numbers & Arithmetic — what a number is, where zero came from, negatives, fractions, powers, logarithms, primes, clock arithmetic, and the different sizes of infinity.
2. Algebra — letters standing for unknowns, equations, polynomials, the quadratic formula derived rather than memorised, and complex numbers.
3. Geometry & Trigonometry — Euclid's game of proof, triangles, circles, coordinates, and the unit circle that turns rotation into waves.
4. Linear Algebra — vectors, matrices as machines that move space, elimination, determinants, eigenvalues, and the decomposition behind image compression and recommendation systems.
5. Calculus — limits, the derivative as instantaneous change, the integral as accumulated total, the theorem that ties them together, and infinite series.
6. Differential Equations — the language of anything that changes: cooling coffee, radioactive decay, springs, resonance, epidemics, chaos.
7. Probability & Statistics — counting, chance, Bayes' theorem in depth, the distribution zoo, the Gaussian and why it is everywhere, and how statistics is honestly done and dishonestly used.
8. Discrete Mathematics & Logic — sets, truth, proof and induction, relations, counting arguments, and graphs.
9. Transforms & Signals — the idea that any signal is a sum of pure tones, Fourier series and transform, convolution, Laplace and Z.
10. Numerical Methods & Optimization — what a computer actually does with these ideas, where it goes wrong, and how machines find the best answer by walking downhill.
11. The Great Stories — Euler, Gauss, Ramanujan, the famous unsolved problems, and the crisis in the foundations that produced Gödel.

How this volume relates to the rest of the book

Volume I leans on this material in several places and forward-referenced it: information theory needs logarithms and probability (Volume I, 1.8), complexity analysis needs recurrences and growth rates (4.1), and machine learning needs vectors, gradients and distributions (12.2). Volume III (Electronics) needs differential equations, Fourier and Laplace throughout. Volume IV (Physics) needs calculus from its first page.

You do not have to read those first. This volume assumes nothing except that you can add, and it builds every tool it uses.

Start here: 1.1 — What a Number Actually Is.