Every concept explained three ways
Each concept opens with an ELI5 (kid-friendly analogy), builds into a student-level intuition, then delivers the full formal definition with proofs and properties. You choose your entry point — or read all three.
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Mathematics
Early Prototype · In active development
Mathematics
Knowledge System
The complete interactive mathematics reference — from arithmetic to topology — where every concept is explained at three levels of depth, verified by machine, and made explorable through interactive widgets.
Learn by reading and doing
Not by asking a chatbot
946 published concepts
946
Published concepts
Coming soon
~2,800
Concepts remaining to full coverage
38
Mathematical domains
125
Concepts with live interactive widgets
10
Widget varieties (plotter, graph, field…)
01Three Levels of Understanding
Every concept explained
at the depth you need.
Not everyone arrives at a concept from the same place. A child, a student, and a researcher all need the same truth — but told differently. Every concept in the system carries all three layers, and you decide where to start.
Example concept — Absolute Value
ELI5
Absolute value is like asking 'how far away from zero is this number?' without caring which direction. If you owe someone 5 dollars (-5) or have 5 dollars (+5), the amount of money involved is still 5 either way.
✦Kid · ELI5
Plain-language analogy. No jargon, no notation — just a mental image that makes the concept stick before the formalism arrives.
◆Student · Intuition
Conceptual intuition with lightweight notation. Builds the mental model that makes the formal definition feel obvious rather than arbitrary.
▲Expert · Formal
Full formal treatment — rigorous definition, proofs, properties, and links to the broader mathematical landscape.
02The Vision
A textbook that grades itself — and meets you where you are.
Most mathematics learning online fails in one of two ways: video lectures that let you feel understood without actually practicing, or AI chatbots that hand you an answer without building the model that would let you derive the next one. The Mathematics Knowledge System is neither.
It is a structured, proof-first, practice-enforced reference where understanding is demonstrated — not assumed. Each concept is verified not by editorial confidence but by symbolic computation. Each explanation is written at the level of the reader. Each practice problem is graded, not guessed.
38 domains covered
FoundationsPre-AlgebraAlgebra IAlgebra IIGeometryTrigonometryAnalytic GeometryCalculus ICalculus IICalculus IIILinear AlgebraDifferential EquationsProbabilityStatisticsNumber TheoryDiscrete MathematicsGraph TheoryCombinatoricsSet TheoryMathematical LogicAbstract Algebra IAbstract Algebra IITopologyReal AnalysisComplex AnalysisTheory of ComputationNumerical AnalysisMeasure TheoryCategory TheoryDifferential GeometryRepresentation TheoryMathematical OptimizationMathematical PhysicsAlgebraic TopologyFunctional AnalysisInformation TheoryDynamical SystemsStochastic Processes
03What it does
Every layer of the learning stack, built from scratch.
Python checks the answers, not a human stamp
A SymPy-based verifier symbolically checks worked examples and mathematical properties — differentiation, integration, limits, series, algebra, ODEs, linear systems, complex arithmetic, and matrices. Every claim either passes, fails, or is honestly marked SKIP — never counted as a pass.
Explore by doing, not by watching
2D function plotter, 3D surface plotter, tangent-line explorer, formula explorer, matrix transforms, vector fields, series convergence visualizer, graph traversal, geometry construction, probability distribution explorer — all on a safe expression parser (no eval). Deployed across 125 concepts.
Every concept covered the same rigorous way
Overview → history → intuition → formal definition → proofs → applications → worked examples → practice problems → quiz → flashcards → references. Enforced by schema. The corpus forms a single connected graph with 0 orphans, checked in CI.
SQLite-backed adaptive learning system
Real progress tracking in a local SQLite database. Adaptive practice, 8 curated audience tracks, placement quiz, and full-screen flashcard review. The system surfaces concepts you're weakest on — not the ones you already know.
Navigate mathematics as a connected graph
Every concept is cross-linked with symmetric return paths — no dead ends. A d3-force visualization lets you see and traverse the entire corpus. Full-text BM25 search (field-weighted, typo-tolerant) runs server-side.
Extractors with mandatory human review
Scrapers pull drafts from Wikipedia, ProofWiki, NIST DLMF, and OpenStax. Every draft lands in staging at confidence < 0.85 and is never auto-published — a human reviews and promotes by hand. A staleness checker flags concepts whose source has changed.
Installable, offline-capable, and embeddable
Full PWA with real installability and offline support via a hand-rolled service worker. An embed API lets third parties drop any concept into their own pages. No AI chatbot — the platform teaches through structured reading and graded practice.
04Audience tracks
Eight curated paths through the corpus.
A placement quiz routes you into the right track. The system then adapts within the track using SM-2 spaced repetition — surfacing concepts you're weakest on, not the ones you already know.
Computer Science Math — Undergraduate
The mathematics a CS undergraduate needs to read algorithms papers, reason about correctness and complexity, and follow the standard core curriculum — discrete math, algorithms, theory of computation, and the linear algebra/probability every later course assumes.
Computer Science Math — Graduate (MS/PhD)
Beyond the undergraduate core: the rigor and depth a master's or PhD student needs for research in algorithms, theory, machine learning, or systems — real analysis for optimization/ML theory, abstract algebra for cryptography and coding theory, advanced complexity theory, and the mathematics of information.
Civil Engineering Math — Undergraduate
The mathematics a civil engineering undergraduate needs — the calculus, linear algebra, differential equations, and probability/statistics behind statics, structural analysis, fluid mechanics, surveying, geotechnics, and transportation engineering.
Civil Engineering Math — Graduate (MS/PhD)
Beyond the undergraduate core: the advanced mathematics for structural dynamics, finite element analysis, computational fluid dynamics, geotechnical modeling, and reliability-based design — partial differential equations, advanced linear algebra, numerical methods, and probabilistic risk analysis.
Pure Mathematics — Undergraduate
The standard pure mathematics undergraduate curriculum: rigorous analysis, algebra, geometry, topology, and number theory — the foundations for graduate study or research in any branch of mathematics.
Pure Mathematics — Graduate (MS/PhD)
Graduate pure mathematics: rigorous analysis, modern algebra, topology, geometry, and their interactions — the mathematical depth required for research in any contemporary pure mathematics field.
Applied Mathematics & Data Science — Undergraduate
The mathematical foundations for data science, machine learning, and applied statistics — linear algebra, probability, calculus, optimization, and the computational techniques that underpin modern data analysis.
Applied Mathematics & Data Science — Graduate (MS/PhD)
Graduate-level applied mathematics for data science and machine learning research — rigorous probability, measure theory, advanced optimization, stochastic processes, and the mathematical frameworks for modern ML theory.
05Tech stack
No external services. No API keys.
The entire system runs locally. Progress tracking is a real SQLite database — not localStorage. Search is a server-side BM25 index, not a hosted service. The content pipeline pulls from live sources but requires human sign-off before anything ships.
Language
TypeScript
Math verifier
Python
Progress tracking
SQLite (local)
Search
BM25 (server-side)
06About
Who's building this?
FH
Fahmy Hassan
Founder & Systems Architect
As an architecture engineer with years of production experience building reliable, high-throughput systems, I bridge rigorous software engineering with modern AI capabilities — currently completing a Master's in Computer Science with a concentration in machine learning at CU Boulder.
With a PhD in self-autonomous systems as my next academic milestone, my research focuses on how AI can make independent, reliable decisions. I teach computer science alongside this work — not as a side activity, but because explaining something clearly is one of the sharpest tests of whether you actually understand it.
About the system
The Mathematics Knowledge System grew out of a straightforward frustration: teaching software engineering well requires solid mathematical foundations, and the available resources force a choice between rigor and interactivity. Textbooks are rigorous but static. Online courses are interactive but shallow. AI chatbots give you an answer without building the model that would let you derive the next one.
The three-level explanation system — kid, student, expert — is the core design decision. It reflects how understanding actually works: you need the analogy before the intuition, and the intuition before the formalism. Skipping levels produces the illusion of understanding, not the real thing.
Design principles
∇
No level-skipping
Every concept has an analogy, an intuition, and a formal definition. You earn the formalism by understanding the intuition first.
∫
Learn by doing
Interactive widgets make abstract structures tangible — plot a function, transform a matrix, visualize a probability distribution.
∑
Machine-verified
Worked examples are checked against live sources. A human curator makes the final call; the pipeline flags what needs review.
Live system
Try the system yourself.
The platform is live. Browse concepts across 38 domains, work through interactive widgets, run the placement quiz, and see how spaced repetition tracks your progress across three levels of depth.
Open the system →