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The Khan Academy
This New Orleans resident has created literally 1,000 of youtube's tutoring on subjects mainly related to math, physics, economics, etc http://www.khanacademy.org/ ...
My wife is working on here MBA which is why/how I found Sal's youtube videos.
Watch his explanation for his website :here:
Info on Sal:
Sal received his MBA from Harvard Business School. He also holds a Masters in electrical engineering and computer science, a BS in electrical engineering and computer science, and a BS in mathematics from the Massachusetts Institute of Technology.
Information from his website (all free all on youtube.com)
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We have taken all 8 math practice tests (432 problems) in "The Offical SAT Study Guide" by the College Board in 100+ videos ... [More]
GMAT Preparation
We have taken all 400+ data sufficiency and math problems in "The Official Guide for GMAT Review" by the GMAC in 100+ videos below. ... [More]
California: Algebra II
Sal has done all 80 problems from the Algebra II Standards Test (click to download) administered as part of the Standardized Testing and Reporting (STAR) Program under policies set by the State Board of Education. ... [More]
Brain Teasers
Liar Truthteller Brain Teaser
Toggler Brain Teaser
Alien Abduction Brain Teaser
Brain Teaser: Blue Forehead Room
Blue Forehead Room Solution
Forehead Numbers Brain Teaser
Light Bulb Switching Brain Teaser
Path Counting Brain Teaser
3-D Path Counting Brain Teaser Current Economics
Economics of a Cupcake Factory
Cupcake Economics 2
Cupcake Economics 3
Inflation, Deflation & Capacity Utilization
Inflation, Deflation & Capacity Utilization 2
Inflation & Deflation 3: Obama Stimulus Plan
Unemployment
CPI Index
Simple Analysis of Cost per Job Saved from Stimulus Banking and Money
Banking 1
Banking 2: A bank's income statement
Banking 3: Fractional Reserve Banking
Banking 4: Multiplier effect and the money supply
Banking 5: Introduction to Bank Notes
Banking 6: Bank Notes and Checks
Banking 7: Giving out loans without giving out gold
Banking 8: Reserve Ratios
Banking 9: More on Reserve Ratios (Bad sound)
Banking 10: Introduction to leverage (bad sound)
Banking 11: A reserve bank
Banking 12: Treasuries (government debt)
Banking 13: Open Market Operations
Banking 14: Fed Funds Rate
Banking 15: More on the Fed Funds Rate
Banking 16: Why target rates vs. money supply
Banking 17: What happened to the gold?
Banking 18: Big Picture Discussion
The Discount Rate
Repurchase Agreements (Repo transactions)
Federal Reserve Balance Sheet
Fractional Reserve Banking Commentary 1
FRB Commentary 2: Deposit Insurance
FRB Commentary 3: Big Picture Venture Capital and Capital Markets
Raising money for a startup
Getting a seed round from a VC
Going back to the till: Series B
An IPO
More on IPOs
Equity vs. Debt
Bonds vs. Stocks
Chapter 7:Bankruptcy Liquidation
Chapter 11: Bankruptcy Restructuring Finance
Introduction to interest
Interest (part 2)
Introduction to Present Value
Present Value 2
Present Value 3
Present Value 4 (and discounted cash flow)
Introduction to Balance Sheets
More on balance sheets and equity
Home equity loans
Renting vs. Buying a home
Renting vs. buying a home (part 2)
Renting vs. Buying (detailed analysis)
The housing price conundrum
Housing price conundrum (part 2)
Housing Price Conundrum (part 3)
Housing Conundrum (part 4)
Raising money for a startup
Getting a seed round from a VC
Going back to the till: Series B
An IPO
More on IPOs
Bonds vs. Stocks
Chapter 7:Bankruptcy Liquidation
Chapter 11: Bankruptcy Restructuring
Return on capital
Mortgage-Backed Securities I
Mortgage-backed securities II
Mortgage-backed securities III
Collateralized Debt Obligation (CDO)
Introduction to the yield curve
Introduction to compound interest and e
Compound Interest and e (part 2)
Compound Interest and e (part 3)
Compound Interest and e (part 4)
Bailout 1: Liquidity vs. Solvency
Bailout 2: Book Value
Bailout 3: Book value vs. market value
Bailout 4: Mark-to-model vs. mark-to-market
Bailout 5: Paying off the debt
Bailout 6: Getting an equity infusion
Bailout 7: Bank goes into bankruptcy
Bailout 8: Systemic Risk
Bailout 9: Paulson's Plan
Bailout 10: Moral Hazard
Credit Default Swaps
Credit Default Swaps 2
Investment vs. Consumption 1
Investment vs. Comsumption 2
Bailout 11: Why these CDOs could be worth nothing
Bailout 12: Lone Star Transaction
Bailout 13: Does the bailout have a chance of working?
Wealth Destruction 1
Wealth Destruction 2
Bailout 14: Possible Solution
Bailout 15: More on the solution
Banking 4: Multiplier effect and the money supply
Banking 3: Fractional Reserve Banking
Banking 2: A bank's income statement
Banking 1
Banking 2: A bank's income statement
Banking 3: Fractional Reserve Banking
Banking 4: Multiplier effect and the money supply Valuation and Investing
Price and Market Capitalization
Introduction to the Income Statement
Earnings and EPS
Introduction to the Price-to-Earnings Ratio
P/E Discussion
ROA Discussion 1
ROA Discussion 2
Depreciation
Amortization
P/E Conundrum
Enterprise Value
EBITDA Credit Crisis
The housing price conundrum
Housing price conundrum (part 2)
Housing Price Conundrum (part 3)
Housing Conundrum (part 4)
Mortgage-Backed Securities I
Mortgage-backed securities II
Mortgage-backed securities III
Collateralized Debt Obligation (CDO)
Bailout 13: Does the bailout have a chance of working?
Credit Default Swaps
Credit Default Swaps 2
Wealth Destruction 1
Wealth Destruction 2 Geithner Plan
Geithner Plan I
Geithner Plan II
Geithner Plan 2.5
Geithner Plan III
Geithner Plan IV
Geithner 5: A better solution Paulson Bailout
Bailout 1: Liquidity vs. Solvency
Bailout 2: Book Value
Bailout 3: Book value vs. market value
Bailout 4: Mark-to-model vs. mark-to-market
Bailout 5: Paying off the debt
Bailout 6: Getting an equity infusion
Bailout 7: Bank goes into bankruptcy
Bailout 8: Systemic Risk
Bailout 9: Paulson's Plan
Bailout 10: Moral Hazard
Bailout 11: Why these CDOs could be worth nothing
Bailout 12: Lone Star Transaction
Bailout 14: Possible Solution
Bailout 15: More on the solution Chemistry
Introduction to the atom
Orbitals
More on orbitals and electron configuration
Electron Configurations
Electron Configurations 2
Valence Electrons
Groups of the Periodic Table
Periodic Table Trends: Ionization Energy
Other Periodic Table Trends
Ionic, Covalent, and Metallic Bonds
Molecular and Empirical Formulas
The Mole and Avogadro's Number
Formula from Mass Composition
Another mass composition problem
Balancing Chemical Equations
Stoichiometry
Stoichiometry: Limiting Reagent
Ideal Gas Equation: PV=nRT
Ideal Gas Equation Example 1
Ideal Gas Equation Example 2
Ideal Gas Equation Example 3
Ideal Gas Example 4
Partial Pressure
States of Matter
States of Matter Follow-Up
Specific Heat, Heat of Fusion and Vaporization
Chilling Water Problem
Phase Diagrams
Van Der Waals Forces
Covalent Networks, Metallic, and Ionic Crystals
Vapor Pressure
Suspensions, Colloids and Solutions
Solubility
Boiling Point Elevation and Freezing Point Supression
Introduction to Kinetics
Reactions in Equilibrium
Mini-Video on Ion Size
Keq Intuition (mathy and not necessary to progress)
Keq derivation intuition (can skip; bit mathy)
Heterogenous Equilibrium
Le Chatelier's Principle
Introduction to pH, pOH, and pKw
Acid Base Introduction
pH, pOH of Strong Acids and Bases
pH of a Weak Acid
pH of a Weak Base
Conjugate Acids and Bases
pKa and pKb Relationship
Buffers
Strong Acid Titration
Weak Acid Titration
Half Equivalence Point
Titration Roundup
Introduction to Oxidation States
More on Oxidation States
Hydrogen Peroxide Correction
Redox Reactions
Galvanic Cells
Types of Decay
Half-Life
Exponential Decay Formula Proof (can skip, involves Calculus)
Introduction to Exponential Decay
More Exponential Decay Examples
Macrostates and Microstates
Quasistatic and Reversible Processes
First Law of Thermodynamics/ Internal Energy
More on Internal Energy
Work from Expansion
PV-diagrams and Expansion Work
Proof: U=(3/2)PV or U=(3/2)nRT
Work Done by Isothermic Process
Carnot Cycle and Carnot Engine
Proof: Volume Ratios in a Carnot Cycle
Proof: S (or Entropy) is a valid state variable
Thermodynamic Entropy Definition Clarification
Reconciling Thermodynamic and State Definitions of Entropy
Entropy Intuition
Maxwell's Demon
More on Entropy
Efficiency of a Carnot Engine
Carnot Efficiency 2: Reversing the Cycle
Carnot Efficiency 3: Proving that it is the most efficient
Enthalpy
Heat of Formation
Hess's Law and Reaction Enthalpy Change
Gibbs Free Energy and Spontaneity
Gibbs Free Energy Example
More rigorous Gibbs Free Energy/ Spontaneity Relationship
A look at a seductive but wrong Gibbs/Spontaneity Proof Arithmetic
Basic Addition
Level 2 Addition
Basic Subtraction
Level 4 Subtraction
Why borrowing works
Level 4 division
Adding Decimals
Subtracting decimals
Dividing decimal
Greatest Common Divisor
Least Common Multiple
Equivalent fractions
Mixed numbers and improper fractions
Converting fractions to decimals
Percent and decimals
Ordering numeric expressions
Why borrowing works
Basic Multiplication
Multiplication 2: The Multiplication Tables
Multiplication 3: 10,11,12 times tables
Multiplication 4: 2-digit times 1-digit number
Multiplication 5: 2-digit times a 2-digit number
Multiplication 6: Multiple Digit Numbers
Multiplication 7: Old video giving more examples
Mulitplication 8: Multiplying decimals (Old video) Pre-algebra
Order of operations
Order of operations II
Adding/Subtracting negative numbers
Multiplying and dividing negative numbers
Adding and subtracting fractions
Multiplying fractions
Dividing fractions
Level 1 Exponents
Level 2 Exponents
Level 3 exponents
Negative Exponent Intuition
Exponent Rules Part 1
Exponent Rules Part 2
Simplifying radicals
Introduction to Logarithms
Unit conversion
Speed translation
Introduction to logarithm properties
Introduction to logarithm properties (part 2)
Scientific Notation
Scientific Notation Examples
Introduction to Ratios (new HD version) Algebra
Algebra: Linear Equations 1
Algebra: Linear Equations 2
Algebra: Linear Equations 3
Algebra: Linear Equations 4
Algebra: Solving Inequalities
Algebra: graphing lines 1
Algebra: Slope and Y-intercept intuition
Algebra: Slope
Algebra: Slope 2
Algebra: Slope 3
Algebra: Equation of a line
Slope and Y-intercept Intuition
Averages
Integer sums
Taking percentages
Growing by a percentage
More percent problems
systems of equations
Introduction to Ratios (new HD version)
Ratio problem with basic algebra (new HD)
More advanced ratio problem--with Algebra (HD version)
Alternate Solution to Ratio Problem (HD Version)
Introduction to Ratios
Advanced ratio problems
Age word problems 1
Age word problems 2
Age word problems 3
Level 1 multiplying expressions
Solving a quadratic by factoring
i and Imaginary numbers
Complex Numbers (part 1)
Complex Numbers (part 2)
Introduction to the quadratic equation
Quadratic Equation part 2
Completing the square
Quadratic Formula (proof)
Quadratic Inequalities
Quadratic Inequalities (Visual Explanation)
Introduction to functions
Functions Part 2
Functions (Part III)
Functions (part 4)
Domain of a function
Proof: log a + log b = log ab
Proof: A(log B) = log (B^A), log A - log B = log (A/B)
Proof: log_a (B) = (log_x (B))/(log_x (A))
Algebraic Long Division
Introduction to Conic Sections
Conic Sections: Intro to Circles
Conic Sections: Intro to Ellipses
Conic Sections: Intro to Hyperbolas
Conic Sections: Hyperbolas 2
Conic Sections: Hyperbolas 3
Identifying Conics 1
Identifying Conics 2
Conic Identification 3
Foci of an Ellipse
Foci of a Hyperbola
Proof: Hyperbola Foci
Partial Fraction Expansion 1
Partial Fraction Expansion 3
Parabola Focus and Directrix 1
Focus and Directrix of a Parabola 2
Two Passing Bicycles Word Problem
Passed Bike Word Problem
Passing Trains
Overtaking Word Problem
Early Train Word Problem
Officer on Horseback California Standards Test: Algebra I
CA Algebra I: Number Properties and Absolute Value
CA Algebra I: Simplifying Expressions
CA Algebra I: Simple Logical Arguments
CA Algebra I: Graphing Inequalities
CA Algebra I: Slope and Y-intercept
CA Algebra I: Systems of Inequalities
CA Algebra I: Simplying Expressions
CA Algebra I: Factoring Quadratics
CA Algebra I: Completing the Square
CA Algebra I: Quadratic Equation
CA Algebra I: Quadratic Roots
CA Algebra I: Rational Expressions
CA Algebra I: Rational Expressions
CA Algebra I: Word Problems
CA Algebra I: More Word Problems
CA Algebra I: Functions California Standards Test: Algebra II
California Standards Test: Algebra II
California Standards Test: Algebra II (Graphing Inequalities
CA Standards: Algebra II (Algebraic Division/Multiplication)
CA Standards: Algebra II
Algebra II: Simplifying Polynomials
Algebra II: Imaginary and Complex Numbers
Algebra II: Complex numbers and conjugates
Algebra II: Quadratics and Shifts
Algebra II: Shifting Quadratic Graphs
Algebra ||: Conic Sections
Algebra II: Circles and Logarithms
Algebra II: Logarithms Exponential Growth
Algebra II: Logarithms and more
Algebra II: Functions, Combinatorics
Algebra II: binomial Expansion and Combinatorics
Algebra II: Binomial Expansions, Geometric Series Sum
Algebra II: Functions and Probability
Algebra II: Probability and Statistics
Algebra II: Mean and Standard Deviation Geometry
Introduction to angles
Angles (part 2)
Angles (part 3)
Angles of parallel lines
The Angle Game
Similar triangles
Similar triangles (part 2)
Angle Game (part 2)
Area and Perimeter
Circles: Radius, Diameter and Circumference
Area of a circle
Introduction to the Pythagorean Theorem
Pythagorean Theorem II
45-45-90 Triangles
Intro to 30-60-90 Triangles
30-60-90 Triangles II California Standards Test: Geometry
CA Geometry: deductive reasoning
CA Geometry: Proof by Contradiction
CA Geometry: More Proofs
CA Geometry: Similar Triangles
CA Geometry: Similar Triangles
CA Geometry: More on cogruent and similar triangles
CA Geometry: Triangles and Parallelograms
CA Geometry: Area, Pythagorean Theorem
CA Geometry: Area, Circumference, Volume
CA Geometry: Pythagorean Theorem, Area
CA Geometry: Exterior Angles
CA Geometry: Deducing Angle Measures
CA Geometry: Pythagorean Theorem, Compass Constructions
CA Geometry: Compass Construction
CA Geometry: Basic Trigonometry
CA Geometry: More Trig
CA Geometry: Circle Area Chords Tangent
CA Geometry: Secants and Translations
Biology
Introduction to Evolution and Natural Selection
Intelligent Design and Evolution
Evolution Clarification
Natural Selection and the Owl Butterfly
DNA
Variation in a Species
Chromosomes, Chromatids, Chromatin, etc.
Mitosis, Meiosis and Sexual Reproduction
Phases of Mitosis
Phases of Meiosis
Embryonic Stem Cells
Cancer
Introduction to Heredity
Punnett Square Fun
Hardy-Weinberg Principle
Sex-Linked Traits
Bacteria
Viruses
ATP: Adenosine Triphosphate
Introduction to Cellular Respiration
Oxidation and Reduction Review From Biological Point-of-View
Oxidation and Reduction in Cellular Respiration
Glycolysis
Krebs / Citric Acid Cycle
Electron Transport Chain
Oxidative Phosphorylation and Chemiosmosis Trigonometry
Basic Trigonometry
Basic Trigonometry II
Radians and degrees
Using Trig Functions
Using Trig Functions Part II
The unit circle definition of trigonometric function
Unit Circle Definition of Trig Functions
Graph of the sine function
Graphs of trig functions
Graphing trig functions
More trig graphs
Determining the equation of a trigonometric function
Trigonometric Identities
Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b)
Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)
Trig identities part 2 (parr 4 if you watch the proofs)
Trig identies part 3 (part 5 if you watch the proofs)
Trigonometry word problems (part 1)
Trigonometry word problems (part 2)
Law of cosines
Navigation Word Problem
Proof: Law of Sines
Ferris Wheel Trig Problem
Ferris Wheel Trig Problem (part 2)
Fun Trig Problem
Polar Coordinates 1
Polar Coordinates 2
Polar Coordinates 3
Inverse Trig Functions: Arcsin
Inverse Trig Functions: Arctan
Inverse Trig Functions: Arccos
Trigonometry Identity Review/Fun Precalculus
Introduction to Limits
Limit Examples (part 1)
Limit Examples (part 2)
Limit Examples (part3)
Limit Examples w/ brain malfunction on first prob (part 4)
Squeeze Theorem
Proof: lim (sin x)/x
More Limits
Sequences and Series (part 1)
Sequences and series (part 2)
Permutations
Combinations
Binomial Theorem (part 1)
Binomial Theorem (part 2)
Binomial Theorem (part 3)
Introduction to interest
Interest (part 2)
Introduction to compound interest and e
Compound Interest and e (part 2)
Compound Interest and e (part 3)
Compound Interest and e (part 4)
Exponential Growth
Polar Coordinates 1
Polar Coordinates 2
Polar Coordinates 3
Parametric Equations 1
Parametric Equations 2
Parametric Equations 3
Parametric Equations 4 Statistics
Statistics: The Average
Statistics: Sample vs. Population Mean
Statistics: Variance of a Population
Statistics: Sample Variance
Statistics: Standard Deviation
Statistics: Alternate Variance Formulas
Introduction to Random Variables
Probability Density Functions
Binomial Distribution 1
Binomial Distribution 2
Binomial Distribution 3
Binomial Distribution 4
Expected Value: E(X)
Expected Value of Binomial Distribution
Poisson Process 1
Poisson Process 2
Law of Large Numbers
Normal Distribution Excel Exercise
Introduction to the Normal Distribution Probability
Probability (part 1)
Probability (part 2)
Probability (part 3)
Probability (part 4)
Probability (part 5)
Probability (part 6)
Probability (part 7)
Probability (part
Permutations
Combinations
Probability using Combinations
Probability and Combinations (part 2)
Conditional Probability and Combinations
Birthday Probability Problem
Introduction to Random Variables
Probability Density Functions
Binomial Distribution 1
Binomial Distribution 2
Binomial Distribution 3
Binomial Distribution 4
Expected Value: E(X)
Expected Value of Binomial Distribution
Poisson Process 1
Poisson Process 2
Law of Large Numbers Calculus
Introduction to Limits
Limit Examples (part 1)
Limit Examples (part 2)
Limit Examples (part3)
Limit Examples w/ brain malfunction on first prob (part 4)
Squeeze Theorem
Proof: lim (sin x)/x
More Limits
Epsilon Delta Limit Definition 1
Epsilon Delta Limit Definition 2
Calculus: Derivatives 1 (new HD version)
Calculus: Derivatives 2 (new HD version)
Calculus: Derivatives 2.5 (new HD version)
Calculus: Derivatives 1
Calculus: Derivatives 2
Calculus: Derivatives 3
Calculus: Derivatives 4
Calculus: Derivatives 5
Calculus: Derivatives 6
Derivatives (part 7)
Derivatives (part
Derivatives (part 9)
Proof: d/dx(x^n)
Proof: d/dx(sqrt(x))
Proof: d/dx(ln x) = 1/x
Proof: d/dx(e^x) = e^x
Proofs of Derivatives of Ln(x) and e^x
Calculus: Derivative of x^(x^x)
Extreme Derivative Word Problem (advanced)
Implicit Differentiation
Implicit Differentiation (part 2)
More implicit differentiation
More chain rule and implicit differentiation intuition
Trig Implicit Differentiation Example
Maxima Minima Slope Intuition
Inflection point intuition
Monotonicity Theorem
Calculus: Maximum and minimum values on an interval
Calculus: Graphing using derivatives
Graphing with Calculus
Hairy inflection point problem
Optimization with Calculus 1
Optimization with Calculus 2
Optimization with Calculus 3
Optimization Example 4
Introduction to rate-of-change problems
Rates-of-change (part 2)
Ladder rate-of-change problem
Mean Value Theorem
The Indefinite Integral or Anti-derivative
Indefinite integrals (part II)
Indefinite Integration (part III)
Indefinite Integration (part IV)
Indefinite Integration (part V)
Indefinite Integration (part 6)
Indefinite Integration (part 7)
Another u-subsitution example
Introduction to definite integrals
Definite integrals (part II)
Definite Integrals (area under a curve) (part III)
Definite Integrals (part 4)
Definite Integrals (part 5)
Definite integral with substitution
Integrals: Trig Substitution 1
Integrals: Trig Substitution 2
Integrals: Trig Substitution 3 (long problem)
Introduction to differential equations
Solid of Revolution (part 1)
Solid of Revolution (part 2)
Solid of Revolution (part 3)
Solid of Revolution (part 4)
Solid of Revolution (part 5)
Solid of Revolution (part 6)
Solid of Revolution (part 7)
Solid of Revolution (part
Sequences and Series (part 1)
Sequences and series (part 2)
Polynomial approximation of functions (part 1)
Polynomial approximation of functions (part 2)
Approximating functions with polynomials (part 3)
Polynomial approximation of functions (part 4)
Polynomial approximations of functions (part 5)
Polynomial approximation of functions (part 6)
Polynomial approximation of functions (part 7)
Taylor Polynomials
Exponential Growth
Equation of a tangent line
AP Calculus BC Exams: 2008 1 a
AP Calculus BC Exams: 2008 1 b&c
AP Calculus BC Exams: 2008 1 c&d
AP Calculus BC Exams: 2008 1 d
Calculus BC 2008 2 a
Calculus BC 2008 2 b &c
Calculus BC 2008 2d
Partial Derivatives
Partial Derivatives 2
Gradient 1
Gradient of a scalar field
Divergence 1
Divergence 2
Divergence 3
Curl 1
Curl 2
Curl 3
Double Integral 1
Double Integrals 2
Double Integrals 3
Double Integrals 4
Double Integrals 5
Double Integrals 6
Triple Integrals 1
Triple Integrals 2
Triple Integrals 3
?(2^ln x)/x dx Antiderivative Example Differential Equations
Introduction to differential equations
Separable Differential Equations
Separable differential equations 2
Exact Equations Intuition 1 (proofy)
Exact Equations Intuition 2 (proofy)
Exact Equations Example 1
Exact Equations Example 2
Exact Equations Example 3
Integrating factors 1
Integrating factors 2
First order homegenous equations
First order homogenous equations 2
2nd Order Linear Homogeneous Differential Equations 1
2nd Order Linear Homogeneous Differential Equations 2
2nd Order Linear Homogeneous Differential Equations 3
2nd Order Linear Homogeneous Differential Equations 4
Complex roots of the characteristic equations 1
Complex roots of the characteristic equations 2
Complex roots of the characteristic equations 3
Repeated roots of the characteristic equation
Repeated roots of the characterisitic equations part 2
Undetermined Coefficients 1
Undetermined Coefficients 2
Undetermined Coefficients 3
Undetermined Coefficients 4
Laplace Transform 1
Laplace Transform 2
Laplace Transform 3 (L{sin(at)})
Laplace Transform 4
Laplace Transform 5
Laplace Transform 6
Laplace Transform to solve an equation
Laplace Transform solves an equation 2
More Laplace Transform tools
Using the Laplace Transform to solve a nonhomogenous eq
Laplace Transform of : L{t}
Laplace Transform of t^n: L{t^n}
Laplace Transform of the Unit Step Function
Inverse Laplace Examples
Laplace/Step Function Differential Equation
Dirac Delta Function
Laplace Transform of the Dirac Delta Function
Introduction to the Convolution
The Convolution and the Laplace Transform
Using the Convolution Theorem to Solve an Initial Value Prob
Linear Algebra
Introduction to matrices
Matrix multiplication (part 1)
Matrix multiplication (part 2)
Inverse Matrix (part 1)
Inverting matrices (part 2)
Inverting Matrices (part 3)
Matrices to solve a system of equations
Matrices to solve a vector combination problem
Singular Matrices
3-variable linear equations (part 1)
Solving 3 Equations with 3 Unknowns
Linear Algebra: Introduction to Vectors
Linear Algebra: Vector Examples
Linear Algebra: Parametric Representations of Lines
Linear Combinations and Span
Linear Algebra: Introduction to Linear Independence
More on linear independence
Span and Linear Independence Example
Linear Subspaces
Linear Algebra: Basis of a Subspace
Vector Dot Product and Vector Length
Proving Vector Dot Product Properties
Proof of the Cauchy-Schwarz Inequality
Linear Algebra: Vector Triangle Inequality
Defining the angle between vectors
Defining a plane in R3 with a point and normal vector
Linear Algebra: Cross Product Introduction
Proof: Relationship between cross product and sin of angle
Dot and Cross Product Comparison/Intuition
Matrices: Reduced Row Echelon Form 1
Matrices: Reduced Row Echelon Form 2
Matrices: Reduced Row Echelon Form 3
Matrix Vector Products
Introduction to the Null Space of a Matrix
Null Space 2: Calculating the null space of a matrix
Null Space 3: Relation to Linear Independence
Column Space of a Matrix
Null Space and Column Space Basis
Visualizing a Column Space as a Plane in R3
Proof: Any subspace basis has same number of elements
Dimension of the Null Space or Nullity
Dimension of the Column Space or Rank
Showing relation between basis cols and pivot cols
Showing that the candidate basis does span C(A)
A more formal understanding of functions
Vector Transformations
Linear Transformations
Matrix Vector Products as Linear Transformations
Linear Transformations as Matrix Vector Products
Image of a subset under a transformation
im(T): Image of a Transformation
Preimage of a set
Preimage and Kernel Example
Sums and Scalar Multiples of Linear Transformations
More on Matrix Addition and Scalar Multiplication
Linear Transformation Examples: Scaling and Reflections
Linear Transformation Examples: Rotations in R2
Rotation in R3 around the X-axis
Unit Vectors
Introduction to Projections
Expressing a Projection on to a line as a Matrix Vector prod
Compositions of Linear Transformations 1
Compositions of Linear Transformations 2
Linear Algebra: Matrix Product Examples
Matrix Product Associativity
Distributive Property of Matrix Products
Linear Algebra: Introduction to the inverse of a function
Proof: Invertibility implies a unique solution to f(x)=y
Surjective (onto) and Injective (one-to-one) functions
Relating invertibility to being onto and one-to-one
Determining whether a transformation is onto
Linear Algebra: Exploring the solution set of Ax=b
Linear Algebra: Matrix condition for one-to-one trans
Linear Algebra: Simplifying conditions for invertibility
Linear Algebra: Showing that Inverses are Linear
Linear Algebra: Deriving a method for determining inverses
Linear Algebra: Example of Finding Matrix Inverse
Linear Algebra: Formula for 2x2 inverse
Linear Algebra: 3x3 Determinant
Linear Algebra: nxn Determinant
Linear Algebra: Determinants along other rows/cols
Linear Algebra: Rule of Sarrus of Determinants
Linear Algebra: Determinant when row multiplied by scalar
Linear Algebra: (correction) scalar muliplication of row
Linear Algebra: Determinant when row is added
Linear Algebra: Duplicate Row Determinant
Linear Algebra: Determinant after row operations
Linear Algebra: Upper Triangular Determinant
Linear Algebra: Simpler 4x4 determinant
Linear Algebra: Determinant and area of a parallelogram
Linear Algebra: Determinant as Scaling Factor
Linear Algebra: Transpose of a Matrix
Linear Algebra: Determinant of Transpose
Linear Algebra: Transposes of sums and inverses
Linear Algebra: Transpose of a Vector
Linear Algebra: Rowspace and Left Nullspace
Lin Alg: Visualizations of Left Nullspace and Rowspace
Linear Algebra: Orthogonal Complements
Linear Algebra: Rank(A) = Rank(transpose of A)
Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
Lin Alg: Representing vectors in Rn using subspace members
Lin Alg: Orthogonal Complement of the Orthogonal Complement
Lin Alg: Orthogonal Complement of the Nullspace
Lin Alg: Unique rowspace solution to Ax=b
Linear Alg: Rowspace Solution to Ax=b example
Linear Alg: Rowspace Solution to Ax=b example
Lin Alg: Showing that A-transpose x A is invertible
Linear Algebra: Projections onto Subspaces
Linear Alg: Visualizing a projection onto a plane
Lin Alg: A Projection onto a Subspace is a Linear Transforma
Linear Algebra: Subspace Projection Matrix Example
Lin Alg: Another Example of a Projection Matrix
Linear Alg: Projection is closest vector in subspace
Linear Algebra: Least Squares Approximation
Linear Algebra: Least Squares Examples
Linear Algebra: Another Least Squares Example
Linear Algebra: Coordinates with Respect to a Basis
Linear Algebra: Change of Basis Matrix
Lin Alg: Invertible Change of Basis Matrix
Lin Alg: Transformation Matrix with Respect to a Basis
Lin Alg: Alternate Basis Tranformation Matrix Example
Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
Lin Alg: Changing coordinate systems to help find a transformation matrix
Linear Algebra: Introduction to Orthonormal Bases
Linear Algebra: Coordinates with respect to orthonormal bases
Lin Alg: Projections onto subspaces with orthonormal bases
Lin Alg: Finding projection onto subspace with orthonormal basis example
Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
Lin Alg: Orthogonal matrices preserve angles and lengths
Linear Algebra: The Gram-Schmidt Process
Linear Algebra: Gram-Schmidt Process Example
Linear Algebra: Gram-Schmidt example with 3 basis vectors
Linear Algebra: Introduction to Eigenvalues and Eigenvectors
Linear Algebra: Proof of formula for determining Eigenvalues
Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
Linear Algebra: Finding Eigenvectors and Eigenspaces example
Linear Algebra: Eigenvalues of a 3x3 matrix
Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
Linear Algebra: Showing that an eigenbasis makes for good coordinate systems Physics
Introduction to motion
Introduction to motion (part 2)
Introduction to motion (part 3)
Projectile motion (part 1)
Projectile motion (part 2)
Projectile motion (part 3)
Projectile motion (part 4)
Projectile motion (part 5)
Projectile motion (part 6)
Projectile motion (part 7)
Projectile motion (part
Projectile motion (part 9)
Projectile motion (part 10)
2 dimensional projectile motion (part 1)
2 dimensional projectile motion (part 2)
2-dimensional projectile motion (part 3)
2 dimensional projectile motion part 4
2-dimensional projectile motion part 5
Newton's First Law of Motion
Newton's Second Law of Motion
Newton's Third Law of Motion
Newton's Laws Problems (part 1)
Newton's Laws Examples (part 2)
Newton's Laws
Newton's Laws and vectors
Force with Vectors
Introduction to Tension
Tension (part 2)
Mass on Inclined Plane
Introduction to friction
Friction on an inclined plane
A more complicated friction/inclined plane problem
Tension in an accelerating system and pie in the face
Moving pulley problem (part 1)
Moving pulley problem (part 2)
Introduction to Momentum
Momentum: Ice skater throws a ball
2-dimensional momentum problem
2-dimensional momentum problem (part 2)
Introduction to work and energy
Work and Energy (part 2)
Conservation of Energy
Work/Energy problem with Friction
Introduction to mechanical advantage
Mechanical Advantage (part 2)
Mechanical Advantage (part 3)
Center of Mass
Introduction to Torque
Moments
Moments (part 2)
Unit Vector Notation
Unit Vector Notation (part 2)
Projectile Motion with Unit Vectors
Projectile Motion with Unit Vectors (part 2)
Projectile Motion with Ordered Set Notation
Introduction to centripetal acceleration (part 1)
Centripetal Acceleration (part 2)
Centripetal Acceleration (part 3)
Visual Proof: a= v^2/r
Calculus Proof that a=v^2/r
Introduction to angular velocity
Conservation of angular momemtum
Introduction to Newton's Law of Gravitation
Gravitation (part 2)
Intro to springs and Hooke's Law
Potential energy stored in a spring
Spring potential energy example (mistake in math)
Introduction to Harmonic Motion
Harmonic Motion Part 2 (calculus)
Harmonic Motion Part 3 (no calculus)
Fluids (part 1)
Fluids (part 2)
Fluids (part 3)
Fluids (part 4)
Fluids (part 5)
Fluids (part 6)
Fluids (part 7)
Fluids (part
Fluids (part 9)
Fluids (part 10)
Fluids (part 11)
Fluids (part 12)
Thermodynamics (part 1)
Thermodynamics (part 2)
Thermodynamics (part 3)
Thermodynamics (part 4)
Thermodynamics (part 5)
Electrostatics (part 1)
Electrostatics (part 2)
Proof (Advanced): Field from infinite plate (part 1)
Proof (Advanced): Field from infinite plate (part 2)
Electric Potential Energy
Electric Potential Energy (part 2-- involves calculus)
Voltage
Capacitance
Circuits (part 1)
Circuits (part 2)
Circuits (part 3)
Circuits (part 4)
Cross product 1
Cross Product 2
Cross Product and Torque
Introduction to Magnetism
Magnetism 2
Magnetism 3
Magnetism 4
Magnetism 5
Magnetism 6: Magnetic field due to current
Magnetism 7
Magnetism 8
Magnetism 9: Electric Motors
Magnetism 10: Electric Motors
Magnetism 11: Electric Motors
Magnetism 12: Induced Current in a Wire
The dot product
Dot vs. Cross Product
Calculating dot and cross products with unit vector notation CNN: Sal on the Credit Crisis
Sal appears on the Rick Sanchez show to explain the Credit Crisis This appearance has led to a paper that Sal has co-authored with Berkeley professor Dr. David Leibweber on a solution to unfreezing credit markets which is available here.
About Salman Khan
Salman Khan (Sal) founded the Khan Academy with the goal of using technology to educate the world Sal received his MBA from Harvard Business School. He also holds a Masters in electrical engineering and computer science, a BS in electrical engineering and computer science, and a BS in mathematics from the Massachusetts Institute of Technology. Frequently Asked Questions.
What people are saying...
"...My eldest kid is dancing around in my room here because she is so excited that she finally found someone that teaches like this. You're awesome. Thank you." "I don't know who you are but in my mind you are a savior. My children really struggle with math, there is an inherited learning disability in my family. They get it but only after seeing it done multiple times. Your videos will allow us to help our children get caught up with their peers. As a parent I have to say, Thank You Thank You Thank You."
"I've never been very successful with algebra, I know this is pretty basic, but I have almost no comprehension of mathematics. I was honestly going to be kicked out of school for being non-productive, I was ready to be thrown out, but now I get this. Thank you so much..."
"This is my first year in college and until today i was completely lost in math. Half an hour with your presentations taught me more than three months of sitting in my professors class. You seriously saved my grade."
"Thank you so much! Keep up the good work :-D This is the most useful channel I've come across on youtube ever! Greetings from Denmark!"
"You're the only math tutor that makes sense to me. Most other ones seem to be speaking some sort of alien math language I cannot understand, assuming I already know formulas and all that. Thanks for keeping it simple and easy."
"You are my savior! I took the SATs during the beginning of the school year and pulled off a 750 in the math part. THANKS TO YOU!!..."
"You are a living legend as appealing to teaching that is
"
"hmmh....With you around...Do I really need to go to math class now?"
"Sal, I just wanted to say thank you for all your help. It is marvellous that someone like yourself gives your time to help better others. This is especially true as you are helping so many people who you do not know. If only there were more people like you in the world. I'm studying physics and I am brushing up on some of my maths skills. You have been a great help to me, and I'm sure many others like me. I am just regretful that I have nothing to offer you in way of a thank you other than this message. If I ever come in to position where I may be able to help, you will undoubtedly hear from me."
"Thanks for educating all of us who're interested on mathematical knowledge. Thanks to you I finally see a light at the end of the tunnel. You're the best. Keep up the good work. If I were an engineer already, I'd donate money to you for this. I'm not going to forget Khan's academy."
"...I just wanted to say that out of every internet learning site I have used yours is by far the best! The maths [are] just so beautifully explained. Thank you so much; now I really think I will pass my maths exams~!"
"...Your videos have become very popular in my school, and my grades have sky-rocketed ever since I became accustomed to your methods of teaching..."
"...Before stumbling across your video's i had no idea how to relate with math problems [math is my extreme weak point] but after watching some of your video's, I can do math without a problem..."
"Thanks so much for your videos, there are immensely helpful. I have got my entire AP Calculus class watching your videos."
"I am a high school Math/Physics Teacher working in Toronto Ontario. I have been referring students to your videos for the past year and have referenced many of them on my website. Those that have used it find it very useful, and have commented that you explain content better than I do..."
"I've been re-teaching myself physics with your videos, and I'd like you to know that you are miles better at keeping my attention than any physics or mathematics professor I had during my university work (and I say that while I'm working on my third degree). I only hope Youtube is still around when I have children so I can use these videos for them as well."
My wife is working on here MBA which is why/how I found Sal's youtube videos.
Watch his explanation for his website :here:
Info on Sal:
Sal received his MBA from Harvard Business School. He also holds a Masters in electrical engineering and computer science, a BS in electrical engineering and computer science, and a BS in mathematics from the Massachusetts Institute of Technology.
Information from his website (all free all on youtube.com)
---------------------------------------------
We have taken all 8 math practice tests (432 problems) in "The Offical SAT Study Guide" by the College Board in 100+ videos ... [More] GMAT Preparation
We have taken all 400+ data sufficiency and math problems in "The Official Guide for GMAT Review" by the GMAC in 100+ videos below. ... [More] California: Algebra II
Sal has done all 80 problems from the Algebra II Standards Test (click to download) administered as part of the Standardized Testing and Reporting (STAR) Program under policies set by the State Board of Education. ... [More]
Brain Teasers
Liar Truthteller Brain Teaser
Toggler Brain Teaser
Alien Abduction Brain Teaser
Brain Teaser: Blue Forehead Room
Blue Forehead Room Solution
Forehead Numbers Brain Teaser
Light Bulb Switching Brain Teaser
Path Counting Brain Teaser
3-D Path Counting Brain Teaser Current Economics
Economics of a Cupcake Factory
Cupcake Economics 2
Cupcake Economics 3
Inflation, Deflation & Capacity Utilization
Inflation, Deflation & Capacity Utilization 2
Inflation & Deflation 3: Obama Stimulus Plan
Unemployment
CPI Index
Simple Analysis of Cost per Job Saved from Stimulus Banking and Money
Banking 1
Banking 2: A bank's income statement
Banking 3: Fractional Reserve Banking
Banking 4: Multiplier effect and the money supply
Banking 5: Introduction to Bank Notes
Banking 6: Bank Notes and Checks
Banking 7: Giving out loans without giving out gold
Banking 8: Reserve Ratios
Banking 9: More on Reserve Ratios (Bad sound)
Banking 10: Introduction to leverage (bad sound)
Banking 11: A reserve bank
Banking 12: Treasuries (government debt)
Banking 13: Open Market Operations
Banking 14: Fed Funds Rate
Banking 15: More on the Fed Funds Rate
Banking 16: Why target rates vs. money supply
Banking 17: What happened to the gold?
Banking 18: Big Picture Discussion
The Discount Rate
Repurchase Agreements (Repo transactions)
Federal Reserve Balance Sheet
Fractional Reserve Banking Commentary 1
FRB Commentary 2: Deposit Insurance
FRB Commentary 3: Big Picture Venture Capital and Capital Markets
Raising money for a startup
Getting a seed round from a VC
Going back to the till: Series B
An IPO
More on IPOs
Equity vs. Debt
Bonds vs. Stocks
Chapter 7:Bankruptcy Liquidation
Chapter 11: Bankruptcy Restructuring Finance
Introduction to interest
Interest (part 2)
Introduction to Present Value
Present Value 2
Present Value 3
Present Value 4 (and discounted cash flow)
Introduction to Balance Sheets
More on balance sheets and equity
Home equity loans
Renting vs. Buying a home
Renting vs. buying a home (part 2)
Renting vs. Buying (detailed analysis)
The housing price conundrum
Housing price conundrum (part 2)
Housing Price Conundrum (part 3)
Housing Conundrum (part 4)
Raising money for a startup
Getting a seed round from a VC
Going back to the till: Series B
An IPO
More on IPOs
Bonds vs. Stocks
Chapter 7:Bankruptcy Liquidation
Chapter 11: Bankruptcy Restructuring
Return on capital
Mortgage-Backed Securities I
Mortgage-backed securities II
Mortgage-backed securities III
Collateralized Debt Obligation (CDO)
Introduction to the yield curve
Introduction to compound interest and e
Compound Interest and e (part 2)
Compound Interest and e (part 3)
Compound Interest and e (part 4)
Bailout 1: Liquidity vs. Solvency
Bailout 2: Book Value
Bailout 3: Book value vs. market value
Bailout 4: Mark-to-model vs. mark-to-market
Bailout 5: Paying off the debt
Bailout 6: Getting an equity infusion
Bailout 7: Bank goes into bankruptcy
Bailout 8: Systemic Risk
Bailout 9: Paulson's Plan
Bailout 10: Moral Hazard
Credit Default Swaps
Credit Default Swaps 2
Investment vs. Consumption 1
Investment vs. Comsumption 2
Bailout 11: Why these CDOs could be worth nothing
Bailout 12: Lone Star Transaction
Bailout 13: Does the bailout have a chance of working?
Wealth Destruction 1
Wealth Destruction 2
Bailout 14: Possible Solution
Bailout 15: More on the solution
Banking 4: Multiplier effect and the money supply
Banking 3: Fractional Reserve Banking
Banking 2: A bank's income statement
Banking 1
Banking 2: A bank's income statement
Banking 3: Fractional Reserve Banking
Banking 4: Multiplier effect and the money supply Valuation and Investing
Price and Market Capitalization
Introduction to the Income Statement
Earnings and EPS
Introduction to the Price-to-Earnings Ratio
P/E Discussion
ROA Discussion 1
ROA Discussion 2
Depreciation
Amortization
P/E Conundrum
Enterprise Value
EBITDA Credit Crisis
The housing price conundrum
Housing price conundrum (part 2)
Housing Price Conundrum (part 3)
Housing Conundrum (part 4)
Mortgage-Backed Securities I
Mortgage-backed securities II
Mortgage-backed securities III
Collateralized Debt Obligation (CDO)
Bailout 13: Does the bailout have a chance of working?
Credit Default Swaps
Credit Default Swaps 2
Wealth Destruction 1
Wealth Destruction 2 Geithner Plan
Geithner Plan I
Geithner Plan II
Geithner Plan 2.5
Geithner Plan III
Geithner Plan IV
Geithner 5: A better solution Paulson Bailout
Bailout 1: Liquidity vs. Solvency
Bailout 2: Book Value
Bailout 3: Book value vs. market value
Bailout 4: Mark-to-model vs. mark-to-market
Bailout 5: Paying off the debt
Bailout 6: Getting an equity infusion
Bailout 7: Bank goes into bankruptcy
Bailout 8: Systemic Risk
Bailout 9: Paulson's Plan
Bailout 10: Moral Hazard
Bailout 11: Why these CDOs could be worth nothing
Bailout 12: Lone Star Transaction
Bailout 14: Possible Solution
Bailout 15: More on the solution Chemistry
Introduction to the atom
Orbitals
More on orbitals and electron configuration
Electron Configurations
Electron Configurations 2
Valence Electrons
Groups of the Periodic Table
Periodic Table Trends: Ionization Energy
Other Periodic Table Trends
Ionic, Covalent, and Metallic Bonds
Molecular and Empirical Formulas
The Mole and Avogadro's Number
Formula from Mass Composition
Another mass composition problem
Balancing Chemical Equations
Stoichiometry
Stoichiometry: Limiting Reagent
Ideal Gas Equation: PV=nRT
Ideal Gas Equation Example 1
Ideal Gas Equation Example 2
Ideal Gas Equation Example 3
Ideal Gas Example 4
Partial Pressure
States of Matter
States of Matter Follow-Up
Specific Heat, Heat of Fusion and Vaporization
Chilling Water Problem
Phase Diagrams
Van Der Waals Forces
Covalent Networks, Metallic, and Ionic Crystals
Vapor Pressure
Suspensions, Colloids and Solutions
Solubility
Boiling Point Elevation and Freezing Point Supression
Introduction to Kinetics
Reactions in Equilibrium
Mini-Video on Ion Size
Keq Intuition (mathy and not necessary to progress)
Keq derivation intuition (can skip; bit mathy)
Heterogenous Equilibrium
Le Chatelier's Principle
Introduction to pH, pOH, and pKw
Acid Base Introduction
pH, pOH of Strong Acids and Bases
pH of a Weak Acid
pH of a Weak Base
Conjugate Acids and Bases
pKa and pKb Relationship
Buffers
Strong Acid Titration
Weak Acid Titration
Half Equivalence Point
Titration Roundup
Introduction to Oxidation States
More on Oxidation States
Hydrogen Peroxide Correction
Redox Reactions
Galvanic Cells
Types of Decay
Half-Life
Exponential Decay Formula Proof (can skip, involves Calculus)
Introduction to Exponential Decay
More Exponential Decay Examples
Macrostates and Microstates
Quasistatic and Reversible Processes
First Law of Thermodynamics/ Internal Energy
More on Internal Energy
Work from Expansion
PV-diagrams and Expansion Work
Proof: U=(3/2)PV or U=(3/2)nRT
Work Done by Isothermic Process
Carnot Cycle and Carnot Engine
Proof: Volume Ratios in a Carnot Cycle
Proof: S (or Entropy) is a valid state variable
Thermodynamic Entropy Definition Clarification
Reconciling Thermodynamic and State Definitions of Entropy
Entropy Intuition
Maxwell's Demon
More on Entropy
Efficiency of a Carnot Engine
Carnot Efficiency 2: Reversing the Cycle
Carnot Efficiency 3: Proving that it is the most efficient
Enthalpy
Heat of Formation
Hess's Law and Reaction Enthalpy Change
Gibbs Free Energy and Spontaneity
Gibbs Free Energy Example
More rigorous Gibbs Free Energy/ Spontaneity Relationship
A look at a seductive but wrong Gibbs/Spontaneity Proof Arithmetic
Basic Addition
Level 2 Addition
Basic Subtraction
Level 4 Subtraction
Why borrowing works
Level 4 division
Adding Decimals
Subtracting decimals
Dividing decimal
Greatest Common Divisor
Least Common Multiple
Equivalent fractions
Mixed numbers and improper fractions
Converting fractions to decimals
Percent and decimals
Ordering numeric expressions
Why borrowing works
Basic Multiplication
Multiplication 2: The Multiplication Tables
Multiplication 3: 10,11,12 times tables
Multiplication 4: 2-digit times 1-digit number
Multiplication 5: 2-digit times a 2-digit number
Multiplication 6: Multiple Digit Numbers
Multiplication 7: Old video giving more examples
Mulitplication 8: Multiplying decimals (Old video) Pre-algebra
Order of operations
Order of operations II
Adding/Subtracting negative numbers
Multiplying and dividing negative numbers
Adding and subtracting fractions
Multiplying fractions
Dividing fractions
Level 1 Exponents
Level 2 Exponents
Level 3 exponents
Negative Exponent Intuition
Exponent Rules Part 1
Exponent Rules Part 2
Simplifying radicals
Introduction to Logarithms
Unit conversion
Speed translation
Introduction to logarithm properties
Introduction to logarithm properties (part 2)
Scientific Notation
Scientific Notation Examples
Introduction to Ratios (new HD version) Algebra
Algebra: Linear Equations 1
Algebra: Linear Equations 2
Algebra: Linear Equations 3
Algebra: Linear Equations 4
Algebra: Solving Inequalities
Algebra: graphing lines 1
Algebra: Slope and Y-intercept intuition
Algebra: Slope
Algebra: Slope 2
Algebra: Slope 3
Algebra: Equation of a line
Slope and Y-intercept Intuition
Averages
Integer sums
Taking percentages
Growing by a percentage
More percent problems
systems of equations
Introduction to Ratios (new HD version)
Ratio problem with basic algebra (new HD)
More advanced ratio problem--with Algebra (HD version)
Alternate Solution to Ratio Problem (HD Version)
Introduction to Ratios
Advanced ratio problems
Age word problems 1
Age word problems 2
Age word problems 3
Level 1 multiplying expressions
Solving a quadratic by factoring
i and Imaginary numbers
Complex Numbers (part 1)
Complex Numbers (part 2)
Introduction to the quadratic equation
Quadratic Equation part 2
Completing the square
Quadratic Formula (proof)
Quadratic Inequalities
Quadratic Inequalities (Visual Explanation)
Introduction to functions
Functions Part 2
Functions (Part III)
Functions (part 4)
Domain of a function
Proof: log a + log b = log ab
Proof: A(log B) = log (B^A), log A - log B = log (A/B)
Proof: log_a (B) = (log_x (B))/(log_x (A))
Algebraic Long Division
Introduction to Conic Sections
Conic Sections: Intro to Circles
Conic Sections: Intro to Ellipses
Conic Sections: Intro to Hyperbolas
Conic Sections: Hyperbolas 2
Conic Sections: Hyperbolas 3
Identifying Conics 1
Identifying Conics 2
Conic Identification 3
Foci of an Ellipse
Foci of a Hyperbola
Proof: Hyperbola Foci
Partial Fraction Expansion 1
Partial Fraction Expansion 3
Parabola Focus and Directrix 1
Focus and Directrix of a Parabola 2
Two Passing Bicycles Word Problem
Passed Bike Word Problem
Passing Trains
Overtaking Word Problem
Early Train Word Problem
Officer on Horseback California Standards Test: Algebra I
CA Algebra I: Number Properties and Absolute Value
CA Algebra I: Simplifying Expressions
CA Algebra I: Simple Logical Arguments
CA Algebra I: Graphing Inequalities
CA Algebra I: Slope and Y-intercept
CA Algebra I: Systems of Inequalities
CA Algebra I: Simplying Expressions
CA Algebra I: Factoring Quadratics
CA Algebra I: Completing the Square
CA Algebra I: Quadratic Equation
CA Algebra I: Quadratic Roots
CA Algebra I: Rational Expressions
CA Algebra I: Rational Expressions
CA Algebra I: Word Problems
CA Algebra I: More Word Problems
CA Algebra I: Functions California Standards Test: Algebra II
California Standards Test: Algebra II
California Standards Test: Algebra II (Graphing Inequalities
CA Standards: Algebra II (Algebraic Division/Multiplication)
CA Standards: Algebra II
Algebra II: Simplifying Polynomials
Algebra II: Imaginary and Complex Numbers
Algebra II: Complex numbers and conjugates
Algebra II: Quadratics and Shifts
Algebra II: Shifting Quadratic Graphs
Algebra ||: Conic Sections
Algebra II: Circles and Logarithms
Algebra II: Logarithms Exponential Growth
Algebra II: Logarithms and more
Algebra II: Functions, Combinatorics
Algebra II: binomial Expansion and Combinatorics
Algebra II: Binomial Expansions, Geometric Series Sum
Algebra II: Functions and Probability
Algebra II: Probability and Statistics
Algebra II: Mean and Standard Deviation Geometry
Introduction to angles
Angles (part 2)
Angles (part 3)
Angles of parallel lines
The Angle Game
Similar triangles
Similar triangles (part 2)
Angle Game (part 2)
Area and Perimeter
Circles: Radius, Diameter and Circumference
Area of a circle
Introduction to the Pythagorean Theorem
Pythagorean Theorem II
45-45-90 Triangles
Intro to 30-60-90 Triangles
30-60-90 Triangles II California Standards Test: Geometry
CA Geometry: deductive reasoning
CA Geometry: Proof by Contradiction
CA Geometry: More Proofs
CA Geometry: Similar Triangles
CA Geometry: Similar Triangles
CA Geometry: More on cogruent and similar triangles
CA Geometry: Triangles and Parallelograms
CA Geometry: Area, Pythagorean Theorem
CA Geometry: Area, Circumference, Volume
CA Geometry: Pythagorean Theorem, Area
CA Geometry: Exterior Angles
CA Geometry: Deducing Angle Measures
CA Geometry: Pythagorean Theorem, Compass Constructions
CA Geometry: Compass Construction
CA Geometry: Basic Trigonometry
CA Geometry: More Trig
CA Geometry: Circle Area Chords Tangent
CA Geometry: Secants and Translations
Biology
Introduction to Evolution and Natural Selection
Intelligent Design and Evolution
Evolution Clarification
Natural Selection and the Owl Butterfly
DNA
Variation in a Species
Chromosomes, Chromatids, Chromatin, etc.
Mitosis, Meiosis and Sexual Reproduction
Phases of Mitosis
Phases of Meiosis
Embryonic Stem Cells
Cancer
Introduction to Heredity
Punnett Square Fun
Hardy-Weinberg Principle
Sex-Linked Traits
Bacteria
Viruses
ATP: Adenosine Triphosphate
Introduction to Cellular Respiration
Oxidation and Reduction Review From Biological Point-of-View
Oxidation and Reduction in Cellular Respiration
Glycolysis
Krebs / Citric Acid Cycle
Electron Transport Chain
Oxidative Phosphorylation and Chemiosmosis Trigonometry
Basic Trigonometry
Basic Trigonometry II
Radians and degrees
Using Trig Functions
Using Trig Functions Part II
The unit circle definition of trigonometric function
Unit Circle Definition of Trig Functions
Graph of the sine function
Graphs of trig functions
Graphing trig functions
More trig graphs
Determining the equation of a trigonometric function
Trigonometric Identities
Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b)
Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)
Trig identities part 2 (parr 4 if you watch the proofs)
Trig identies part 3 (part 5 if you watch the proofs)
Trigonometry word problems (part 1)
Trigonometry word problems (part 2)
Law of cosines
Navigation Word Problem
Proof: Law of Sines
Ferris Wheel Trig Problem
Ferris Wheel Trig Problem (part 2)
Fun Trig Problem
Polar Coordinates 1
Polar Coordinates 2
Polar Coordinates 3
Inverse Trig Functions: Arcsin
Inverse Trig Functions: Arctan
Inverse Trig Functions: Arccos
Trigonometry Identity Review/Fun Precalculus
Introduction to Limits
Limit Examples (part 1)
Limit Examples (part 2)
Limit Examples (part3)
Limit Examples w/ brain malfunction on first prob (part 4)
Squeeze Theorem
Proof: lim (sin x)/x
More Limits
Sequences and Series (part 1)
Sequences and series (part 2)
Permutations
Combinations
Binomial Theorem (part 1)
Binomial Theorem (part 2)
Binomial Theorem (part 3)
Introduction to interest
Interest (part 2)
Introduction to compound interest and e
Compound Interest and e (part 2)
Compound Interest and e (part 3)
Compound Interest and e (part 4)
Exponential Growth
Polar Coordinates 1
Polar Coordinates 2
Polar Coordinates 3
Parametric Equations 1
Parametric Equations 2
Parametric Equations 3
Parametric Equations 4 Statistics
Statistics: The Average
Statistics: Sample vs. Population Mean
Statistics: Variance of a Population
Statistics: Sample Variance
Statistics: Standard Deviation
Statistics: Alternate Variance Formulas
Introduction to Random Variables
Probability Density Functions
Binomial Distribution 1
Binomial Distribution 2
Binomial Distribution 3
Binomial Distribution 4
Expected Value: E(X)
Expected Value of Binomial Distribution
Poisson Process 1
Poisson Process 2
Law of Large Numbers
Normal Distribution Excel Exercise
Introduction to the Normal Distribution Probability
Probability (part 1)
Probability (part 2)
Probability (part 3)
Probability (part 4)
Probability (part 5)
Probability (part 6)
Probability (part 7)
Probability (part
Permutations
Combinations
Probability using Combinations
Probability and Combinations (part 2)
Conditional Probability and Combinations
Birthday Probability Problem
Introduction to Random Variables
Probability Density Functions
Binomial Distribution 1
Binomial Distribution 2
Binomial Distribution 3
Binomial Distribution 4
Expected Value: E(X)
Expected Value of Binomial Distribution
Poisson Process 1
Poisson Process 2
Law of Large Numbers Calculus
Introduction to Limits
Limit Examples (part 1)
Limit Examples (part 2)
Limit Examples (part3)
Limit Examples w/ brain malfunction on first prob (part 4)
Squeeze Theorem
Proof: lim (sin x)/x
More Limits
Epsilon Delta Limit Definition 1
Epsilon Delta Limit Definition 2
Calculus: Derivatives 1 (new HD version)
Calculus: Derivatives 2 (new HD version)
Calculus: Derivatives 2.5 (new HD version)
Calculus: Derivatives 1
Calculus: Derivatives 2
Calculus: Derivatives 3
Calculus: Derivatives 4
Calculus: Derivatives 5
Calculus: Derivatives 6
Derivatives (part 7)
Derivatives (part
Derivatives (part 9)
Proof: d/dx(x^n)
Proof: d/dx(sqrt(x))
Proof: d/dx(ln x) = 1/x
Proof: d/dx(e^x) = e^x
Proofs of Derivatives of Ln(x) and e^x
Calculus: Derivative of x^(x^x)
Extreme Derivative Word Problem (advanced)
Implicit Differentiation
Implicit Differentiation (part 2)
More implicit differentiation
More chain rule and implicit differentiation intuition
Trig Implicit Differentiation Example
Maxima Minima Slope Intuition
Inflection point intuition
Monotonicity Theorem
Calculus: Maximum and minimum values on an interval
Calculus: Graphing using derivatives
Graphing with Calculus
Hairy inflection point problem
Optimization with Calculus 1
Optimization with Calculus 2
Optimization with Calculus 3
Optimization Example 4
Introduction to rate-of-change problems
Rates-of-change (part 2)
Ladder rate-of-change problem
Mean Value Theorem
The Indefinite Integral or Anti-derivative
Indefinite integrals (part II)
Indefinite Integration (part III)
Indefinite Integration (part IV)
Indefinite Integration (part V)
Indefinite Integration (part 6)
Indefinite Integration (part 7)
Another u-subsitution example
Introduction to definite integrals
Definite integrals (part II)
Definite Integrals (area under a curve) (part III)
Definite Integrals (part 4)
Definite Integrals (part 5)
Definite integral with substitution
Integrals: Trig Substitution 1
Integrals: Trig Substitution 2
Integrals: Trig Substitution 3 (long problem)
Introduction to differential equations
Solid of Revolution (part 1)
Solid of Revolution (part 2)
Solid of Revolution (part 3)
Solid of Revolution (part 4)
Solid of Revolution (part 5)
Solid of Revolution (part 6)
Solid of Revolution (part 7)
Solid of Revolution (part
Sequences and Series (part 1)
Sequences and series (part 2)
Polynomial approximation of functions (part 1)
Polynomial approximation of functions (part 2)
Approximating functions with polynomials (part 3)
Polynomial approximation of functions (part 4)
Polynomial approximations of functions (part 5)
Polynomial approximation of functions (part 6)
Polynomial approximation of functions (part 7)
Taylor Polynomials
Exponential Growth
Equation of a tangent line
AP Calculus BC Exams: 2008 1 a
AP Calculus BC Exams: 2008 1 b&c
AP Calculus BC Exams: 2008 1 c&d
AP Calculus BC Exams: 2008 1 d
Calculus BC 2008 2 a
Calculus BC 2008 2 b &c
Calculus BC 2008 2d
Partial Derivatives
Partial Derivatives 2
Gradient 1
Gradient of a scalar field
Divergence 1
Divergence 2
Divergence 3
Curl 1
Curl 2
Curl 3
Double Integral 1
Double Integrals 2
Double Integrals 3
Double Integrals 4
Double Integrals 5
Double Integrals 6
Triple Integrals 1
Triple Integrals 2
Triple Integrals 3
?(2^ln x)/x dx Antiderivative Example Differential Equations
Introduction to differential equations
Separable Differential Equations
Separable differential equations 2
Exact Equations Intuition 1 (proofy)
Exact Equations Intuition 2 (proofy)
Exact Equations Example 1
Exact Equations Example 2
Exact Equations Example 3
Integrating factors 1
Integrating factors 2
First order homegenous equations
First order homogenous equations 2
2nd Order Linear Homogeneous Differential Equations 1
2nd Order Linear Homogeneous Differential Equations 2
2nd Order Linear Homogeneous Differential Equations 3
2nd Order Linear Homogeneous Differential Equations 4
Complex roots of the characteristic equations 1
Complex roots of the characteristic equations 2
Complex roots of the characteristic equations 3
Repeated roots of the characteristic equation
Repeated roots of the characterisitic equations part 2
Undetermined Coefficients 1
Undetermined Coefficients 2
Undetermined Coefficients 3
Undetermined Coefficients 4
Laplace Transform 1
Laplace Transform 2
Laplace Transform 3 (L{sin(at)})
Laplace Transform 4
Laplace Transform 5
Laplace Transform 6
Laplace Transform to solve an equation
Laplace Transform solves an equation 2
More Laplace Transform tools
Using the Laplace Transform to solve a nonhomogenous eq
Laplace Transform of : L{t}
Laplace Transform of t^n: L{t^n}
Laplace Transform of the Unit Step Function
Inverse Laplace Examples
Laplace/Step Function Differential Equation
Dirac Delta Function
Laplace Transform of the Dirac Delta Function
Introduction to the Convolution
The Convolution and the Laplace Transform
Using the Convolution Theorem to Solve an Initial Value Prob
Linear Algebra
Introduction to matrices
Matrix multiplication (part 1)
Matrix multiplication (part 2)
Inverse Matrix (part 1)
Inverting matrices (part 2)
Inverting Matrices (part 3)
Matrices to solve a system of equations
Matrices to solve a vector combination problem
Singular Matrices
3-variable linear equations (part 1)
Solving 3 Equations with 3 Unknowns
Linear Algebra: Introduction to Vectors
Linear Algebra: Vector Examples
Linear Algebra: Parametric Representations of Lines
Linear Combinations and Span
Linear Algebra: Introduction to Linear Independence
More on linear independence
Span and Linear Independence Example
Linear Subspaces
Linear Algebra: Basis of a Subspace
Vector Dot Product and Vector Length
Proving Vector Dot Product Properties
Proof of the Cauchy-Schwarz Inequality
Linear Algebra: Vector Triangle Inequality
Defining the angle between vectors
Defining a plane in R3 with a point and normal vector
Linear Algebra: Cross Product Introduction
Proof: Relationship between cross product and sin of angle
Dot and Cross Product Comparison/Intuition
Matrices: Reduced Row Echelon Form 1
Matrices: Reduced Row Echelon Form 2
Matrices: Reduced Row Echelon Form 3
Matrix Vector Products
Introduction to the Null Space of a Matrix
Null Space 2: Calculating the null space of a matrix
Null Space 3: Relation to Linear Independence
Column Space of a Matrix
Null Space and Column Space Basis
Visualizing a Column Space as a Plane in R3
Proof: Any subspace basis has same number of elements
Dimension of the Null Space or Nullity
Dimension of the Column Space or Rank
Showing relation between basis cols and pivot cols
Showing that the candidate basis does span C(A)
A more formal understanding of functions
Vector Transformations
Linear Transformations
Matrix Vector Products as Linear Transformations
Linear Transformations as Matrix Vector Products
Image of a subset under a transformation
im(T): Image of a Transformation
Preimage of a set
Preimage and Kernel Example
Sums and Scalar Multiples of Linear Transformations
More on Matrix Addition and Scalar Multiplication
Linear Transformation Examples: Scaling and Reflections
Linear Transformation Examples: Rotations in R2
Rotation in R3 around the X-axis
Unit Vectors
Introduction to Projections
Expressing a Projection on to a line as a Matrix Vector prod
Compositions of Linear Transformations 1
Compositions of Linear Transformations 2
Linear Algebra: Matrix Product Examples
Matrix Product Associativity
Distributive Property of Matrix Products
Linear Algebra: Introduction to the inverse of a function
Proof: Invertibility implies a unique solution to f(x)=y
Surjective (onto) and Injective (one-to-one) functions
Relating invertibility to being onto and one-to-one
Determining whether a transformation is onto
Linear Algebra: Exploring the solution set of Ax=b
Linear Algebra: Matrix condition for one-to-one trans
Linear Algebra: Simplifying conditions for invertibility
Linear Algebra: Showing that Inverses are Linear
Linear Algebra: Deriving a method for determining inverses
Linear Algebra: Example of Finding Matrix Inverse
Linear Algebra: Formula for 2x2 inverse
Linear Algebra: 3x3 Determinant
Linear Algebra: nxn Determinant
Linear Algebra: Determinants along other rows/cols
Linear Algebra: Rule of Sarrus of Determinants
Linear Algebra: Determinant when row multiplied by scalar
Linear Algebra: (correction) scalar muliplication of row
Linear Algebra: Determinant when row is added
Linear Algebra: Duplicate Row Determinant
Linear Algebra: Determinant after row operations
Linear Algebra: Upper Triangular Determinant
Linear Algebra: Simpler 4x4 determinant
Linear Algebra: Determinant and area of a parallelogram
Linear Algebra: Determinant as Scaling Factor
Linear Algebra: Transpose of a Matrix
Linear Algebra: Determinant of Transpose
Linear Algebra: Transposes of sums and inverses
Linear Algebra: Transpose of a Vector
Linear Algebra: Rowspace and Left Nullspace
Lin Alg: Visualizations of Left Nullspace and Rowspace
Linear Algebra: Orthogonal Complements
Linear Algebra: Rank(A) = Rank(transpose of A)
Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
Lin Alg: Representing vectors in Rn using subspace members
Lin Alg: Orthogonal Complement of the Orthogonal Complement
Lin Alg: Orthogonal Complement of the Nullspace
Lin Alg: Unique rowspace solution to Ax=b
Linear Alg: Rowspace Solution to Ax=b example
Linear Alg: Rowspace Solution to Ax=b example
Lin Alg: Showing that A-transpose x A is invertible
Linear Algebra: Projections onto Subspaces
Linear Alg: Visualizing a projection onto a plane
Lin Alg: A Projection onto a Subspace is a Linear Transforma
Linear Algebra: Subspace Projection Matrix Example
Lin Alg: Another Example of a Projection Matrix
Linear Alg: Projection is closest vector in subspace
Linear Algebra: Least Squares Approximation
Linear Algebra: Least Squares Examples
Linear Algebra: Another Least Squares Example
Linear Algebra: Coordinates with Respect to a Basis
Linear Algebra: Change of Basis Matrix
Lin Alg: Invertible Change of Basis Matrix
Lin Alg: Transformation Matrix with Respect to a Basis
Lin Alg: Alternate Basis Tranformation Matrix Example
Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
Lin Alg: Changing coordinate systems to help find a transformation matrix
Linear Algebra: Introduction to Orthonormal Bases
Linear Algebra: Coordinates with respect to orthonormal bases
Lin Alg: Projections onto subspaces with orthonormal bases
Lin Alg: Finding projection onto subspace with orthonormal basis example
Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
Lin Alg: Orthogonal matrices preserve angles and lengths
Linear Algebra: The Gram-Schmidt Process
Linear Algebra: Gram-Schmidt Process Example
Linear Algebra: Gram-Schmidt example with 3 basis vectors
Linear Algebra: Introduction to Eigenvalues and Eigenvectors
Linear Algebra: Proof of formula for determining Eigenvalues
Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
Linear Algebra: Finding Eigenvectors and Eigenspaces example
Linear Algebra: Eigenvalues of a 3x3 matrix
Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
Linear Algebra: Showing that an eigenbasis makes for good coordinate systems Physics
Introduction to motion
Introduction to motion (part 2)
Introduction to motion (part 3)
Projectile motion (part 1)
Projectile motion (part 2)
Projectile motion (part 3)
Projectile motion (part 4)
Projectile motion (part 5)
Projectile motion (part 6)
Projectile motion (part 7)
Projectile motion (part
Projectile motion (part 9)
Projectile motion (part 10)
2 dimensional projectile motion (part 1)
2 dimensional projectile motion (part 2)
2-dimensional projectile motion (part 3)
2 dimensional projectile motion part 4
2-dimensional projectile motion part 5
Newton's First Law of Motion
Newton's Second Law of Motion
Newton's Third Law of Motion
Newton's Laws Problems (part 1)
Newton's Laws Examples (part 2)
Newton's Laws
Newton's Laws and vectors
Force with Vectors
Introduction to Tension
Tension (part 2)
Mass on Inclined Plane
Introduction to friction
Friction on an inclined plane
A more complicated friction/inclined plane problem
Tension in an accelerating system and pie in the face
Moving pulley problem (part 1)
Moving pulley problem (part 2)
Introduction to Momentum
Momentum: Ice skater throws a ball
2-dimensional momentum problem
2-dimensional momentum problem (part 2)
Introduction to work and energy
Work and Energy (part 2)
Conservation of Energy
Work/Energy problem with Friction
Introduction to mechanical advantage
Mechanical Advantage (part 2)
Mechanical Advantage (part 3)
Center of Mass
Introduction to Torque
Moments
Moments (part 2)
Unit Vector Notation
Unit Vector Notation (part 2)
Projectile Motion with Unit Vectors
Projectile Motion with Unit Vectors (part 2)
Projectile Motion with Ordered Set Notation
Introduction to centripetal acceleration (part 1)
Centripetal Acceleration (part 2)
Centripetal Acceleration (part 3)
Visual Proof: a= v^2/r
Calculus Proof that a=v^2/r
Introduction to angular velocity
Conservation of angular momemtum
Introduction to Newton's Law of Gravitation
Gravitation (part 2)
Intro to springs and Hooke's Law
Potential energy stored in a spring
Spring potential energy example (mistake in math)
Introduction to Harmonic Motion
Harmonic Motion Part 2 (calculus)
Harmonic Motion Part 3 (no calculus)
Fluids (part 1)
Fluids (part 2)
Fluids (part 3)
Fluids (part 4)
Fluids (part 5)
Fluids (part 6)
Fluids (part 7)
Fluids (part
Fluids (part 9)
Fluids (part 10)
Fluids (part 11)
Fluids (part 12)
Thermodynamics (part 1)
Thermodynamics (part 2)
Thermodynamics (part 3)
Thermodynamics (part 4)
Thermodynamics (part 5)
Electrostatics (part 1)
Electrostatics (part 2)
Proof (Advanced): Field from infinite plate (part 1)
Proof (Advanced): Field from infinite plate (part 2)
Electric Potential Energy
Electric Potential Energy (part 2-- involves calculus)
Voltage
Capacitance
Circuits (part 1)
Circuits (part 2)
Circuits (part 3)
Circuits (part 4)
Cross product 1
Cross Product 2
Cross Product and Torque
Introduction to Magnetism
Magnetism 2
Magnetism 3
Magnetism 4
Magnetism 5
Magnetism 6: Magnetic field due to current
Magnetism 7
Magnetism 8
Magnetism 9: Electric Motors
Magnetism 10: Electric Motors
Magnetism 11: Electric Motors
Magnetism 12: Induced Current in a Wire
The dot product
Dot vs. Cross Product
Calculating dot and cross products with unit vector notation CNN: Sal on the Credit Crisis
Sal appears on the Rick Sanchez show to explain the Credit Crisis This appearance has led to a paper that Sal has co-authored with Berkeley professor Dr. David Leibweber on a solution to unfreezing credit markets which is available here. About Salman Khan
Salman Khan (Sal) founded the Khan Academy with the goal of using technology to educate the world Sal received his MBA from Harvard Business School. He also holds a Masters in electrical engineering and computer science, a BS in electrical engineering and computer science, and a BS in mathematics from the Massachusetts Institute of Technology. Frequently Asked Questions. What people are saying...
"...My eldest kid is dancing around in my room here because she is so excited that she finally found someone that teaches like this. You're awesome. Thank you." "I don't know who you are but in my mind you are a savior. My children really struggle with math, there is an inherited learning disability in my family. They get it but only after seeing it done multiple times. Your videos will allow us to help our children get caught up with their peers. As a parent I have to say, Thank You Thank You Thank You."
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"You're the only math tutor that makes sense to me. Most other ones seem to be speaking some sort of alien math language I cannot understand, assuming I already know formulas and all that. Thanks for keeping it simple and easy."
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"You are a living legend as appealing to teaching that is
" "hmmh....With you around...Do I really need to go to math class now?"
"Sal, I just wanted to say thank you for all your help. It is marvellous that someone like yourself gives your time to help better others. This is especially true as you are helping so many people who you do not know. If only there were more people like you in the world. I'm studying physics and I am brushing up on some of my maths skills. You have been a great help to me, and I'm sure many others like me. I am just regretful that I have nothing to offer you in way of a thank you other than this message. If I ever come in to position where I may be able to help, you will undoubtedly hear from me."
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