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Foundations

Grade 6–9

Intermediate

Grade 10–12

Advanced

College

Numbers are everywhere around us: small numbers, large numbers, fractional numbers, negative numbers, percentages, ratios, and many more. These chapters will help you understand different kinds of numbers, what you can do with them, and what interesting properties they have.

Coming Soon### The Integers

UpdatedNegative NumbersAbsolute ValueOrder of OperationsDistributive LawDivision by Zero

Multiples and FactorsDivisibility RulesPrime Numbers and Prime FactorisationLCM and GCD

Coming Soon### Fractions

IntroductionComparing FractionsSimplifying FractionsAddition and SubtractionMultiplication and DivisionMixed Numbers and Improper Fractions

Coming Soon### Decimals and Percentages

Decimal NotationDecimals on a number lineAddition and SubtractionMultiplication and DivisionConverting between Fractions and DecimalsPercentagesRatios, Proportion and Rates

Coming Soon### Place Value and Rounding

Place ValueBig and Small NumbersOrderingRounding Integers and DecimalsDifferent BasesRational and Irrational Numbers

Triangle, Square and Cube NumbersFibonacci NumbersPerfect NumbersArithmetic and Geometric SequencesRecursive EquationsPascal’s TriangleConvergence

Coming Soon### Exponents and Roots

ExponentsSquare and Cube RootsExponent LawsNegative and Fractional ExponentsSimplifying ExpressionsRationalising the DenominatorScientific Notation

Geometry is one of the oldest areas of mathematics: the laws of angles, triangles and circles were discovered by Greek scientists more than 2500 years ago. Geometry is also one of the most fun and beautiful parts of mathematics, and it has applications everywhere from architecture to astronomy, engineering or computer games.

Coming Soon### Euclidean Geometry

Introduction and HistoryLines, Segments and RaysAngles, Angle Types and Relationships between AnglesClassifying ShapesPrisms and Pyramids

Coming Soon### Measuring

Space, Time and Weight UnitsUnit ConversionArea, Circumference and Volume

Coming Soon### Triangles and Trigonometry

UpdatedTriangle InequalityPerpendicular Bisectors, Angle Bisectors, Medians, Centroids and AltitudesPythagoras’ TheoremTrigonometric FunctionsSine and Cosine Rules

Polygons and Internal AnglesQuadrilateralsPolyhedra and Platonic SolidsCross Section of 3D Objects

Circles and PiTangentsCircle TheoremsRadiansSectors and ArcsSpheres, Cones and Cylinders

Similar and Congruent ShapesReflection, Rotation, Translation and ScalingInverse and CombinationsSymmetry

Constructing Bisectors of Lines and AnglesConstructing TrianglesConstructing Regular PolygonsConstructing Circumcircles and IncirclesConstructing Tangent

Coming Soon### Coordinate Geometry

The Coordinate PlaneDistances, Midpoints and GradientsParallel and Perpendicular LinesEquations of Circles

Mathematics is not just about numbers or shapes - it is about abstract concepts and relationships. Algebra is the language which allows us to describe and understand these ideas. It is the foundation for all mathematics you will learn in the future, and has infinite applications.

Coming Soon### Introduction to Algebra

Introduction and HistoryVariablesManipulating and Evaluating ExpressionsDirect and Inverse Proportions

Coming Soon### Linear Equations

Equations and IdentitiesSolving Linear EquationsLinear InequalitiesSystems of Linear EquationsAbsolute Value Equations, Functions and Inequalities

Coming Soon### Functions

Graphing Two-Variable EquationsIntercepts and SlopeFunction NotationDomain and RangeMaxima and Minima, Increasing and Decreasing FunctionsRates of Change

Coming Soon### Polynomials

Polynomial AlrithmeticDividing Polynomials and Remainder TheoremBinomial Theorem and Difference of SquaresPolynomial FactorisationSolving Polynomial Equations

Coming Soon### Quadratics

Solving Quadratic EquationsCompleting the Square and the Quadratic FormulaParabolaeFinding Zeros and the Number of SolutionsTransforming Quadratic Functions

Coming Soon### Exponential and Rational Functions

Exponential Growth and DecayLimitsSquare Root Functions and Equations

Every day we use the topics from this section in some way - often unconsciously. Data lets us analyse and understand the world around us. Probability helps us make decisions. Codes let us to communicate with each other. And Graphs make up all transportation networks.

Updated### Probability

### Combinatorics

### Graphs and Networks

### Game Theory

IntroductionCoins and DiceRouletteVenn Diagrams and Set OperationsProbability TreesMonty Hall ProblemRandomness

FactorialsPermutations and CombinationsProbability and CombinatoricsRandom Walks

UpdatedBirthday ParadoxKönigsberg BridgesPlanar GraphsEuler’s FormulaMap ColouringTravelling SalesmanSocial Networks and Applications

Coming Soon### Codes and Ciphers

Simple Codes and MorseError Detection and CorrectionOne Time PadsCaesar and Vigenère CipherThe EnigmaRSA Cryptography

Combinatorial Games and NimTree DiagramsCoins, Cards and DicePrisoners’ Paradox

Coming Soon### Statistics and Data

Questions about DataUnderstanding ChartsMean, Meadian and ModeBox-and-whisker plotsHistogramsVariance and Standard Deviation

Building up on what you’ve already learned, these chapters will expand and formalise the core tools and concepts of mathematics: logic and proof, algebra and functions, sequences and series, complex numbers and polynomials.

Number SystemsLogic and ProofSets, Notation and Venn DiagramsFunctions, Domain and Range

Arithmetic and Geometric SequencesSigma NotationConvergence and DivergenceMathematical Induction

Coming Soon### Advanced Algebra

Solving Equations by GraphingQuadratic inequalitiesManipulating Rational ExpressionsRational InequalitiesSystems with Three VariablesPartial fraction expansion

Coming Soon### Functions

Combining and Composing FunctionsOven and Odd FunctionsShifting and Stretching FunctionsISketching FunctionsInverses of FunctionsFinding Domain and Range of FunctionsSymmetry and Periodicity3-Dimensional Functions

Coming Soon### Factors and Polynomials

Binomial TheoremFactor and Remainder TheoremAdvanced Polynomial FactorisationFundamental Theorem of AlgebraCubic EquationsZeros of PolynomialsPolynomial Functions

Coming Soon### Exponentials and Logarithms

Exponential EquationsIntroduction to LogarithmsThe Natural Logarithm and eChange of Base of LogarithmsLogarithmic EquationsGraphs of Exponential and Logarithmic FunctionsLogarithmic Scale

iThe Complex PlaneComplex Number ArithmeticModulus and ArgumentConjugate and InversePolar FormComplex Powers and n-th RootsDe Moivre’s Theorem

Coming Soon### Numeric Methods

InterationInterval BisectionNewton-Raphson Method

Hilbert’s HotelCountabilityCantor’s DiagonalThe Continuum Hypothesis

There is much more to geometry than just angles, shapes and solids. Here you can discover incredibly powerful tools, like trigonometry, vectors or matrices, as well as beautiful patterns like fractals or conic sections. Prepare to be amazed…

Coming Soon### Trigonometry

Trigonometric IdentitiesSpecial TrianglesUnit Circle DefinitionsInverse Trig FunctionsPythagorean IdentityDouble Angle and Addition FormulaeSine and Cosine RulesExponential RepresentationHyperbolic functions

Equation of a CircleEllipsesParabolae and HyperbolaeKepler’s laws and Planetary Orbits

Coming Soon### Parametric and Polar Coordinates

Parametric EquationsPolar Coordinates

Coming Soon### Vectors

IntroductionMagnitude and DirectionVector ArithmeticScalar ProductUnit Vectors

Coming Soon### Matrices

IntroductionMatrix ArithmeticMatrix MultiplictionLinear Systems of EquationsGaussian EleminationDeterminantsMatrix InversesMatrices as Transformations

Metric SpacesSpherical GeometryHyperbolic GeometryProjectionsHigher DimensionsTopologyMöbius Strip and Klein Bottle

Dimension of FractalsThe Sierpinski GasketMandelbrot SetFractals in Nature and Technology

Calculus is a completely new way of thinking in mathematics, invented in the 17th century by Newton and Leibnitz. We want to understand the change of functions and variables – by looking at them in infinitely small pieces. Calculus is also the foundation of most modern physics and science.

Coming Soon### Differentiation

LimitsEpsilon Delta DefinitionDifferentiation as Gradient of TangentsDerivativesPower Rule and Chain RuleProduct and Quotient RuleImplicit DifferentiationDerivatives of Inverse FunctionsEquations of Tangents and NormalsStationary Points and OptimisationMean Value TheoremL’Hôpital’s rule

Coming Soon### Integration

Integrals as Anti-DerivativesIntegrals as Areas under a CurveFundamental Theorem of CalculusTrapezium RuleIntegration by PartsIntegration by SubstitutionReverse Chain RuleIntegration using Trigonometric IdentitiesPartial Fraction ExpansionArc LengthSolids of RevolutionPolar GraphsApplications in Physics

Coming Soon### Power Series

Power SeriesMaclaurin Series and Euler’s IdentityTaylor Series

Coming Soon### Differential Equations

First Order Differential EquationsSeparable EquationsLogistic MapEuler’s MethodHomogenous EquationsThe Characteristic EquationMethod of Undetermined Coefficients

Coming Soon### Mechanics

Position Vectors, Velocity and AccelerationProjectilesNewton’s LawsEquilibriumApplications of CalculusMomentum, Energy and ConservationCircular Motion and TorqueSimple Harmonic Motion

From rolling dice, to weather forecasts to playing the lottery – probability and randomness is everywhere around us. We can use probability to predict the future, or statistics to analyse data from the past.

Coming Soon### Probability

Birthday ParadoxLaw of Large NumbersRandom Variables and DistributionsExpectation and VarianceBinomial distributionPoisson ProcessNormal Distribution

Coming Soon### Statistics

Sampling and EstimationMean and VarianceFrequency TablesScatter PlotsLinear CorrelationHypothesis TestsConfidence IntervalsChi-square TestsAnalysis of Variance

Coming Soon### Data Representation

Understanding and Interpreting ChartsCollecting DataCreating Charts and Data Visualisations

Coming Soon### Algorithms and Computing

Introduction to ComputingComputational Complexity and O() NotationSorting AlgorithmsLinear programming and the Simplex AlgorithmData StructuresGraphs, Trees and Networks

While the purpose of most sciences is to solve real world problems, pure mathematics is studied for its own sake. We want to investigate abstract patterns, structures and principles, and prove new results using a consistent logical framework. But of course, even pure mathematics has many real applications.

Coming Soon### Introduction

Notation and LogicSets, Relations and MapsInduction and Well-Ordering PrincipleBinomial TheoremModular ArithmeticEuclid’s Algorithm

Coming Soon### Linear Algebra

Systems of Equations and Gaussian EleminationVector SpacesIndependence and Change of BasisSubspacesLinear Maps and MatricesKernel and ImageEigenvalues and EigenvectorsMatrix Determinants and InversesBilinear FormsDual SpacesInner Product SpacesModules and Jordan Normal Form

Coming Soon### Groups, Rings and Modules

Group AxiomsSymmetry GroupsMatrix GroupsPermutationsSubgroups and ProductsLagrange’s theoremGroup Actions and Orbit Stabiliser TheoremQuotion Groups and Isomorphism TheoremsConjugationSylow TheoremsRing AxiomsPolynomial RingsDivision AlgorithmModule AxiomsPrimary Decomposition Theorem and Jordan Normal Form

Coming Soon### Geometry

Complex Numbers and De Moivre’s TheoremCoordinate Geometry and TransformationsVector EquationsConic SectionsIsometries of R3

Coming Soon### Analysis

Real NumbersSequences and ConvergenceSeries and Convergence TestsContinuityDifferentiabilityPower SeriesRiemann IntegrationNormed Spaces and Metric SpacesHomeomorphismsCompleteness and Contraction Mapping TheoremConnectedness and CompactnessTopological Spaces

Coming Soon### Complex Analysis

Complex DifferentiationHolomorphic FunctionsPower SeriesCauchy’s Theorem and Contour TntegrationExpansions and SingularitiesResidue Theorem and Jordan’s LemmaRiemann sphere and Mobius TransformationsConformal Mappings and Riemann Mapping Theorem

Coming Soon### Number Theory

Prime Numbers and Modular ArithmeticQuadratic ResiduesBinary Quadratic FormsDistribution of the PrimesContinued FractionsPrimality TestingFactorisation

Coming Soon### Logic and Set Theory

Countability and UncountabilityOrdinals and CardinalsPosets and Zorn’s LemmaPropositional Logic

Coming Soon### Graph Theory

IntroductionConnectivity and MatchingsExtremal Graph TheoryMatrix MethodsGraph Colouring Ramsey TheoryProbabilistic Methods

In applied mathematics we want to model and explain the real world - using methods like differential equations, linear algebra and vector calculus. Applications include fluid dynamics, electromagnetism and mechanics, but we often transform realistic conditions into more ideal and abstract analogues.

Coming Soon### Differential Equations

Limits and CalculusFirst Order ODEs (homogenous, inhomogenous, non-linear)Discrete EquationsHigher Order Linear ODEs (complimentary functions, Wronskian, resonance and damping)Difference EquationsSystems of Linear Differential EquationsPartial Derivatives

Coming Soon### Vector Calculus

Surface IntegralsVolume Integrals and JacobiansVector Differential OperatorsDivergence TheoremStokes’s theoremGauss’ Flux TheoremLaplace’s EquationCartesian Tensors

Coming Soon### Mathematical Methods

Fourier SeriesSturm-Liouville Theory and Green’s FunctionWave Equation and Bessel’s EquationDiffusion EquationLaplace EquationDirac Delta Function and Green’s FunctionsFourier Transforms and DFTsPartial DIfferential Equations on Unbounded DomainsPhase plane analysis

Coming Soon### Numerical Analysis

Polynomial ApproximationsnOrdinary Differential Equations: Euler’s method, Multistep methods, A-stabilitySystems of Equations and Gaussian EliminationQR factorizationLeast Squares Calculations

Coming Soon### Mechanics and Relativity

KinematicsDimensional AnalysisEnergy and MomentumCircular Motion and Rotating FramesKepler’s Laws and GravitySystems of Particles and Centre of MassRigid Bodies and InertiaLorentz Transformations and Special Relativity

Problems in applied mathematics are mostly related to physics and nature, whereas problems in applicable mathematics are usually related to conscious, human activities. This includes probability, statistics and data analysis, optimisation, networks and algorithms.

Coming Soon### Introduction to Probability

Axiomatic ProbabilityConditional Probability and Bayes TheoremDiscrete and Continuous Random VariablesProbability Generating FunctionsJoint, Marginal and Conditional DistributionsFunctions of Random VariablesLaw of Large NumbersCentral Limit Theorem

Coming Soon### Applied Probability and Markov Chains

Discrete Time Markov ChainsHitting Probabilities and Stopping TimesRecurrence and TransienceInvariant DistributionsTime ReversalContinuous Time Markov ChainsPoisson ProcessesQueueing TheoryApplications: Genetics, Renewal-Reward and Epidemics

Coming Soon### Statistics

LikelihoodEstimation and Confidence IntervalsBayesian InferenceHypothesis TestingGoodness-of-fit tests and Contingency TablesSimple Linear RegressionAnalysis of VarianceUnsupervised Learning and Principal Components AnalysisClustering Algorithms

Coming Soon### Optimisation

Lagrange Multipliers and Sufficiency TheoremLinear Progamming, Simplex Algorithm and the Duality TheoremFord-Fulkerson Algorithm and Max-flow Min-cut TheoremDynamic ProgrammingInfinite-horizon ProblemsStopping ProblemsAverage-cost ProgrammingLQG Systems and Quadratic Costs

These websites, apps and magazines showcase the incredible breadth and beauty of mathematics.

Learn about the countless hidden uses and applications which mathematics has in everyday life: From computers to traffic control, from weather prediction to video games, construction, medicine, sports, music or gambling…

Explore the beautiful world of Origami and Mathematics. Be amazed by stunning photographs, and discover folding instructions and mathematical explanations.

Eureka, published by the mathematical society of Cambridge University, is one of the oldest recreational mathematics magazines in the world. Authors include Stephen Hawking, Martin Gardner, Paul Dirac and Ian Stewart.

A selection of our favourite mathematical puzzles and problems. Most are very simple to explain, but the solutions require clever and unconventional thinking.

The key to successful teaching is captivating storytelling – through real life applications, curious examples, historic background, or even fictional characters. These interactive slideshows combine an engaging narrative with beautiful graphics – explaining mathematical ideas in the context of popular stories and movies. They can be watched individually or be presented in classrooms.

When Alice falls down the rabbit hole, she discovers curious and wonderful mathematics she could have never imagined: Pascal’s triangle on a colour changing floor, sequences of rabbit generations, and beautiful, never-ending fractals and golden spirals…

How do you rob an infinite hotel, getting infinitely rich without anyone noticing? Only Danny Ocean knows. Did you know that there are things bigger than infinity? And that some things in mathematics can never be proven?

Coming Soon

One day at Hogwarts School of Mathematics: planetary orbits and conic sections in Astronomy, crystal polyhedra in Divination, Möbius transformations in Transfiguration, and spherical and hyperbolic geometry in the Room of Requirements.

Coming Soon