lessons delivered
View standard site
• Math
• Arithmetic and pre-algebra
• Intro to addition and subtraction

Adding and subtracting is the basis of all mathematics. This tutorial introduces you to one-digit addition and subtraction. You might become pretty familiar with the number line too!

• Two digit addition and subtraction

In this tutorial, we'll start adding and subtracting numbers that have two (yes, two!) digits. We won't be doing any carrying or borrowing (you'll learn what those are shortly) so you can see that adding or subtracting two digit numbers is really just an extension of what you already know.

You're somewhat familar with adding, say, 17+12 or 21+32, but what happens for 13+19? Essentially, what happens when I max out the "ones place". In this tutorial, we'll introduce you to the powerful tool of carrying and why it works.

• Subtraction with borrowing (regrouping)

You can subtract 21 from 45, but are a bit perplexed trying to subtract 26 from 45 (how do you subtract the 6 in 26 from the 5 in 45). This tutorial is your answer. You'll see that we can essentially "regroup" the value in a number from one place to another to solve your problem. This is also often called borrowing (although it is like "borrowing" sugar from your neighbor in that you never give it back).

• Addition and subtraction word problems

You feel comfortable with adding and subtracting multi-digit numbers. Now you can apply some of your skills to solve problems that arise in the real world (often called "word problems").

• Multiplication and division
• Multiplication fun

If 3 kids each have two robot possums, how many total robot possums do we have? You liked addition, but now you're ready to go to the next level. Depending on how you view it, multiplication is about repeated addition or scaling a number or seeing what number you get when you have another number multiple times. If that last sentence made little sense, you might enjoy this tutorial.

• Delightful division

Every time you split your avocado harvest with your 10 pet robot possums, you've been dividing. You don't farm avocados? You only have 8 robot possums? No worries. I'm sure you've divided as well. Multiplication is awesome, but you're ready for the next step. Division is the art of trying to split things into equal groups. Like subtraction undoes addition, division also undoes multiplication. After this tutorial, you'll have a basic understanding of all of the core operations in arithmetic!

• The distributive property

The distributive property is an idea that shows up over and over again in mathematics. It is the idea that 5 x (3 + 4) = (5 x 3) + (5 x 4). If that last statement made complete sense, no need to watch this tutorial. If it didn't or you don't know why it's true, then this tutorial might be a good way to pass the time :)

• Order of operations

If you have the expression "3 + 4 x 5", do you add the 3 to the 4 first or multiply the 4 and 5 first? To clear up confusion here, the math world has defined which operation should get priority over others. This is super important. You won't really be able to do any involved math if you don't get this clear. But don't worry, this tutorial has your back.

• Place value

You've been counting for a while now. It's second nature to go from "9" to "10" or "99" to "100", but what are you really doing when you add another digit? How do we represent so many numbers (really as many as we want) with only 10 number symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)? In this tutorial you'll learn about place value. This is key to better understanding what you're really doing when you count, carry, regroup, multiply and divide with mult-digit numbers. If you really think about it, it might change your worldview forever!

• Multi-digit multiplication

You know your multiplication tables and are ready to learn how to multiply *any* number (actually, any whole number). Imagine the possibilities! This tutorial will make you unstoppable.

• Loooong division!

You know your multiplication tables and are getting the hang of basic division. In this tutorial, we will journey into the world of loooong division (sometimes, referred to as "long division", but that's not as much fun to say). After this tutorial, you'll be able to divide any whole number by any other. The fun will not stop!

• Rounding whole numbers

If you're looking to create an army of robot dogs, will it really make a difference if you have 10,300 dogs, 9,997 dogs or 10,005 dogs? Probably not. All you really care about is how many dogs you have to, say, the nearest thousand (10,000 dogs). In this tutorial, you'll learn about conventions for rounding whole numbers. Very useful when you might not need to (or cannot) be completely precise.

• Lattice multiplication

Tired of "standard multiplication". In this tutorial we'll explore a different way. Not only is lattice multiplication interesting, it'll help us appreciate that there are many ways to do things. We'll also try to grasp why it works in the first place. Enjoy!

• Partial quotient division

Feeling constrained by traditional long division? Want to impress your friends, family and even your enemies? Well, partial quotient division may be for you (or it might not). This very optional tutorial will show you that there are many ways to slice a walnut (just made up that colloquialism).

• Factors and multiples
• Divisibility tests

Whether you are trying to impress your dog's friends (who are obsessed with figuring out number divisibility) or quickly factor a number, it can be very useful to tell whether a number is divisible by another. This tutorial walks through some of the more standard divisibility methods and describes why they work.

• Divisibility and factors

In this tutorial, we'll begin to think about the numbers that "make up" the number. This will be useful throughout our study of math. Whether we are adding fractions, exploring mystical number patterns, or breaking computer codes, factoring numbers are key! Eye of the tiger!

• Prime numbers

Prime numbers have been studied by mathematicians and mystics for ages (seriously). They are both basic and mysterious. The more you explore them, the more you will realize that the universe is a fascinating place. This tutorial will introduce you to the magical world of prime numbers.

• Prime factorization

You know what prime numbers are and how to identify them. In this tutorial, we'll see that *all* positive whole numbers can be broken down into products of prime numbers (In some way, prime numbers are the "atoms" of the number world that can be multiplied to create any other number). Besides being a fascinating idea, it is also extremely useful. Prime factorization can be used to decrypt encrypted information!

• Least common multiple

Life is good, but it can always get better. Just imagine being able to find the smallest number that is a multiple of two other numbers! Other than making your life more fulfilling, it will allow you to do incredible things like adding fractions.

• Greatest common divisor

You know how to find factors of a number. But what about factors that are common to two numbers? Even better, imagine the largest factors that are common to two numbers. I know. Too exciting!

• Negative numbers and absolute value
• Negative number basics

What is a "negative number"? What happens when I add or subtract one of these? If you come to this tutorial armed with the basics of adding an subtracting numbers, you'll learn about what happens in the world below zero! You will learn what negative numbers are and how we can add and subtract them.

• Adding and subtracting negative numbers

You understand that negative numbers represent how far we are "below zero". Now you are ready to add and subtract them! In this tutorial, we will explain, give examples, and give practice adding and subtracting negative numbers. This is a super-important concept for the rest of your mathematical career so, no pressure, learn it as well as you can!

• Multiplying and dividing negative numbers

It is starting to dawn on you that negative numbers are incredibly awesome. So awesome that you are feeling embarrassed to think how excited you are about them! Well, the journey is just beginning. In this tutorial we will think about multiplying and dividing numbers throughout the number line!

• Absolute value

You'll find absolute value absolutely straightforward--it is just the "distance from zero". If you have a positive number, it is its own absolute value. If you have a negative number, just make it positive to get the absolute value. As you see as you develop mathematically, this idea will eventually extended to more contexts and dimensions, so super important that you understand this core concept now.

• Decimals and percent
• Conceptualizing decimals and place notation

You've been using decimals all of your life. When you pay \$0.75 at a vending machine, 0.75 is a decimal. When you see the ratings of gymnastics judges at the Olympics ("9.5, 9.4, 7.5 (booooo)"), those are decimals. This tutorial will help you understand this powerful tool all the better. Before you know it, you'll be representing numbers that are in-between whole numbers all the time!

You get the general idea of decimal is and what the digits in different places represent (place value). Now you're ready to do something with the decimals. Adding and subtracting is a good place to start. This will allow you to add your family's expenses to figure out if your little brother is laundering money (perhaps literally). Have fun!

• Multiplying decimals

The real world is seldom about whole numbers. If you precisely measure anything, you're likely to get a decimal. If you don't know how to multiply these decimals, then you won't be able to do all the powerful things that multiplication can do in the real world (figure out your commission as a robot possum salesperson, determining how much shag carpet you need for your secret lair, etc.).

• Dividing decimals

You can add, subtract and multiply decimals. You know you'd feel a bit empty if you couldn't divide them as well. But something more powerful is going to happen. If you are like us, you never quite liked those pesky remainders when dividing whole numbers. Well, those pesky remainders better watch out because they are going to be divided too!!!! Ah ha ha ha ha!!!!!

• Converting between fractions and decimals

Both fractions and decimals are desperate to capture that little part of our heart that desires to express non-whole numbers. But must we commit? Can't we have business in the front and party in the back (younger people should look up the word "mullet" to see a hair-style worth considering for your next trip to the barber)? Can't it look like a pump, but feel like a sneaker? Well, if 18-wheelers can turn into self-righteous robots, then why can't decimals and fractions turn into each other?

• Intro to percentages

At least 50% of the math that you end up doing in your real life will involve percentages. We're not really sure about that figure, but it sounds authoritative. Anyway, unless you've watched this tutorial, you're really in no position to argue otherwise. As you'll see "percent" literally means "per cent" or "per hundred". It's a pseudo-decimally thing that our society likes to use even though decimals or fractions alone would have done the trick. Either way, we're 100% sure you'll find this useful.

• Percent word problems

Whether you're calculating a tip at your favorite restaurant or figuring out how many decades you'll be paying your student debt because of the interest, percents will show up again and again and again in your life. This tutorial will expose you to some of these problems before they show up in your actual life so you can handle them with ease (kind of like a vaccine for the brain). Enjoy.

• Estimating and rounding with decimals

Laziness is usually considered a bad thing. But sometimes, it is useful to be lazy in a smart way. Why do a big, hairy calculation if you just need a rough estimate? Why keep track of 2.345609 when you only need 2.35? This tutorial will get you comfortable with sometimes being a little rough with numbers. By being able to round and estimate them, it'll only add one more tool to your toolkit.

• Significant figures

There is a strong temptation in life to appear precise, even when you are aren't accurate. If you precisely measure one dimension of a carpet to be 3.256 meters and eyeball the other dimensional to be "roughly 2 meters", can you really claim that the area is 6.512 square meters (3.256 x 2)? Isn't that a little misleading? This tutorial gets us thinking about this conundrum and gives us the best practices that scientists and engineers use to not mislead each other.

• Fractions
• Understanding fractions

If you don't understand fractions, you won't be even 1/3 educated. Glasses will seem half empty rather than half full. You'll be lucky to not be duped into some type of shady real-estate scheme or putting far too many eggs in your cake batter. Good thing this tutorial is here. You'll see that fractions allow us to view the world in entirely new ways. You'll see that everything doesn't have to be a whole. You'll be able to slice and dice and then put it all back together (and if you order now, we'll throw in a spatula warmer for no extra charge).

• Equivalent fractions and simplified form

There are literally infinite ways to represent any fraction (or number for that matter). Don't believe us? Let's take 1/3. 2/6, 3/9, 4/12 ... 10001/30003 are all equivalent fractions (and we could keep going)! If you know the basics of what a fraction is, this is a great tutorial for recognizing when fractions are equivalent and then simplifying them as much as possible!

You've already got 2 cups of sugar in the cupboard. Your grandmother's recipe for disgustingly-sweet-fudge-cake calls for 3 and 1/3 cups of sugar. How much sugar do you need to borrow from you robot neighbor? Adding and subtracting fractions is key. It might be a good idea to look at the equivalent fractions tutorial before tackling this one.

• Multiplying and dividing fractions

What is 2/3 of 2/3? If I divide a dozen donuts into donut thirds, how many donut pieces will I have? Multiplying and dividing fractions is not only super-useful, but super-fun as well.

• Mixed numbers and improper fractions

We can often have fractions whose numerators are not less than the denominators (like 23/4 or 3/2 or even 6/6). These top-heavy friends are called improper fractions. Since they represent a whole or more (in absolute terms), they can also be expressed as a combination of a whole number and a "proper fraction" (one where the numerator is less than the denominator) which is called a "mixed number." They are both awesome ways of representing a number and getting acquainted with both (as this tutorial does) is super useful in life!

• Mixed number addition and subtraction

You know the basics of what mixed numbers are. You're now ready to add and subtract them. This tutorial gives you plenty of examples and practice in this core skill!

• Mixed number multiplication and division

My recipe calls for a cup and a half of blueberries and serves 10 people. But I have 23 people coming over. How many cups of blueberries do I need? You know that mixed numbers and improper fractions are two sides of the same coin (and you can convert between the two). In this tutorial we'll learn to multiply and divide mixed numbers (mainly by converting them into improper fractions first).

• Decimals and fractions

If you already know a bit about both decimals and fractions, this tutorial will help build a bridge between the two. Through a bunch of examples and practice, you'll be able operate in both worlds. Have fun!

• Number sets

The world of numbers can be split up into multiple "sets", many of which overlap with each other (integers, rational numbers, irrational numbers, etc.). This tutorial works through examples that expose you to the terminology of the various sets and how you can differentiate them.

• Ratios, proportions, units and rates
• Ratios and proportions

Would you rather go to a college with a high teacher-to-student ratio or a low one? What about the ratio of girls-to-boys? What is the ratio of eggs to butter in your favorite dessert? Ratios show up EVERYWHERE in life. This tutorial introduces you to what they (and proportions) are and how to make good use of them!

• Rates

How fast can a robot possum fly? What is the rate at which a hungry you can consume avocados? This tutorial helps you make sense of these fundamental questions in life.

• Unit conversion

Wait, I'm in Europe and my car only tells my distance traveled in kilometers! But I'm used to a units of distance devised by the Romans to measure the average length of 1000 paces of a soldier (the "mile")! How do I operate? This tutorial is about the fundamental skill of unit conversion. Sal's cousin Nadia being a bit confused with this was actually the reason why he started tutoring her (which led to the creation of the Khan Academy).

• Exponents, radicals, and scientific notation
• The world of exponents

Addition was nice. Multiplication was cooler. In the mood for a new operation that grows numbers even faster? Ever felt like expressing repeated multiplication with less writing? Ever wanted to describe how most things in the universe grow and shrink? Well, exponents are your answer! This tutorial covers everything from basic exponents to negative and fractional ones. It assumes you remember your multiplication, negative numbers and fractions.

A strong contender for coolest symbol in mathematics is the radical. What is it? How does it relate to exponents? How is the square root different than the cube root? How can I simplify, multiply and add these things? This tutorial assumes you know the basics of exponents and exponent properties and takes you through the radical world for radicals (and gives you some good practice along the way)!

• Exponent properties

Tired of hairy exponent expressions? Feel compelled to clean them up? Well, this tutorial might just give you the tools you need. If you know a bit about exponents, you'll learn a ton more in this tutorial as you learn about the rules for simplifying exponents.

• Scientific notation

Scientists and engineers often have to deal with super huge (like 6,000,000,000,000,000,000,000) and super small numbers (like 0.0000000000532) . How can they do this without tiring their hands out? How can they look at a number and understand how large or small it is without counting the digits? The answer is to use scientific notation. If you come to this tutorial with a basic understanding of positive and negative exponents, it should leave you with a new appreciation for representing really huge and really small numbers!

• Applying mathematical reasoning

This tutorial is less about statistics and more about interpreting data--whether it is presented as a table, pictograph, bar graph or line graph. Good for someone new to these ideas. For a student in high school or college looking to learn statistics, it might make sense to skip (although it might not hurt either).

• Multistep word problems

The world seldom gives you two numbers and tells you which operation to perform. More likely, you'll be presented with a bunch of information and you (yes, YOU) need to make sense of them. This tutorial gives you practice doing exactly that. When watching videos, pause and attempt it before Sal. Then work on as many problems as you want in the exercise at the end of the tutorial.

• Arithmetic properties
• Arithmetic properties

2 + 3 = 3 + 2, 6 x 4 = 4 x 6. Adding zero to a number does not change the number. Likewise, multiplying a number by 1 does not change it. You may already know these things from working through other tutorials, but some people (not us) like to give these properties names that sound far more complicated than the property themselves. This tutorial (which we're not a fan of), is here just in case you're asked to identify the "Commutative Law of Multiplication". We believe the important thing isn't the fancy label, but the underlying idea (which isn't that fancy).

• Algebra
• Introduction to algebra
• Overview and history of algebra

Did you realize that the word "algebra" comes from Arabic (just like "algorithm" and "al jazeera" and "Aladdin")? And what is so great about algebra anyway? This tutorial doesn't explore algebra so much as it introduces the history and ideas that underpin it.

• The why of algebra

Much of algebra seems obsessed with "doing the same thing to both sides". Why is this? How can we develop an intuition for which algebraic operations are valid and which ones aren't? This tutorial takes a high-level, conceptual walk-through of what an equation represents and why we do the same thing to both sides of it.

• Yoga plans

This tutorial is a survey of all the core ideas in a traditional first-year algebra course. It is by no means comprehensive (that's what the other 600+ videos are for), but it will hopefully whet your appetite for more algebra!

• Linear equations
• Variables and expressions

Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? I must figure this out!!! Yes, you must. This tutorial is great if you're just beginning to delve into the world of algebraic variables and expressions.

• The why of algebra

Algebra seems mysterious to me. I really don't "get" what an equation represents. Why do we do the same thing to both sides? This tutorial is a conceptual journey through the basics of algebra. It is made for someone just beginning their algebra adventure. But even folks who feel pretty good that they know how to manipulate equations might pick up a new intuition or two.

• Super Yoga plans

This tutorial is a survey of the major themes in basic algebra in five videos! From basic equations to graphing to systems, it has it all. Great for someone looking for a gentle, but broad understanding of the use of algebra. Also great for anyone unsure of which gym plan they should pick!

• Manipulating expressions

Using the combined powers of Chuck Norris and polar bears (which are much less powerful than Mr. Norris) to better understand what expressions represent and how we can manipulate them. Great tutorial if you want to understand that expressions are just a way to express things!

• Equations for beginners

Like the "Why of algebra" and "Super Yoga plans" tutorials, we'll introduce you to the most fundamental ideas of what equations mean and how to solve them. We'll then do a bunch of examples to make sure you're comfortable with things like 3x – 7 = 8. So relax, grab a cup of hot chocolate, and be on your way to becoming an algebra rockstar. And, by the way, in any of the "example" videos, try to solve the problem on your own before seeing how Sal does it. It makes the learning better!

• More fancy equations for beginners

You've been through "Equation examples for beginners" and are feeling good. Well, this tutorial continues that journey by addressing equations that are just a bit more fancy. By the end of this tutorial, you really will have some of the core algebraic tools in your toolkit!

• Percent word problems

I paid \$5.00 for some tanning lotion (ok, I've never really bought tanning lotion) after a 35% discount. How can we find the full price? You know how to take a percentage. In this tutorial, we use our newfound powers to solve equations to tackle fascinating percentage problems.

• Solving for a variable

You feel comfortable solving for an unknown. But life is all about stepping outside of your comfort zone--it's the only way you can grow! This tutorial takes solving equations to another level by making things a little more abstract. You will now solve for a variable, but it will be in terms of other variables. Don't worry, we think you'll find it quite therapeutic once you get the hang of it.

• Converting repeating decimals to fractions

You know that converting a fraction into a decimal can sometimes result in a repeating decimal. For example: 2/3 = 0.666666..., and 1/7 = 0.142857142857... But how do you convert a repeating decimal into a fraction? As we'll see in this tutorial, a little bit of algebra magic can do the trick!

• Age word problems

In 72 years, Sal will be 3 times as old as he is today (although he might not be... um... capable of doing much). How old is Sal today? These classic questions have plagued philosophers through the ages. Actually, they haven't. But they have plagued algebra students! Even though few people ask questions like this in the real-world, these are strangely enjoyable problems.

• Absolute value equations

You are absolutely tired of not knowing how to deal with equations that have absolute values in them. Well, this tutorial might help.

• Simplifying complicated equations

You feel good about your rapidly developing equation-solving ability. Now you're ready to fully flex your brain. In this tutorial, we'll explore equations that don't look so simple at first, but that, with a bit of skill, we can turn into equations that don't cause any stress! Have fun!

• Evaluating expressions with unknown variables

When solving equations, there is a natural hunger to figure out what an unknown is equal to. This is especially the case if we want to evaluate an expression that the unknown is part of. This tutorial exposes us to a class of solvable problems that challenges this hunger and forces us to be the thinking human beings that we are! In case you're curious, these types of problems are known to show up on standardized exams to see if you are really a thinking human (as opposed to a robot possum).

• More equation practice

This tutorial is for you if you already have the basics of solving equations and are looking to put your newfound powers to work in more examples.

• Old school equations with Sal

Some of Sal's oldest (and roughest) videos on algebra. Great tutorial if you want to see what Khan Academy was like around 2006. You might also like it if you feel like Sal has lost his magic now that he doesn't use the cheapest possible equipment to make the videos.

• Linear inequalities
• Basic inequalities

In this tutorial you'll discover that much of the logic you've used to solve equations can also be applied to think about inequalities!

• Compound and absolute value inequalities

You're starting to get comfortable with a world where everything isn't equal. In this tutorial, we'll add more constraints to think of at the same time. You may not realize it, but the ability to understand and manipulate compound and absolute value inequalities is key to many areas of science, engineering, and manufacturing (especially when tolerances are concerned)!

• Graphing points, equations and inequalities
• Coordinate plane

How can we communicate exactly where something is in two dimensions? Who was this Descartes character? In this tutorial, we cover the basics of the coordinate plane. We then delve into graphing points and determining whether a point is a solution of an equation. This will be a great tutorial experience if you are just starting to ramp up your understanding of graphing or need some fundamental review.

• Graphing solutions to equations

In this tutorial, we'll work through examples that show how a line can be viewed as all of coordinates whose x and y values satisfy a linear equation. Likewise, a linear equation can be viewed as describing a relationship between the x and y values on a line.

• Graphing lines using x and y intercepts

There are many ways to graph a line and this tutorial covers one of the simpler ones. Since you only need two points for a line, let's find what value an equation takes on when x = 0 (essentially the y-intercept) and what value it takes on when y = 0 (the x-intercept). Then we can graph the line by going through those two points.

• Slope

If you've ever struggled to tell someone just how steep something is, you'll find the answer here. In this tutorial, we cover the idea of the slope of a line. We also think about how slope relates to the equation of a line and how you can determine the slope or y-intercept given some clues. This tutorial is appropriate for someone who understands the basics of graphing equations and want to dig a bit deeper. After this tutorial, you will be prepared to start thinking deeper about the equation of a line.

• Equation of a line

You know a bit about slope and intercepts, but want to know more about all the ways you can represent the equation of a line including slope-intercept form, point-slope form, and standard form. This tutorial will satisfy that curiosity!

• More analytic geometry

You're familiar with graphing lines, slope and y-intercepts. Now we are going to go further into analytic geometry by thinking about distances between points, midpoints, parallel lines and perpendicular ones. Enjoy!

• Graphing linear inequalities

In this tutorial we'll see how to graph linear inequalities on the coordinate plane. We'll also learn how to determine if a particular point is a solution of an inequality.

• Systems of equations and inequalities
• A system for solving the King's problems

Whether in the real world or a cliche fantasy one, systems of equations are key to solving super-important issues like "the make-up of change in a troll's pocket" or "how can order the right amount of potato chips for a King's party." Join us as we cover (and practice with examples and exercises) all of the major ways of solving a system: graphically, elimination, and substitution. This tutorial will also help you think about when system might have no solution or an infinite number of solutions. Very, very exciting stuff!

• Super fast systems of equations

Have no time for trolls, kings and parrots and just want to get to the essence of system. This might be a good tutorial for you. As you can see, this stuff is so important that we're covering it in several tutorials!

• Solving systems graphically

This tutorial focuses on solving systems graphically. This is covered in several other tutorials, but this one gives you more examples than you can shake a chicken at. Pause the videos and try to do them before Sal does.

• Thinking about solutions to systems

You know how to solve systems of equations (for the most part). This tutorial will take things a bit deeper by exploring cases when you might have no solutions or an infinite number of them.

• Solving systems with substitution

This tutorial is focused on solving systems through substitution. This is covered in several other tutorials, but this one focuses on substitution with more examples than you can shake a dog at. As always, pause the video and try to solve before Sal does.

• Solving systems with elimination (addition-elimination)

You can solve a system of equations with either substitution or elimination. This tutorial focuses with a ton of examples on elimination. It is covered in other tutorials, but we give you far more examples here. You'll learn best if you pause the videos and try to do the problem before Sal does.

• Systems of equations word problems

This tutorial doesn't involve talking parrots and greedy trolls, but it takes many of the ideas you might have learned in that tutorial and applies them to word problems. These include rate problems, mixture problems, and others. If you can pause and solve the example videos before Sal does, we'd say that you have a pretty good grasp of systems. Enjoy!

• Systems of inequalities

You feel comfortable with systems of equations, but you begin to realize that the world is not always fair. Not everything is equal! In this short tutorial, we will explore systems of inequalities. We'll graph them. We'll think about whether a point satisfies them. We'll even give you as much practice as you need. All for 3 easy installments of... just kidding, it's free (although the knowledge obtained in priceless). A good deal if we say so ourselves!

• Systems with three variables

Two equations with two unknowns not challenging enough for you? How about three equations with three unknowns? Visualizing lines in 2-D too easy? Well, now you're going to visualize intersecting planes in 3-D, baby. (Okay, we admit that it is weird for a website to call you "baby.") Tired of linear systems? Well, we might just bring a little nonlinearity into your life, honey. (You might want to brush up on your solving quadratics before tackling the non-linear systems.) As always, try to pause the videos and do them before Sal does!

• Non-linear systems of equations

Tired of linear systems? Well, we might just bring a little nonlinearity into your life, honey. (You might want to brush up on your solving quadratics before tackling the non-linear systems.) As always, try to pause the videos and do them before Sal does!

• Functions
• Function introduction

Relationships can be any association between sets of numbers while functions have only one output for a given input. This tutorial works through a bunch of examples of testing whether something is a valid function. As always, we really encourage you to pause the videos and try the problems before Sal does!

• Domain and range

What values can you and can you not input into a function? What values can the function output? The domain is the set of values that the function is defined for (i.e., the values that you can input into a function). The range is the set of values that the function output can take on. This tutorial covers the ideas of domain and range through multiple worked examples. These are really important ideas as you study higher mathematics.

• Analyzing linear functions

Linear functions show up throughout life (even though you might not realize it). This tutorial will have you thinking much deeper about what a linear function means and various ways to interpret one. Like always, pause the video and try the problem before Sal does. Then test your understanding by practicing the problems at the end of the tutorial.

• Linear and nonlinear functions

Not every relationship in the universe can be represented by a line (in fact, most can't be). We call these "nonlinear". In this tutorial, you'll learn to tell the difference between a linear and nonlinear function! Have fun!

• Direct and inverse variation

Whether you are talking about how force relates to acceleration or how the cost of movie tickets relates to the number of people going, it is not uncommon in this universe for things to vary directly. Similarly, when you are, say, talking about how hunger might relate to seeing roadkill, things can vary inversely. This tutorial digs deeper into these ideas with a bunch of examples of direct and inverse variation.

• Graphing functions

You've already graphed functions when you graphed lines and curves in other topics so this really isn't anything new. Now we'll do a few more examples in this tutorial, but we'll use the function notation to make things a bit more explicit.

• Evaluating function expressions

This is a super fun tutorial where we'll evaluate expressions that involve functions. We'll add, subtract, multiply and divide them. We'll also do composite functions which involves taking the output of one function to be the input of another one! As always, pause the video and try the problem before Sal does!

• Function inverses

Functions associate a set of inputs with a set of outputs (in fancy language, they "map" one set to another). But can we go the other way around? Are there functions that can start with the outputs as inputs and produce the original inputs as outputs? Yes, there are! They are called function inverses! This tutorial works through a bunch of examples to get you familiar with the world of function inverses.

• New operator definitions

Are you bored of the traditional operators of addition, subtraction, multiplication and division? Do even exponents seem a little run-of-the-mill? Well in this tutorial, we will--somewhat arbitrarily--define completely new operators and notation (which are essentially new function definitions without the function notation). Not only will this tutorial expand your mind, it could be the basis of a lot of fun at your next dinner party!

• Classic function videos

These oldie-but-maybe-goodies are the original function videos that Sal made years ago for his cousins. Despite the messy handwriting, some people claim that they like these better than the new ones (they claim that there is a certain charm to them). We'll let you decide.

Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer. This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!

• Completing the square

You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula. Welcome to the world of completing the square!

Probably one of the most famous tools in mathematics, the quadratic formula (a.k.a. quadratic equation) helps you think about the roots of ANY quadratic (even ones that have no real roots)! As you'll see, it is just the by-product of completing the square, but understanding and applying the formula will take your algebra skills to new heights. In theory, one could apply the quadratic formula in a brainless way without understanding factoring or completing the square, but that's lame and uninteresting. We recommend coming to this tutorial with a solid background in both of those techniques. Have fun!

Tired of lines? Not sure if a parabola is a disease of the gut or a new mode of transportation? Ever wondered what would happen to the graph of a function if you stuck an x² someplace? Well, look no further. In this tutorial, we will study the graphs of quadratic functions (parabolas), including their foci and whatever the plural of directrix is.

You are familiar with factoring quadratic expressions and solving quadratic equations. Well, as you might guess, not everything in life has to be equal. In this short tutorial we will look at quadratic inequalities.

This tutorial has a bunch of extra, but random, videos on quadratics. A completely optional tutorial that you may or may not want to look at. If you do, watch it last. There are some Sal oldies here and some random examples.

• Exponent expressions and equations
• Exponent properties

You first learned about exponents in pre-algebra. You're now ready to apply many of those same principles using abstract variables. Welcome to exponents in algebra! In this tutorial, you will learn about how to manipulate expressions with exponents in them. We'll give lots of example to make sure you see a lot of scenarios. For optimal learning (and fun), pause the video before Sal does an example.

You're enjoying algebra and equations, but you miss radicals. Wouldn't it be unbelievably awesome if you could solve equations with radicals in them. Well, your dreams can come true. In this tutorial, we work through a bunch of examples to help you understand how to solve radical equations. As always, pause the videos and try to solve the example before Sal does.

• Polynomials
• Polynomial basics

"Polynomials" sound like a fancy word, but you just have to break down the root words. "Poly" means "many". So we're just talking about "many nomials" and everyone knows what a "nomial" is. Okay, most of us don't. Well, a polynomials has "many" terms. From understanding what a "term" is to basic simplification, addition and subtraction of polynomials, this tutorial will get you very familiar with the world of many "nomials." :)

• Multiplying monomials, binomials and polynomials in general

No algebraic arsenal is complete without the ability to multiply expressions. In this tutorial, we'll review some basic exponent properties and the distributive property when multiplying single terms. We'll then see that multiplying anything fancier (binomials, polynomials, etc.) is just an extension of the distributive property that you first learned in elementary school. We do cover the FOIL method as well because some schools teach this, but we think it is lame and not real learning. The distributive property is FAR more powerful and general.

• Dividing polynomials

You know what polynomials are. You know how to add, subtract, and multiply them. Unless you are completely incurious, you must be wondering how to divide them! In this tutorial we'll explore how we divide polynomials--both through algebraic long division and synthetic division. (We like classic algebraic long division more since you can actually understand what you're doing.)

• Ratios and rational expressions
• Ratios with algebra

You remember a thing or two about ratios and proportions from you pre-algebra days. Well, how can we use these same ideas to solve problems in algebra. This tutorial re-introduces ratios in an algebraic context and helps us solve some awesome problems!

• Simplifying rational expressions

You get a rational expression when you divide one polynomial by another. If you have a good understanding of factoring quadratics, you'll be able to apply this skill here to help realize where a rational expression may not be defined and how we can go about simplifying it.

• Rational expressions and equations

Have you ever wondered what would happen if you divide one polynomial by another? What if you set that equal to something else? Would it be as unbelievably epic as you suspect it would be? Well, rational expressions are just algebraic expressions formed by dividing one expression by another. We get a rational equation if we set that equal to something else. In this tutorial, we work through examples to understand and apply rational expressions and equations.

• Graphing rational functions

Rational functions are often not defined at certain points and have very interesting behavior has the input variable becomes very large in magnitude. This tutorial explores how to graph these functions, paying attention to these special features. We'll talk a lot about vertical and horizontal asymptotes.

• Partial fraction expansion

If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions.

• Logarithms
• Logarithm basics

If you understand how to take an exponent and you're looking to take your mathematical game to a new level, then you've found the right tutorial. Put simply and confusingly, logarithms are inverse operators to exponents (just as subtraction to addition or division to multiplication). As you'll see, taking a logarithm of something tells you what exponent you need to raise a base to to get that number.

• Logarithm properties

You want to go deeper in your understanding of logarithms. This tutorial does just that by exploring properties of logarithms that will help you manipulate them in entirely new ways (mostly falling out of exponent properties).

• Natural logarithms

e is a special number that shows up throughout nature (you will appreciate this more and more as you develop your mathematical understanding). Given this, logarithms with base e have a special name--natural logarithms. In this tutorial, we will learn to evaluate and graph this special function.

• Conic sections
• Conic section basics

What is a conic other than a jazz singer from New Orleans? Well, as you'll see in this tutorial, a conic section is formed when you intersect a plane with cones. You end up with some familiar shapes (like circles and ellipses) and some that are a bit unexpected (like hyperbolas). This tutorial gets you set up with the basics and is a good foundation for going deeper into the world of conic sections.

• Circles

You've seen circles your entire life. You've even studied them a bit in math class. Now we go further, taking a deep look at the equations of circles.

• Ellipses

What would you call a circle that isn't a circle? One that is is is taller or fatter rather than being perfectly round? An ellipse. (All circles are special cases of ellipses.) In this tutorial we go deep into the equations and graphs of ellipses.

• Parabolas

You've seen parabolas already when you graphed quadratic functions. Now we will look at them from a conic perspective. In particular we will look at them as the set of all points equidistant from a point (focus) and a line (directrix). Have fun!

• Hyperbolas

It is no hyperbole to say that hyperbolas are awesome. In this tutorial, we look closely at this wacky conic section. We pay special attention to its graph and equation.

• Identifying conics from equations

You're familiar with the graphs and equations of all of the conic sections. Now you want practice identifying them given only their equations. You, my friend, are about to click on exactly the right tutorial.

• Matrices
• Basic matrix operations

Keanu Reeves' virtual world in the The Matrix (I guess we can call all three movies "The Matrices") have more in common with this tutorial than you might suspect. Matrices are ways of organizing numbers. They are used extensively in computer graphics, simulations and information processing in general. The super-intelligent artificial intelligences that created The Matrix for Keanu must have used many matrices in the process. This tutorial introduces you to what a matrix is and how we define some basic operations on them.

• Matrix multiplication

You know what a matrix is, how to add them and multiply them by a scalar. Now we'll define multiplying one matrix by another matrix. The process may seem bizarre at first (and maybe even a little longer than that), but there is a certain naturalness to the process. When you study more advanced linear algebra and computer science, it has tons of applications (computer graphics, simulations, etc.)

• Inverting matrices

Multiplying by the inverse of a matrix is the closest thing we have to matrix division. Like multiplying a regular number by its reciprocal to get 1, multiplying a matrix by its inverse gives us the identity matrix (1 could be thought of as the "identity scalar"). This tutorial will walk you through this sometimes involved process which will become bizarrely fun once you get the hang of it.

• Reduced row echelon form

You've probably already appreciated that there are many ways to solve a system of equations. Well, we'll introduce you to another one in this tutorial. Reduced row echelon form has us performing operations on matrices to get them in a form that helps us solve the system.

• Imaginary and complex numbers
• The imaginary unit i

This is where math starts to get really cool. It may seem strange to define a number whose square is negative one. Why do we do this? Because it fits a nice niche in the math ecosystem and can be used to solve problems in engineering and science (not to mention some of the coolest fractals are based on imaginary and complex numbers). The more you think about it, you might realize that all numbers, not just i, are very abstract.

• Complex numbers

Let's start constructing numbers that have both a real and imaginary part. We'll call them complex. We can even plot them on the complex plane and use them to find the roots of ANY quadratic equation. The fun must not stop!

• Geometry
• Points, lines, and planes
• Introduction to Euclidean geometry

Roughly 2400 years ago, Euclid of Alexandria wrote Elements which served as the world's geometry textbook until recently. Studied by Abraham Lincoln in order to sharpen his mind and truly appreciate mathematical deduction, it is still the basis of what we consider a first year course in geometry. This tutorial gives a bit of this background and then lays the conceptual foundation of points, lines, circles and planes that we will use as we journey through the world of Euclid.

• Angles and intersecting lines
• Angle basics and measurement

This tutorial will define what an angle is and help us think about how to measure them. If you're new to angles, this is a great place to start.

• Angles between intersecting and parallel lines

Welcome. I'd like to introduce you to Mr. Angle. Nice to meet you. So nice to meet you. This tutorial introduces us to angles. It includes how we measure them, how angles relate to each other and properties of angles created from various types of intersecting lines. Mr. Angle is actually far more fun than you might initially presume him to be.

• Angles with triangles and polygons

Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out.

• Sal's old angle videos

These are some of the classic, original angle video that Sal had done way back when (like 2007). Other tutorials are more polished than this one, but this one has charm. Also not bad if you're looking for more examples of angles between intersected lines, transversals and parallel lines.

• Congruent triangles
• Congruence postulates

We begin to seriously channel Euclid in this tutorial to really, really (no, really) prove things--in particular, that triangles are congruents. You'll appreciate (and love) what rigorous proofs are. It will sharpen your mind and make you a better friend, relative and citizen (and make you more popular in general). Don't have too much fun.

• Congruence and isosceles and equilateral triangles

This tutorial uses our understanding of congruence postulates to prove some neat properties of isosceles and equilateral triangles.

• Perimeter, area and volume
• Perimeter and area of rectangles

How long of a fence do you need? How big is your house? How big is your waistline? What's your hat size? These are fundamentally important questions that need to be answered! This is a tutorial to give you the basics of what perimeter, circumference (really the perimeter of a circle) and area are and then applies the ideas to triangles, rectangles and circles. This is more of review for students who are going through the main geometry narrative and can be skipped if yo u remember it from grade-school.

• Perimeter and area of triangles

You first learned about perimeter and area when you were in grade school. In this tutorial, we will revisit those ideas with a more mathy lense. We will use our prior knowledge of congruence to really start to prove some neat (and useful) results (including some that will be useful in our study of similarity).

• Triangle inequality theorem

The triangle inequality theorem is, on some level, kind of simple. But, as you'll see as you go into high level mathematics, it is often used in fancy proofs. This tutorial introduces you to what it is and gives you some practice understanding the constraints on the dimensions of a triangle.

• Koch snowflake fractal

Named after Helge von Koch, the Koch snowflake is one of the first fractals to be discovered. It is created by adding smaller and smaller equilateral bumps to an existing equilateral triangle. Quite amazingly, it produces a figure of infinite perimeter and finite area!

• Heron's formula

Named after Heron of Alexandria, Heron's formula is a power (but often overlooked) method for finding the area of ANY triangle. In this tutorial we will explain how to use it and then prove it!

• Circumference and area of circles

Circles are everywhere. How can we measure how big they are? Well, we could think about the distance around the circle (circumference). Another option would be to think about how much space it takes up on our paper (area). Have fun!

• Perimeter and area of non-standard shapes

Not everything in the world is a rectangle, circle or triangle. In this tutorial, we give you practice at finding the perimeters and areas of these less-than-standard shapes!

• Volume and surface area

Tired of perimeter and area and now want to measure 3-D space-take-upness. Well you've found the right tutorial. Enjoy!

• Similarity
• Triangle similarity

This tutorial explains a similar (but not congruent) idea to congruency (if that last sentence made sense, you might not need this tutorial). Seriously, we'll take a rigorous look at similarity and think of some reasonable postulates for it. We'll then use these to prove some results and solve some problems. The fun must not stop!

• Old school similarity

These videos may look similar (pun-intended) to videos in another playlist but they are older, rougher and arguably more charming. These are some of the original videos that Sal made on similarity. They are less formal than those in the "other" similarity tutorial, but, who knows, you might like them more.

• Right triangles
• Pythagorean theorem

Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful. In this tutorial, we will cover what it is and how it can be used. We have another tutorial that gives you as many proofs of it as you might need. In thi

• Pythagorean theorem proofs

The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful.

• Special right triangles

We hate to pick favorites, but there really are certain right triangles that are more special than others. In this tutorial, we pick them out, show why they're special, and prove it! These include 30-60-90 and 45-45-90 triangles (the numbers refer to the measure of the angles in the triangle).

• Special properties and parts of triangles
• Perpendicular bisectors

In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices. This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.

• Angle bisectors

This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").

• Medians and centroids

You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).

• Altitudes

Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).

• Bringing it all together

This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!

Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals!

• Circles
• Angles

Identifying, measuring, and calculating angles.

• Triangles

Identifying types of triangles, using principles of similarity and congruence, and applying and proving triangle postulates.

• Worked Examples

Sal does the 80 problems from the released questions from the California Standards Test for Geometry. Basic understanding of Algebra I necessary.

• Trigonometry and precalculus
• Graphing lines
• The coordinate plane

How can we communicate exactly where something is in two dimensions? Who was this Descartes character? In this tutorial, we cover the basics of the coordinate plane. We then delve into graphing points and determining whether a point is a solution of an equation. This will be a great tutorial experience if you are just starting to ramp up your understanding of graphing or need some fundamental review.

• Slope

If you've ever struggled to tell someone just how steep something is, you'll find the answer here. In this tutorial, we cover the idea of the slope of a line. We also think about how slope relates to the equation of a line and how you can determine the slope or y-intercept given some clues. This tutorial is appropriate for someone who understands the basics of graphing equations and want to dig a bit deeper. After this tutorial, you will be prepared to start thinking deeper about the equation of a line.

• Equation of a line

You know a bit about slope and intercepts, but want to know more about all the ways you can represent the equation of a line including slope-intercept form, point-slope form, and standard form. This tutorial will satisfy that curiosity!

• Midpoint and distance

This tutorial covers some of the basics of analytic geometry: the distance between two points and the coordinate of the midpoint of two points.

• Equations of parallel and perpendicular lines

You're familiar with graphing lines, slope and y-intercepts. Now we are going to go further into analytic geometry by thinking about the equations of parallel and perpendicular lines. Enjoy!

• Graphing inequalities

In this tutorial we'll see how to graph linear inequalities on the coordinate plane. We'll also learn how to determine if a particular point is a solution of an inequality.

• Functions and their graphs
• Introduction to functions

You've already been using functions in algebra, but just didn't realize it. Now you will. By introducing a little more notation and a few new ideas, you'll hopefully realize that functions are a very, very powerful tool. This tutorial is an old one that Sal made in the early days of Khan Academy. It is rough on the edges (and in between the edges), but it does go through the basic idea of what a function is and how we can define and evaluate functions.

• Domain and range

What values can you and can you not input into a function? What values can the function output? The domain is the set of values that the function is defined for (i.e., the values that you can input into a function). The range is the set of values that the function output can take on. This tutorial covers the ideas of domain and range through multiple worked examples. These are really important ideas as you study higher mathematics.

• Function inverses

Functions associate a set of inputs with a set of outputs (in fancy language, they "map" one set to another). But can we go the other way around? Are there functions that can start with the outputs as inputs and produce the original inputs as outputs? Yes, there are! They are called function inverses! This tutorial works through a bunch of examples to get you familiar with the world of function inverses.

• Analyzing functions

You know a function when you see one, but are curious to start looking deeper at their properties. Some functions seem to be mirror images around the y-axis while others seems to be flipped mirror images while others are neither. How can we shift and reflect them? This tutorial addresses these questions by covering even and odd functions. It also covers how we can shift and reflect them. Enjoy!

In second grade you may have raised your hand in class and asked what you get when you divide by zero. The answer was probably "it's not defined." In this tutorial we'll explore what that (and "indeterminate") means and why the math world has left this gap in arithmetic. (They could define something divided by 0 as 7 or 9 or 119.57 but have decided not to.)

• More mathy functions

In this tutorial, we'll start to use and define functions in more "mathy" or formal ways.

• Polynomial and rational functions

Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer. This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!

• Completing the square and the quadratic formula

You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula. Welcome to the world of completing the square!

Tired of lines? Not sure if a parabola is a disease of the gut or a new mode of transportation? Ever wondered what would happen to the graph of a function if you stuck an x² someplace? Well, look no further. In this tutorial, we will study the graphs of quadratic functions (parabolas), including their foci and whatever the plural of directrix is.

You are familiar with factoring quadratic expressions and solving quadratic equations. Well, as you might guess, not everything in life has to be equal. In this short tutorial we will look at quadratic inequalities.

• Polynomials

"Polynomials" sound like a fancy word, but you just have to break down the root words. "Poly" means "many". So we're just talking about "many nomials" and everyone knows what a "nomial" is. Okay, most of us don't. Well, a polynomials has "many" terms. From understanding what a "term" is to basic simplification, addition and subtraction of polynomials, this tutorial will get you very familiar with the world of many "nomials." :)

• Binomial theorem

You can keep taking the powers of a binomial by hand, but, as we'll see in this tutorial, there is a much more elegant way to do it using the binomial theorem and/or Pascal's Triangle.

• Simplifying rational expressions

You get a rational expression when you divide one polynomial by another. If you have a good understanding of factoring quadratics, you'll be able to apply this skill here to help realize where a rational expression may not be defined and how we can go about simplifying it.

• Rational functions

Have you ever wondered what would happen if you divide one polynomial by another? What if you set that equal to something else? Would it be as unbelievably epic as you suspect it would be?

• Asymptotes and graphing rational functions
• Partial fraction expansion

If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions. This has many uses throughout mathematics. In particular, it is key when taking inverse Laplace transforms in differential equations (which you'll take, and rock, after calculus).

• Exponential and logarithmic functions
• Exponential growth and decay

From compound interest to population growth to half lives of radioactive materials, it all comes down to exponential growth and decay.

• Logarithmic functions

This tutorial shows you what a logarithmic function is. It will then go on to show the many times in nature and science that these type of functions are useful to describe what is happening.

• Continuous compounding and e

This tutorial introduces us to one of the derivations (from finance and continuously compounding interest) of the irrational number 'e' which is roughly 2.71...

• Basic Trigonometry
• Basic trigonometric ratios

In this tutorial, you will learn all the trigonometry that you are likely to remember in ten years (assuming you are a lazy non-curious, non-lifelong learner). But even in that non-ideal world where you forgot everything else, you'll be able to do more than you might expect with the concentrated knowledge you are about to get.

Most people know that you can measure angles with degrees, but only exceptionally worldly people know that radians can be an exciting alternative. As you'll see, degrees are somewhat arbitrary (if we lived on a planet that took 600 days to orbit its star, we'd probably have 600 degrees in a full revolution). Radians are pure. Seriously, they are measuring the angle in terms of how long the arc that subtends them is (measured in radiuseseses). If that makes no sense, imagine measuring a bridge with car lengths. If that still doesn't make sense, watch this tutorial!

• Unit circle definition of trigonometric functions

You're beginning to outgrow SOH CAH TOA. It breaks down for angles greater than or equal to 90. It breaks down for negative angles. Sometimes in life, breaking a bad relationship early is good for both parties. Lucky for you, you don't have to stay lonely for long. We're about to introduce you to a much more robust way to define trigonometric functions. Don't want to get too hopeful, but this might be a keeper.

• Graphs of trig functions

The unit circle definition allows us to define sine and cosine over all real numbers. Doesn't that make you curious what the graphs might look like? Well this tutorial will scratch that itch (and maybe a few others). Have fun.

• Inverse trig functions

Someone has taken the sine of an angle and got 0.85671 and they won't tell you what the angle is!!! You must know it! But how?!!! Inverse trig functions are here to save your day (they often go under the aliases arcsin, arccos, and arctan).

• Long live Tau

Pi (3.14159...) seems to get all of the attention in mathematics. On some level this is warranted. The ratio of the circumference of a circle to the diameter. Seems pretty pure. But what about the ratio of the circumference to the radius (which is two times pi and referred to as "tau")? Now that you know a bit of trigonometry, you'll discover in videos made by Sal and Vi that "tau" may be much more deserving of the throne!

• Trig identities and examples
• Trigonometric identities

If you're starting to sense that there may be more to trig functions than meet the eye, you are sensing right. In this tutorial you'll discover exciting and beautiful and elegant and hilarious relationships between our favorite trig functions (and maybe a few that we don't particularly like). Warning: Many of these videos are the old, rougher Sal with the cheap equipment!

• More trig examples

This tutorial is a catch-all for a bunch of things that we haven't been able (for lack of time or ability) to categorize into other tutorials :(

• Parametric equations and polar coordinates
• Parametric equations

Here we will explore representing our x's and y's in terms of a third variable or parameter (often 't'). Not only can we describe new things, but it can be super useful for describing things like particle motion in physics.

• Polar coordinates

Feel that Cartesian coordinates are too "square". That they bias us towards lines and away from cool spirally things. Well polar coordinates be just what you need!

• Conic sections
• Conic section basics

What is a conic other than a jazz singer from New Orleans? Well, as you'll see in this tutorial, a conic section is formed when you intersect a plane with cones. You end up with some familiar shapes (like circles and ellipses) and some that are a bit unexpected (like hyperbolas). This tutorial gets you set up with the basics and is a good foundation for going deeper into the world of conic sections.

• Circles

You've seen circles your entire life. You've even studied them a bit in math class. Now we go further, taking a deep look at the equations of circles.

• Ellipses

What would you call a circle that isn't a circle? One that is is is taller or fatter rather than being perfectly round? An ellipse. (All circles are special cases of ellipses.) In this tutorial we go deep into the equations and graphs of ellipses.

• Parabolas

You've seen parabolas already when you graphed quadratic functions. Now we will look at them from a conic perspective. In particular we will look at them as the set of all points equidistant from a point (focus) and a line (directrix). Have fun!

• Hyperbolas

It is no hyperbole to say that hyperbolas are awesome. In this tutorial, we look closely at this wacky conic section. We pay special attention to its graph and equation.

• Conics from equations

You're familiar with the graphs and equations of all of the conic sections. Now you want practice identifying them given only their equations. You, my friend, are about to click on exactly the right tutorial.

• Conics in the IIT JEE

Do you think that the math exams that you have to take are hard? Well, if you have the stomach, try the problem(s) in this tutorial. They are not only conceptually difficult, but they are also hairy. Don't worry if you have trouble with this. Most of us would. The IIT JEE is an exam administered to 200,000 students every year in India to select which 2000 go to the competitive IITs. They need to make sure that most of the students can't do most of the problems so that they can really whittle the applicants down.

• Systems of equations and inequalities
• Solving systems of equations for the king

Whether in the real world or a cliche fantasy one, systems of equations are key to solving super-important issues like "the make-up of change in a troll's pocket" or "how can order the right amount of potato chips for a King's party." Join us as we cover (and practice with examples and exercises) all of the major ways of solving a system: graphically, elimination, and substitution. This tutorial will also help you think about when system might have no solution or an infinite number of solutions. Very, very exciting stuff!

• Systems of inequalities

You feel comfortable with systems of equations, but you begin to realize that the world is not always fair. Not everything is equal! In this short tutorial, we will explore systems of inequalities. We'll graph them. We'll think about whether a point satisfies them. We'll even give you as much practice as you need. All for 3 easy installments of... just kidding, it's free (although the knowledge obtained in priceless). A good deal if we say so ourselves!

• Systems with three variables

Two equations with two unknowns not challenging enough for you? How about three equations with three unknowns? Visualizing lines in 2-D too easy? Well, now you're going to visualize intersecting planes in 3-D, baby. (Okay, we admit that it is weird for a website to call you "baby.")

• Non-linear systems of equations

Tired of linear systems? Well, we might just bring a little nonlinearity into your life, honey. (You might want to brush up on your solving quadratics before tackling the non-linear systems.) As always, try to pause the videos and do them before Sal does!

• Sequences and induction
• Induction

Proof by induction is a core tool. This tutorial walks you through the general idea that if 1) something is true for a base case (say when n=1) and 2) if it is true for n, then it is also true for n+1, then it must be true for all n! Amazing!

• Basic sequences and series

This sequence (pun intended) of videos and exercises will help us explore ordered lists of objects--even infinite ones--that often have some pattern to them. We will then explore constructing sequences where the nth term is the sum of the first n terms of another sequence (series). This is surprisingly useful in a whole series (pun intended) of applications from finance to drug dosage.

• Deductive and inductive reasoning

You will hear the words "deductive reasoning" and "inductive reasoning" throughout your life. This very optional tutorial will give you context for what these mean.

You understand what sequences and series are and the mathematical notation for them. This tutorial takes things further by exploring ideas of convergence divergence and other, more challenging topics.

• Probability and combinatorics
• Basic probability
• Venn diagrams and the addition rule

What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).

• Compound, independent events

What is the probability of making three free throws in a row (LeBron literally asks this in this tutorial). In this tutorial, we'll explore compound events happening where the probability of one event is not dependent on the outcome of another (compound, independent, events).

• Dependent events

What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.

• Permutations and combinations

If want to display your Chuck Norris dolls on your desk at school and there is only room for five of them. Unfortunately, you own 50. How many ways can you pick the dolls and arrange them on your desk? What if you don't what order they are in or how they are posed (okay, of course you care about their awesome poses)?

• Probability using combinatorics

This tutorial will apply the permutation and combination tools you learned in the last tutorial to problems of probability. You'll finally learn that there may be better "investments" than poring all your money into the Powerball Lottery.

• Imaginary and complex numbers
• The imaginary unit i

This is where math starts to get really cool. It may see strange to define a number whose square is negative one. Why do we do this? Because it fits a nice niche in the math ecosystem and can be used to solve problems in engineering and science (not to mention some of the coolest fractals are based on imaginary and complex numbers). The more you think about it, you might realize that all numbers, not just i, are very abstract.

• Complex numbers

Let's start constructing numbers that have both a real and imaginary part. We'll call them complex. We can even plot them on the complex plane and use them to find the roots of ANY quadratic equation. The fun must not stop!

• Intro to complex analysis

You know what imaginary and complex numbers are, but want to start digging a bit deeper. In this tutorial, we will explore different ways of representing a complex number and finding its roots.

• Challenging complex number problems

This tutorial goes through a fancy problem from the IIT JEE exam in India (competitive exam for getting into their top engineering schools). Whether or not you live in India, this is a good example to test whether you are a complex number rock star.

• Hyperbolic trig functions
• Intro to hyperbolic trigonometric functions

You know your regular trig functions that are defined with the help of the unit circle. We will now define a new class of functions constructed from exponentials that have an eery resemblance to those classic trig functions (but are still quite different).

• Limits
• Limit basics

Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can thinking about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to a curve. If you have a decent background in algebra (graphing and functions in particular), you'll hopefully enjoy this tutorial!

• Calculus
• Limits
• Limits

Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can think about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to a curve. If you have a decent background in algebra (graphing and functions in particular), you'll hopefully enjoy this tutorial!

• Old limits tutorial

This tutorial covers much of the same material as the "Limits" tutorial, but does it with Sal's original "old school" videos. The sound, resolution or handwriting isn't as good, but some people find them more charming.

• Limits and infinity

You have a basic understanding of what a limit is. Now, in this tutorial, we can explore situation where we take the limit as x approaches negative or positive infinity (and situations where the limit itself could be unbounded).

• Squeeze theorem

If a function is always smaller than one function and always greater than another (i.e. it is always between them), then if the upper and lower function converge to a limit at a point, then so does the one in between. Not only is this useful for proving certain tricky limits (we use it to prove lim (x → 0) of (sin x)/x, but it is a useful metaphor to use in life (seriously). :) This tutorial is useful but optional. It is covered in most calculus courses, but it is not necessary to progress on to the "Introduction to derivatives" tutorial.

• Epsilon delta definition of limits

This tutorial introduces a "formal" definition of limits. So put on your ball gown and/or tuxedo to party with Mr. Epsilon Delta (no, this is not referring to a fraternity). This tends to be covered early in a traditional calculus class (right after basic limits), but we have mixed feelings about that. It is cool and rigorous, but also very "mathy" (as most rigorous things are). Don't fret if you have trouble with it the first time. If you have a basic conceptual understanding of what limits are (from the "Limits" tutorial), you're ready to start thinking about taking derivatives.

• Taking derivatives
• Introduction to differential calculus

The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.

• Introduction to derivatives

Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.

• Visualizing derivatives

You understand that a derivative can be viewed as the slope of the tangent line at a point or the instantaneous rate of change of a function with respect to x. This tutorial will deepen your ability to visualize and conceptualize derivatives through videos and exercises. We think you'll find this tutorial incredibly fun and satisfying (seriously).

• Power rule

Calculus is about to seem strangely straight forward. You've spent some time using the definition of a derivative to find the slope at a point. In this tutorial, we'll derive and apply the derivative for any term in a polynomial. By the end of this tutorial, you'll have the power to take the derivative of any polynomial like it's second nature!

• Chain rule

You can take the derivatives of f(x) and g(x), but what about f(g(x)) or g(f(x))? The chain rule gives us this ability. Because most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), you'll be able to take the derivative of almost any function after this tutorial. Just imagine.

• Product and quotient rules

You can figure out the derivative of f(x). You're also good for g(x). But what about f(x) times g(x)? This is what the product rule is all about. This tutorial is all about the product rule. It also covers the quotient rule (which really is just a special case of the product rule).

• Implicit differentiation

Like people, mathematical relations are not always explicit about their intentions. In this tutorial, we'll be able to take the derivative of one variable with respect to another even when they are implicitly defined (like "x^2 + y^2 = 1").

• Proofs of derivatives of common functions

We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!

• Derivative applications
• Minima, maxima, inflection points and critical points

Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.

• Optimization with calculus

Using calculus to solve optimization problems

• Rates of change

Solving rate-of-change problems using calculus

• Mean value theorem

If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).

• L'Hôpital's Rule

Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.

• Indefinite and definite integrals
• Indefinite integral as anti-derivative

You are very familiar with taking the derivative of a function. Now we are going to go the other way around--if I give you a derivative of a function, can you come up with a possible original function. In other words, we'll be taking the anti-derivative!

• Riemann sums and definite integration

In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.

• Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.

• U-substitution

U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverise chain rule. U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

• Definite integrals

Until now, we have been viewing integrals as anti-derivatives. Now we explore them as the area under a curve between two boundaries (we will now construct definite integrals by defining the boundaries). This is the real meat of integral calculus!

• Trigonometric substitution

We will now do another substitution technique (the other was u-substitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the anti-derivative of.

• Fundamental Theorem of Calculus

You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. We'll explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.

• Improper integrals

Not everything (or everyone) should or could be proper all the time. Same is true for definite integrals. In this tutorial, we'll look at improper integrals--ones where one or both bounds are at infinity! Mind blowing!

• Solid of revolution
• Disc method

You know how to use definite integrals to find areas under curves. We now take that idea for "spin" by thinking about the volumes of things created when you rotate functions around various lines. This tutorial focuses on the "disc method" and the "washer method" for these types of problems.

• Shell method

You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.

• Solid of revolution volume

Using definite integration, we know how to find the area under a curve. But what about the volume of the 3-D shape generated by rotating a section of the curve about one of the axes (or any horizontal or vertical line for that matter). This in an older tutorial that is now covered in other tutorials. This tutorial will give you a powerful tool and stretch your powers of 3-D visualization!

• Sequences, series and function approximation
• Sequences and series review

You want to learn about Maclaurin and Taylor series but are a little rough on your sequences and series. This tutorial will get you brushed up on the concepts, vocabulary and ideas behind sequences and series.

• Maclaurin and Taylor series

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

• Sal's old Maclaurin and Taylor series tutorial

Everything in this tutorial is covered (with better resolution and handwriting) in the "other" Maclaurin and Taylor series tutorial, but this one has a bit of old-school charm so we are keeping it here for historical reasons.

• AP Calculus practice questions
• Calculus AB example questions

Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).

• Calculus BC sample questions

The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle!

• Double and triple integrals
• Double integrals

A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface!

• Triple integrals

This is about as many integrals we can use before our brains explode. Now we can sum variable quantities in three-dimensions (what is the mass of a 3-D wacky object that has variable mass)!

• Partial derivatives, gradient, divergence, curl
• Partial derivatives

Let's jump out of that boring (okay, it wasn't THAT boring) 2-D world into the exciting 3-D world that we all live and breath in. Instead of functions of x that can be visualized as lines, we can have functions of x and y that can be visualized as surfaces. But does the idea of a derivative still make sense? Of course it does! As long as you specify what direction you're going in. Welcome to the world of partial derivatives!

Ever walk on hill (or any wacky surface) and wonder which way would be the fastest way up (or down). Now you can figure this out exactly with the gradient.

• Divergence

Is a vector field "coming together" or "drawing apart" at a given point in space. The divergence is a vector operator that gives us a scalar value at any point in a vector field. If it is positive, then we are diverging. Otherwise, we are converging!

• Curl

Curl measures how much a vector field is "spinning". A bit of a pain to calculate, but could come in handy when we work with Stokes' and Greens' theorems later on.

• Line integrals and Green's theorem
• Line integrals for scalar functions

With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).

• Position vector functions and derivatives

In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).

• Line integrals in vector fields

You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

• Green's theorem

It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.

• 2-D Divergence theorem

Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.

• Surface integrals and Stokes' theorem
• Parameterizing a surface

You can parameterize a line with a position vector valued function and understand what a differential means in that context already. This tutorial will take things further by parametrizing surfaces (2 parameters baby!) and have us thinking about partial differentials.

• Surface integrals

Finding line integrals to be a bit boring? Well, this tutorial will add new dimension to your life by explore what surface integrals are and how we can calculate them.

• Flux in 3-D and constructing unit normal vectors to surface

Flux can be view as the rate at which "stuff" passes through a surface. Imagine a next placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.

• Stokes' theorem intuition and application

Stokes' theorem relates the line integral around a surface to the curl on the surface. This tutorial explores the intuition behind Stokes' theorem, how it is an extension of Green's theorem to surfaces (as opposed to just regions) and gives some examples using it. We prove Stokes' theorem in another tutorial. Good to come to this tutorial having experienced the tutorial on "flux in 3D".

• Proof of Stokes' theorem

You know what Stokes' theorem is and how to apply it, but are craving for some real proof that it is true. Well, you've found the right tutorial!

• Divergence theorem
• Divergence theorem (3D)

An earlier tutorial used Green's theorem to prove the divergence theorem in 2-D, this tutorial gives us the 3-D version (what most people are talking about when they refer to the "divergence theorem"). We will get an intuition for it (that the flux through a close surface--like a balloon--should be equal to the divergence across it's volume). We will use it in examples. We will prove it in another tutorial.

• Types of regions in three dimensions

This tutorial classifies regions in three dimensions. Comes in useful for some types of double integrals and we use these ideas to prove the divergence theorem.

• Divergence theorem proof

You know what the divergence theorem is, you can apply it and you conceptually understand it. This tutorial will actually prove it to you (references types of regions which are covered in the "types of regions in 3d" tutorial.

• Probability and statistics
• Independent and dependent events
• Basic probability

Can I pick a red frog out of a bag that only contains marbles? Is it smart to buy a lottery ticket? Even if we are unsure about whether something will happen, can we start to be mathematical about the "chances" of an event (essentially realizing that some things are more likely than others). This tutorial will introduce us to the tools that allow us to think about random events.

• Venn diagrams and adding probabilities

What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).

• Compound, independent events

What is the probability of making three free throws in a row (LeBron literally asks this in this tutorial). In this tutorial, we'll explore compound events happening where the probability of one event is not dependent on the outcome of another (compound, independent, events).

• Dependent probability

What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.

• Basic set operations

Whether you are learning computer science, logic, or probability (or a bunch of other things), it can be very, very useful to have this "set" of skills. From what a set is to how we can operate on them, this tutorial will have you familiar with the basics of sets!

• Old school probability (very optional)

Sal's old videos on probability. Covered better in other tutorials but here because some people actually like these better.

• Probability and combinatorics
• Permutations and combinations

If want to display your Chuck Norris dolls on your desk at school and there is only room for five of them. Unfortunately, you own 50. How many ways can you pick the dolls and arrange them on your desk? What if you don't care what order they are in or how they are posed (okay, of course you care about their awesome poses)?

• Probability using combinatorics

This tutorial will apply the permutation and combination tools you learned in the last tutorial to problems of probability. You'll finally learn that there may be better "investments" than poring all your money into the Powerball Lottery.

• Random variables and probability distributions
• Descriptive statistics
• Measures of central tendency

This is the foundational tutorial for the rest of statistics. We start thinking about how you can represent a set of numbers with one number that somehow represents the "center". We then talk about the differences between populations, samples, parameters and statistics.

• Box-and-whisker plots

Whether you're looking at scientific data or stock price charts, box-and-whisker plots can show up in your life. This tutorial covers what they are, how to read them and how to construct them. We'd consider this tutorial very optional, but it is a good application of dealing with medians and ranges.

• Variance and standard deviation

We have tools (like the arithmetic mean) to measure central tendency and are now curious about representing how much the data in a set varies from that central tendency. In this tutorial we introduce the variance and standard deviation (which is just the square root of the variance) as two commonly used tools for doing this.

• Sal's old statistics videos

This tutorial covers central tendency and dispersion. It is redundant with the other tutorials on this topic, but it has the benefit of messy handwriting and a cheap microphone. This is Sal circa 2007 so take it all with a grain of salt (or just skip it altogether).

• Regression
• Linear regression and correlation

Even when there might be a rough linear relationship between two variables, the data in the real-world is never as clean as you want it to be. This tutorial helps you think about how you can best fit a line to the relationship between two variables.

• Inferential statistics
• Normal distribution

The normal distribution (often referred to as the "bell curve" is at the core of most of inferential statistics. By assuming that most complex processes result in a normal distribution (we'll see why this is reasonable), we can gauge the probability of it happening by chance. To best enjoy this tutorial, it is good to come to it understanding what probability distributions and random variables are. You should also be very familiar with the notions of population and sample mean and standard deviation.

• Sampling distribution

In this tutorial, we experience one of the most exciting ideas in statistics--the central limit theorem. Without it, it would be a lot harder to make any inferences about population parameters given sample statistics. It tells us that, regardless of what the population distribution looks like, the distribution of the sample means (you'll learn what that is) can be normal. Good idea to understand a bit about normal distributions before diving into this tutorial.

• Confidence intervals

We all have confidence intervals ("I'm the king of the world!!!!") and non-confidence intervals ("No one loves me"). That is not what this tutorial is about. This tutorial takes what you already know about the central limit theorem, sampling distributions, and z-scores and uses these tools to dive into the world of inferential statistics. It may seem magical at first, but from our sample, we can now make inferences about the probability of our population mean actually being in an interval.

• Bernoulli distributions and margin of error

Ever wondered what pollsters are talking about when they said that there is a 3% "margin of error" in their results. Well, this tutorial will not only explain what it means, but give you the tools and understanding to be a pollster yourself!

• Hypothesis testing with one sample

This tutorial helps us answer one of the most important questions not only in statistics, but all of science: how confident are we that a result from a new drug or process is not due to random chance but due to an actual impact. If you are familiar with sampling distributions and confidence intervals, you're ready for this adventure!

• Hypothesis testing with two samples

You're already familiar with hypothesis testing with one sample. In this tutorial, we'll go further by testing whether the difference between the means of two samples seems to be unlikely purely due to chance.

• Chi-square probability distribution

You've gotten good at hypothesis testing when you can make assumptions about the underlying distributions. In this tutorial, we'll learn about a new distribution (the chi-square one) and how it can help you (yes, you) infer what an underlying distribution even is!

• Analysis of variance

You already know a good bit about hypothesis testing with one or two samples. Now we take things further by making inferences based on three or more samples. We'll use the very special F-distribution to do it (F stands for "fabulous").

• Differential equations
• First order differential equations
• Intro to differential equations

How is a differential equation different from a regular one? Well, the solution is a function (or a class of functions), not a number. How do you like me now (that is what the differential equation would say in response to your shock)!

• Separable equations

Arguably the 'easiest' class of differential equations. Here we use our powers of algebra to "separate" the y's from the x's on two different sides of the equation and then we just integrate!

• Exact equations and integrating factors

A very special class of often non-linear differential equations. If you know a bit about partial derivatives, this tutorial will help you know how to 'exactly' solve these!

• Homogeneous equations

In this equations, all of the fat is fully mixed in so it doesn't collect at the top. No (that would be homogenized equations). Actually, the term "homogeneous" is way overused in differential equations. In this tutorial, we'll look at equations of the form y'=(F(y/x)).

• Second order linear equations
• Linear homogeneous equations

To make your life interesting, we'll now use the word "homogeneous" in a way that is not connected to the way we used the term when talking about first-order equations. As you'll see, second order linear homogeneous equations can be solved with a little bit of algebra (and a lot of love).

• Complex and repeated roots of characteristic equation

Thinking about what happens when you have complex or repeated roots for your characteristic equation.

• Method of undetermined coefficients

Now we can apply some of our second order linear differential equations skills to nonhomogeneous equations. Yay!

• Laplace transform
• Laplace transform

We now use one of the coolest techniques in mathematics to transform differential equations into algebraic ones. You'll also learn about transforms in general!

• Properties of the Laplace transform

You know how to use the definition of the Laplace transform. In this tutorial, we'll explore some of the properties of the transform that will start to make it clear why they are so useful for differential equations. This tutorial is paired well with the tutorial on using the "Laplace transform to solve differential equations". In fact you might come back to this tutorial over and over as you solve more and more problems.

• Laplace transform to solve a differential equation

You know a good bit about taking Laplace transform and useful properties of the transform. You are dying to actually apply these skills to an actual differential equation. Wait no longer!

• The convolution integral

This tutorial won't be as convoluted as you might suspect. We'll see what multiplying transforms in the s-domain give us in the time domain.

• Linear algebra
• Vectors and spaces
• Vectors

We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication).

• Linear combinations and spans

Given a set of vectors, what other vectors can you create by adding and/or subtracting scalar multiples of those vectors. The set of vectors that you can create through these linear combinations of the original set is called the "span" of the set.

• Linear dependence and independence

If no vector in a set can be created from a linear combination of the other vectors in the set, then we say that the set in linearly independent. Linearly independent sets are great because there aren't any extra, unnecessary vectors lying around in the set. :)

• Subspaces and the basis for a subspace

In this tutorial, we'll define what a "subspace" is --essentially a subset of vectors that has some special properties. We'll then think of a set of vectors that can most efficiently be use to construct a subspace which we will call a "basis".

• Vector dot and cross products

In this tutorial, we define two ways to "multiply" vectors-- the dot product and the cross product. As we progress, we'll get an intuitive feel for their meaning, how they can used and how the two vector products relate to each other.

• Matrices for solving systems by elimination

This tutorial is a bit of an excursion back to you Algebra II days when you first solved systems of equations (and possibly used matrices to do so). In this tutorial, we did a bit deeper than you may have then, with emphasis on valid row operations and getting a matrix into reduced row echelon form.

• Null space and column space

We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.

• Matrix transformations
• Functions and linear transformations

People have been telling you forever that linear algebra and matrices are useful for modeling, simulations and computer graphics, but it has been a little non-obvious. This tutorial will start to draw the lines by re-introducing you functions (a bit more rigor than you may remember from high school) and linear functions/transformations in particular.

• Linear transformation examples

In this tutorial, we do several examples of actually constructing transformation matrices. Very useful if you've got some actual transforming to do (especially scaling, rotating and projecting) ;)

• Transformations and matrix multiplication

You probably remember how to multiply matrices from high school, but didn't know why or what it represented. This tutorial will address this. You'll see that multiplying two matrices can be view as the composition of linear transformations.

• Inverse functions and transformations

You can use a transformation/function to map from one set to another, but can you invert it? In other words, is there a function/transformation that given the output of the original mapping, can output the original input (this is much clearer with diagrams). This tutorial addresses this question in a linear algebra context. Since matrices can represent linear transformations, we're going to spend a lot of time thinking about matrices that represent the inverse transformation.

• Finding inverses and determinants

We've talked a lot about inverse transformations abstractly in the last tutorial. Now, we're ready to actually compute inverses. We start from "documenting" the row operations to get a matrix into reduced row echelon form and use this to come up with the formula for the inverse of a 2x2 matrix. After this we define a determinant for 2x2, 3x3 and nxn matrices.

• More determinant depth

In the last tutorial on matrix inverses, we first defined what a determinant is and gave several examples of computing them. In this tutorial we go deeper. We will explore what happens to the determinant under several circumstances and conceptualize it in several ways.

• Transpose of a matrix

We now explore what happens when you switch the rows and columns of a matrix!

• Alternate coordinate systems (bases)
• Orthogonal complements

We will know explore the set of vectors that is orthogonal to every vector in a second set (this is the second set's orthogonal complement).

• Orthogonal projections

This is one of those tutorials that bring many ideas we've been building together into something applicable. Orthogonal projections (which can sometimes be conceptualized as a "vector's shadow" on a subspace if the light source is above it) can be used in fields varying from computer graphics and statistics! If you're familiar with orthogonal complements, then you're ready for this tutorial!

• Change of basis

Finding a coordinate system boring. Even worse, does it make certain transformations difficult (especially transformations that you have to do over and over and over again)? Well, we have the tool for you: change your coordinate system to one that you like more. Sound strange? Watch this tutorial and it will be less so. Have fun!

• Orthonormal bases and the Gram-Schmidt Process

As we'll see in this tutorial, it is hard not to love a basis where all the vectors are orthogonal to each other and each have length 1 (hey, this sounds pretty much like some coordinate systems you've known for a long time!). We explore these orthonormal bases in some depth and also give you a great tool for creating them: the Gram-Schmidt Process (which would also be a great name for a band).

• Eigen-everything

Eigenvectors, eigenvalues, eigenspaces! We will not stop with the "eigens"! Seriously though, eigen-everythings have many applications including finding "good" bases for a transformation (yes, "good" is a technical term in this context).

• Applied math
• Recreational mathematics
• Vi Hart
• Spirals, Fibonacci and being a plant

You're feeling spirally today, and math class today is taking place in greenhouse #3...

• Doodling in Math

Let's say you're me and you're in math class…

• Hexaflexagons

Since it's shaped like a hexagon and flex rhymes with hex, hexaflexagon it is!

When you want to make a circle, how is it done? Well you probably will start with the radius one.

• Singing (and noises)

The title says it all...

• Mobius strips

Playing mathematically with strips!

• Thanksgiving math

Mathed potatoes, Borromean onion rings, green bean matheroles and Turduckenen-duckenen (yes, you read that right)

• Other cool stuff

Pythagoras, snakes, fractals, snowflakes...

• Brain teasers

Random logic puzzles and brain teasers. Fun to do and useful for many job interviews!

• Science & Economics
• Biology
• Chemistry
• Physics
• One-dimensional motion
• Displacement, velocity and time

This tutorial is the backbone of your understanding of kinematics (i.e., the motion of objects). You might already know that distance = rate x time. This tutorial essentially reviews that idea with a vector lens (we introduce you to vectors here as well). So strap your belts (actually this might not be necessary since we don't plan on decelerating in this tutorial) and prepare for a gentle ride of foundational physics knowledge.

• Acceleration

In a world full of unbalanced forces (which you learn more about when you study Newton's laws), you will have acceleration (which is the rate in change of velocity). Whether you're thinking about how fast a Porsche can get to 60mph or how long it takes for a passenger plane to get to the necessary speed for flight, this tutorial will help.

• Kinematic formulas and projectile motion

We don't believe in memorizing formulas and neither should you (unless you want to live your life as a shadow of your true potential). This tutorial builds on what we know about displacement, velocity and acceleration to solve problems in kinematics (including projectile motion problems). Along the way, we derive (and re-derive) some of the classic formulas that you might see in your physics book.

• Old videos on projectile motion

This tutorial has some of the old videos that Sal first did around 2007. This content is covered elsewhere, but some folks like the raw (and masculine) simplicity of these first lessons (Sal added the bit about "masculine").

• Two-dimensional motion
• Two-dimensional projectile motion

Let's escape from the binds of one-dimension (where we were forced to launch things straight up) and start launching at angles. With a little bit of trig (might want to review sin and cos) we'll be figuring out just how long and far something can travel.

• Optimal angle for a projectile

This tutorial tackles a fundamental question when trying to launch things as far as possible (key if you're looking to capture a fort with anything from water balloons to arrows). With a bit of calculus, we'll get to a fairly intuitive answer.

• Centripetal acceleration

Why do things move in circles? Seriously. Why does *anything* ever move in a circle (straight lines seem much more natural). ? Is something moving in a circle at a constant speed accelerating? If so, in what direction? This tutorial will help you get mind around this super-fun topic.

• Forces and Newton's Laws of Motion
• Newton's laws of motion

This tutorial will expose you to the foundation of classical mechanics--Newton's laws. On one level they are intuitive, on another lever they are completely counter-intuitive. Challenge your take on reality and watch this tutorial. The world will look very different after you're done.

• Normal force and contact force

A dog is balancing on one arm on my head. Is my head applying a force to the dog's hand? If it weren't, wouldn't there be nothing to offset the pull of gravity causing the acrobatic dog to fall? What would we call this force? Can we have a general term from the component of a contact force that acts perpendicular to the plane of contact? These are absolutely normal questions to ask.

• Balanced and unbalanced forces

You will often hear physics professors be careful to say "net force" or "unbalanced force" rather than just "force". Why? This tutorial explains why and might give you more intuition about Newton's laws in the process.

• Slow sock on Lubricon VI

This short tutorial will have you dealing with orbiting frozen socks in order to understand whether you understand Newton's Laws. We also quiz you a bit during the videos just to make sure that you aren't daydreaming about what you would do with a frozen sock.

• Inclined planes and friction

We've all slid down slides/snow-or-mud-covered-hills/railings at some point in our life (if not, you haven't really lived) and noticed that the smoother the surface the more we would accelerate (try to slide down a non-snow-or-mud-covered hill). This tutorial looks into this in some depth. We'll look at masses on inclined planes and think about static and kinetic friction.

• Tension

Bad commute? Baby crying? Bills to pay? Looking to take a bath with some Calgon (do a search on YouTube for context) to ease your tension? This tutorial has nothing (actually little, not nothing) to do with that. So far, most of the forces we've been dealing with are forces of "pushing"--contact forces at the macro level because of atoms not wanting to get to close at the micro level. Now we'll deal with "pulling" force or tension (at a micro level this is the force of attraction between bonded atoms).

• Work and energy
• Work and energy

You're doing a lot more work than you realize (most of which goes unpaid). This tutorial will have you seeing the world in terms of potentials and energy and work (which is more fun than you can possibly imagine).

If you have every used a tool of any kind (including the bones in your body), you have employed mechanical advantage. Whether you used an incline plane to drag something off of a pick-up truck or the back of a hammer to remove a nail, the world of mechanical advantage surrounds us.

• Springs and Hooke's Law

Weighing machines of all sorts employ springs that take a certain amount of force to keep compressed or stretched to a certain point. Hooke's law will give us all the tools to weigh in on the subject ourselves and spring into action (yes, the puns are annoying us too)!

• Impacts and linear momentum
• Momentum

Depending on your view of things, this may be the most violent of our tutorials. Things will crash and collide. We'll learn about momentum and how it is transferred. Whether you're playing pool (or "billiards") or deciding whether you want to get tackled by the 300lb. guy, this tutorial is of key importance.

• Moments, torque and angular momentum
• Torque, moments and angular momentum

Until this tutorial, we have been completely ignoring that things rotate. In this tutorial, we'll explore why they rotate and how they do it. It will leave your head spinning (no, it won't, but seemed like a fun thing to say given the subject matter).

• Gravitation
• Newton's law of gravitation

Why are you sticking to your chair (ignoring the spilled glue)? Why does the earth orbit the sun (or does it)? How high could I throw my dog on the moon? Gravitation defines our everyday life and the structure of the universe. This tutorial will introduce it to you in the Newtonian sense.

• Oscillatory motion
• Harmonic motion

Every watch a slinky gyrate back and forth. This is harmonic motion (a special class of oscillatory motion). In this tutorial we'll see how we can model and deal with this type of phenomena.

• Fluids
• Thermodynamics
• Electricity and magnetism
• Waves and optics
• Cosmology and astronomy
• Scale of the Universe
• Scale of the small and large

We humans have trouble comprehending something larger than, say, our planet (and even that isn't easy to conceptualize) and smaller than, say, a cell (once again, still not easy to think about). This tutorial explores the scales of the universe well beyond that of normal human comprehension, but does so in a way that makes them at least a little more understandable. How does a bacteria compare to an atom? What about a galaxy to a star? Turn on your inertial dampeners. You're in store for quite a ride!

• Light and fundamental forces

This tutorial gives an overview of light and the fundamental four forces. You won't have a degree in physics after this, but it'll give you some good context for understanding cosmology and the universe we are experiencing. It should be pretty understandable by someone with a very basic background in science.

• Scale of earth, sun, galaxy and universe

The Earth is huge, but it is tiny compared to the Sun (which is super huge). But the Sun is tiny compared to the solar system which is tiny compared to the distance to the next star. Oh, did we mention that there are over 100 billion stars in our galaxy (which is about 100,000 light years in diameter) which is one of hundreds of billions of galaxies in just the observable universe (which might be infinite for all we know). Don't feel small. We find it liberating. Your everyday human stresses are nothing compared to this enormity that we are a part of. Enjoy the fact that we get to be part of this vastness!

• Time scale of the cosmos

Not only is the universe unimaginable large (possibly infinite), but it is also unimaginably old. If you were feeling small in space, wait until you realize that all of human history is but a tiny blip in the history of the universe.

• Big bang and expansion of the universe

What does it mean for the universe to expand? Was the "big bang" an explosion of some sort or a rapid expansion of space-time (it was the latter)? If the universe was/is expanding, what is "outside" it? How do we know how far/old things are? This tutorial addresses some of the oldest questions known to man.

• Stars, black holes and galaxies
• Life and death of stars

Stars begin when material drifting in space condenses due to gravity to be dense enough for fusion to occur. Depending on the volume and make-up of this material, the star could then develop into very different things--from supernovae, to neutron stars, to black holes. This tutorial explores the life of stars and will have you appreciating the grand weirdness of our reality.

• Quasars and galactive collisions

Quasars are the brightest objects in the universe. The gamma rays from them could sterilize a solar system (i.e. obliterate life). What do we think these objects are? Why don't we see any close by (which we should be thankful for)? Could they tell us what our own galaxy may have been like 1 billion or so years ago?

• Stellar parallax

We've talked a lot about distances to stars, but how do we know? Stellar parallax--which looks at how much a star shifts in the sky when Earth is at various points in its orbit--is the oldest technique we have for measuring how far stars are. It is great for "nearby" stars even with precise instruments (i.e, in our part of our galaxy). To measure distance further, we have to start thinking about Cepheid variables (other tutorial).

• Cepheid variables

Stellar parallax can be used for "nearby" stars, but what if we want to measure further out? Well this tutorial will expose you to a class of stars that helps us do this. Cepheids are large, bright, variable stars that are visible in other galaxies. We know how bright they should be and can gauge how far they are by how bright they look to us.

• Earth geological and climatic history
• Plate tectonics

Is it a coincidence that Africa and South America could fit like puzzle pieces? Why do earthquakes happen where they do? What about volcanoes and mountains? Are all of these ideas linked? Yes, they are. This tutorial on plate-tectonics explains how and why the continents have shifted over time. In the process, we also explore the structure of the Earth, all the way down to the core.

• Seismic waves and how we know Earth's structure

How do we know what the Earth is made up of? Has someone dug to the core? No, but we humans have been able to see how earthquake (seismic) waves have been bent and reflected through our planet to get a reasonable idea of what is down there.

• Earth's rotation and tilt

What causes the seasons? Even more, can Earth's climate change over long period just to "wobbles" in its orbit? This tutorial explains it all. You'll know more about orbits (and precession and Milankovitch cycles) than you ever thought possible. Have fun!

• Life on Earth and in the Universe
• History of life on Earth

Earth is over 4.5 billion years old. How do we know this? When did life first emerge? From the dawn of Earth as a planet to the first primitive life forms to our "modern" species, this tutorial is an epic journey of the history of life on Earth.

• Humanity on Earth

Where do we think humans come from? How and why have we developed as a species. This tutorial attempts to give an overview of these truly fundamental questions. From human evolution (which is covered in more depth in the biology playlist) to the development of agriculture, this tutorial will give you an appreciation of where we've been (and maybe where we're going).

• Measuring age on Earth

Geologists and archaeologists will tell you how old things are or when they happened, but how do they know? This tutorial answers this question by covering some of the primary techniques of "dating" (not in the romantic sense).

• Life in the Universe

Are dolphins the only intelligent life in the universe? We don't know for sure, but this tutorial gives a framework for thinking about the problem.

• Organic chemistry
• General chemistry review
• Hybridization

It's the liger of orbitals and essential to organic chemistry! In this tutorial, Sal discusses the hybridization of orbitals.

• Dot structures

In this tutorial, Jay reminds you how to draw the dot structures of simple organic molecules.

• Hybrid orbitals

In this tutorial, Jay goes over sp3, sp2, and sp hybridization.

• Electronegativity

What is the most important concept to understand in undergraduate organic chemistry? Of course, it's electronegativity! In this tutorial, Jay explains the concept of electronegativity and shows how it applies to polarity, intermolecular forces, and physical properties.

• Organic structures and acid/base
• Bond-line structures

Call them Bonds. Covalent Bonds. Smart chemists need time to stir (and shake) their solutions. In this tutorial, Jay explains how chemists use bond-line structures as a form of organic shorthand to skip time-consuming carbon and hydrogen atoms labeling. Watch this tutorial so you too can be in the Dr. Know.

• Functional groups

In this tutorial, Jay puts the "fun" back into recognizing functional groups.

• Formal charge and resonance

Positive and negative charges are everywhere in orgo! In this tutorial, Jay shows you how to assign formal charges to molecules and how to draw resonance structures.

• Oxidation and reduction

LEO the Lion goes GERRRR!!!! If that is all that you remember about redox from general chemistry, then this tutorial is for you! Jay shows you how to assign oxidation states in organic molecules.

• Acid/base

Do you remember the basics of acid/base chemistry? In this tutorial, Jay reminds you of a few definitions, shows you how the stability of the conjugate base affects the acidity of the molecule, and demonstrates the importance of pKa values.

• Alkanes and cycloalkanes
• Naming alkanes

In this tutorial, Sal shows how to name alkanes.

• Conformations

In this tutorial, Sal draws Newman projections and also explains chair and boat conformations for cyclohexane.

• Naming alkanes, cycloalkanes, and bicyclic compounds

Do you speak the language of organic chemistry? In this tutorial, Jay shows you how to be fluent in naming alkanes, cycloalkanes, and bicyclic compounds.

• Conformations of alkanes and cycloalkanes

In this tutorial, Jay shows the different conformations of straight chain alkanes and cyclohexane.

In this tutorial, Sal introduces free radical reactions by showing the reaction of methane with chlorine.

• Stereochemistry
• Chirality and the R,S system

Are you right handed or sinister-handed? Have you ever thought that you might not be as attractive as you look in the mirror? Welcome to the world of chirality. In this tutorial, Sal explores molecules that have the same composition and bonding, but are fundamentally different because they are mirror images of each other (kind of like Tomax and Xamot--the Crimson Guard Commanders from GI Joe).

• Chirality and absolute configuration

Mirror, mirror on the wall . . . who is the fairest stereoisomer of all? In this tutorial, Jay explains chirality and how to determine the absolute configuration at a chirality center.

• Optical activity

In this tutorial, Jay explains the concept of optical activity and demonstrates how to calculate specific rotation and enantiomeric excess.

• Diastereomers and meso compounds

In this tutorial, Sal and Jay define stereoisomers, diastereomers, and meso compounds.

• Fischer projections

In this tutorial, Jay shows how to draw a fischer projection and how to assign an absolute configuration to a chirality center in a fischer projection.

• Substitution and elimination reactions
• SN1 vs SN2

In this tutorial, Sal analyzes the differences between SN1 and SN2 reactions.

• Nucleophilicity and basicity

In this tutorial, Sal discusses the difference between nucleophilicity and basicity.

• Elimination reactions

In this tutorial, Sal explains the difference between an E1 and an E2 elimination reaction.

• SN1/SN2/E1/E2

In this tutorial, Sal compares the differences between E2, E1, SN2, and SN1 reactions.

• SN1 and SN2

In this tutorial, Jay covers the definitions of nucleophile/electrophile, The Schwartz Rules (may the Schwartz be with you!), and the differences between SN1 and SN2 reactions.

• E1 and E2 reactions

In this tutorial, Jay covers the E1 elimination mechanism, carbocation rearrangements, and the details of the E2 elimination reaction.

• SN1/SN2/E1/E2

In this tutorial, Jay discusses the strength of a nucleophile and the differences between SN1, SN2, E1, and E2 reactions.

• Alkenes and alkynes
• Naming alkenes

In this tutorial, Sal names alkenes and discusses the E-Z system.

• Alkene reactions

In this tutorial, Sal introduces reaction mechanisms and demonstrates a few reactions of alkenes.

• Alkene nomenclature

In this tutorial, Jay names alkenes, discusses the stability of alkenes, and introduces the E/Z system.

• Alkene reactions

In this tutorial, Jay explains the addition reactions of alkenes.

• Naming and preparing alkynes

In this tutorial, Jay covers the nomenclature and preparation of alkynes, the acidity of terminal alkynes, and the alkylation of alkynes.

• Alkyne reactions

In this tutorial, Jay shows the reactions of alkynes.

• Synthesis using alkynes

In this tutorial, Jay demonstrates how to use Dr. Schwartz's organic flowsheet to solve synthesis problems involving alkynes. Always remember, pain is temporary, orgo is forever!

• Alcohols, ethers, epoxides, sulfides
• Alcohol nomenclature and properties

It can clean a wound or kill your liver. Some religions ban it, others use it in their sacred rites. Some of the most stupid acts humanity every committed were done under its influence. It is even responsible for some of our births. In this tutorial, Sal and Jay name alcohols and discuss their properties.

• Synthesis of alcohols

In this tutorial, Jay shows how to synthesize alcohols using sodium borohydride, lithium aluminum hydride, and grignard reagents.

• Reactions of alcohols

In this tutorial, Jay assigns oxidation states to alcohols, shows an oxidation mechanism using the Jones reagent, shows the formation of nitrate esters from alcohols, and demonstrates how to make alkyl halides from alcohols. Biochemical redox reactions are also discussed.

• Nomenclature and properties of ethers

In this tutorial, Sal and Jay name ethers and discuss the physical properties of ethers.

• Synthesis and cleavage of ethers

In this tutorial, Jay shows how to synthesize ethers using the Williamson ether synthesis and how to cleave an ether linkage using acid.

• Nomenclature and preparation of epoxides

In this tutorial, Sal and Jay name epoxides. Jay also shows the preparation of epoxides and includes the stereochemistry of the reaction.

• Ring-opening reactions of epoxides

In this tutorial, Sal and Jay show the SN1 and SN2 ring opening reactions of epoxides.

• Thiols and sulfides

In this tutorial, Jay shows how to prepare sulfides from thiols.

• Conjugation, Diels-Alder, and MO theory
• Addition reactions of conjugated dienes

In this tutorial, Jay shows the possible products for an addition to a conjugated diene and how the end product can be controlled by changing the reaction conditions.

• Diels-alder reaction

In this tutorial, Jay shows the mechanism, stereochemistry, and regiochemistry for the classic Diels-Alder reaction.

• Molecular orbital theory

In this tutorial, Jay introduces molecular orbital (MO) theory and shows how MO theory explains the experimental observations of the Diels-Alder reaction.

• Color in organic molecules

In this tutorial, Jay explains basic color theory and shows how conjugation determines the color of organic molecules. Basic knowledge of MO theory is assumed.

• Aromatic compounds
• Naming benzene derivatives

Would a cyclohexatriene by any other name smell as sweet? In this tutorial, Sal and Jay explain how to name benzene derivatives, the sometimes sweet-smelling cyclic molecules that can be used in the synthesis of explosives and plastics.

• Aromatic stability

In this tutorial, Sal and Jay explain the concept of aromatic stabilization and show how to determine if a compound or an ion exhibits aromaticity. Knowledge of MO theory is assumed.

• Reactions of benzene

In this tutorial, Sal shows the mechanism of Electrophilic Aromatic Substitution and the reactions of bromination and Friedel-Crafts Acylation.

• Electrophilic Aromatic Substitution

In this tutorial, Jay shows several electrophilic aromatic substitution reactions.

• Aldehydes and ketones
• Carboxylic acids and derivatives
• Amines
• Naming amines

In this tutorial, Sal shows how to name amines.

• Amines in reactions

In this tutorial, Sal shows amines acting as nucleophiles in SN1 and SN2 reactions.

• Finance and capital markets
• Interest and debt
• Compound interest basics

Interest is the basis of modern capital markets. Depending on whether you are lending or borrowing, it can be viewed as a return on an asset (lending) or the cost of capital (borrowing). This tutorial gives an introduction to this fundamental concept, including what it means to compound. It also gives a rule of thumb that might make it easy to do some rough interest calculations in your head.

• Interest basics

This is a good introduction to the basic concept of interest. We will warn you that it is an older video so Sal's sound and handwriting weren't quite up to snuff then.

• Credit cards and loans

Most of us have borrowed to buy something. Credit cards, in particular, can be quite convenient (but dangerous if not used in moderation). This tutorial explains credit card interest, how credit card companies make money and a far more silly way of borrowing money called "payday" loans.

• Continuous compound interest and e

This is an older tutorial (notice the low-res, bad handwriting) about one of the coolest numbers in reality and how it falls out of our innate desire to compound interest continuously.

• Present value

If you gladly pay for a hamburger on Tuesday for a hamburger today, is it equivalent to paying for it today? A reasonable argument can be made that most everything in finance really boils down to "present value". So pay attention to this tutorial.

• Personal bankruptcy

Back in the day (like medieval Europe), you would actually be thrown in jail if you couldn't pay your debts (debtor's prison). That seemed like a pretty awful thing to do (not to mention that lenders are much less likely to be paid by someone rotting in prison), so governments created an "out" called bankruptcy (which, as you'll see, is a pseudo-painful "reset" button on your finances).

• Housing
• Home equity and personal balance sheets

This old and badly drawn tutorial covers a topic essential to anyone planning to not live in the woods -- your personal balance sheet. Since homes are usually the biggest part of these personal balance sheets, we cover that too.

• Mortgages

Most people buying a home need a mortgage to do so. This tutorial explains what a mortgage is and then actually does some math to figure out what your payments are (the last video is quite mathy so consider it optional).

• Renting vs. buying a home

Is it always better to buy than rent? What if home prices go up dramatically and rents don't? How can we compare home prices to rents to figure out what to do. This older tutorial (low-res, bad handwriting) walks us through this. It is about housing but similar thinking can be applied to any rent-vs-buy decision (spoiler alert, Sal did eventually buy a home).

• Housing price conundrum

Back before the 2008 credit crisis, Sal was perplexed by why housing prices were going up so fast and theorized that it was a bubble forming (he was right). These pre-2008 videos are fun from a historical point-of-view since they were made before all the poo poo hit the fan.

• Credit Crisis

This tutorial talks about how the housing-bubble-induced credit crisis unfolded with a focus on the derivative securities that helped pump the bubble.

• Paulson Bailout

In the fall of 2008, it became clear that a cascade of bank failures was happening because of shoddy loans and exotic securities (both which fueled a now popping housing bubble). In an attempt to avoid a depression, the Treasury Secretary (Hank Paulson) wanted to pour \$1 Trillion into the same banks that had created the mess. This tutorial walks us through the beginnings of the mess and possible solutions. Historical note: it was created as the crisis was unfolding.

• Investment and consumption

When are you using capital to create more things (investment) vs. for consumption (we all need to consume a bit to be happy). When you do invest, how do you compare risk to return? Can capital include human abilities? This tutorial hodge-podge covers it all.

• Inflation
• Inflation basics

\$1 went a lot further in 1900 than today (you could probably buy a good meal for the family for \$1 back then). Why? And how to we measure how-more-expensive-things-have-gotten (i.e., inflation)?

• Inflation scenarios

You know about inflation, but now want to look at how thing might play out in different scenarios. This tutorial focuses on when inflation is "acceptable" and when it isn't (and the causes and repercussions).

• Real and nominal return

If the value of money is constantly changing, can we compare investment return in the future or past to that earned in the present? This tutorial focuses on how to do this (another good tutorial to watch is the one on "present value").

• Capacity utilization and inflation

This tutorial starts with a very "micro" view of when firms decide to raise (or lower prices). It then jumps back to the macro view to discuss how capacity utilization can impact prices.

• Deflation

Prices don't always go up. They often go down. This might seem like a good thing, but it could be disastrous for a modern economy is it goes too far. This tutorial explains what deflation is, how it happens and what the effects of it might be.

• Taxes
• Personal taxes

Benjamin Franklin (and several other writers/philosophers) tells us that "In this world nothing can be said to be certain, except death and taxes." He's right. This tutorial focus on personal income tax. Very important to watch if you ever plan on earning money (some of which the government will take for itself).

• Corporate taxation

In exchange for being treated as a person-like-legal entity (and the limited liability this gives for its owners), most corporations pay taxes. This tutorial focuses on what corporations are, "double taxation" and a few ways that multinationals might try to get out of paying taxes.

• Accounting and financial statements
• Cash versus accrual accounting

Just keeping track of cash that goes in and out of a business doesn't always reflect what's going on. This tutorial compares cash and accrual accounting. Very valuable if you ever plan on starting or investing in any type of business (you might also discover a nascent passion for accounting)!

• Three core financial statements

Corporations use three financial statements to report what's going on: balance sheets, cash flow statements and income statements. They can be derived from each other and each give a valuable lens on the operations and condition of a business. After you know the basics of accrual accounting (available in another tutorial), this tutorial will give you tools you need to responsibly understand any business.

• Depreciation and amortization

How do you account for things that get "used up" or a cost that should be spread over time. This tutorial has your answer. Depreciation and amortization might sound fancy, but you'll hopefully find them to be quite understandable.

• Stocks and bonds
• Introduction to stocks

Many people own stocks, but, unfortunately, most of them don't really understand what they own. This tutorial will keep you from being one of those people (not keep you from owning stock, but keep you from being ignorant about your investments).

• Shorting stock

Can you sell something that you borrowed from someone else? Why, yes, you can and it is called "shorting". Why would you do this? Well, you can now make money if the price goes down. Is this bad? This tutorial has your answers.

• Understanding company statements and capital structure

If you understand what a stock is (also a good idea to look at the topic on accounting and financial statements), then you're ready to dig in a bit on a company's actual financials. This tutorial does this to help you understand what the price of a company really is.

• Corporate metrics and valuation

Life is full of people who will try to convince you that something is a good or bad idea by spouting technical jargon. Most of them have no idea what they are talking about. Don't be one of those people or their victims when it comes to stocks. From P/E rations to EV/EBITDA, we've got your back!

• Life of a company--from birth to death

This is an old set of videos, but if you put up with Sal's messy handwriting (it has since improved) and spotty sound, there is a lot to be learned here. In particular, this tutorial walks through starting, financing and taking public a company (and even talks about what happens if it has trouble paying its debts).

• Dilution

When companies issue new shares, many people consider this a share "dilution"--implying that the value of each share has been "watered down" a bit. This tutorial walks through the mechanics and why--assuming management isn't doing something stupid--the shares might not be diluted at all.

• Mergers and acquisitions

Companies often buy or merge with other companies using shares (which is sometimes less intuitive than when they use cash). This tutorial walks through the mechanics of how this happens and details what is likely to happen in the public markets because of the transaction (including opportunities for arbitrage).

Private equity firms often borrow money (use leverage) to buy companies. This tutorial explains how they do it and pay the debt.

• Bonds

Both corporations and governments can borrow money by selling bonds. This tutorial explains how this works and how bond prices relate to interest rates. In general, understanding this not only helps you with your own investing, but gives you a lens on the entire global economy.

• Corporate bankruptcy

Anybody or anything (you can decide if a corporation is a person) can have trouble paying its debts. This tutorial walks through what happens to a corporation in these circumstances.

• Investment vehicles, insurance and retirement
• Mutual funds and ETFs

If we're not in the mood to research and pick our own stocks, mutual funds and/or ETFs might be a good option. This tutorial explains what they are and how they are different.

• Retirement accounts: IRAs and 401ks

The government apparently wants us to save for retirement (not always obvious because it also wants us to spend as much as possible to pump the economy going into the next election cycle). To encourage this, it has created some ways to save that avoid or defer taxes: IRAs and 401ks.

• Life insurance

It is a bit of a downer to think about, but we are all going to die. Do we care what happens to our loved ones (if they really are "loved" than the answer is obvious). This tutorial walks us through the options to insure our families against losing us. The reason why we stuck it in the "investment vehicles" topic is because it can also be an investment that we can use before we die.

• Hedge funds

Hedge funds have absolutely nothing to do with shrubbery. Their name comes from the fact that early hedge funds (and some current ones) tried to "hedge" their exposure to the market (so they could, in theory, do well in an "up" or "down" market as long as they were good at picking the good companies). Today, hedge funds represent a huge class investment funds. They are far less regulated than, say, mutual funds. In exchange for this, they aren't allowed to market or take investments from "unsophisticated" investors. Some use their flexibility to mitigate risk, other use it to amplify it.

• Money, banking and central banks
• Banking and Money

We all use money and most of us use banks. Despite this, the actual working of the banking system is a bit of a mystery to most (especially fractional reserve banking). This older tutorial (bad handwriting and resolution) starts from a basic society looking to do more than barter and incrementally builds to a modern society with fraction reserve banking. Through this process, you will hopefully gain a deep understanding of how money and banking works in our modern world.

• Quantative easing

You know that the Federal Reserve (or central banks in general) controls the money supply and short-term interest rates. But how exactly do they do this. Even more, how is "quantitative easing" different than regular open market operations. This tutorial explains it all in the context of the Federal Reserves attempts to stave off deflation during the 2008-2012 recession.

• 2008 Bank bailout

In 2008, the entire financial system was at a potential breaking point because of a popping housing bubble. This tutorial breaks down how the government attempted to address this (historical note: Sal made these videos as the crisis was unfolding).

• Geithner Plan

The poop really started to hit the fan in the fall of 2008. When the new administration took office in early 2009, the poop was still there. This is tutorial explains an attempt--probably not a well thought out one--to clean the poop and slow the fan. Videos on the Geithner Plan to solve the continuing banking crisis in early 2009.

This tutorial walks through how China's undervaluing of its currency impacts trade and prices (which also fuels cheap borrowing for the U.S.).

• Chinese currency and U.S. debt

This tutorial contains short videos that explain how China and the United States are intertwined through currency and debt. This is key for understanding the current global macro picture.

• 2011-2012 Greek Debt Crisis

The Greek government incurred debt beyond its means but didn't have control over its own currency to inflate away its obligations. From austerity, to a bailout, to leaving the Eurozone, none of the options looked great. In this tutorial, Sal walks through the situation Greece was in and its options (these videos were made as the crisis was unfolding).

• Bitcoin

Learn about bitcoins and how they work. Videos by Zulfikar Ramzan. Zulfikar is a world-leading expert in computer security and cryptography and is currently the Chief Scientist at Sourcefire. He received his Ph.D. in computer science from MIT.

• Options, swaps, futures, MBSs, CDOs and other derivatives
• Put and call options

Options allow investors and speculators to hedge downside (or upside). It allows them to trade on a belief that prices will change a lot--just not clear about direction. It allows them to benefit in any market (with leverage) if they speculate correctly. This tutorial walks through option basics and even goes into some fairly sophisticated option mechanics.

• Forward and futures contracts

In many commodities markets, it is very helpful for buyers or sellers to lock-in future prices. This is what both forwards and futures allow for. This tutorial explains how they work and what the difference is between the two.

• Mortgage-backed securities

What started out as a creative way to spread risk ended up fueling a monster housing bubble. This tutorial explains what mortgage-backed securities are and how they work.

• Collateralized debt obligations
• Credit default swaps
• Interest rate swaps
• Current Economics
• Unemployment

Unemployment is a key metric for judging the health of an economy (and even political stability). This tutorial is a primer on what it is and how it's measured (which you might find surprising).

• Microeconomics
• Supply, demand and market equilibrium
• Introduction to economics

This tutorial (that only has one video) is an overview of what economics is. In particular it will tell you the difference between microeconomics (the subject you're in right now) and macroeconomics. Really good first watch to give you some context on the world of economics.

• The demand curve

You've probably heard of supply and demand. Well, this tutorial focuses on the demand part. All else equal, do people want more or less of something if the price goes down (what would you do)? Not only will you get an intuition for the way we typically depict a demand curve, you'll get an understanding for what might shift it.

• The supply curve

Now we'll focus on the "supply" part of supply and demand. Supply curves (as we typically depict them) come out of the idea that producers will make more if they get paid more.

• Market equilibrium

You understand demand and supply. This tutorial puts it all together by thinking about where the two curves intersect. This point represents the equilibrium price and quantity which is, in an ideal world, where the market would transact.

• What drives oil prices

This tutorial tries to address a very important question in the real world--what drives oil prices? And we will do it using the tools of the supply and demand curves.

• Elasticity
• Price elasticity

You're familiar with supply and demand curves already. In this tutorial we'll explore what implications their steepness (or lack of) implies. Price elasticity is a measure of how sensitive something is to price.

• Consumer and producer surplus
• Consumer and producer surplus

Many times, the equilibrium price is lower than the highest price some folks are willing to pay. For all consumers, this is called consumer surplus. Similarly, the price might be higher than the minimum price at which some are willing to produce. For all the producers, this is called producer surplus. This tutorial covers them both with an emphasis on the visual.

We can often lose economic efficiency because of things like price floors, ceilings and taxes. This loss in surplus (people who have more marginal benefit than marginal cost are not buying or people who have more marginal cost than benefit are buying) is called deadweight loss.

• Public goods and externalities

In many scenarios thinking only about producers' marginal cost or consumers' marginal benefit does not fully capture *all* of the costs or benefits from the production/use of a product. In this tutorial, we explore these externalities (negative and positive ones) to think a bit deeper about ways to maximize total surplus not just for producers and consumers, but for society as a whole.

• Scarcity, possibilities, preferences and opportunity cost
• Production possibilities frontier

This tutorial goes back to the basics. You are a hunter-gatherer with only so much time to hunt or gather. How do you allocate your time and energy to maximize you happiness? This is what we try to understand through our study of the production possibilities frontier and opportunity cost.

Should you try to produce everything yourself or only what you are best at and trade for everything else? What if you're better than your trading partners at everything? This tutorial focuses on comparative advantage, specialization and gains from trade with a microeconomic lens.

• Marginal utility and budget lines

In this tutorial we look at the utility of getting one more of something and put numbers to it. We then use this to construct a budget line and think about indifference curves.

• Production decisions and economic profit
• Economic profit and opportunity cost

Economic profit and accounting profit are two different things (the difference being that economic profit takes into account opportunity cost). Confused? This tutorial lays it all out with the example of a restaurant.

• Average costs (ATC, MC) and marginal revenue (MR)

In this tutorial, Sal uses the example of an orange juice business to help us understand the ideas of average total cost (ATC), marginal cost (MC) and marginal revenue (MR). We then use this understanding to answer the age-old question, "how much orange juice should I produce?" Finally, we use these ideas to construct a long-run supply curve. A must watch if you're interested in making juice!

• Average fixed, variable and marginal costs

Using a spreadsheet, Sal walks through an example of average costs per line of code as a firm hires more engineers. Really good primer to understand what average fixed costs, average variable costs, average total costs (ATC) and average marginal costs (MC) are (and how they are calculated).

• Labor and marginal product revenue

Constructing a demand curve for an individual firm by thinking about how much increment benefit they get from an incremental employee (marginal product of labor (MPL) and marginal product revenue (MPR). We later think about how we can add these "demand" curves to construct a "demand" curve for the market for labor in this industry.

• Price discrimination

This short tutorial explores how a wine business can utilize first-degree price discrimination to maximize economic profit (it uses many of the ideas we've explored in the rest of this tutorial).

• Forms of competition
• Perfect competition

This tutorial looks at markets that are deemed to have "perfect competition." This means that there are many players with identical products, no barriers to entry, no advantage for existing players and good pricing information. Few to no real market completely matches this theoretical ideal, but many are close. Even the example we use in this tutorial (the airline industry) isn't quite perfect (you should think about why).

• Monopoly

No, we aren't talking about the board game although the game does try to approximate what this tutorial is about--notice that you can charge more rent at either Boardwalk or Park Place if you own both (you have a "monopoly" in the navy blue market). The opposite of perfect competition is when you have only one firm operating. This tutorial explores what this firm would do to maximize economic profit.

• Between perfect competition and monopoly

Most markets sit somewhere in-between perfect competition and monopolies. This tutorial explores some of those scenarios--from monopolistic competition to oligopolies and duopolies.

• Game theory and Nash equilibrium
• Nash equilibrium

If you haven't watched the movie "A Beautiful Mind", you should. It is about John Nash (played by Russell Crowe) who won the Nobel Prize in economics for his foundational contributions to game theory. This is what this tutorial is about. Nash put some structure around how players in a "game" can optimize their outcomes (if the movie is to be fully believed, this insight struck him when he realize that if all his friends hit on the most pretty girl, he should hit on the second-most pretty one). In this tutorial, we use the classic "prisoner's dilemma" to highlight this concept.

• Why parties in a cartel will cheat

You know what Nash equilibrium is (from the other tutorial). Now we apply it to a scenario that is fairly realistic--parties to a cartel cheating. A cartel is a group of actors that agree (sometimes illegally) to coordinate their production/pricing to maximize their collective economic profit. What we will see, however, is that this is not a "Pareto optimal" state and they will soon start producing more than agreed on.

• Macroeconomics
• GDP - measuring national income
• Introduction to economics

This very short tutorial gives us the big picture of what economics is all about and, in particular, compares macroeconomics (where you are now) to microeconomics.

• GDP and the circular flow of income and expenditures

Economics can some times get confusing because one person's expenditure is another person's income which can then be used for expenditure and on and on and on. Seems very circular. It is. This tutorial helps us grapple with this and introduces us to the primary tool economists use to measure a nations productivity/income/expenditure--GDP (gross domestic product).

• Components of GDP

You already understand the circular nature of the economy and how GDP is defined from the last tutorial. Now let's think about how economists define the composition of GDP. In particular, we'll focus on consumption (C), investment (I), government spending (G) and net exports.

• Real and nominal GDP

The value of a currency is constantly changing (usually going down in terms of what you can buy). Given this, how can we compare GDP measured in dollars in one year to another year? This tutorial answers that question by introducing you to real GDP and GDP deflators.

• Inflation - measuring the cost of living
• Measuring cost of living --inflation and the consumer price index

We might generally sense that our cost of living is going up (inflation), but how can we measure it? This tutorial shows how it is done in the United States with the consumer price index (CPI).

• Real and nominal return

We think we're getting a certain return on our investments, but can we put it in terms of real purchasing power since the value of money is constantly changing? The answer is yes and this tutorial shows you how.

• Deflation

Prices don't always go up. Sometimes they go down (we call this deflation). This tutorial explains how this happens.

• Inflationary and deflationary scenarios

This tutorial walks through various scenarios of moderate and extreme price changes. Very good way to understand how activity in the economy may impact price (and vice versa).

• The Phillips Curve - Inflation and unemployment

Economists have notices a correlation between unemployment and correlation (you may wan to guess what type of correlation). On some level, this tutorial is common sense, but it will give you fancy labels for this relation so that you can sound fancy at fancy parties.

• Aggregate demand and aggregate supply
• Aggregate demand and aggregate supply

This tutorial looks at supply and demand in aggregate-from the perspective of the entire economy (not just the market for one good or service). Instead of thinking of quantity of one good, we think of total output (GDP). Very useful model for thinking through macroeconomic events.

• Historical circumstances explained by AD/AS

In the last tutorial, we claimed that the aggregate demand and aggregate supply model (AD-AS) would be useful for analyzing macroeconomic events. Well, in this tutorial, we'll do exactly that.

Economies never have a long steady march upwards. They constantly oscillate between growth and recession. This tutorial gives a little intuition for why that is.

• Monetary and fiscal policy

Governments (and pseudo government entities like central banks) have two tools at their disposal to try to impact the business cycle --monetary and fiscal policy. This will help you understand what they are.

• Keynesian thinking

Whether you love him or hate him (or just consider him a friend that you respect but disagree with every-now-and-then), Keynes has helped define how many modern governments think about their economies. This tutorial explains how his thinking was a fundamental departure from classical economics.

• The monetary system
• Fractional reserve banking

Most modern economies use a counter-intuitive model of banking called "fractional reserve banking." It is counter-intuitive (and some people would say wrong) because it allows banks to lend out money that it tells depositors is available at any time and essentially involves private banks in money creation. It also creates the possibility of mass instability through bank runs that tend to be mitigated through government regulation and insurance (some would say government subsidy of banks). This tutorial explains how fractional reserve lending works and outlines the good and bad. It also talks about the alternative of full reserve banking.

• Money supply

This short tutorial explains how we measure how much "money" there is out there. As we'll see, this isn't as straightforward as counting dollars in people's pockets, especially because there are multiple type of money.

• Fractional reserve accounting

If you already know a bit of what fractional reserve banking involves, this tutorial will take you deeper by looking at the actual accounting of central banks and banks.

• Interest as the price of money
• Income and expenditure - Keynesian cross and IS-LM model
• Marginal propensity to consume (MPC)

If you earn a \$1, you might spend some fraction of it. This can then be income for someone else. This can keep going. In this tutorial, we'll explore how the incremental spend per incremental earnings (marginal propensity to consume) and the multiplier effect based on it can drive economic activity.

• Consumption function

We are steadily building up the tools to understand the Keynesian Cross and the IS-LM model. In this tutorial, we begin to model consumption as a linear function of disposable income. Seems reasonable to me.

• Keynesian Cross

We now build on our consumption function models and start to explore ideas of planned expenditures as a function of output. When plotted with the actual output line, we get our Keynesian Cross which helps us think about whether the economy is operating at its potential.

• IS-LM Model

In this tutorial, we begin thinking about the impact of real interest rates on planned investment and output. We then use this to help us plot the IS curve. We then think about how, assuming a fixed money supply, as there is more economic activity, people are willing to pay more for money (helps us plot the LM curve). Finally, we use the IS-LM model to think about how fiscal policy can impact both GDP and real interest rates. You should watch the Keynesian Cross tutorial before this one.

• Currency reserves

This tutorial delves into how and why countries (usually their central banks) would want to keep other countries' currency in reserve. It then goes into why this sometime leaves the reserve-holding country open to a speculative attack (this is seriously high drama).

• Computer science

Introduction to programming and computer science

• Healthcare and medicine
• MCAT Video Competition
• MCAT Video Competition

• The Heart
• Heart introduction

No organ quite symbolizes love like the heart. One reason may be that your heart helps you live, by moving ~5 liters (1.3 gallons) of blood through almost 100,000 kilometers (62,000 miles) of blood vessels every single minute! It has to do this all day, everyday, without ever taking a vacation! Now that is true love. Learn about how the heart works, how blood flows through the heart, where the blood goes after it leaves the heart, and what your heart is doing when it makes the sound “Lub Dub”.

• Blood vessels

Where does your blood go after it leaves the heart? Your body has a fantastic pipeline system that moves your blood around to drop off oxygen and food to those hungry cells, and removes cell waste. Learn how arteries carry blood away from the heart, how veins bring blood back to the heart, and about the different layers of cells that make up these blood vessels.

• Blood pressure

Using the stethoscope to check blood pressure is a technique that’s been used for >100 years! Blood pressure is one of the major vital signs frequently measured by health care workers, and it tells us a lot about our blood circulation. Learn what blood pressure is, how it relates to resistance in a tube, why it is necessary to get oxygen to your cells, and how it can change as you age. We’ll finally put it all together by relating pressure, flow, and resistance in one awesome equation!

• Blood pressure control

The human body enjoys stability. For example, if your blood pressure changes, the body puts a couple of brilliant systems into motion in order to respond and bring your blood pressure back to normal. There are some quick responses using nerves and some slower responses using hormones. The system using hormones is sometimes called the renin-angiotensin-aldosterone-system (RAAS), which is the main system in the body for controlling blood pressure. When your blood pressure drops too low or gets too high, your kidneys, liver, and pituitary gland (part of your brain) talk to each other to solve the problem. They do this without you even noticing! Learn how the body knows when the blood pressure has changed, and how hormones like angiotensin 2, aldosterone, and ADH help return blood pressure to back to normal.

• The Lungs
• Influenza
• Influenza Symptoms

Getting the flu is awful! You get respiratory symptoms (Example: stuffy nose, sore throat, or cough) and constitutional symptoms (Example: fevers, chills, or body aches), and you’re usually in bed for 3-7 days. The flu virus spreads from person to person through tiny little droplets and is really common during the winter. Learn how we’re getting smart about tracking the flu, and how you can avoid getting sick…

• Influenza Pathology

The flu is caused and spread by a virus called influenza, which has proteins on its outer coat (think of a person wearing a jacket) called Hemagglutinin and Neuraminidase. The flu uses these proteins to enter and exit cells, and we actually name these proteins H and N (easy to remember) and number them to keep track of all the different types that we have found. The flu is a sneaky little bugger though, and can avoid our immune system by making subtle genetic mutations over time (drift) or shuffling up its genetic material completely (shift)!

• Influenza Diagnosis

Most of the time, we don’t test for the flu, but it can be useful. Rapid flu tests are done with a quick nose swab or wash, and can detect Type A or Type B Flu, but beware – like all tests, sometimes they make mistakes.

• Influenza Prevention and Treatment

Want to avoid getting sick with the flu? If so, get a flu vaccine, it’s 60-70% effective! You have a couple of options: TIV (dead virus, injection) or LAIV (weak virus, spray). They can cause some side effects like a sore arm (TIV) or a runny nose (LAIV), but isn’t that better then lying in bed with a cough and fever for a week? Don’t worry though, if you do get really sick with the flu, we have some medications that can help.

• Influenza Special Topics

The fact is that many folks don’t get the flu shot. Find out some of the common reasons why this is the case, and some of the common myths that continue to circulate on the internet!

• Blood

Find out what's inside of blood

• Blood Vessel Diseases

Learn about how arteries differ from veins and how arteries can get damaged over time.

• Fetal Circulation

Find out how the heart and circulatory system work in the fetus!

• Arterial Stiffness

Find out why having "stiff arteries" can increase blood pressure and cause uneven blood flow. It all comes back to the idea of "total energy"!

• Healthy Lifestyle

Learn some of the fundamentals behind staying healthy: Reducing your salt, keeping your weight in a healthy range, and exercising regularly.

• Endocrinology and Diabetes

Learn about hormones and diabetes, as well as some important information on glucose control.

• Health Care System

Learn some basics about the Health Care system in the US

• Colon Disease

Get a ring-side seat as a pathologist talks about colon cells under the microscope.

• Cervical Spine

Learn about the cervical (neck) spine through an Xray!

• Hypertension

Learn all about hypertension: symptoms, diagnosis, and how lifestyle changes can make a difference!

• Lab Values and Concentrations

Ever wonder about your lab values and what they mean? Lab values measure amounts of electrolytes or cells in your blood and occasionally tell you about how hormones and enzymes are working! Dive deeper and get a good understanding of concentrations as well!

• Heart Disease and Stroke

Find out what can cause a heart disease and stroke, and how the two are related!

• Heart Muscle Contraction

Find out more about heart muscles and what happens when they contract.

• Heart Depolarization

Check out the flow of positive charge through the heart that happens every time your heart beats! Learn how the heart is able to keep a regular rate and rhythm throughout your life.

• Nerve Regulation of the Heart

Learn how the sympathetic and parasympathetic nerves regulate your heart in so many different ways!

• Pressure Volume Loops

Watch as the pressure and amount of blood inside the left ventricle go up and down within fractions of a second!

• Changing the PV Loop

Find out how preload, afterload, and contractility work together to change your PV Loop (and affect stroke volume and cardiac output)!

Find out exactly what preload and afterload mean, and why most of us use a handy shortcut to guess-timating what they are!

• Miscellaneous

Enjoy!

• Projects
• Humanities
• World History
• 1900 - Present: The Recent Past
• Beginning of World War I

Called the Great War (before World War II came about), World War I was the bloody wake-up call that humanity was entering into a new stage of civilization. Really the defining conflict that took Europe from 19th Century Imperial states that saw heroism in war into a modern shape. Unforunately, it had to go through World War II as well (that some would argue was due to imbalances created by World War I).

Naval blockades in World War I to starve enemy nation of trade. Contrary to what many think, American entry into WWI was not due purely to the sinking of the Lusitania. Learn more about what caused the United States to play its first major direct role in a European conflict.

• Western and Eastern fronts of World War I

This tutorial goes into some detail to describe the tactics and battles of the two major fronts of World War I--the Western Front and the Eastern Front.

• Other fronts of World War I

Contrary to what some history books and movies would have you believe, World War I was not just fought on the Western and/or Eastern fronts. Because of the empires involved, it was a truly global conflict. This tutorial will cover some of the campaigns that your history book might not (but are important to understanding the War).

• World War I shapes the Middle East

The Middle East is a center of cultures, religions, and, unfortunately, conflict in our modern world. This tutorial takes us from a declining Ottoman Empire to the modern Middle East which is still the center of many religions, cultures and conflicts.

• Ottoman Empire and Turkey

The end of World War I would mean the end of the centuries-old Ottoman Empire (which had been in decline since the 18th Century). Along the way, we'll also look at some of the major military campaigns involving the Ottomans during World War I. We finish in the aftermath of the war with the independence of Turkey.

• Aftermath of World War I

World war I (or the Great War) was a defining event for the 20th Century. It marked the end (or beginning of the end) of centuries-old empires and the dawn-of newly independent states based on ethnic and linguistic commonality. It didn't just change the face of Europe, it changed the face of the world. From the Paris Peace Conference and Treaty of Versailles, we'll see how the end of World War I may have been just the set up for even more conflict in Europe and the world.

• World War I Quiz
• Rise of Hitler and the Nazis

How did the National Socialists (Nazis) go from being a tiny, marginal party in the early 1920s to having full control of Germany and catalyzing World War II? Who was Hitler and what was his philosophy and how did he come to power?

• Rise of Mussolini and Fascism

The word "Fascist" is now a pejorative term ("pejorative" means "negative" or "derogatory") to describe leaders or states that have absolute control and are aggressively nationalistic. The terms "fascism" and "fascist", however, were first embraced by Benito Mussolini in Italy in the 1920s and 1930s to describe their party and policies (that were absolutist and aggressively nationalistic). This tutorial described Mussolini and the Fascists' rapid rise to power and the influence it had on the rest of the world (including providing a model for Hitler in Germany).

• The Cold War

The cold war between the United States and the Soviet and their respective allies never involved direct conflict (which might have ended the world). Instead, it involved posturing, brinksmanship and proxy wars in far-flung regions of the world.

• 1700-1900: Enlightenment and the Industrial Revolution
• French Revolution

"Let them eat cake!" "No, how about we cut your head off instead!" The French Revolution was ugly, bloody and idealistic. This tutorial covers the beginning of the end of the Bourbon rule (actually doesn't really go away for 60 years) and birth of France as a Republic (which will really take about 80 years).

• Napoleon Bonaparte

A man with such a huge "Napoleonic complex", that they named it after him. A military genius with a ginormous ego, some people consider him a hero or a tyrant or both. France has successfully overthrown Louis XVI in 1789. It has gone through a many-year period of bloodshed and instability. The monarch's of Europe are not happy about this "overthrow-your-king" business. A 5'6'' Corsican establishes himself as a strong military tactician during the wars with other European powers and soon comes to power in France. This tutorial covers the rise and fall of one of the most famous men in all of history: Napoleon Bonaparte (Napoleon I).

• France's many revolutions and republics

Unlike the American Revolution which fairly cleanly transitioned the United States from British rule to a republic, France's process of democratization was much longer and more painful. This tutorial gives a scaffold of that (and gives some context for the book/musical/movie "Les Miserables").

• Haitian Revolution

Yes, you are right. Haiti is not in Europe. We put the tutorial here because it was a French colony and its own revolution is closely linked to that of France's. Possibly one of the saddest histories that a nation can have, this tutorial tries to give as much context as possible for the birth of Haiti.

• 1500-1600: Renaissance & Reformation
• The Protestant Reformation

In 1517 a German theologian and monk, Martin Luther, challenged the authority of the Pope and sparked the Protestant Reformation. His ideas spread quickly, thanks in part to the printing press. By challenging the power of the Church, and asserting the authority of individual conscience (it was increasingly possible for people to read the bible in the language that they spoke), the Reformation laid the foundation for the value that modern culture places on the individual.

• Before 1300: Ancient and Medieval History
• Ancient

This tutorial includes the Ancient Near East, and Ancient Greece and Rome.

• Medieval
• Surveys of History
• Art History
• Introduction to Art History
• The Basics

If you’ve ever found yourself saying “How can that be art?,” then this tutorial is for you! We’ve tried to answer the real questions that come up during a typical visit to a modern art museum. Learn how describing can help you slow down and better understand what a work of art is communicating. Learn how important historical context is and how meaning can change over time. This tutorial asks: Why is art important? How can a snow shovel be art? How can we find meaning in abstract art?

• Big Questions in Modern & Contemporary Art

Why is that art? Can art be an idea? Does art have to represent the world we see?

• Media

Watching a soccer game is a lot more interesting if you’ve actually played the game yourself. Similarly, art makes a lot more sense if you understand how it was made. Different materials can have a profound effect on how a work of art looks and how we respond to it. Learn about how artist’s made their own supplies before there were art supply stores! Understand the differences between oil and tempera and how marble is quarried and carved.

• - 400 C.E. Ancient Cultures
• Prehistoric

You’ve seen the drawing of human evolution showing a procession of monkey, ape, primitive and modern man? Well, somewhere along that line early man could be shown holding a paint brush and a chisel. Mankind has been making art for at least 100,000 years. Why was the earliest art made? What might it have once meant? This tutorial focuses on what we do and do not know about one of the earliest representations of the human body, The Venus of Willendorf.

• Ancient Near East

Was writing invented to record poetry or great literature? Nope. Writing was invented to help keep track of beer and other goods and services! The ancient Sumerians were nothing if not practical. Ancient cultures established the first cities and large scale architecture. From the Gates of the City of Ancient Babylon, to the ancient code of laws instituted by King Hammurabi, the Ancient Near East is strangely distant but also remarkably familiar.

• Ancient Egypt

Woody Allen famously said, “I don't want to achieve immortality through my work. I want to achieve it through not dying.” The ancient Egyptians on the other hand, confronted death head-on. In fact, the art of the ancient Egyptians was (for the most part) never meant to be seen by the eyes of the living—it was meant to benefit the dead in the afterlife. Throughout human history, art has been recognized for its ability to outlive us and has been used as a receptacle for our fears and hopes about our own mortality.

• Ancient Greece

The ancient Greeks kept busy. They produced painting and sculpture that was copied by the ancient Romans, by Renaissance and Baroque masters, and by the royal academies up until the 19th century. We still copy ancient Greek architecture, refer to their philosophy, use their geometry, perform their theatre, hold olympic games, and redefine their democracy.

• Ancient Rome

The Romans weren’t very original (they borrowed the Greek’s and Egyptian’s gods, architecture, etc.), but they sure knew the political value of art and they were brilliant engineers and administrators. This tutorial traces Roman art and architecture from the Republic through the collapse of the Empire and the rise Christianity. Fly over a reconstruction of the ancient city of Rome. Understand how the Colosseum was built to appease a population angry at the excesses of the former Emperor Nero, and uncover the secret initiation rites buried by the ash of Mount Vesuvius.

• Buddhist Art

Buddhism began when Siddhārtha renounced his princely life and succeeded in attaining enlightenment. As the philosophy and practice of Buddhism traveled, it was shaped and reshaped first across Northern India, Pakistan and Afghanistan and later in China, Japan and South East Asia. Learn how we trace the history of Buddhism through the representation of the historical Buddha and the many other buddhas that are still widely venerated today.

• 400-1300 Medieval Era
• A Beginner's Guide to the Medieval Period

An overview of Europe in the middle ages.

• Early Christian

The first Christians were often persecuted in the Ancient Roman empire, but when—thanks to Constantine—Christianity became legal in the 4th century, Christians could worship openly and the first churches were built in Rome and Christians could be buried in tombs sculpted with Christian imagery. This tutorial takes you on a tour of early Christian churches, and takes a close look at the sarcophagus of a Roman senator who died in 359.

• Anglo Saxon England

Arthur, Guinevere, Merlin and Robin Hood come to mind when England’s rather murky early Medieval history is recalled. But what do we really know about this period? What artifacts have been found? In this small tutorial (so far there is only one video), we explore the great Sutton Hoo ship burial and the treasures from the era of the Anglo Saxon King Aethelberht.

• Byzantine

Today, we know the city of Constantinople as Istanbul (in fact there’s a song about that!). But even before it was Constantinople, it was the ancient city of Byzantium, and it was renamed Constantinople (city of Constantine) by none other than the Emperor Constantine himself (it’s good to be the emperor!). From there a succession of emperors ruled the Byzantine empire as the Roman empire dissolved. For two centuries, the Byzantine empire even included the Italian city of Ravenna, where many churches decorated with astoundingly beautiful mosaics can still be found. In Byzantine art we see a departure from the naturalism of the ancient Greek and Roman world. Figures float in ethereal gold heavenly spaces in mosaics, and we find intricate carvings made from ivory, a luxurious material imported from Africa.

• Romanesque

Visogoths, Ostrogoths, and Vikings, oh my! Western Europe was not a peaceful place during the 600 years after the fall of the Roman Empire. Western Europe (what is now Italy, France, Spain, England, etc.) had been repeatedly invaded. The result was a fractured feudal society with little stability and little economic growth. Charlemagne and the Ottonians had partially and briefly unified the West, and of course the Church was a stabilizing institution, but it was only in the 11th Century that everything changed. Now there was finally enough peace and prosperity to allow for travel and for the widespread construction of large buildings. These were, with rare exceptions, the first large structures to be built in the West since the fall of the Romans so many centuries before. We call the period Romanesque (Roman-like) because the masons of this period looked back to the architecture of ancient Rome. The relative calm of the Romanesque period also meant it was possible to travel, and the faithful set out on pilgrimages in great numbers to visit holy relics in churches across Europe. This meant that ideas and styles also traveled, towns grew and churches were built and enlarged.

• Gothic

No, we’re not talking about the dark subculture we know as Goth! We’re talking about the style of art and architecture In Europe from the 1100s to the beginnings of the Renaissance at about 1400. Hopefully by the end of this tutorial when someone says Gothic, you’ll think of enormous stained-glass windows in churches whose vaulted ceilings reach toward heaven and not black clothing and dark eyeliner!

• Islamic Art

More than 1 billion people call themselves muslim and this monotheistic religion is now estimated to be the second largest in the world. Islamic culture was among the most advanced and tolerant during the Medieval era. Cities from Isfahan in the East to Grenada in the West became important centers of art and learning. This tutorial looks at the sculpture, tilework, costume and interior spaces of this brilliant culture.

• Art of the Americas

If you are an art historian with expertise in this area and would like to contribute content, please contact Beth or Steven.

• 1300-1400 Proto-Renaissance
• Siena

When we think of the Renaissance, we tend to think of Florence (and Rome). But the city of Siena also deserves our attention. Today, the lovely walled city of Siena is one of the best preserved Medieval cities in Europe and it was chosen by the United Nations as a World Heritage Site. In the 14th Century, Siena was an independent nation and often at war with its neighbor, Florence. Some of the most important art of the 14th Century was commissioned for Siena’s Cathedral and town hall. Duccio and his students, the Lorenzetti Brothers and Simone Martini produced large-scale painting with an intricacy and subtle coloration that is unique in the Renaissance.

• Florence

When Vasari wrote his enormously influential book, Lives of the Artists, in the 16th century, he credited Giotto, the 14th century Florentine artist with beginning "the great art of painting as we know it today, introducing the technique of drawing accurately from life, which had been neglected for more than two hundred years." In other words, for Vasari, Giotto was the first artist to leave behind the medieval practice of painting what one knows and believes, for the practice of painting what one sees. This tutorial looks at painting and sculpture in both Pisa and Florence to highlight some of the most forward-thinking art of this century.

• 1400-1500 Renaissance in Italy and the North
• A Beginner's Guide to the Renaissance
• Burgundy

No, not the color and not the wine! In the 15th century, the duchy of Burgundy was one of the most powerful regions in Europe, and stretched from what is today central France up to what is today Belgium and Holland. The home of the Dukes of Burgundy was Dijon, and Duke Philip the Bold commissioned some of the century's greatest works there, including a Carthusian monastery just outside the city walls, where he hoped to be buried so the monks could pray for his soul for eternity. He hired some of the most brilliant artists in Europe to work for him there, including Claus Sluter. Very little of the monastery survives today, but thankfully Sluter’s great work, The Well of Moses, can still be seen there.

• Flanders

In the 15th century, Flanders (an area that is today part of the Netherlands and Belgium) was ruled by the Dukes of Burgundy. The enormous wealth of the court of the Dukes and the wealth of the merchant class (in the thriving cities of Bruges and Tournai) made for a new interest in art that mimicked the material world, though in a way that was quite different from what was happening concurrently in Florence. Oil paint comes into its own in the work of Jan van Eyck and Rogier van der Weyden, and we see how the artists of the North used it to create stunning illusions of fur, wood, wool, glass, and jewels.

• Florence

Walking through the streets of Florence in the summer isn’t always easy, it’s jam packed with students and tourists who have come to study and admire the city’s treasures including Michelangelo’s David, Botticelli’s Birth of Venus, and the engineering brilliance of Brunelleschi. Florence in the 15th Century was fabulously wealthy thanks to its banking and manufacturing families. The Medici, the de-facto rulers of the city, were bankers who commissioned some of the most important art of the Renaissance. Florence was also a center of the Humanist revival that in many ways is what separates our Modern world from the Medieval era.

• Venice

Renaissance Venice was a city of merchants that traded with the Byzantine and Islamic Empires to the East, with the Germanic nations North of the Alps, and with Kingdoms to the West. In the 15th Century, Venice was at the height of its power with colonies and a well equipped navy to protect its merchant fleet abroad. Its fabulous wealth financed the construction of sumptuous churches and palaces—and art to fill them. The city’s salty, humid air meant that frescos faded quickly and so in the late 15th century, artists adopted oil, a medium they had seen used on panels from the north. By the late 15th century, they had also adopted canvas as their support of choice and artists like Giovanni Bellini created some of the most subtle and engaging art ever made.

• Tyrol

Currently this tutorial contains the work of only one artist, Michael Pacher. It would have been so much easier if we could have included him in Venice, since Pacher was influenced by the work of the great Venetian painter Mantegna. But Pacher spoke German, and the specific area he was from (today in the north-east of Italy), was known as Tyrol (though there is a state of Tyrol today in Austria). Confusing, we know. In this tutorial, we take a look at Pacher’s amazing St. Wolfgang Altarpiece which is still in the church it was made for—a church on a lake surrounded by mountains in Austria, still visited by religious pilgrims. Most altarpieces in the Renaissance were made of many interconnected panels that were later sold and ended up in different collections, and this means that to see one altarpiece you usually have to travel to many museums. But this is not true of the St. Wolfgang altarpiece. Seeing a Renaissance work of art in the space it was made for helps us to travel back in time to the late 15th Century.

• England

The International Gothic Style persists into the 15th Century in many courts in Europe including England.

• 1500-1600 End of the Renaissance and the Reformation
• High Renaissance: Florence and Rome

We don’t call this the “High” Renaissance for nothing! This period sees hugely ambitious projects, from Michelangelo’s painting of the Sistine Chapel ceiling and the rebuilding of St. Peter’s in Rome, to Raphael’s frescoes in the papal palace. We use this term to refer to the art of the Italian Renaissance, beginning with Leonardo, whose great masterpiece the Last Supper, actually dates to the last decades of the 15th century (history is never neat!). In the painting and sculpture of this period, ideally beautiful figures who move gracefully, often in complex, multi-figure compositions conveying the sense that human beings are an echo of the perfection of God.

• Venice

The light is different in Venice, the sun glints off the city’s watery streets illuminating Renaissance palaces and churches filled with the art of Bellini, Giorgione, Tintoretto, Veronese and Titian. Perhaps due to the complex play of light reflected off the canals and Byzantine mosaics, Venetian art is known for its brilliant color and subtle tone. The great Venetian artists produced many of the most important paintings of the Renaissance, but their work was different from their colleagues in Florence and Rome who found inspiration in ancient ruins. The Venetians favored the sumptuous, the exotic and the poetic and created art for a society that had grown wealthy trading with distant lands.

• Mannerism

You could say that High Renaissance painters had achieved it all—ideally beautiful, graceful figures, rational spaces, and unified compositions with dozens of figures. If you were a young artist in the early decades of the 1500s you might have felt that there was nothing left to accomplish! Renaissance art was always based on the visual world—on representing things as we see them, but Mannerist art was more artificial, it looked to other art rather than to nature, and Mannerist artists purposely looked for complexity and difficulty to showcase their skills. Figures are elongated, the illusion of space that was so important for the Renaissance no longer makes sense, and the human body is often impossibly twisted.

• Northern Renaissance

The Renaissance in the North continues, but now with the impact of the Protestant Reformation, where there was growing concern that images in the church violated the commandment against making likenesses, as part of the prohibition against worshipping idols. The Reformation had a direct impact on some of the greatest painters of this period, including the German artists Durer, who converted, and Cranach, who was a close friend of Luther. There was increasing exchange during this period between artists in Italy and those in Northern Europe in terms of both methods and style, though the two styles remain distinct. Here we see some of the most complex painting in the work of Holbein, some of the most playful in the work of Bruegel and some of the most terrifying in the work of Bosch.

• The Protestant Reformation

In 1517 a German theologian and monk, Martin Luther, challenged the authority of the Pope and sparked the Protestant Reformation. His ideas spread quickly, thanks in part to the printing press. By challenging the power of the Church, and asserting the authority of individual conscience (it was increasingly possible for people to read the bible in the language that they spoke), the Reformation laid the foundation for the value that modern culture places on the individual.

• 1600-1700 The Baroque
• Italy

Baroque painting, sculpture and architecture in Italy seems to miraculously unite the heavenly and the earthly to deepen the faith of believers. We’ll look at the utterly convincing illusions of heaven on the painted ceilings of Il Gesu and St. Ignazio, and the way Caravaggio painted biblical scenes with a gritty realism that makes them look as though they are taking place on the streets of Rome. And of course we explore the great genius of the period, Bernini, who used every means at his disposal—painting, sculpture and architecture—in works like The Ecstasy of St. Theresa and the Cathedra Petri to bring the viewer closer to the divine.

• Flanders

This tutorial focuses on the art of Peter Paul Rubens, whose work was in high demand by nearly every King, Queen and aristocrat in Catholic Europe (good thing he had a huge workshop!). Rubens was a master of color, dramatic compositions, and movement. Although he was from Northern Europe, he traveled to Italy and absorbed the art of the Renaissance, of classical antiquity, and of Caravaggio. He painted nearly every type of subject—landscapes, portraits, mythology, and history paintings.

• Holland

In the Protestant Dutch Republic of the 17th century there was an enormous demand for art from a wide cross-section of the public. This was a very good thing, since the institution that had been the main patron for art—the Church—was no longer in the business of commissioning art due to the Protestant Reformation. Dutch artists sought out new subjects of interest to their new clientele, scenes of everyday life (genre paintings), landscapes and still-lifes. There was also an enormous market for portraits. One of the greatest artist of this period, Rembrandt, made his name as a portrait painter, but was also a printmaker, and his work also includes moving interpretations of biblical subjects (though from a Protestant perspective).

• Spain

The main focus of this tutorial, and a leading artist at this time is the great Diego Velazquez, who spent most of his career as the court painter to the King of Spain painting official portraits. But in the hands of Velazquez, even mundane portraits became masterpieces of brushwork and color. His early work was influenced by the realism of Caravaggio. Get up close to the princess in his later masterpiece, Las Meninas (The Maids of Honor), and you’ll see broad brushstrokes of red, pink, black and white, but step back and they magically resolve to create a perfect illusion of the silk of her dress and the light moving across her face and hair. No other artist, except perhaps Titian and Rubens, revealed so honestly the alchemy of painting—how paint can be turned into reality.

• France

In France, the LeNain Brothers painted scenes of every-day life (genre paintings), often depicting peasants. There was a renewal of interest in their art in the mid-Nineteenth Century, when the art critic, Champfluery wrote that the brothers “considered men in tatters more interesting than courtiers in embroidered garments.” At the same time, Poussin created a very different style—one that was highly intellectual and looked back to Renaissance, and ancient Greek and Roman art.

• 1700-1800 Age of Enlightenment
• Rococo

It’s hard not to like Rococo art. After all, it’s subjects are often about luxury and pleasure, which makes sense since its patrons were the extremely wealthy French aristocracy. This tutorial features two romantic liaisons—Fragonard’s The Swing and The Meeting, portraits and a mythological subject, “Venus Consoling Love.” You get the idea.

• Neo-Classicism

Jacques Louis David, an active supporter of the Revolution of 1789, is the star of this tutorial. David served in the revolutionary government, used his art in the service of its cause—and voted to behead King Louis XVI. He captured the patriotism of the revolution’s early phase and later, memorialized its dead heroes. And when the revolution failed, and Napoleon came to power, David used his great talents to present a heroic image of that military general-turned emperor. David invents a new style for the democratic values of the Enlightenment—one that is the very opposite of the luxuriousness of the Rococo—and that looks back to Renaissance and ancient Greek art, hence the name—Neo-Classicism (new classicism).

• Britain & America in the Age of Revolution

Britain and America in the Age of Revolutions (Reynolds, Copley, Peale) It was hard to be an artist in America during the colonial period, and for decades after too. There were no real art schools, no grand tradition of art, and no wealthy aristocratic patrons to commission heroic subjects. Americans were practical, and they wanted portraits—and not paintings of classical mythology (which didn’t always make American artists, at least those with wider ambitions, happy). As you’ll learn in this tutorial, Copley was the greatest American portrait painter of the period, and Peale, who studied with Copley, painted portraits of American heroes such as George Washington, Thomas Jefferson and Benjamin Franklin, and founded what became the first real American art museum. American artists looked to England for support and inspiration, often to the older Sir Joshua Reynolds, who was the painter to the King, and first President of the Royal Academy in London.

• 1800-1848 Industrial Revolution I
• Romanticism in France

Romanticism begins in France with the violent and exotic battle scenes of Gros and the famous shipwreck, the Raft of the Medusa, painted by Gericault. Soon after, two distinct trends emerge in French painting, one—represented by the artist Delacroix—was rebellious, and emphasized emotion, color and loose brushwork. The other—which can be seen in the art of Ingres—upheld tradition, and emphasized line and a highly finished surface. Of course, things were more complicated—but those were battle lines!

• Romanticism in Spain

The great artist Francisco Goya is the focus of this tutorial. Goya began his career designing tapestries for the royal residences, and eventually became court painter to the King of Spain. But after Napoleon’s army occupied Spain and deposed the King, Goya documented the horrors he witnessed. His work following the occupation, including the Third of May 1808, remains some of the most powerful anti-war images ever created. His later years were spent largely in a house outside Madrid which he painted with haunting scenes. Saturn Devouring his sons belongs to this late series, known as the “Black Paintings.”

• Romanticism in England

As the industrial revolution transformed the British countryside, replacing fields with factories, painters turned to landscape. Constable painted his native suffolk, where he spent his childhood, and imbued it with a sense of affection for rural life. Turner, on the other hand, created dramatic and sublime landscapes with a sense of the heroic or even the tragic. What both of these artists have in common is a desire to make landscape painting—understood as a low subject by the Academy which dictated official views on art—carry serious meaning.

• Romanticism in Germany

This tutorial focuses exclusively on the art of Caspar David Friedrich, whose work best exemplifies Romanticism’s interest in the big questions of man’s mortality and place in the universe. The world had changed dramatically since the time of Michelangelo, Bernini and Rembrandt, and as a result, Friedrich approached these big questions without the Christian narratives that dominated the art of the past. And like his English counterparts during this period, he imbues nature and the landscape with symbolic and often spiritual meaning.

• Romanticism in the United States
• 1848-1907 Industrial Revolution II
• Pre-Raphaelites

In 1848, a small group of young artists banded together and formed “The Pre-Raphaelite Brotherhood,” a name which sounds intentionally backward-looking and medieval. The Pre-Raphaelites looked back to art before the time of Raphael (before about 1500 that is)—before the art of the Renaissance was reduced to formulas followed for centuries by artists associated with the art academies of Europe. Their idea was that the art before Raphael was more sincere, more true to nature and how we see, and therefore less formulaic. These artists also embraced a wide range of subjects, including modern life, biblical and literary subjects, and even history. By looking backwards, the Pre-Raphaelites led British art into the modern era.

• Late Victorian

British art saw a return to the classical after the 1860s, not just in terms of style, but also subject matter. Alma Tadema created sensual Victorian visions of the ancient Greeks and Romans, and Leighton too rendered classicizing figures and subjects. Both of these artists, together with Sargent, were influenced by the Aesthetic Movement, where the subject or narrative of a work of art was minimized in favor of a focus on issues of form (color harmonies, line, composition).

• Realism

In the mid-Nineteenth Century, great art was still defined as art that took it’s subjects from religion, history or mythology and its style from ancient Greece and Rome. Hardly what we would consider modern and appropriate for an industrial, commercial, urban culture! Courbet agreed, and so did his friend, the writer Charles Baudelaire who called for an art that would depict, as he called it, the beauty of modern life. Courbet painted the reality of life in the countryside—not the idealized peasants that were the usual fare at the exhibits in Paris. The revolution of 1848, in which both the working class and the middle class played a significant role, set the stage for Realism. Later, Manet and then Degas painted modern life in Paris, a city which was undergoing rapid modernization in the period after 1855 (the Second Empire).

• Art & the French State

Despite the brief dismantling of the Royal Academy during the French Revolution, art remained an extension of the power of the French State which regularly purchased art that it favored (often art that supported its political objectives). Through the Royal Academy (originally been founded by Louis XIV), the state extended its reach to the official exhibitions (salons) and to matters of style and subject matter through the École des Beaux Arts (School of Fine Arts). These were not just the official institutions of art, they were, in essence, the only institutions available for living artists to train and to make their work known. This tutorial looks at a crucial moment for painting, on the eve of the Revolution of 1848. We also examine one of the great State commissions of the Second Empire, The Opera House, as well as The Dance, a sculpture that adorned its façade.

• Impressionism

Impressionism is both a style, and the name of a group of artists who did something radical—in 1874 they banded together and held their own independent exhibition. These artists described, in fleeting sensations of light, the new leisure pastimes of the city and its suburbs It’s hard to imagine, but at this time in France, the only place of consequence that artists could exhibit their work was the official government-sanctioned exhibitions (called salons), held just once a year, and controlled by a conservative jury. The Impressionists painted modern Paris and landscapes with a loose open brushstrokes, bright colors, and unconventional compositions—none of which was appreciated by the salon jury!

• Post-Impressionism

The work of van Gogh, Gauguin, Cézanne and Seurat together constitute Post-Impressionism and yet their work is so varied and unrelated, we might never otherwise think of these four artists as a group. Certainly van Gogh and Gauguin were friends and they briefly painted together, but each of these artists was concerned with solving particular issues that had to do with their own individual sensibility. Ironically, if anything ties these artists together it is this focus on subjectivity. This tutorial explores the sketchy multiperspectival views of Cézanne, Seurat’s systematized critiques of upper middle-class Paris, Gauguin’s fascination with the primitive and exotic, and van Gogh’s unerring ability to convey deeply human experiences.

• Symbolism & Art Nouveau

The 1880s saw a shift away from the modern-life focus of Impressionism, as artists turned toward the interior self, to dreams, and myth. There was a sense that Impressionism had been too tied up with the materialism of middle-class culture. In some ways, van Gogh and Gauguin can also be seen as Symbolists. Many Symbolist belonged to groups of artists who broke away (or seceded) from the art establishment in their respective countries, to hold their own exhibitions. For example, Klimt belonged to the Vienna Secession (he was its first president), Khnopff to a similar group in Belgium called Lex XX (The Twenty), and Stuck co-founded the Munich Secession.

• 1907-1960 Age of Global Conflict
• Expressionism

Wild Beasts! Les Fauve (wild beasts) is what one critic called the brilliant expressive canvases of Matisse and other artists who exhibited together in 1905. This tutorial traces the work of Henri Matisse from his early Fauvist work with its jarringly bright colors to the stricter geometries he introduced during the First World War. It also tracks Expressionist developments in Germany and Austria with videos on Kirchner, Kandinsky and Jawlensky, artists who adopted a rough, “primitive” style, and on Egon Schiele’s taut, sexually charged paintings from Vienna.

• Cubism and its Impact

The Spaniard Picasso changed the way we see the world. He could draw with academic perfection at a very young age but he gave it up in order to create a language of representation suited to the modern world. Together with the French artist George Braque, Picasso undertook an analysis of form and vision that would inspire radical new visual forms across Europe and in America. This tutorial explains the underlying principles of Cubism and the abstract experiments that followed including Italian Futurism, Russian Suprematism, and the Dutch movement, de Stijl.

Do we know who we really are? What parts of our mind do we know and what parts are hidden from us? Should art only focus on the rational, the conscious, or should we also pay attention to the irrational, the uncanny, the powerful impulses that remain unarticulated and just beyond the reach of our awareness. Dada was born during WWI when poets, artists, and actors, sickened by the violence around them, chose to celebrate the irrational. They created an anti-art that challenged the cultural assumptions that they felt supported the ruling elite that had, in turn, caused the war. In the years after the war, Dada gave way to Surrealism which reinstituted traditional forms of art-making but focused on Freud’s theories of the unconscious.

• German Art Between the Wars

Germany was defeated and exhausted in 1918 at the end of WWI. The equally exhausted victors imposed harsh terms on Germany. It was forced to forfeit its overseas colonial possessions, to cede land to its neighbors, and to pay reparations. As demobilized troops returned, German cities filled with unemployed, often maimed veterans. The Socialists briefly seized power and by the early 1920s hyperinflation further destabilized the nation. Neue Sachlichkeit or the New Objectivity cast a cold sharp eye on Modern Germany’s hypocrisy, aggression, and destitution even as extremists on the political right consolidated power. The National Socialists or Nazi Party won the chancellorship in 1933 and quickly used art and architecture as a means build the myth of a pure German people shaped by the land and unsullied by modern industrial culture. This tutorial looks at the ways that competing political ideologies each used art for its own purposes.

• International Style Architecture

Towers of glass and steel from the mid-20th Century suggest, for many people, the rationalization of urban space that dehumanized our cities with empty plazas, rigorous geometries and uniformity. But International style architecture was born of the utopian idea that innovative design could improve the lives millions and its forms recall the clarity and harmony of ancient Greek architecture. This tutorial treats the late work of Ludwig Mies van der Rohe, Gordon Bunshaft, and Frank Lloyd Wright.

• American Modernism

Art had never been especially important in America. Before the Civil War, many of America’s best artists went to Europe and stayed. Even after the war, American artists found little enthusiasm for their work unless it was directly informed by European precedents. By the first years of the 20th Century, a small group of American artists began to paint the gritty streets of New York and were called the Ashcan School for their portrayal of life in the tenements. In 1913 however, the Armory Show exhibited advanced American and European art and helped to create a market for the work of Georgia O’Keeffe and other members of modern galleries like Alfred Steiglitz’s 291 and Peggy Guggenheim’s Art of This Century. During the Great Depression artists such as Grant Wood portrayed rural life in the south and midwest and became known as regionalists while other realists such as Edward Hopper rendered the alienation of the modern city. Meanwhile, Surrealist ideas infused a younger generation of artists’ work in Mexico and the US which would result, by the end of WWII, in the first internationally important American art movement, Abstract Expressionism.

• 1960 - Age of Post-Colonialism
• A Beginner's Guide to Contemporary Art
• The Postwar Figure

Artists first represented the human body more than 30,000 years ago and haven’t stopped since. Figurative art has been a continuous tradition through human history. Even in societies where the biblical law forbidding the graven image is most strictly interpreted (Judaism and Islam for example), there have always been instances of figural art. The same is true for the era of modernist abstraction when artists found new ways to portray the body on canvas or with the lens of a camera that could profoundly describe the human condition in , and abstraction the post-holocaust era.

• Pop & After

When people walk into an art museum they often expect to see treasures of their cultural history—beautifully crafted precious objects that express profound truths—images of God, nature, man’s heroism, but Campbell soup cans hawked on TV? Pop art sought to upend our comfortable understanding of what art is and it did just that. Warhol, Lichtenstein, Oldenburg, and others confronted the visual reality of our commercial consumer culture by focusing on the mechanics of representation and the subject matter of daily life in the middle of the 20th Century.

• Minimalism & the Land

“Primary Structures,” “ABC Art,” and “Minimalism” are terms that attempted to categorize the work of Donald Judd, Dan Flavin, Sol Lewitt and other artists who produced hard edged, often geometric and seemingly machined objects in the later 1960s and early 1970s. These stark, often cold and cerebral forms were the very antithesis of the deeply emotive gestural art of the Abstract Expressionists and their followers who had dominated the New York art scene since the 1940s. Here was an art that renounced the authentic “hand’ of the artist and sought instead to create forms without reference to the world beyond the object’s own logic except perhaps as Platonic expressions of a pure ideal.

• Process Art

By the second half of the 20th Century, the avant-garde had an avant-garde of its own. Even as Pop, Minimal and Concept artists renounced the handmade work of art, a small group of women recognized the subversive value of handicraft in an age of industrial manufacturing and in an art world dominated by male artists and critics who sought theoretical purity. During her short career, Eva Hesse resituated the ancient question of how to meaningfully represent the human body and unleashed decades of experimentation that continues to this day. Judy Chicago, Linda Benglis, Jackie Windsor, Faith Ringgold, and others used the lowly status of craft and its historical association with female artisans to contrast with “high art.” By focusing of the act of making, on process and craft, these women began the process of fracturing Modernism’s reductive and largely male narratives.

• Conceptual Art

Can art be an enactment? Can the “art” be relocated from the object crafted by an artist to the more ephemeral reaction of the viewer? Must an artist actually “make” a work of art at all? Conceptual artists recognized that when the 19th Century avant-garde broke with the academies and their emphasis on technical execution (the blending color or compositional clarity for example), they were freer to focus on more conceptual issues such as modern urban life, subjectivity, or pictorial language. Fast forward to the second half of the Twentieth Century when intellectual content became the defining characteristic of art and concept fully eclipsed craft. In the 1960 and 70s the art of Beuys, Haacke and others became increasingly conceptual. These artists used found objects, performance, and installation while de-emphasizing the act of “object making.”

• Postmodernism

If you’ve done the tutorials on Nineteenth and Twentieth Century art then you likely have an idea that by “Modernism,” art historians are referring to a set of ideas that characterize art and culture after about 1848. Key to understanding Modern art are the ideas of a heroic “avant-garde” that challenges authority and the expression of the individual (one that is invariably white and male). The term “Post-Modernism,” was initially used in the art world in 1979 for architecture that arbitrarily borrowed historical styles with little regard for original meaning or context. The term was quickly and broadly adopted, and came to refer to a strategy to undermine Modernism’s utopian and heroizing tendencies by using multiple yet simultaneous critical perspectives. Post-Modernism was not anti-Modernism, it was instead, an effort to destabilize Modernist narratives with deeply skeptical critical strategies that emphasized the plurality of gender, race, nationality, politics, and economic inequality.

• American Civics

Videos about how government works in the United States.

• Test Prep
• SAT Math

Sal works through every problem in the first edition of the College Board "Official SAT Study Guide" (ISBN Number: 0-87447-718-2 published in 2004). You should take the practice tests on your own, grade them and then use these videos to understand the problems you didn't get or to review. Have fun! If you're using the second edition of the study guide with 10 practice tests, you can still use some of these videos. Practice tests 4-10 in the newer book correspond to tests 2-8 below.

• GMAT
• Problem Solving

Sal works through the 249 problem solving questions in chapter 5 of the the 11th edition of the official GMAC GMAT Review (ISBN Number: 0-9765709-0-4 published in 2005)

• Data Sufficiency

Sal works through the 155 data sufficiency questions in chapter 6 of the the 11th edition of the official GMAC GMAT Review (ISBN Number: 0-9765709-0-4 published in 2005)

• CAHSEE

Sal working through the 53 problems from the practice test available at http://www.cde.ca.gov/ta/tg/hs/documents/mathpractest.pdf for the CAHSEE (California High School Exit Examination). Clearly useful if you're looking to take that exam. Probably still useful if you want to make sure you have a solid understanding of basic high school math.

• California Standards Test
• Algebra I

Sal works through the problems from the CA Standards released questions (http://www.cde.ca.gov/ta/tg/sr/documents/rtqalg1.pdf). It is best to go through the "algebra" topic first as it is more comprehensive.

• Algebra II

Sal works through 80 questions taken from the California Standards Test for Algebra II (http://www.cde.ca.gov/ta/tg/sr/documents/rtqalg2.pdf). If you struggle with these you can get more help by viewing the "algebra" topic and completing its exercises.

• Geometry

Sal does the 80 problems from the released questions from the California Standards Test for Geometry (http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf). Basic understanding of algebra is necessary. The "geometry" topic has comprehensive treatment of the subject.

• Competition Math

Example problems from random math competitions

• IIT JEE

Questions from previous IIT JEEs

• Partner Content
• Stanford School of Medicine
• Breastfeeding

Learn how a mother is able to nourish a baby through breastfeeding

• Asthma

Learn how asthma causes breathing difficulties in adults and children

• Growth and Metabolism

Find out what helps you grow and how we can measure growth

• Influenza

• Tuberculosis

Learn about tuberculosis and how it can be treated.

• MIT+K12
• Physics

Watch fun, educational videos on all sorts of Physics questions.

• Biology

What makes living things tick?