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Pure Mathematics

Study Unit 1 Pure Mathematics through original explanations, accessible notation, worked examples, graphs, proof and modelling tasks, formative practice, and objective-level mastery.

3920 minestimated study time ENcourse language

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Unit 1

Pure Mathematics I

Develop logic, algebra, functions, trigonometry, coordinate geometry, vectors, limits, differentiation, integration, and differential equations through traceable mathematical reasoning.
1

Basic Algebra and Functions

Free

Develop rigorous logic, algebra, proof, polynomial, function, exponential, logarithmic, absolute-value, and cubic reasoning.

592 min · 27 published activities
  1. Basic Algebra and Functions Readiness Check 14 min
  2. Propositions and Connectives 20 min
  3. Truth Tables with Purpose 22 min
  4. Conditionals, Converses, and Contrapositives 20 min
  5. Equivalence, Tautology, and Contradiction 22 min
  6. Real-Number Laws and Inverses 22 min
  7. Binary Operations 21 min
  8. Surds and Rationalisation 21 min
  9. Summation Notation 21 min
  10. Direct Proof and Counterexample 20 min
  11. Induction for Sums 21 min
  12. Induction for Divisibility 22 min
  13. Factor Theorem 22 min
  14. Remainder Theorem and Unknown Coefficients 22 min
  15. Differences of Powers and Polynomial Order 21 min
  16. Function Language and Mappings 23 min
  17. Inverse and Composite Functions 23 min
  18. Graphs of Functions and Inverses 21 min
  19. Logarithm Laws 21 min
  20. Exponential Functions and Models 22 min
  21. Logarithmic Functions and Natural Logs 22 min
  22. Exponential and Logarithmic Equations 23 min
  23. Absolute-Value Definition and Properties 22 min
  24. Absolute-Value Graph Transformations 21 min
  25. Absolute-Value and Rational Inequalities 23 min
  26. Cubic Roots and Coefficients 24 min
  27. Basic Algebra and Functions Review Challenge 36 min
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2

Trigonometry, Coordinate Geometry and Vectors

Subscription required · from XCD 25.00 / month, all courses

Connect exact trigonometric reasoning, coordinate geometry, and vector models in two and three dimensions.

566 min · 25 published activities
  1. Trigonometry, Coordinate Geometry and Vectors Readiness Check 14 min
  2. Reciprocal Trigonometric Functions 22 min
  3. Reciprocal-Function Graphs and Asymptotes 22 min
  4. Derive Compound-Angle Formulae 23 min
  5. Use Compound and Double Angles 23 min
  6. Multiple-Angle Formulae 22 min
  7. Factor Formulae 21 min
  8. Prove Trigonometric Identities 23 min
  9. Simple Trigonometric General Solutions 24 min
  10. Interval Solutions and Branch Control 24 min
  11. Phase-Amplitude Form 23 min
  12. Phase-Amplitude Equations and Extrema 22 min
  13. Equations and Graphs of Circles 22 min
  14. Circle Tangents and Normals 22 min
  15. Line-Curve Intersections 22 min
  16. Curve-Curve Intersections 22 min
  17. Parametric to Cartesian Curves 22 min
  18. Geometric Loci 21 min
  19. Vector Representation and Operations 22 min
  20. Magnitude, Unit, Position, and Displacement Vectors 22 min
  21. Dot Product and Angle 22 min
  22. Equations of Lines in Space 23 min
  23. Relationships Between Lines 23 min
  24. Equations of Planes 24 min
  25. Trigonometry, Coordinate Geometry and Vectors Review Challenge 36 min
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3

Calculus I

Subscription required · from XCD 25.00 / month, all courses

Reason with limits, derivatives, integrals, curve behaviour, accumulation, and first-order differential equations.

630 min · 27 published activities
  1. Calculus I Readiness Check 14 min
  2. Limits from Graphs 22 min
  3. Limit Notation and Theorems 23 min
  4. The Fundamental Sine Limit 22 min
  5. Continuity and One-Sided Limits 23 min
  6. Derivative as a Limit 23 min
  7. First Principles for Powers 23 min
  8. First Principles for Sine and Cosine 23 min
  9. Sum, Product, and Quotient Rules 24 min
  10. Differentiate Trigonometric Expressions 22 min
  11. Chain Rule 24 min
  12. Parametric Differentiation 22 min
  13. Rates of Change 24 min
  14. Increasing, Decreasing, and Stationary Points 23 min
  15. Second Derivative and Classification 24 min
  16. Sketch Polynomial Curves 24 min
  17. Rational and Trigonometric Curves 23 min
  18. Tangents and Normals 22 min
  19. Reverse Differentiation and Indefinite Integrals 23 min
  20. Constants of Integration and Conditions 22 min
  21. Integrate Polynomial Functions 23 min
  22. Trigonometric Integrals and Substitution 24 min
  23. Definite Integrals and Properties 24 min
  24. Areas Under and Between Curves 24 min
  25. Volumes of Revolution 24 min
  26. First-Order Differential Equations 25 min
  27. Calculus I Review Challenge 36 min
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Unit 2

Complex Calculus and Modelling

Develop complex-number, advanced-calculus, approximation, probability, matrix, and differential-equation methods with accessible mathematical notation.
1

Complex Numbers and Calculus II

Subscription required · from XCD 25.00 / month, all courses

Represent and solve problems with complex numbers, advanced differentiation, and advanced integration.

790 min · 33 published activities
  1. Complex Numbers and Calculus II readiness 12 min
  2. Cartesian complex arithmetic 22 min
  3. Argand diagrams 24 min
  4. Polar and exponential forms 26 min
  5. Geometric meaning of multiplication 22 min
  6. De Moivre powers 24 min
  7. Complex roots 26 min
  8. Roots of unity 22 min
  9. Complex loci 24 min
  10. Polynomial roots and factors 26 min
  11. Real trigonometric identities 22 min
  12. Exponential definitions of trigonometric functions 24 min
  13. Hyperbolic functions and identities 26 min
  14. Implicit differentiation 22 min
  15. Parametric differentiation 24 min
  16. Logarithmic differentiation 26 min
  17. Higher derivatives 22 min
  18. Stationary points and classification 24 min
  19. Concavity and points of inflection 26 min
  20. Related rates 22 min
  21. Differentials and local approximation 24 min
  22. Trigonometric and algebraic substitution 26 min
  23. Integration by parts 22 min
  24. Partial-fraction integration 24 min
  25. Trigonometric integrals 26 min
  26. Reduction formulae 22 min
  27. Improper integrals 24 min
  28. Trapezium and Simpson rules 26 min
  29. Area between curves 22 min
  30. Volumes of revolution 24 min
  31. Arc length 26 min
  32. Mean values and accumulated change 22 min
  33. Complex Numbers and Calculus II review 36 min
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2

Sequences, Series and Approximations

Subscription required · from XCD 25.00 / month, all courses

Analyse convergence, construct series, approximate functions, and refine roots with controlled error.

574 min · 24 published activities
  1. Sequences, Series and Approximations readiness 12 min
  2. Explicit and recursive sequences 22 min
  3. Monotonicity and bounds 24 min
  4. Sequence limits 26 min
  5. Recurrence fixed points 22 min
  6. Inductive sequence arguments 24 min
  7. Sigma notation and finite sums 26 min
  8. Arithmetic and geometric series 22 min
  9. Convergence and divergence 24 min
  10. Telescoping series 26 min
  11. Power-series convergence 22 min
  12. Maclaurin expansions 24 min
  13. Taylor expansions about a point 26 min
  14. Series approximation and error 22 min
  15. Integer binomial expansion 24 min
  16. General binomial series 26 min
  17. Binomial approximation 22 min
  18. Change-of-sign root location 24 min
  19. Bisection method 26 min
  20. Fixed-point iteration 22 min
  21. Newton-Raphson method 24 min
  22. Iteration comparison 26 min
  23. Accuracy and stopping criteria 22 min
  24. Sequences, Series and Approximations review 36 min
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3

Counting, Matrices and Differential Equations

Subscription required · from XCD 25.00 / month, all courses

Use discrete counting, probability, matrices, and differential equations to analyse structured mathematical models.

768 min · 32 published activities
  1. Counting, Matrices and Differential Equations readiness 12 min
  2. Addition and multiplication principles 22 min
  3. Permutations 24 min
  4. Repeated and circular arrangements 26 min
  5. Combinations 22 min
  6. Mixed selection and arrangement 24 min
  7. Probability laws and complements 26 min
  8. Conditional probability 22 min
  9. Independence 24 min
  10. Tree diagrams and sequential events 26 min
  11. Bayes theorem 22 min
  12. Discrete random variables 24 min
  13. Expectation and variance 26 min
  14. Binomial distribution 22 min
  15. Matrix arithmetic 24 min
  16. Determinants and inverses 26 min
  17. Simultaneous equations 22 min
  18. Two-dimensional transformations 24 min
  19. Composite transformations 26 min
  20. Eigenvalues 22 min
  21. Eigenvectors 24 min
  22. Diagonalisation and powers 26 min
  23. Matrix models 22 min
  24. Differential-equation formulation 24 min
  25. Separable first-order equations 26 min
  26. Linear first-order equations 22 min
  27. First-order modelling 24 min
  28. Homogeneous second-order equations 26 min
  29. Non-homogeneous second-order equations 22 min
  30. Initial and boundary conditions 24 min
  31. Second-order modelling and interpretation 26 min
  32. Counting, Matrices and Differential Equations review 36 min
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