Core Mathematics

The pure mathematics track. Every subject here is built to be worked through in order, with proofs, worked examples and a quiz on every lesson. Start wherever you are honest about being, not where you think you should be.

Each group below assumes the ones above it. If a lesson feels like it skipped a step, the step is usually in the previous subject.

Foundations

Arithmetic through the first taste of algebra. No prior knowledge assumed.

ArithmeticThe fundamentals: addition, subtraction, fractions, decimals, and more. Everyone starts somewhere, and this is a great place.Pre-AlgebraVariables, expressions, equations, and inequalities. Build the bridge from arithmetic to algebra.Algebra BasicsLinear equations, functions, graphing, systems of equations, and intro to polynomials.

Intermediate

Shape, higher algebra and triangles. This is where the tools start compounding.

GeometryShapes, angles, area, circles, coordinate geometry, and the Pythagorean Theorem. See math in the world around you.Algebra 2Quadratics, complex numbers, logarithms, conic sections, and sequences. The gateway to higher mathematics.TrigonometrySine, cosine, tangent, the unit circle, identities, graphing, and the Laws of Sines and Cosines. Triangles have never been this interesting.

The Calculus Sequence

Limits through multivariable calculus. Pre-Calculus is the on-ramp; do not skip it.

Pre-CalculusAdvanced functions, vectors, polar coordinates, limits, and the toolkit you need before calculus.Calculus 1Limits, derivatives, and an introduction to integrals.Calculus 2Integration techniques, applications, sequences, and series.Calculus 3Multivariable calculus: partial derivatives, multiple integrals, vector fields, and the big theorems of vector calculus.

Upper Division

Where mathematics stops being computation and starts being proof. These can largely be taken in any order once you have the calculus sequence and some comfort with writing an argument.

Linear AlgebraVectors, matrices, transformations, and eigenvalues. The language of modern mathematics and computer science.Differential EquationsODEs, systems of equations, Laplace transforms, and modeling. The math that describes how things change in the real world.Discrete MathematicsLogic, sets, proof techniques, combinatorics, and graph theory. The math behind every algorithm and every airtight argument.StatisticsData, probability, distributions, and inference. The math for reading the world’s numbers without being fooled by them.Abstract AlgebraGroups, rings, and fields. The structure underneath every algebra you have already met, and the reason cryptography works.Real AnalysisRigorous foundations: completeness, epsilon-delta limits, continuity, and the theorems of calculus finally proved.Complex AnalysisFunctions of a complex variable: analyticity, contour integrals, residues, and conformal maps. Where calculus turns unreasonably powerful.TopologyShape without distance: open sets, continuity, connectedness, compactness, surfaces, and the fundamental group.