🧮 Calculators
Quadratic, derivative, integral, statistics, matrix, finance and risk tools.
Open →Mathematics Lab
Calculation, visualisation, proof study, problem solving, applied modelling and research preparation in one browser-based workspace.
Technology & privacy: calculations, equation solving and progress records run locally in your browser. PDF export uses the version-pinned html2pdf.js library loaded from cdnjs; the page does not send your entered equations or salary-like personal data to a calculation API.
📈 Visualisation
Local function graphing and a trigonometric geometry visualiser.
Open →🧠 Problem Bank
Foundation through professional practice with solutions.
Practice →📖 Theorem & Proof Library
Core statements, hypotheses and proof sketches.
Study →💼 Applied Projects
Finance, risk, AI/ML, engineering, cryptography and optimisation.
Explore →🔬 Research Hub
Research workflow, reproducibility and open-question exercises.
Explore →🧮 Calculators & Solvers
Enter values.
🧠 Solution Logic
🧑🏫 Equation Tutor — Enter a problem, learn the method
Enter an equation such as 3x+5=20, x^2-5x+6=0, or 2(x+3)=14. The tutor identifies the structure, chooses a method, shows the calculation trail, gives the answer, and verifies it.
Enter an equation and press Solve & Teach.
📈 Function & Geometry Visualiser
45°
🧠 Graph / Geometry Logic
- Represent: choose the function or angle and define the coordinate/triangle model.
- Compute: evaluate the selected function across x-values or calculate sin, cos and tan from θ.
- Visualise: map numerical values to canvas coordinates and draw axes and the curve.
- Interpret: connect the graph/ratio to the underlying mathematical relationship.
- Verify: compare behaviour with known properties such as symmetry, periodicity or domain restrictions.
🧠 Practice Studio — Problems, Logic & Self-Check
International Mathematics Practice Framework
Every worked solution follows this reasoning chain:
- Understand — identify the givens, constraints and target.
- Represent — express the problem with equations, diagrams, tables, graphs or symbols.
- Plan — select the appropriate theorem, formula, algorithm or strategy.
- Execute — carry out every algebraic/computational step explicitly.
- Justify — explain why the chosen formula or transformation is valid.
- Verify — use substitution, units, bounds, an alternative method or a reasonableness check.
- Reflect — interpret the result and identify how the method generalises.
📖 Theorem & Proof Library
Pythagorean Theorem
Statement: In a right triangle, a²+b²=c².
Proof sketch: Compare the area of a square of side a+b with four copies of the triangle; expanding both expressions yields the identity.
Fundamental Theorem of Arithmetic
Statement: Every integer greater than 1 has a unique prime factorisation, apart from ordering.
Proof sketch: Existence follows by repeated factorisation; uniqueness follows from Euclid's lemma.
Intermediate Value Theorem
Statement: A continuous function on [a,b] attains every value between f(a) and f(b).
Proof sketch: Use the supremum of the set where f is below the target and continuity at the boundary point.
Mean Value Theorem
Statement: If f is continuous on [a,b] and differentiable on (a,b), some c has f′(c)=(f(b)−f(a))/(b−a).
Proof sketch: Subtract the secant line and apply Rolle's theorem.
Bayes' Theorem
Statement: P(A|B)=P(B|A)P(A)/P(B), P(B)>0.
Proof sketch: Express P(A∩B) in two conditional forms and equate them.
Rank–Nullity Theorem
Statement: dim(V)=rank(T)+nullity(T) for a linear map on finite-dimensional V.
Proof sketch: Extend a basis of ker(T) to a basis of V; images of the added vectors form a basis of im(T).
💼 Professional & Applied Projects
Finance — Discounted Cash Flow
Build a cash-flow table, apply PV=FV/(1+r)^n, then test sensitivity to r and timing.
Risk — Expected Loss
For PD=2%, LGD=40%, EAD=500,000 calculate EL=PD×LGD×EAD and vary one input at a time.
AI/ML — Gradient Descent
For J(θ)=(θ−3)² use θₜ₊₁=θₜ−η∇J(θₜ); compare learning rates.
Engineering — Exponential Model
Use y(t)=y₀e^{kt}; estimate k from two observations and inspect residuals.
Cryptography — Modular Arithmetic
Compute congruences and modular inverses; explain why coprimality controls invertibility.
Optimisation — Linear Programming
Define an objective and resource constraints, then inspect feasible solutions and boundary cases.
🔬 Research Hub
Workflow: Question → Definitions → Literature map → Hypothesis/Conjecture → Method → Computation → Validation → Limitations → Reproducible record.
Open-question exercises: investigate prime-gap patterns; test sequence convergence; compare numerical methods for one differential equation; vary constraints in an optimisation model.
Reproducibility checklist: record inputs, formulae, parameters, units, software/version, tolerance, random seed where relevant, expected result and observed result.
Live feeds, cloud notebooks, MATLAB/Mathematica execution and discussion forums are not connected because they require external services/API or persistent backend connectivity.
📊 Local Mathematics Dashboard
Topics completed0
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Course progress remains in the existing course storage; the Lab keeps its own local record.
🤖 Local Mathematics Study Assistant
Rule-based and dependency-free; it is not an AI/API service.
🎓 Micro-Lessons, Computation & Academic Workshops
▶ Animated Concept Explorer
Watch a local animation showing a point moving along y=x². Use it to connect coordinates, slope and function shape.
💻 Computation Starter Shelf
API-free starter snippets for Python/NumPy/SymPy, MATLAB and Mathematica. Copy and run them in your own environment.
Python: import sympy as sp; x=sp.symbols('x'); sp.diff(x**3-2*x,x)
NumPy: import numpy as np; np.linalg.det(np.array([[2,1],[3,4]]))
MATLAB: det([2 1; 3 4])
Mathematica: Det[{{2,1},{3,4}}]📚 Reference Shelf
Suggested study anchors: OpenStax Calculus; standard undergraduate texts in Linear Algebra, Probability, Real Analysis and Abstract Algebra; AMS and SIAM publications for advanced study.
🎤 Seminar Track
Self-run seminar sequence: Mathematical Modelling → Numerical Methods → Probability & Risk → AI/ML Mathematics → Cryptography → Research Reproducibility.
🛠 Workshop Track
Workshop sequence: Formula Derivation → Proof Writing → Graph Interpretation → Computational Experiment → Case Study → Research Presentation.
📝 Mathematical Writing
Practice definitions, theorem statements, proof structure, assumptions, notation, citations and reproducible computational notes.