Sunday, 28 April 2013

Algebra




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Introduction to algebra



Introduction to algebra
Videos exploring why algebra was developed and how it helps us explain our world.
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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!
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Linear equations
Linear equations
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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.
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Linear inequalities
Linear inequalities
Exploring a world where both sides aren't equal anymore!
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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)!
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Graphing points, equations and inequalities




Graphing points, equations and inequalities
Understanding the coordinate plane, slope, and intercept. Writing, solving, and graphing linear equations and inequalities.
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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.
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Systems of equations and inequalities
Systems of equations and inequalities
Solving a system of equations or inequalities in two variables by elimination, substitution, and graphing.
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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!

Fancier systems of equations

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!
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Functions
Functions
Identifying, solving, and graphing various types of functions.
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Introduction to functions (old tutorial)

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.

Relationships and valid functions

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!

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.

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.

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!

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.
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Quadratics
Quadratics
Solving quadratics through factoring, completing the square, graphing, and the quadratic equation.
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Factoring quadratics

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!

The quadratic formula (quadratic equation)

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!

Graphing quadratics

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.

Quadratic inequalities

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.

Quadratic odds and ends

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.
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Exponent expressions and equations
Exponent expressions and equations
Solving exponential and radical expressions and equations. Using scientific notation and significant figures.
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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.

Radical equations

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.
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Polynomials
Polynomials
Using polynomial expressions and factoring polynomials.
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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.)
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Ratios and rational expressions
Ratios and rational expressions
You have probably been wondering whether our powers of algebraic problem solving break down if we divide by the variable or we have entire expressions in denominator of a fraction. Well, they don't! In this topic, you'll learn how to interpret and manipulate algebraic ratios and rational expressions (when you have one algebraic expression divided by another)!
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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.
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Logarithms
Logarithms
Log-a-what? No, this tutorial is about neither chopped wood nor music (actually logarithms do have applications in music), but it is fascinating nonetheless. You know how to take an exponent. Now you can think about what exponent you have to raise a number to to get another number. Yes, I agree--unstoppable fun for the whole family. No, seriously, logarithms are used everywhere (including to measure earthquakes and sound).
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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.
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Conic sections
Conic sections
Identifying and graphing circles, ellipses, parabolas, and hyperbolas.
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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.
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Matrices
Matrices
Understanding and solving matrices.
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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.
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Imaginary and complex numbers
Imaginary and complex numbers
Understanding and solving equations with imaginary numbers.
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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!

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