"Core" Subject Taxonomy for Mathematical Sciences Education 4/29/2005 1.0 Numbers and Computation 1.1 Number Concepts 1.1.1 Natural 1.1.2 Integers 1.1.3 Rational 1.1.4 Irrational 1.1.5 Algebraic 1.1.6 Real 1.1.7 Complex 1.1.8 Famous Numbers 1.1.8.1 0 1.1.8.2 pi 1.1.8.3 e 1.1.8.4 i 1.1.8.5 Golden Mean 1.2 Arithmetic 1.2.1 Operations 1.2.1.1 Addition 1.2.1.2 Subtraction 1.2.1.3 Multiplication 1.2.1.4 Division 1.2.1.5 Roots 1.2.1.6 Factorials 1.2.1.7 Factoring 1.2.1.8 Properties of Operations 1.2.1.9 Estimation 1.2.2 Fractions 1.2.2.1 Addition 1.2.2.2 Subtraction 1.2.2.3 Multiplication 1.2.2.4 Division 1.2.2.5 Ratio and Proportion 1.2.2.6 Equivalent Fractions 1.2.3 Decimals 1.2.3.1 Addition 1.2.3.2 Subtraction 1.2.3.3 Multiplication 1.2.3.4 Division 1.2.3.5 Percents 1.2.4 Comparison of numbers 1.2.5 Exponents 1.2.5.1 Multiplication 1.2.5.2 Division 1.2.5.3 Powers 1.2.5.4 Integer Exponents 1.2.5.5 Rational Exponents 1.3 Patterns and Sequences 1.3.1 Number Patterns 1.3.2 Fibonacci Sequence 1.3.3 Arithmetic Sequence 1.3.4 Geometric Sequence 1.4 Measurement 1.4.1 Units of Measurement 1.4.1.1 Metric System 1.4.1.2 Standard Units 1.4.1.3 Nonstandard Units 1.4.2 Linear Measure 1.4.2.1 Distance 1.4.2.2 Circumference 1.4.2.3 Perimeter 1.4.3 Area 1.4.3.1 Area of Polygons 1.4.3.2 Area of Circles 1.4.3.3 Surface Area 1.4.3.4 Nonstandard Shapes 1.4.4 Volume 1.4.5 Weight and Mass 1.4.6 Temperature 1.4.7 Time 1.4.8 Speed 1.4.9 Money 1.4.10 Scale 2.0 Logic and Foundations 2.1 Logic 2.1.1 Venn Diagrams 2.1.2 Propositional and Predicate Logic 2.1.3 Methods of Proof 2.2 Set Theory 2.2.1 Sets and Set Operations 2.2.2 Relations and Functions 2.2.3 Cardinality 2.2.4 Axiom of Choice 2.3 Computability and Decidability 2.4 Model Theory 3.0 Algebra and Number Theory 3.1 Algebra 3.1.1 Graphing Techniques 3.1.2 Algebraic Manipulation 3.1.3 Functions 3.1.3.1 Linear 3.1.3.2 Quadratic 3.1.3.3 Polynomial 3.1.3.4 Rational 3.1.3.5 Exponential 3.1.3.6 Logarithmic 3.1.3.7 Piece-wise 3.1.3.8 Step 3.1.4 Equations 3.1.4.1 Linear 3.1.4.2 Quadratic 3.1.4.3 Polynomial 3.1.4.4 Rational 3.1.4.5 Exponential 3.1.4.6 Logarithmic 3.1.4.7 Systems 3.1.5 Inequalities 3.1.6 Matrices 3.1.7 Sequences and Series 3.1.8 Algebraic Proof 3.2 Linear Algebra 3.2.1 Systems of Linear Equations 3.2.2 Matrix algebra 3.2.3 Vectors in R3 3.2.4 Vector Spaces 3.2.5 Linear Transformations 3.2.6 Eigenvalues and Eigenvectors 3.2.7 Inner Product Spaces 3.3 Abstract Algebra 3.3.1 Groups 3.3.2 Rings and Ideals 3.3.3 Fields 3.3.4 Galois Theory 3.3.5 Multilinear Algebra 3.4 Number Theory 3.4.1 Integers 3.4.2 Primes 3.4.2.1 Divisibility 3.4.2.2 Factorization 3.4.2.3 Distributions of Primes 3.4.3 Congruences 3.4.4 Diophantine Equations 3.4.5 Irrational Numbers 3.4.6 Famous Problems 3.4.7 Coding Theory 3.4.8 Cryptography 3.5 Category Theory 3.6 K-Theory 3.7 Homological Algebra 3.8 Modular Arithmetic 4.0 Discrete Mathematics 4.1 Cellular Automata 4.2 Combinatorics 4.2.1 Combinations 4.2.2 Permutations 4.3 Game Theory 4.4 Algorithms 4.5 Recursion 4.6 Graph Theory 4.7 Linear Programming 4.8 Order and Lattices 4.9 Theory of Computation 4.10 Chaos 5.0 Geometry and Topology 5.1 Geometric Proof 5.2 Plane Geometry 5.2.1 Measurement 5.2.2 Lines and Planes 5.2.3 Angles 5.2.4 Triangles 5.2.4.1 Properties 5.2.4.2 Congruence 5.2.4.3 Similarity 5.2.4.4 Pythagorean Theorem 5.2.5 Polygons 5.2.5.1 Properties 5.2.5.2 Regular 5.2.5.3 Irregular 5.2.5.4 Congruence 5.2.5.5 Similarity 5.2.6 Circles 5.2.7 Patterns 5.2.7.1 Geometric Patterns 5.2.7.2 Tilings and Tessellations 5.2.7.3 Symmetry 5.2.7.4 Golden Ratio 5.2.8 Transformations 5.2.8.1 Translation 5.2.8.2 Rotation 5.2.8.3 Reflection 5.2.8.4 Scaling 5.3 Solid Geometry 5.3.1 Dihedral Angles 5.3.2 Spheres 5.3.3 Cones 5.3.4 Cylinders 5.3.5 Pyramids 5.3.6 Prisms 5.3.7 Polyhedra 5.4 Analytic Geometry 5.4.1 Cartesian Coordinates 5.4.2 Lines 5.4.3 Circles 5.4.4 Planes 5.4.5 Conics 5.4.6 Polar Coordinates 5.4.7 Parametric Curves 5.4.8 Surfaces 5.4.9 Distance Formula 5.5 Projective Geometry 5.6 Differential Geometry 5.7 Algebraic Geometry 5.8 Topology 5.8.1 Point Set Topology 5.8.2 General Topology 5.8.3 Differential Topology 5.8.4 Algebraic Topology 5.9 Trigonometry 5.9.1 Angles 5.9.2 Trigonometric Functions 5.9.3 Inverse Trigonometric Functions 5.9.4 Trigonometric Identities 5.9.5 Trigonometric Equations 5.9.6 Roots of Unity 5.9.7 Spherical Trigonometry 5.10 Fractal Geometry 6.0 Calculus 6.1 Single Variable 6.1.1 Functions 6.1.2 Limits 6.1.3 Continuity 6.1.4 Differentiation 6.1.5 Integration 6.1.6 Series 6.2 Several Variables 6.2.1 Functions of Several Variables 6.2.2 Limits 6.2.3 Continuity 6.2.4 Partial Derivatives 6.2.5 Multiple integrals 6.2.6 Taylor Series 6.3 Advanced Calculus 6.3.1 Vector Valued Functions 6.3.2 Line Integrals 6.3.3 Surface Integrals 6.3.4 Stokes Theorem 6.3.5 Curvilinear Coordinates 6.3.6 Linear spaces 6.3.7 Fourier Series 6.3.8 Orthogonal Functions 6.4 Tensor Calculus 6.5 Calculus of Variations 6.6 Operational Calculus 7.0 Analysis 7.1 Real Analysis 7.1.1 Metric Spaces 7.1.2 Convergence 7.1.3 Continuity 7.1.4 Differentiation 7.1.5 Integration 7.1.6 Measure Theory 7.2 Complex Analysis 7.2.1 Convergence 7.2.2 Infinite Series 7.2.3 Analytic Functions 7.2.4 Integration 7.2.5 Contour Integrals 7.2.6 Conformal Mappings 7.2.7 Several Complex Variables 7.3 Numerical Analysis 7.3.1 Computer Arithmetic 7.3.2 Solutions of Equations 7.3.3 Solutions of Systems 7.3.4 Interpolation 7.3.5 Numerical Differentiation 7.3.6 Numerical Integration 7.3.7 Numerical Solutions of ODEs 7.3.8 Numerical Solutions of PDEs 7.4 Integral Transforms 7.4.1 Fourier Transforms 7.4.2 Laplace Transforms 7.4.3 Hankel Transforms 7.4.4 Wavelets 7.4.5 Other Transforms 7.5 Signal Analysis 7.5.1 Sampling Theory 7.5.2 Filters 7.5.3 Noise 7.5.4 Data Compression 7.5.5 Image Processing 7.6 Functional Analysis 7.6.1 Hilbert Spaces 7.6.2 Banach Spaces 7.6.3 Topological Spaces 7.6.4 Locally Convex Spaces 7.6.5 Bounded Operators 7.6.6 Spectral Theorem 7.6.7 Unbounded Operators 7.7 Harmonic Analysis 7.8 Global Analysis 8.0 Differential and Difference Equations 8.1 Ordinary Differential Equations 8.1.1 First Order 8.1.2 Second Order 8.1.3 Linear Oscillations 8.1.4 Nonlinear Oscillations 8.1.5 Systems of Differential Equations 8.1.6 Sturm Liouville Problems 8.1.7 Special Functions 8.1.8 Power Series Methods 8.1.9 Laplace Transforms 8.2 Partial Differential Equations 8.2.1 First Order 8.2.2 Elliptic 8.2.3 Parabolic 8.2.4 Hyperbolic 8.2.5 Integral Transforms 8.2.6 Integral Equations 8.2.7 Potential Theory 8.2.8 Nonlinear Equations 8.2.9 Symmetries and Integrability 8.3 Difference Equations 8.3.1 First Order 8.3.2 Second Order 8.3.3 Linear Systems 8.3.4 Z Transforms 8.3.5 Orthogonal Polynomials 8.4 Dynamical Systems 8.4.1 1D Maps 8.4.2 2D Maps 8.4.3 Lyapunov Exponents 8.4.4 Bifurcations 8.4.5 Fractals 8.4.6 Differentiable Dynamics 8.4.7 Conservative Dynamics 8.4.8 Chaos 8.4.9 Complex Dynamical Systems 9.0 Statistics and Probability 9.1. Data Collection 9.1.1. Experimental Design 9.1.2. Sampling and Surveys 9.1.3. Data and Measurement Issues 9.2. Data Summary and Presentation 9.2.1. Summary Statistics 9.2.1.1. Measures of Central Tendency 9.2.1.2. Measures of Spread 9.2.2. Data Representation 9.2.2.1. Graphs and Plots 9.2.2.2. Tables 9.3. Statistical Inference and Techniques 9.3.1. Sampling Distributions 9.3.2. Regression and Correlation 9.3.3. Confidence Intervals 9.3.4. Hypothesis Tests 9.3.5. Statistical Quality Control 9.3.6. Non-parametric Techniques 9.3.7. Multivariate Techniques 9.3.8. Survival Analysis 9.3.9. Bayesian Statistics 9.4. Probability 9.4.1. Elementary Probability 9.4.1.1. Sample Space and Events 9.4.1.2. General Rules 9.4.1.3. Combinations and Permutations 9.4.1.4. Random Variables 9.4.2. Univariate Distributions 9.4.2.1. Discrete Distributions 9.4.2.2. Continuous Distributions 9.4.2.3. Expected Value 9.4.3. Limit Theorems 9.4.3.1. Central Limit Theorem 9.4.3.2. Law of Large Numbers 9.4.4. Multivariate Distributions 9.4.4.1. Joint 9.4.4.2. Conditional 9.4.4.3. Expectations 9.4.5. Stochastic Processes 9.4.5.1. Brownian Motion 9.4.5.2. Markov Chains 9.4.5.3. Queuing Theory 9.4.6. Probability Measures 9.4.7. Simulation 10.0 Applied Mathematics 10.1 Mathematical Physics 10.2 Mathematical Economics 10.3 Mathematical Biology 10.4 Mathematics for Business 10.5 Engineering Mathematics 10.6 Mathematical Sociology 10.7 Mathematics for Social Sciences 10.8 Mathematics for Computer Science 10.9 Mathematics for Humanities 10.10 Consumer Mathematics 11.0 Mathematics History 11.1 General 11.2 Famous Problems 11.3 Biographies of Mathematicians