Mathematics

Every later module assumes you can do this. It is the one thing school mathematics almost never teaches, and the one thing university mathematics assumes on day one.

• What a proof isWhy is 'it works for the first thousand cases' not enough?
• Propositional and predicate logicWhat does 'if...then' actually commit me to?
• Sets, functions and cardinalityAre some infinities bigger than others?
• Induction and recursionHow can a finite argument establish infinitely many facts?

The school syllabus, rebuilt as a coherent story about transformation rather than a list of procedures to memorise.

• Functions as machines and as objectsWhat is a function, if not a formula?
• Polynomials and the fundamental theorem of algebraWhy do we need complex numbers to solve real equations?
• Exponentials and logarithmsWhat makes e special?
• Trigonometry as circular motionWhy does the same function describe a triangle and a wave?

Newton and Leibniz's machine for handling change. Learn the idea from 3Blue1Brown, then earn it with MIT's problem sets.

• Limits and continuityWhat does it mean to get arbitrarily close without arriving?
• The derivativeWhy is the slope of a curve at a single point even meaningful?
• Optimisation, related rates, approximationHow does a derivative let me find a best answer?
• Integration and the fundamental theoremWhy is area the opposite of slope?
• Sequences, series and Taylor expansionHow can an infinite sum equal a finite number?

Arguably the single highest-return module in this entire curriculum. Quantum mechanics, machine learning, computer graphics and optimisation are all applied linear algebra.

• Vectors, spans and basisWhat is a vector, if not an arrow?
• Matrices as linear transformationsWhat is a matrix actually doing to space?
• Determinant, rank and invertibilityWhat does the determinant measure?
• Eigenvalues and eigenvectorsWhich directions does a transformation leave alone?
• Orthogonality, projection and the SVDWhat is the best low-dimensional summary of a high-dimensional thing?

Calculus in the world as it actually is: more than one dimension, with fields instead of functions.

• Partial derivatives and the gradientIn which direction does a surface rise fastest?
• Multiple integrals and change of variablesWhy does the Jacobian appear when I change coordinates?
• Divergence, curl and vector fieldsWhat are divergence and curl actually measuring?
• Green, Stokes and the divergence theoremWhy are these three theorems the same theorem?

The form almost every law of nature actually takes. Physics gives you the equation; this module tells you what it does.

• First-order equations and modellingHow do I turn a described process into an equation?
• Linear systems and matrix exponentialsWhy does linear algebra suddenly matter here?
• Fourier series and partial differential equationsHow do I solve an equation with more than one independent variable?
• Nonlinear dynamics and stabilityWhat happens when superposition fails?

Probability is where careful people are wrong most often. Build it axiomatically and the paradoxes stop being paradoxes.

• Sample spaces and the axiomsWhat is a probability a probability of?
• Conditional probability and Bayes' theoremHow should evidence change what I believe?
• Random variables and distributionsWhy do so few distributions describe so much?
• Law of large numbers and the central limit theoremWhy is the bell curve everywhere?

Probability reasons forward from model to data. Statistics reasons backward — which is much harder, and much more often done badly.

• Estimation, bias and varianceHow much can I claim to know from a sample?
• Hypothesis testing and what p-values are notWhat question does a p-value actually answer?
• Regression and the danger of causal languageWhen does a fitted line license a causal claim?
• Bayesian inferenceShould a parameter have a probability distribution?

The mathematics of things you can list. Computer science is built almost entirely from this module.

• Counting, pigeonhole and generating functionsHow do I count without enumerating?
• Graph theoryWhat problems become easy once you draw them as a network?
• Recurrences and asymptoticsHow do I describe growth without solving exactly?

Calculus, rebuilt from the definition of a limit upward. This is the module where mathematics stops being computation.

• Construction of the reals and completenessWhat is a real number, exactly?
• Sequences, series and convergenceWhen is an infinite process legitimate?
• Measure theory and the Lebesgue integralWhy does the Riemann integral fail, and what replaces it?

Groups, rings and fields: the study of symmetry itself. Noether's theorem, the Standard Model and public-key cryptography all live here.

• Groups and symmetryWhat is symmetry, stated precisely?
• Rings, fields and polynomialsWhich familiar rules survive when you drop the familiar numbers?
• Galois theoryWhy is there no formula for the quintic?

The most surprisingly beautiful subject in mathematics: assume differentiability once in the complex plane and infinitely many consequences follow.

• Complex functions and analyticityWhy is complex differentiability so much stronger than real?
• Contour integration and the residue theoremHow does a loop in the plane evaluate an integral on the line?
• Analytic continuation and the zeta functionIn what sense does 1+2+3+... equal −1/12?

The geometry general relativity needs, and the reason a coffee cup and a doughnut are the same object.

• Topological spaces and continuityWhat survives when you delete distance from geometry?
• Manifolds and tangent spacesHow do I do calculus on a curved thing?
• Curvature and the Gauss–Bonnet theoremCan a surface know it is curved without leaving itself?

The purest branch of mathematics, and — to everyone's surprise, including its practitioners' — the foundation of modern cryptography.

• Divisibility, primes and modular arithmeticWhy are the primes distributed so irregularly?
• Fermat, Euler and RSAHow does arithmetic make a secret?
• The prime number theorem and the Riemann hypothesisWhat is the deepest unsolved question in mathematics actually asking?

How mathematics is actually executed: approximately, iteratively, and on a machine that cannot represent most numbers.

• Convex optimisationWhy is convexity the line between easy and hard?
• Gradient descent and its descendantsWhy does the simplest possible algorithm dominate machine learning?
• Numerical linear algebra and stabilityWhen does correct mathematics give wrong answers on a computer?