Table of Contents

MathPebbles

  1. Arithmetic
    1. Counting
      1. Number as Matching
      2. Pebbles of Various Sizes
      3. Digits
      4. Carry Overs
      5. Units
      6. Base Representations
      7. Largest Number
    2. Addition
      1. Notions of Additions
      2. Adding Single Digits
      3. Complements of Ten
      4. Left to Right Addition
      5. Significant Addition
      6. Tricks and Checks of Addition
      7. Properties of Addition
    3. Multiplication
      1. Notions of Multiplication
      2. Single Digit Multiplication
      3. No Carry Left to Right Multiplication
      4. Carry Left to Right Multiplication
      5. Significant Multiplication
      6. Tricks and Checks of Multiplication
      7. Properties of Multiplication
    4. Subtraction
      1. Removal
      2. Complement Subtraction
      3. Left to Right Subtraction
      4. Negative Numbers
      5. Significant Subtraction
      6. Tricks and Checks of Subtraction
      7. Properties of Subtraction
    5. Division
      1. Notions of Division
      2. Foundational Divisions
      3. Long Division
      4. Significant Division
      5. Fractions
      6. Decimals
      7. Tricks and Checks of Division
    6. Powers
      1. Notions of Exponents
      2. Rules of Exponents
      3. Scientific Notation
      4. Significant Powers
      5. Guesstimation
      6. Undoing Exponentiation
      7. Prime Factorization
    7. Constructing Numbers
      1. Containers
      2. Natural Numbers
      3. Addition and Multiplication
      4. Integers
      5. Rationals
      6. Properties Revisited
      7. Relations
  2. Algebra
    1. Lines
      1. Shortest Distance
      2. Basic Distance Shapes
      3. Cartesian Coordinate Plane
      4. Slope
      5. Solving a Linear Equation
      6. Families of Lines
      7. Linear Inequalities
    2. Quadratics
      1. Notions of Quadratics
      2. The Vertex of a Quadratic
      3. Square Roots
      4. Solving Quadratics
      5. Complex Numbers
      6. Common Quadratic Problems
      7. Quadratic Inequalities
    3. Polynomials
      1. Arithmetic of Polynomials
      2. The Questions of Polynomials
      3. Synthetic Division
      4. Zeros of Polynomials
      5. Calculus of Polynomials
      6. Graphing polynomials
      7. Powering Polynomials
    4. Interest Accumulations
      1. Percentages
      2. Simple Interest and Discount
      3. Compound Interest
      4. Effective Rate
      5. Payments
      6. Personal Finances
      7. Tricks of Business Mathematics
    5. Inequalities and Absolute Values
      1. Basic Inequalities
      2. Polynomial Inequalities
      3. Triangle Inequality
      4. Absolute Value
      5. Solving Absolute Value Equations
      6. Absolute Value Inequalities
      7. Distance
    6. Linear Systems
      1. Two Variable Systems
      2. Systems as Matrices
      3. Solving Systems
      4. Applications
      5. Not the Right Amount of Information
      6. Linear Programming
      7. Simplex Method
    7. Constructing the Real and Complex Numbers
      1. Square Root 2
      2. Zeroing in on Zeros
      3. Notions of Real Numbers
      4. Polynomial Zeros Beyond Radicals
      5. Complex Number Construction
      6. Arithmetic and Geometry of Complex Numbers
      7. Sizes of Infinity
  3. Geometry
    1. Plane Geometry
      1. Euclidean Lines
      2. Euclidean Triangles
      3. Similarity
      4. Polygons
      5. Euclidean Circles
      6. Transformations
      7. Coordinate Proofs
    2. Constructions
      1. Ruler and Compass
      2. Constructions with Lines
      3. Constructing Angles
      4. Drawing Triangles
      5. Constructions with Circles
      6. Trisecting the Angle
      7. Constructing Solutions to Equations
    3. Trigonometry
      1. Right Triangle
      2. Any Triangle
      3. Regular Polygons
      4. Areas
      5. Measuring from a Distance
      6. Parallax
      7. Circumference of the Earth
    4. Conic Sections
      1. Notions of a Conic Section
      2. Ellipses
      3. Parabolas
      4. Hyperbolas
      5. Cones
      6. Unifying Pictures
      7. Generalizations
    5. Parametric Curves
      1. Notions of Parametric Curves
      2. Functions of x as a Function of t
      3. Spirals
      4. Tangent Vectors
      5. Arc Length
      6. Reparametrizations
      7. Geodesics
    6. Higher Dimensions
      1. Three Dimensions
      2. Volumes
      3. Surface area
      4. Even Higher Dimensions
      5. Many Variable Inequalities
      6. Norms and Angles
      7. Transforming Space
    7. Other Geometries
      1. Taxicab Geometry
      2. Spherical Geometry
      3. Hyperbolic Geometry
      4. Discrete Geometries
      5. Metric Geometries
      6. Infinite Dimensional Geometries
      7. Compact Geometry
  4. Functions
    1. Rational and Power Functions
      1. Rational Functions
      2. Graphing Rational Functions
      3. Calculus of Rational Functions
      4. Rational Approximations
      5. Powering Functions
      6. Graphing Radical Functions
      7. Calculus of Radical Functions
    2. Infinite Processes
      1. Sequences
      2. Sums and products
      3. Off to Infinity
      4. Limits
      5. Iterative Processes
      6. Errors
      7. The Reality of Infinity
    3. Differential Calculus
      1. Notions of Derivatives
      2. Using Derivatives
      3. Derivative Rules
      4. Newtons Method
      5. Optimization Problems
      6. Taylor Polynomials
      7. Implicit Differentiation
    4. Integral Calculus
      1. Notions of Integrals
      2. Fundamental Theorem of Calculus
      3. Integral Rules
      4. Numerical Integration
      5. Arc Length of Functions
      6. Volumes of Revolution
      7. Error Estimates
    5. Exponentials and Logarithms
      1. Paths to the Exponential
      2. Properties of the Exponential
      3. Paths to the Logarithm
      4. Properties of the Logarithm
      5. Dealing with Large Numbers
      6. Applications of Exponentials and Logarithms
      7. Asymptotics
    6. Trigonometric Functions
      1. Notions of Trigonometric Functions
      2. Properties of Trigonometric Functions
      3. Approximating Trigonometric Functions
      4. Inverse Trigonometric Functions
      5. Applications of Trigonometric Functions
      6. Sums and Products of Trigonometric Functions
      7. Complex Exponentials
    7. Defining Functions
      1. Taylor Series
      2. Functional Equations
      3. Differential Equations
      4. Special Functions
      5. Laplace Transforms
      6. Fourier Approximations
      7. Tricks
  5. Many Variables
    1. Linear Algebra
      1. Vectors
      2. Subspaces
      3. Matrices
      4. Solving Linear Systems
      5. Changing the Basis
      6. Diagonalizations
      7. Complex Linear Spaces
    2. Systems of Ordinary Differential Equations
      1. Predator Prey
      2. Two Dimensional Systems
      3. Matrix Solutions
      4. Numerical Systems
      5. Non-linear Differential Systems
      6. Chaos
      7. Critical Points
    3. Multivariable Functions
      1. One to Many
      2. Many to One
      3. Complex to Complex
      4. Many to Many
      5. Graphs
      6. Questions to Ask
      7. Multivariable Polynomials
    4. Multivaribale Differential Calculus
      1. Total Derivatives
      2. Partial Derivatives
      3. Tangentials
      4. Extrema
      5. Lagrange Multiplies
      6. High Dimensional Newtons Method
      7. Local Approximations
    5. Multivariable Integral Calculus
      1. Volume Integrals
      2. Surface Area Integrals
      3. Line Integrals
      4. Integral Relations
      5. Change of Basis in Integrals
      6. Antiderivatives
      7. Generalizing Multivariable Integrals
    6. Partial Differential Equations
      1. Partial versus Ordinary
      2. Verifying Solutions
      3. Laplacian Equations
      4. Boundary Conditions
      5. Time versus Space
      6. Approximations in PDEs
      7. Generalized Solutions
    7. Curved Spaces
      1. Surfaces in 3D
      2. Curves
      3. Coordinates
      4. Tangent Spaces
      5. Metrics
      6. Tensors and Differential Forms
      7. Topological Concerns
  6. Probability and Statistics
    1. Descriptive Statistics
      1. Gathering Data
      2. Notions of Centrality
      3. Notions of Spread
      4. Charts
      5. Other Measures
      6. Stories
      7. Lies
    2. Fitting Functions
      1. Exact Fits
      2. Least Squares
      3. Linear Regression
      4. Polynomial Regression
      5. Logging Data for Regression
      6. Trust and Regression
      7. General Linear Model
    3. Probability
      1. Counting Outcomes
      2. Combinatorial Probability
      3. Random variables
      4. Special Random Variables
      5. Conditional Probabilities
      6. Central Limit Theorem
      7. To Ponder
    4. Frequentist Statistics
      1. Notions of Frequentist Statistics
      2. Varieties of Statistics
      3. Proper Data Collection
      4. z-Test
      5. t-Test
      6. ANOVA
      7. Categorical Tests
    5. Bayesian Statistics
      1. Using Bayes Theorem
      2. Parameter Estimation
      3. Priors
      4. Posteriors
      5. Sampling
      6. Monte Carlo
      7. Comparing Methods
    6. Simulations
      1. Exploring Probability
      2. Randomness
      3. Simulating Random Variables
      4. Central Limit Theorem Explorations
      5. z-Test Explorations
      6. t-Test Explorations
      7. Bayesian Explorations
    7. Multivariate Statistics
      1. Random Multivariates
      2. Joint Distribution
      3. Multivariate Distributions
      4. Multivariate Means
      5. Multivariate Normals
      6. Multivariate Analysis
      7. Multivariate Regression
  7. Practitioners
    1. Algebraists
      1. Abstract Algebra
      2. Number Theory
      3. Cryptography
      4. Algebraic Geometry
      5. Algebraic Topology
      6. Computer Algebra
      7. Category Theory
    2. Analysts
      1. Real Analysis
      2. Complex Analysis
      3. Numerical Analysis
      4. Functional Analysis
      5. Calculus of Variations
      6. Delta Functions
      7. Optimization
    3. Geometers
      1. Toplogy
      2. Differential Geometry
      3. Soap Bubbles
      4. Symplectic Geometry
      5. Control Theory
      6. Mathematical Physics
      7. Infinite Dimensional Spaces
    4. Booleans
      1. Combinatorics
      2. Graph Theory
      3. Mathematical Logic
      4. Game Theory
      5. Computer Scientists
      6. Politicians
      7. Risk Managers
    5. Scientists
      1. Physics
      2. Chemistry
      3. Biology
      4. Health
      5. Ecology
      6. Economics
      7. Sociology
    6. Engineers
      1. Mechanical Engineering
      2. Electrical Engineering
      3. Chemical Engineering
      4. Aerospace Engineering
      5. Actuarial Science
      6. Meterology
      7. Finance
    7. Artists
      1. Music
      2. Writing
      3. Architecture
      4. Visualizations
      5. Computer Generation
      6. Entertainment
      7. Fractals
Arithmetic Algebra Geometry Functions Many Variables Probability and Statistics Practitioners