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To add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂). Here's a concrete example: the sum of (2,4) and (1,5) is (2+1,4+5), which is (3,9). There's also a nice graphical way to add vectors, and the two ways will always result in the same vector.​​

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Salman Khan
07/12/2018
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This site teaches Arithmetic with Polynomials and Rational Expressions to High Schoolers through a series of 4333 questions and interactive activities aligned to 26 Common Core mathematics skills.

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This site teaches High Schoolers how to create equations through a series of 298 questions and interactive activities aligned to 5 Common Core mathematics skills.

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This site teaches Reasoning with Equations and Inequalities to High Schoolers through a series of 5909 questions and interactive activities aligned to 36 Common Core mathematics skills.

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This site teaches Structure in Algebraic Expressions to High Schoolers through a series of 3482 questions and interactive activities aligned to 26 Common Core mathematics skills.

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07/12/2018
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This arithmetic lesson demonstrates more long division problems with remainders.[Arithmetic playlist: Lesson 26 of 38]

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Salman Khan
07/12/2018
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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! Common Core Standard: 5.NBT.B.7

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Salman Khan
07/12/2018
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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! Common Core Standard: 5.NBT.B.7

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Mathematics
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Author:
Salman Khan
07/12/2018
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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! Common Core Standard: 5.NBT.B.7

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Salman Khan
07/12/2018
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In this tutorial, we'll extend our division skills to include decimals! Common Core Standard 5.NBT.B.7

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07/12/2018
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In this tutorial, we'll extend our division skills to include decimals! Common Core Standard 5.NBT.B.9

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07/12/2018
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In this tutorial, we'll extend our division skills to include decimals! Common Core Standard 5.NBT.B.10

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07/12/2018
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In this tutorial, we'll extend our division skills to include decimals! Common Core Standard 5.NBT.B.11

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07/12/2018
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Let's learn to divide any whole number by any other! Common Core Standard: 5.NBT.B.6

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07/12/2018
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Let's learn to divide any whole number by any other! Common Core Standard: 5.NBT.B.7

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07/12/2018
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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.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

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Salman Khan
07/12/2018
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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.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

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Mathematics
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Author:
Salman Khan
07/12/2018
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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.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

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Mathematics
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Author:
Salman Khan
07/12/2018
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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.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

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Mathematics
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Author:
Salman Khan
07/12/2018
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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.). Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

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Salman Khan
07/12/2018
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Anything you can do with whole numbers, you can do with decimals. Subtraction is no exception. In this tutorial, you'll get some good practice subtracting decimals! Common Core Standard: 5.NBT.B.7

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Salman Khan
07/12/2018
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Anything you can do with whole numbers, you can do with decimals. Subtraction is no exception. In this tutorial, you'll get some good practice subtracting decimals! Common Core Standard: 5.NBT.B.7

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07/12/2018
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Describe a bivariate relationship's linearity, strength, and direction. In other words, plotting things that take two variables into consideration and trying to see whether there's a pattern with how they relate.

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07/12/2018
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This is a brief conditional probability examplediscussing probabilities like P(A | B) using breakfast and lunch as examples.

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07/12/2018
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This series of videos focusing on calculus covers sample questions from the A.P, Calculus AB and AC exams (both multiple choice and free answer).

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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!

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Salman Khan
07/12/2018
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This series of videos focusing on calculus covers minima, maxima, and critical points, rates of change, optimization, rates of change, L'Hopital's Rule, mean value theorem.

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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!

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07/12/2018
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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)!

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07/12/2018
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This series of videos focusing on calculus covers indefinite integral as anti-derivative, definite integral as area under a curve, integration by parts, u-substitution, trig substitution.

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This 9-minute video lesson provides an introduction to L'Hopital's Rule. [Calculus playlist: Lesson 36 of 156]

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Salman Khan
02/20/2011
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This series of videos focusing on calculus covers limit introduction, squeeze theorem, and epsilon-delta definition of limits.

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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.

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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.

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This series of videos focusing on calculus covers line integral of scalar and vector-valued functions, Green's Theorem and 2-D Divergence Teorem.

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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.

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07/12/2018
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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).

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07/12/2018
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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.

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07/12/2018
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This 17-minute video lesson looks at the iIntuition behind the Mean Value Theorem. [Calculus playlist: Lesson 55 of 156]

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Salman Khan
02/20/2011
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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!

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Mathematics
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