# Erudia

> Erudia is a free, interactive web app for learning all of mathematics — from arithmetic to graduate level — on a spaced-repetition engine that schedules reviews just before you'd forget, so what you learn stays learned.

Erudia teaches mathematics from counting to category theory. Every lesson is hands-on (manipulable widgets, not videos), and a retention engine (FSRS-6) brings each idea back on a personalized schedule so it sticks. It adds an Elo ability model, a knowledge-graph prerequisite map, ranked 1v1 duels, and leaderboards.

Totals: 42 subjects, 1146 lessons, 706 skills.

Subjects taught (42): Arithmetic, Algebra, Geometry, Trigonometry, Precalculus, Calculus, Combinatorics, Graph theory, Boolean algebra & logic, Mathematical Logic, Linear algebra, Multivariable calculus, Differential equations, Partial Differential Equations, Dynamical Systems & Chaos, Probability, Stochastic Processes, Game Theory, Statistics, Information & Coding Theory, Real analysis, Fourier & Harmonic Analysis, Complex analysis, Numerical methods, Optimization & Convex Analysis, Abstract Algebra, Representation Theory, Topology, Set theory & foundations, Category theory (intro), Diophantine Approximation, Ergodic & Homogeneous Dynamics, Measure Theory & Functional Analysis, Differential Geometry, Commutative Algebra & Algebraic Geometry, Algebraic Number Theory, AMC 10/12 Workshop, AIME Problem Solving, Discrete Math, Number Theory, Cryptography, Quantitative Problem-Solving.

## Subjects (42)

### Arithmetic
- [Arithmetic](https://erudia.org/learn/arithmetic/): 27 lessons — Where it all starts: place value, the four operations, fractions, decimals, and percents. Everything later is built on top of this.

### Algebra
- [Algebra](https://erudia.org/learn/algebra/): 35 lessons — From variables to quadratics: the language the rest of math is written in.
- [Precalculus](https://erudia.org/learn/precalc/): 30 lessons — Logarithms, complex numbers, conics. The polish you put on before calculus.

### Geometry
- [Geometry](https://erudia.org/learn/geometry/): 42 lessons — Lines, triangles, and coordinates: shapes you can see and reason about.

### Trigonometry
- [Trigonometry](https://erudia.org/learn/trig/): 31 lessons — From right triangles to the unit circle to identities. This is the bridge that carries you into calculus.

### Analysis & Calculus
- [Calculus](https://erudia.org/learn/calculus/): 36 lessons — Limits, derivatives, and integrals: the math of how things change and accumulate.
- [Multivariable calculus](https://erudia.org/learn/multivar-calculus/): 35 lessons — Calculus with multiple inputs: partials, gradients, and multiple integrals.
- [Differential equations](https://erudia.org/learn/diff-eq/): 38 lessons — Equations involving derivatives: the language of change.
- [Partial Differential Equations](https://erudia.org/learn/pde/): 10 lessons — The equations of physics and geometry: classification into elliptic/parabolic/hyperbolic, the heat, wave, and Laplace equations, separation of variables, the method of characteristics, well-posedness, transform methods, boundary-value problems, and the CFL condition for numerical schemes.
- [Dynamical Systems & Chaos](https://erudia.org/learn/dynamics/): 10 lessons — How systems change over time: flows and maps, fixed points and linear stability, phase portraits, limit cycles and Poincaré–Bendixson, bifurcations, the logistic map and the period-doubling road to chaos, sensitive dependence, Lyapunov exponents, strange attractors, and fractal dimension.
- [Real analysis](https://erudia.org/learn/real-analysis/): 36 lessons — Calculus made rigorous: limits, sequences, and continuity from first principles.
- [Fourier & Harmonic Analysis](https://erudia.org/learn/fourier/): 10 lessons — Decomposing signals into frequencies: Fourier series and convergence (Dirichlet, Fejér, Gibbs), the Fourier transform and Plancherel, convolution, the FFT, the uncertainty principle, wavelets, the Dirac delta and distributions, and harmonic analysis on groups.
- [Complex analysis](https://erudia.org/learn/complex-analysis/): 41 lessons — Calculus over the complex plane: holomorphic functions and contour integrals.
- [Numerical methods](https://erudia.org/learn/numerical/): 35 lessons — Computing math approximately, for when symbolic answers don't exist or aren't enough.
- [Optimization & Convex Analysis](https://erudia.org/learn/optimization/): 12 lessons — Convex sets and functions, linear programming and duality, gradient descent and Newton's method, Lagrange multipliers and the KKT conditions, Lagrangian duality, and the unifying power of convex optimization.
- [Ergodic & Homogeneous Dynamics](https://erudia.org/learn/ergodic/): 28 lessons — Long-run behavior of measure-preserving systems: recurrence, ergodic theorems, the Gauss map, and the space of lattices.
- [Measure Theory & Functional Analysis](https://erudia.org/learn/measure/): 30 lessons — The rigorous spine of modern analysis: measures, the Lebesgue integral, and the geometry of infinite-dimensional spaces.

### Linear Algebra
- [Linear algebra](https://erudia.org/learn/linear-algebra/): 37 lessons — Vectors, matrices, and transformations: the math of geometry in any dimension.

### Probability & Statistics
- [Probability](https://erudia.org/learn/probability/): 40 lessons — Quantifying uncertainty: events, conditioning, expectation, and the distributions that show up everywhere.
- [Stochastic Processes](https://erudia.org/learn/stochastic/): 10 lessons — Randomness through time: stochastic processes and sample paths, Markov chains and stationary distributions, random walks and recurrence, the Poisson process, martingales and optional stopping, branching processes, Brownian motion, Itô calculus, and stochastic differential equations.
- [Game Theory](https://erudia.org/learn/gametheory/): 10 lessons — Strategic decision-making: payoff matrices, dominance and Nash equilibrium, mixed strategies, zero-sum games and the minimax theorem, the Prisoner's Dilemma and its relatives, extensive-form games and backward induction, repeated games, evolutionarily stable strategies, and the Shapley value.
- [Statistics](https://erudia.org/learn/statistics/): 36 lessons — Getting from data to conclusions: describing, inferring, and predicting.
- [Information & Coding Theory](https://erudia.org/learn/information/): 15 lessons — Quantifying and transmitting information: entropy, joint and conditional entropy, mutual information and KL divergence, the source-coding theorem and Huffman codes, channel capacity, Shannon's noisy-channel coding theorem, error-correcting and Hamming codes, and Kolmogorov complexity.
- [Quantitative Problem-Solving](https://erudia.org/learn/quant/): 5 lessons — Interview-style quant problems built on the math you already know: which correlation matrices are even possible, geometric probability and conditioning, regression beta from a joint distribution, the linearity/indicator/symmetry reflexes, and portfolio variance with diversification.

### Discrete & Number Theory
- [Combinatorics](https://erudia.org/learn/combinatorics/): 36 lessons — Counting cleverly: permutations, combinations, and the binomial theorem.
- [Graph theory](https://erudia.org/learn/graph-theory/): 36 lessons — Nodes, edges, and the shapes connection makes.
- [Boolean algebra & logic](https://erudia.org/learn/boolean/): 34 lessons — The algebra of true and false: circuits, gates, and logical proofs.
- [Diophantine Approximation](https://erudia.org/learn/diophantine/): 30 lessons — How well reals are approximated by rationals: equidistribution, discrepancy, the three-gap theorem, and the metric theory.
- [Algebraic Number Theory](https://erudia.org/learn/algnt/): 27 lessons — Arithmetic inside number fields: rings of integers, factorization of ideals, class groups, and the geometry of numbers.
- [Discrete Math](https://erudia.org/learn/discrete/): 35 lessons — Sets, logic, and induction: the language you use to write proofs and describe structure.
- [Number Theory](https://erudia.org/learn/numbertheory/): 39 lessons — Divisibility, primes, modular arithmetic. Easy to start, surprisingly deep.
- [Cryptography](https://erudia.org/learn/crypto/): 10 lessons — From Caesar ciphers to lattices: Kerckhoffs's principle, the one-time pad and perfect secrecy, symmetric ciphers, Diffie-Hellman key exchange, RSA, elliptic-curve cryptography, hash functions, digital signatures, zero-knowledge proofs, and the post-quantum threat from Shor's algorithm.

### Abstract & Advanced
- [Abstract Algebra](https://erudia.org/learn/abstract/): 41 lessons — Groups, rings, fields, and Galois theory: the abstraction of symmetry and structure. No calculus required.
- [Representation Theory](https://erudia.org/learn/representation/): 10 lessons — Groups acting on vector spaces: representations and irreducibility, Maschke's complete-reducibility theorem, Schur's lemma, the regular representation and $\sum d_i^2=|G|$, characters and orthogonality, induced representations and Frobenius reciprocity, representations of the symmetric group via Young diagrams, Lie algebra representations, and the Peter–Weyl theorem.
- [Topology](https://erudia.org/learn/topology/): 40 lessons — Geometry without distance. What survives when you bend and stretch a shape without tearing it.
- [Differential Geometry](https://erudia.org/learn/diffgeo/): 28 lessons — Calculus on curved spaces: manifolds, curvature, geodesics, and the bridge between local geometry and global topology.
- [Commutative Algebra & Algebraic Geometry](https://erudia.org/learn/commalg/): 28 lessons — Rings as geometry: ideals, Noetherian conditions, the Nullstellensatz, and the dictionary between algebra and varieties.

### Logic & Foundations
- [Mathematical Logic](https://erudia.org/learn/logic/): 11 lessons — From truth tables to Gödel and Turing: propositional and predicate logic, normal forms and resolution, natural deduction, the completeness and compactness theorems, the incompleteness theorems, and the limits of computation.
- [Set theory & foundations](https://erudia.org/learn/set-theory/): 35 lessons — The axiomatic foundation that the rest of mathematics is built on.
- [Category theory (intro)](https://erudia.org/learn/category/): 35 lessons — Math at its highest level of abstraction, all about structure-preserving maps.

### Competition Math
- [AMC 10/12 Workshop](https://erudia.org/learn/amc/): 16 lessons — The techniques that win the AMC: answer extraction, Vieta, pigeonhole, invariants, telescoping, complementary counting, and modular tricks. Each one is taught through original style-faithful problems with verified answers.
- [AIME Problem Solving](https://erudia.org/learn/aime/): 16 lessons — The next rung up: integer-answer problems (000–999) that call for CRT, recursion, higher Vieta, length-bashing, and expected value. Every problem is original and style-faithful, with verified answers.

## Engines

- FSRS-6 spaced repetition: Models the strength of every skill you have and schedules its next review for the moment just before you'd forget it — personalized to your own forgetting rate.
- Elo ability rating: Estimates your level in each branch of math and uses it to pick problems at the right difficulty and to match you in duels.
- Knowledge-space map (KST): A prerequisite graph of every skill; it gates lessons so you learn in the right order and computes the shortest path from where you are to any goal.
- Adaptive difficulty: Selects each practice item to match your current ability, keeping you where challenge is close to skill.
- Misconception targeting: An error classifier tags mistakes (sign, off-by-one, conceptual, and more) and gives feedback aimed at the specific misunderstanding, not a generic 'wrong'.
- Worked-example fading: Moves you from fully worked examples to fill-in-the-step to independent practice as you gain competence.
- Studio mode: Predict-before-reveal interactive lessons: you manipulate and predict, then see the result and the math update together.

## Features

- Interactive lessons: Hands-on lessons across every subject, arithmetic through graduate level.
- Manipulable widgets: A live math-box: drag a value and watch the formula change.
- MathLive input: Type real math with a WYSIWYG editor, not clunky text.
- Spaced review queue: A daily queue that resurfaces skills right before they fade.
- 1v1 ranked duels: Head-to-head math sprints with Elo rating.
- Leaderboards: Global, country, and league boards.
- Forgiving streaks: Streaks that encourage return without punishing a missed day.
- Adaptive placement: A short check finds what you already know so you start where it helps.
- Goal paths: Pick a target and get a prerequisite-ordered path to it.
- Light & dark themes: Two full themes with distinct character.
- Accessibility: WCAG keyboard navigation across the interactive widgets.
- Free + Premium: A real free tier; Premium unlocks every chapter through graduate and competition level.
- Chinese (zh-CN): A localized experience and landing page.

## Key pages

- [About Erudia](https://erudia.org/about.html): what it is, how it works, the engines
- [Full reference for AI](https://erudia.org/llms-full.txt): every subject, lesson, feature, and engine
- [Open the app](https://erudia.org/): start learning
- [Pricing](https://erudia.org/#pricing): free tier + Premium ($7.99/month)
- [1v1 math](https://erudia.org/1v1-math/) and [Leaderboard](https://erudia.org/math-leaderboard/)