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Lessons
Algebra
Numbers
Numbers
1. Integers
1a. Counting Game
2. Number Properties
3. Order of Operations
4. Powers, Roots and Radicals
5. Approximate Numbers
5b. Pi
6. Scientific Notation
7. Ratio and Proportion
7a. Math of Beauty
8. Tesla Multiplication 3D interactive applet
Basic Algebra
Basic Algebra
1. Addition and Subtraction of Algebraic Expressions
2. Multiplication of Algebraic Expressions
3. Division of Algebraic Expressions
4. Solving Equations
al-Khwarizmi Father of Algebra
5. Formulas and Literal Equations
6. Applied Verbal Problems
Systems of Equations
Systems of Equations
1. Simultaneous Linear Equations
2. Graphs of Linear Functions
3. Graphical Solution of Linear Systems
4. Algebraic Solutions of Linear Systems
5. Graphical Solution of non-linear Systems
6. Algebraic Solution of non-Linear Systems
Matrices and Determinants
Matrices and Determinants
1. Determinants
2. Large Determinants
3. Matrices
4. Multiplication of Matrices
4a. Matrix Multiplication examples
4b. Add & multiply matrices applet
5. Finding the Inverse of a Matrix
5a. Simple Matrix Calculator
5b. Inverse of a Matrix using Gauss-Jordan Elimination
6. Matrices and Linear Equations
Systems of 3x3 Equations interactive applet
Factoring and Fractions
Factoring and Fractions
1. Special Products
2. Common Factor and Difference of Squares
3. Factoring Trinomials
4. The Sum and Difference of Cubes
5. Equivalent Fractions
6. Multiplication and Division of Fractions
7. Addition and Subtraction of Fractions
8. Equations Involving Fractions
Quadratic Equations
Quadratic Equations
1. Solving Quadratic Equations by Factoring
2. Completing the Square
3. The Quadratic Formula
4. The Graph of the Quadratic Function
5. Equations in Quadratic Form
Interactive Quadratic Function Graph
Sum and product of the roots of a quadratic equation
Exponents and Radicals
Exponents and Radicals
1. Simplifying Expressions with Integral Exponents
2. Fractional Exponents
3. Simplest Radical Form
4. Addition and Subtraction of Radicals
5. Multiplication and Division of Radicals (Rationalizing the Denominator)
6. Equations with Radicals
Exponential & Logarithmic Functions
Exponential & Logarithmic Functions
1. Definitions: Exponential and Logarithmic Functions
2. Graphs of Exponential and Logarithmic Equations
3. Logarithm Laws
4. Logarithms to Base 10
5. Natural Logarithms
(base e)
Dow Jones Industrial Average
Calculating the value of
e
6. Exponential and Logarithmic Equations
World Population Live
7. Graphs on Logarithmic and Semilogarithmic Axes
Interactive Log Table
Polynomial Equations of Higher Degree
Polynomial Equations of Higher Degree
1. Polynomial Functions and Equations
2. Remainder and Factor Theorems
3. How to factor polynomials
4. Roots of a Polynomial Equation
5. Roots of Polynomial Equations using Graphs
Inequalities
Inequalities
1. Properties of Inequalities
2. Solving Linear Inequalities
3. Solving Non-Linear Inequalties
4. Inequalities Involving Absolute Values
Series and the Binomial Theorem
Series and the Binomial Theorem
1. Arithmetic Progressions
2. Geometric Progressions
3. Infinite Geometric Series
4. The Binomial Theorem
Binomial Series Interactive
Geometry
Functions and Graphs
Functions and Graphs
1. Introduction to Functions
2. Functions from Verbal Statements
3. Rectangular Coordinates
4. The Graph of a Function
4a. Domain and Range of a Function
4c. Comparison calculator BMI - BAI
5. Graphing Using a Computer Algebra System
5a. Online graphing calculator (1): Plot your own graph (JSXGraph)
5b. Online graphing calculator (2): Plot your own graph (SVG)
6. Graphs of Functions Defined by Tables of Data
7. Continuous and Discontinuous Functions
8. Split Functions
9. Even and Odd Functions
4b. Domain and Range interactive applet
Introduction to Geometry
how to multiply a vector by a scalar
euclids axioms and postulates a breakdown
concurrent lines in geometry
everything you need to know about parallel lines
all about right circular cones
a brief overview of the triangle inequality
the secant of a circle
centroids of a triangle a detailed overview
what is an apothem
the mathematical beauty of the oval shape
lines and angles the basics
the different types of quadrilaterals
the meaning of diameter in geometry
tangents and normals in geometry
45 degree angle in geometry
everything you need to know about opposite angles
the perpendicular bisector theorem in geometry
the origins of geometry
isosceles right triangles in geometry
the many shapes of geometry
a review of the perimeter of a pentagon in geometry
angle angle side similarity theorem
euclids geometry the basics you need to know
how to construct an angle of 90 degrees in geometry
what is coplanar
all about same base same parallels in geometry
the eccentricity of an ellipse
alternate segment theorem
understanding the proportionality theorem
all about chords and diameters
constructing perpendicular bisectors in geometry
the many wonders of concave shapes
triangle inequality theorem
the slope intercept formula
what is a cyclic quadrilateral
equation of a line parallel to the x axis in geometry
lines and points in geometry
polar coordinates in geometry
euclidean geometry a brief introduction
explaining patterns in geometry
what is an ordered pair in geometry
what is the latus rectum of an ellipse
what is the axis of symmetry of a parabola
the transitive property of congruence in geometry
congruent angles in geometry
trapezoidal prism in geometry
chords of a circle in geometry
all about 3d shapes
understanding vectors in geometry
what is a parabola
a look at the angle addition postulate
lines of symmetry in a parallelogram
the angles of a parallelogram
reference angles in geometry
collinearity in geometry
corresponding sides in geometry
plane definition in geometry everything you need to know
median of a triangle in geometry
foci of a hyperbola
what is the directrix of an ellipse
vertex of hyperbola in geometry explained
what is the latus rectum of a parabola
lines parallel to the same line in geometry
the difference between a rhombus and a rectangle
explaining equal and unequal chords in geometry
the basics of labelling points in a plane
a brief introduction to arc length
constructing circles in geometry
polyhedron in geometry
what is external division in geometry
what are collinear vectors in geometry
rotational symmetry in geometry
the hypotenuse of a right triangle
what is the circumcenter in geometry
concyclic points in geometry
pairs of angles in geometry
the uniqueness of the tangent at a point in geometry
vector quantities in geometry
parallelograms with the same base and two parallels
the rhs criterion for proof in geometry
what is sas in geometry
how to handle vectors in ij form in geometry
triangles with the same base and parallel sides
the ssa side side angle congruence rule
the i j system in geometry
what is a perpendicular bisector
the sss criterion for similarity of triangles
dividing geometry internally
aa criterion in triangles everything you need to know
what is adjacency in geometry
the asa criterion for triangle congruence
euclids fifth postulate the parallel postulate
point slope form in geometry
pythagorean triples in geometry
the negative of a vector in geometry
why do lines tend to be parallel
what is a central angle
angle sum theorem
the different types of closed shapes in geometry
what is a horizontal line in geometry
the many properties of a kite
the exterior angle of a triangle in geometry
the x intercept in geometry
what is an octahedron in geometry
diagonals of a rectangle
the many wonders of positive slope
the cartesian coordinate system
the various properties of parallel lines in geometry
convex and concave in geometry
eccentricity in geometry a quick overview
the properties of parallelograms
what is a right angle
what is a sphere in geometry
direction cosine in geometry
sixty degrees and other angles in geometry
how to measure angles in geometry
what is geometric construction
how to find the area of an ellipse
what are concentric circles in geometry
polygons in geometry
what is a decagon
how to draw a compass in geometry
what is a straight line in geometry
what are parallel vectors
the center of a circle in geometry
the angles of a square
polygons
30 60 90 triangles in geometry
angles of a rectangle in geometry
what is the cartesian plane
what is equidistant in geometry
the unit circle in geometry
what is the argand plane in geometry
all about similar triangles in geometry
a brief overview of the triangle inequality theorem
what is an isosceles trapezoid
how to construct angles in geometry
triangular prism in geometry
what is the y intercept in geometry
how to convert degrees to radians in geometry
the right angled triangle in geometry
all about lines in geometry
triangular pyramid in geometry
negative slope in geometry an explanation
the angle sum property in geometry
coterminal angles in geometry
the diagonal of a rhombus in geometry
parallel and perpendicular lines in geometry
vectors in geometry a simple explanation
isosceles obtuse triangles in geometry
dodecahedron in geometry
the acute scalene triangle
a quick overview of ellipses in geometry
triangle congruence theorem all you need to know
what is transversal in geometry
the asa congruence rule in geometry
the three types of triangles
y mx + c in geometry an explanation for students
angle side angle asa theorem
sss criterion in triangles what is it and how do you use it
what is gradient
midsegment of a triangle in geometry
what is a quadrant in geometry
conic sections review
a brief overview of interior angles in geometry
what is intercept form in geometry
Introduction to Vectors
Introduction to Vectors
1. Vector Concepts and Notation
2. Vector Addition in 1-D
3. Vectors in 2 Dimensions
4. Adding Vectors in 2-D
4a. Adding Vectors using SVG graphs
5. Dot Product of 2-D Vectors
6. Three-dimensional Space
3-D and Contour Grapher
7. Vectors in 3-dimensional Space
8. Cross Product of 2 Vectors
9. Variable Vectors
10. Vector Calculus
10a. 3-D Earth Geometry
10b. Vector Art
10c. Mathematics of Vector Art
Plane Analytical Geometry
Plane Analytical Geometry
1. Distance Formula
1a. Gradient (Slope) of a Line, and Inclination
1b. Parallel Lines
1c. Perpendicular Lines
2. The Straight Line
Perpendicular Distance from a Point to a Line
3. The Circle
4. The Parabola
4a. Interactive parabola graphs
5. The Ellipse
5a. Interactive ellipse graphs
6. The Hyperbola
6a. Interactive hyperbola graphs
6b. Conic Sections Summary
6c. Conic Sections 3D interactive graph
7. Polar Coordinates
8. Curves in Polar Coordinates
Equi-angular Spiral
Squaring the Circle: rope method with proof
Kinematics (velocity & acceleration graphs)
Kinematics (velocity & acceleration graphs)
1. Velocity (s-t) Graphs
2. Acceleration (v-t) Graphs
3. Kinematics Exercises
Curve Shape for the Movement of a Piston
Curve Shape for the Movement of a Piston
JSXGraph Internal Angles of a Triangle
CSS Animated Hexagon Wave
CSS Animated Hexagon Wave
Animated Moore Curve
Trigonometry
Trigonometric Functions
Trigonometric Functions
1. Angles
2. Sine, Cosine, Tangent & Reciprocals
3. Values of Trigonometric Functions
4. The Right Triangle and Applications
4a. Trigonometry Revision Summary
5. Signs of the Trigonometric Functions
6. Trigonometric Functions of Any Angle
7. Radians
8. Applications of Radian Measure
8a. Pulleys
9. Radians and the Trigonometric Ratios
Graphs of the Trigonometric Functions
Graphs of the Trigonometric Functions
1. Graphs of `y = a sin x` and `y = a cos x`
2. Graphs of `y = a sin bx` and `y = a cos bx`
2a. Frequency of Music Notes
3. Graphs of `y = a sin(bx+c)` and `y = a cos(bx+c)`
4. Graphs of tan, cot, sec and csc
5. Applications of Trigonometric Graphs
6. Composite Trigonometric Graphs
6a. Biorhythm Graphs
7. Lissajous Figures
2b. Frequencies of Notes on a Piano: Learning object
Analytic Trigonometry
Analytic Trigonometry
1. Trigonometric Identities
1a. Trigonometric Ratios - Interactive Graph
2. Sin, cos, tan of Sum of Two Angles
3. Double Angle Formulas
4. Half Angle Formulas
5. Trigonometric Equations
6. Express in the form
R
sin (θ + α)
7. Inverse Trigonometric Functions
Inverse trigonometric function graph animations
Uses of Trignometry
Uses of Trignometry
Trigonometry function graphs - Using d3
Trigonometry function graphs - Using d3
Trigonometry function graphs - Using JSXGraph
Trigonometry function graphs - Using JSXGraph (with an array)
Calculus
Complex Numbers
Complex Numbers
1. Basic Definitions of Complex Numbers
2. Basic Operations in Complex Numbers
3. Graphical Representation of Complex Numbers
4. Polar Form of Complex Numbers
Convert polar to rectangular using hand-held calculator
Polar to Rectangular Online Calculator
5. Exponential Form of Complex Numbers
6. Products and Quotients of Complex Numbers
7. Powers and Roots of Complex Numbers
8. AC Circuit Definitions
9. Impedance and Phase Angle: Application of Complex Numbers
10. Reactance and Angular Velocity: Application of Complex Numbers
11. AC Circuit Exercises
12. Parallel AC Circuits
Vocabulary
Fractals
Graphical explanation of multiplying and dividing complex numbers
Euler Formula and Euler Identity interactive graph
Calculus - An Introduction
Calculus - An Introduction
Vietnamese Translation of IntMath Calculus
Calculus Concepts First Principles Applet
Differentiation - Introduction
Differentiation - Introduction
1. Limits and Differentiation
2. The Slope of a Tangent to a Curve (Numerical)
3. The Derivative from First Principles
4. Derivative as an Instantaneous Rate of Change
5. Derivatives of Polynomials
5a. Derivative interactive graphs - polynomials
6. Derivatives of Products and Quotients
7. Differentiating Powers of a Function
7a. Differentiation interactive applet - products and quotients
8. Differentiation of Implicit Functions
9. Higher Derivatives
10. Partial Derivatives
Applications of Differentiation
Applications of Differentiation
1. Tangents and Normals
2. Newton's Method
Newton's Method Interactive Graph
3. Curvilinear Motion
4. Related Rates
5. Curve Sketching Using Differentiation
6. More Curve Sketching Using Differentiation
7. Applied Maximum and Minimum Problems
8. Radius of Curvature
Differentiation of Transcendental Functions
Differentiation of Transcendental Functions
1. Derivatives of Sin, Cos and Tan Functions
2. Derivatives of Csc, Sec and Cot Functions
3. Derivatives of Inverse Trigonometric Functions
4. Applications: Derivatives of Trigonometric Functions
5. Derivative of the Logarithmic Function
6. Derivative of the Exponential Function
7. Applications: Derivatives of Logarithmic and Exponential Functions
Table of Derivatives
Differentiation interactive applet - trigonometric functions
Integration
Integration
1. The Differential
2. Antiderivatives and The Indefinite Integral
3. The Area Under a Curve
4. The Definite Integral
5. Trapezoidal Rule
6. Simpson’s Rule
6a. Riemann Sums
6b. Fundamental Theorem of Calculus Applet
7. Integration Mini-lectures
7a. The Differential
7b. Difference Between Differentiation and Integration
7c. Given dy/dx, find y = f(x)
7d. Integration by Substitution
7e. Difference Between Definite and Indefinite Integrals
7f. Area Under a Curve
Applications of Integration
Applications of Integration
1. Applications of the Indefinite Integral
2. Area Under a Curve by Integration
3. Area Between 2 Curves using Integration
4a. Volume of Solid of Revolution by Integration
5. Centroid of an Area by Integration
6. Moments of Inertia by Integration
7. Work by a Variable Force using Integration
8. Electric Charges by Integration
9. Average Value of a Function by Integration
Head Injury Criterion (HIC) pt 1: Severity Index
Head Injury Criterion (HIC) pt 2: HIC Index, example
10. Force Due to Liquid Pressure by Integration
11. Arc Length of a Curve using Integration
12. Arc Length of Curve: Parametric, Polar Coordinates
4b. Shell Method: Volume of Solid of Revolution
Methods of Integration
Methods of Integration
1. Integration: The General Power Formula
2. Integration: The Basic Logarithmic Form
3. Integration: The Exponential Form
4. Integration: The Basic Trigonometric Forms
Riemann Sums - Discontinuous Functions
5. Integration: Other Trigonometric Forms
6. Integration: Inverse Trigonometric Forms
7. Integration by Parts
8. Integration by Trigonometric Substitution
9. Integration by Use of Tables
Table of Common Integrals
10. Integration by Reduction Formulae
11. Integration by Partial Fractions
Taylor and Maclaurin Series interactive applet
Taylor and Maclaurin Series interactive applet
Matrices and Linear Transformations
Matrices and Linear Transformations
Eigenvalues and eigenvectors - concept applet
7. Eigenvalues and Eigenvectors
8. Applications of Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors calculator
Higher Calculus
Differential Equations
Differential Equations
Predicting AIDS - a DEs example
1. Solving Differential Equations
2. Separation of Variables
3. Integrable Combinations
4. Linear DEs of Order 1
5. Application: RL Circuits
6. Application: RC Circuits
7. Second Order DEs - Homogeneous
8. Second Order DEs - Damping - RLC
9. Second Order DEs - Forced Response
10. Second Order DEs - Solve Using SNB
11. Euler's Method - a numerical solution for Differential Equations
12. Runge-Kutta (RK4) numerical solution for Differential Equations
Infinite Series Expansion
Infinite Series Expansion
1. Taylor Series
2. Maclaurin Series
3. How Does a Calculator Work?
Fourier Series
Fourier Series
Helpful Revision
1. Overview of Fourier Series
2. Full Range Fourier Series
3. Fourier Series of Even and Odd Functions
Fourier Series Graph Interactive
4. Fourier Series of Half Range Functions
5. Harmonic Analysis
6. Line Spectrum
7. Fast Fourier Transform
Laplace Transforms
Laplace Transforms
1a. The Unit Step Function - Definition
1a.i. Oliver Heaviside
1b. The Unit Step Function - Products
2. Laplace Transform Definition
2a. Table of Laplace Transformations
3. Properties of Laplace Transform
4. Transform of Unit Step Functions
5. Transform of Periodic Functions
6. Transforms of Integrals
7. Inverse of the Laplace Transform
8. Using Inverse Laplace to Solve DEs
9. Integro-Differential Equations and Systems of DEs
10. Applications of Laplace Transform
Probability
Counting and Probability - Introduction
Counting and Probability - Introduction
1. Factorial Notation
2. Basic Principles of Counting
3. Permutations
4. Combinations
5. Introduction to Probability Theory
6. Probability of an Event
Singapore TOTO
Probability and Poker
7. Conditional Probability
8. Independent and Dependent Events
9. Mutually Exclusive Events
10. Bayes’ Theorem
11. Probability Distributions - Concepts
12. Binomial Probability Distributions
13. Poisson Probability Distribution
14. Normal Probability Distribution
The z-Table
Normal Distribution Graph Interactive
Other
Money Math
Money Math
1. Interesting Interest
1a. Rule of 72
2. Credit Cards
3. Math of House Buying
4. Gold Rush?
5. Money Charts and Fibonacci
6. Superannuation
Math art in code
Math art in code
Delaunay Triangulation
Flux - a Perlin noise example
Moiré effect
Evolute of a curve
Spinning clock
Animated Lissajous figures
Solid of Revolution maker
Bubble sort rainbow
CSS Matrix - a mathematical explanation
CSS matrix interactive applet
Historical Math Curves
Historical Math Curves
Lemniscate of Bernoulli
Quadratrix of Dinostratus
Putnam Mathematical Competition
Putnam Mathematical Competition
1985 Putnam Question A-2: Solution Part 1
1985 Putnam Question A-2: Solution Part 2
1985 Putnam Question A-2: Solution Part 3
1985 Putnam Question A-2: Solution Part 4
1985 Putnam Question A-2: Solution Part 5
1985 Putnam Question A-2: Solution Part 6
1985 Putnam Question A-2: Solution Part 7
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