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.
Negative NumbersAbsolute ValueOrder of OperationsDistributive LawDivision by Zero
Multiples and FactorsDivisibility RulesPrime Numbers and Prime FactorisationLCM and GCD
IntroductionComparing FractionsSimplifying FractionsAddition and SubtractionMultiplication and DivisionMixed Numbers and Improper Fractions
Decimal NotationDecimals on a number lineAddition and SubtractionMultiplication and DivisionConverting between Fractions and DecimalsPercentagesRatios, Proportion and Rates
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
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.
Introduction and HistoryLines, Segments and RaysAngles, Angle Types and Relationships between AnglesClassifying ShapesPrisms and Pyramids
Space, Time and Weight UnitsUnit ConversionArea, Circumference and Volume
Triangle 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
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.
Introduction and HistoryVariablesManipulating and Evaluating ExpressionsDirect and Inverse Proportions
Equations and IdentitiesSolving Linear EquationsLinear InequalitiesSystems of Linear EquationsAbsolute Value Equations, Functions and Inequalities
Graphing Two-Variable EquationsIntercepts and SlopeFunction NotationDomain and RangeMaxima and Minima, Increasing and Decreasing FunctionsRates of Change
Polynomial AlrithmeticDividing Polynomials and Remainder TheoremBinomial Theorem and Difference of SquaresPolynomial FactorisationSolving Polynomial Equations
Solving Quadratic EquationsCompleting the Square and the Quadratic FormulaParabolaeFinding Zeros and the Number of SolutionsTransforming Quadratic 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.
IntroductionCoins and DiceRouletteVenn Diagrams and Set OperationsProbability TreesMonty Hall ProblemRandomness
FactorialsPermutations and CombinationsProbability and CombinatoricsRandom WalksUpdated
Birthday ParadoxKönigsberg BridgesPlanar GraphsEuler’s FormulaMap ColouringTravelling SalesmanSocial Networks and Applications
Simple Codes and MorseError Detection and CorrectionOne Time PadsCaesar and Vigenère CipherThe EnigmaRSA Cryptography
Combinatorial Games and NimTree DiagramsCoins, Cards and DicePrisoners’ Paradox
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
Solving Equations by GraphingQuadratic inequalitiesManipulating Rational ExpressionsRational InequalitiesSystems with Three VariablesPartial fraction expansion
Combining and Composing FunctionsOven and Odd FunctionsShifting and Stretching FunctionsISketching FunctionsInverses of FunctionsFinding Domain and Range of FunctionsSymmetry and Periodicity3-Dimensional Functions
Binomial TheoremFactor and Remainder TheoremAdvanced Polynomial FactorisationFundamental Theorem of AlgebraCubic EquationsZeros of PolynomialsPolynomial Functions
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
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…
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
Parametric EquationsPolar Coordinates
IntroductionMagnitude and DirectionVector ArithmeticScalar ProductUnit Vectors
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.
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
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
Power SeriesMaclaurin Series and Euler’s IdentityTaylor Series
First Order Differential EquationsSeparable EquationsLogistic MapEuler’s MethodHomogenous EquationsThe Characteristic EquationMethod of Undetermined Coefficients
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.
Birthday ParadoxLaw of Large NumbersRandom Variables and DistributionsExpectation and VarianceBinomial distributionPoisson ProcessNormal Distribution
Sampling and EstimationMean and VarianceFrequency TablesScatter PlotsLinear CorrelationHypothesis TestsConfidence IntervalsChi-square TestsAnalysis of Variance
Understanding and Interpreting ChartsCollecting DataCreating Charts and Data Visualisations
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.
Notation and LogicSets, Relations and MapsInduction and Well-Ordering PrincipleBinomial TheoremModular ArithmeticEuclid’s Algorithm
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
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
Complex Numbers and De Moivre’s TheoremCoordinate Geometry and TransformationsVector EquationsConic SectionsIsometries of R3
Real NumbersSequences and ConvergenceSeries and Convergence TestsContinuityDifferentiabilityPower SeriesRiemann IntegrationNormed Spaces and Metric SpacesHomeomorphismsCompleteness and Contraction Mapping TheoremConnectedness and CompactnessTopological Spaces
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
Prime Numbers and Modular ArithmeticQuadratic ResiduesBinary Quadratic FormsDistribution of the PrimesContinued FractionsPrimality TestingFactorisation
Countability and UncountabilityOrdinals and CardinalsPosets and Zorn’s LemmaPropositional Logic
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.
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
Surface IntegralsVolume Integrals and JacobiansVector Differential OperatorsDivergence TheoremStokes’s theoremGauss’ Flux TheoremLaplace’s EquationCartesian Tensors
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
Polynomial ApproximationsnOrdinary Differential Equations: Euler’s method, Multistep methods, A-stabilitySystems of Equations and Gaussian EliminationQR factorizationLeast Squares Calculations
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.
Axiomatic ProbabilityConditional Probability and Bayes TheoremDiscrete and Continuous Random VariablesProbability Generating FunctionsJoint, Marginal and Conditional DistributionsFunctions of Random VariablesLaw of Large NumbersCentral Limit Theorem
Discrete Time Markov ChainsHitting Probabilities and Stopping TimesRecurrence and TransienceInvariant DistributionsTime ReversalContinuous Time Markov ChainsPoisson ProcessesQueueing TheoryApplications: Genetics, Renewal-Reward and Epidemics
LikelihoodEstimation and Confidence IntervalsBayesian InferenceHypothesis TestingGoodness-of-fit tests and Contingency TablesSimple Linear RegressionAnalysis of VarianceUnsupervised Learning and Principal Components AnalysisClustering Algorithms
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 weekly collection of fun and challenging exercises, designed to encourage young mathematicians to explore the world of mathematics, written by bestselling author Simon Singh.
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?
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.