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Mathematics

0 completedProgress saved on this device
01 · LEARNBuild the conceptRead the guided lesson and identify the core idea.
02 · PRACTICEApply itUse the worked example and practice the topic.
03 · SELF-CHECKTest yourselfComplete the quick practice and review feedback.
04 · EVIDENCEDocument learningMark completion and keep your study record on this device.
LEVELS · SCHOOL → PROFESSIONAL

Learn mathematics at the depth that fits your stage

Mathematics changes from intuitive foundations to rigorous proof, research and industry modelling. Choose a level without losing the connections between them.

LEVEL 01 · FOUNDATION / SCHOOL

Build intuition

Confidence, concepts and mathematical reasoning.

  • Arithmetic, algebra, geometry and basic statistics
  • Visual explanations, puzzles and real-life examples
  • Step-by-step “why and how” solutions
LEVEL 02 · SECONDARY

Build analytical fluency

Move from formulas to functions, models and structured problem solving.

  • Trigonometry, calculus, vectors and probability
  • Formula revision and graph-based exploration
  • Exam-style and timed practice
LEVEL 03 · UNDERGRADUATE

Think mathematically

Develop definitions, theorems, proof and applied mathematics.

  • Analysis, linear algebra, algebra and discrete mathematics
  • Definitions → lemmas → theorems → proofs
  • Problem sets and mathematical writing
LEVEL 04 · GRADUATE / MASTER'S

Deepen abstraction

Work with advanced structures, rigorous arguments and computation.

  • Measure, functional analysis, topology and advanced algebra
  • LaTeX, symbolic and numerical experimentation
  • Advanced references and proof-oriented exercises
LEVEL 05 · RESEARCH / PHD

Investigate new questions

Use mathematics as a research language and computational laboratory.

  • Papers, preprints, conjectures and open problems
  • Python, SymPy, NumPy, Jupyter and other research tools
  • Literature review, modelling and reproducible experiments
LEVEL 06 · PROFESSIONAL & APPLIED

Apply mathematics

Connect mathematical methods to real operational and industry problems.

  • Finance, risk, AI/ML, engineering and cryptography
  • Case studies, modelling and simulation
  • Decision-focused quantitative analysis
KNOWLEDGE DOMAINS

Mathematics subject map

Explore mathematics by discipline, then follow the prerequisite chain to the level you need.

Foundations

Numbers · Arithmetic · Algebra · Geometry · Logic · Sets

Core & Analytical

Functions · Trigonometry · Calculus · Vectors · Matrices · Probability · Statistics

Pure Mathematics

Analysis · Number Theory · Abstract Algebra · Discrete Mathematics · Topology · Logic

Applied Mathematics

Differential Equations · Numerical Analysis · Optimization · Modelling · Operations Research

Advanced & Research

Complex Analysis · Measure Theory · Functional Analysis · Differential Geometry · PDE · Algebraic Geometry

Professional Mathematics

Financial Mathematics · Quantitative Risk · AI/ML Mathematics · Cryptography · Engineering Mathematics

PREREQUISITE ROADMAPS

Know what comes before what

Use prerequisite chains to prevent learners from jumping into an advanced topic without the required mathematical foundation.

Arithmetic→Algebra→Functions→Calculus→Analysis / Optimization→AI / Quant / Research
Second pathway: Arithmetic → Number Theory → Modular Arithmetic → Finite Fields → Cryptography.
STANDARD TOPIC MODEL

Every topic should answer more than “what is the formula?”

1 · ConceptPlain-language intuition and formal definition.
2 · PrerequisitesWhat the learner should know first.
3 · Formula & DerivationFormula, variables, conditions and derivation.
4 · Theorem & ProofRigorous statement, proof and counterexample where appropriate.
5 · Worked ExamplesSimple → intermediate → advanced applications.
6 · PracticeHints, solutions, difficulty and verification.
7 · ComputationVisualisation, symbolic or numerical experimentation.
8 · ApplicationReal-world use, limitations and assumptions.
MATHEMATICS LAB

Tools for learning, checking and experimenting

🧮 Calculators & Solvers

Open Mathematics Lab →

Scientific, equation, matrix, derivative, integral, probability and statistics tools.

📈 Visualisation

Open Visualisation →

Function graphs, geometry, vectors, surfaces and probability distributions.

💻 Computational Mathematics

Open Computation Shelf →

Python, NumPy, SymPy, Jupyter, MATLAB/Mathematica and reproducible numerical experiments.

📐 Formula Library

Formulae organised by topic, conditions, derivation, examples and related concepts.

Open Formula Library →

🧠 Problem Bank

Open Problem Bank →

Practice by level, difficulty, method, proof requirement and application context.

RESEARCH & PROFESSIONAL

From research mathematics to industry mathematics

🔬 Research Hub

Papers, preprints, open problems, literature maps, computational notebooks, seminars and research-writing resources.

💼 Professional Case Studies

Financial modelling, risk analysis, optimisation, AI/ML, engineering, cryptography and quantitative decision problems.

🤖 AI Mathematics Assistant

Explain, hint, solve, check a solution, check a proof, compare methods, generate practice and identify mistakes.

Guided lesson

Concept

How to study

Example / application

Start Your Mathematics Course

Choose a topic below to open the guided study lesson. After learning, mark it complete and practice the concept.

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Learning sources: Use the authoritative or original source linked by the course where a factual claim, standard, definition or technical requirement needs verification. Third-party availability may change.