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Sal shows examples of intersection and union of sets and introduces some set notation.

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Basic set notation (practice) | Probability | Khan Academy

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- [Voiceover] So we have a bunch of numbers listed up here, and my goal, in this video, is to see if we can classify them into different types of number categories, and let me draw the categories. So this circle, over here, this represents all of the numbers that can be represented as the fraction of two integers, and, of course, the denominator can't be equal to zero, because we don't know what it means to put a zero in the denominator. So, let's call these, or the standard way of calling these things. These things can be represented as a fraction of two integers, we call these rational numbers. Rational numbers. And if something cannot be represented as a fraction of two integers, we call irrational numbers. Irrational numbers. Irrational numbers. And the size of these circles don't show how large these sets are. There's actually an infinite number of rational and an infinite number of irrational numbers. So, these are the irrational numbers. Irrational. So, these cannot be represented as a fraction of two integers. And then, within rational numbers, you have integers themselves. So, I'll do that in, let me do that in this blue color. Integers. So, integers are numbers that don't have to be represented as a fraction or a decimal. So, these are integers, right over here. Integers. And then a subset of integers are whole numbers. So, if you essentially say the non-negative integers, you're then talking about whole numbers. So let me do that subset, right over here. So, these are going to be the whole numbers. So, whole numbers. Whole numbers, right over here. And, actually, let me just label it all. These are rational... Let me do that in the same color. Rational numbers. And, of course, irrational numbers. Irrational numbers. Irrational numbers. An integer. Well, if I could say, "Look, that is an integer. "Let's think about the integers." But I wouldn't say, "Let's just think about the rational." I'd say, "Let's think about the rational numbers." All right, now that we have these categories in place, let's categorize them. Like always, pause the video. See if you can figure out what category these numbers fall into. Where would you put them on this diagram? So, let's start off with three. This is positive three. It can be definitely represented as a fraction. You can represent it as three over one. But, it doesn't have to be represented as a fraction. It, literally, could be just a three, right over there, but it's also non-negative. So three is a whole number. So three, and maybe I'll do it in the color of the category. So, three is a whole number. So, it's a member of that set. But if you're a whole number, you're also an integer, and you're also a rational number. So, three is a whole number, it's an integer, and it's a rational number. Now, let's think about negative five. Now, negative five, once again, it can be represented as a fraction, but it doesn't have to be, but it is negative. So, it's not gonna be a whole number. So, negative five is going to sit right over here. It's an integer, and if you're an integer, you're definitely going to be a rational number, but it's not a whole number because it is negative. Now we have 0.25. Well, this, for sure, can be represented as a fraction. This is 25-hundredths, right over here. So, we can represent that as a fraction of two integers, I should say. It's 25-hundredths. But there's no way to represent this except using a fraction of two integers. So, 0.25 is a rational number, but it's not an integer and not a whole number. Now what about 22 over seven. Well, here it's clearly represented, already, as a fraction of two integers, but I don't think I can represent this any other, except as a fraction of two integers. I can't somehow make this without using a fraction or some type of decimal that might repeat. So, this, right over here, this would also be a rational number, but it's not an integer, not a whole number. Now this over here. 0.2713. Now the 13 repeats. This is the same thing as 0.27131313, that's what line up there represents. Now, you might not realize it yet, but any number that repeats eventually, this one does repeat eventually, you have the .1313, or you have the 0.27131313, any number like this can be represented as a fraction. For example, and I'm not going to do it here, just for the sake of time, but, for example, 0.3, repeating, that's the same thing as one-third. And later on, we're gonna see techniques of how do you convert this to a fraction of two integers. But, for our sake, we just know that this can be represented as a fraction of two integers just the way that 0.3, repeating, can be. And so, we would put this under rational numbers. 0.2713, repeating. But you have to represent it either as a decimal or a fraction of integers. If you didn't have to, then it could have been an integer, but we'll throw it up there in rational numbers. Now the square root of ten. Square root of ten. This is interesting. So, any square root of a non-perfect square is going to be irrational. So, this is gonna be irrational. I'm not proving it to you here, but you cannot represent this as the ratio of two integers, or a fraction with two integers, with an integer in the numerator and an integer in the denominator. This will be, if you were to represent it as a decimal, it will not repeat. It'll just keep being new and new digits. It will not repeat over time. So, this, right over here, is an irrational number. It's not rational. It cannot be represented as the ratio of two integers. All right, 14 over seven. This is the ratio of two integers. So, this, for sure, is rational. But if you think about it, 14 over seven, that's another way of saying, 14 over seven is the same thing as two. These two things are equivalent. So, 14 over seven is the same thing as two. So, this is actually a whole number. It doesn't look like a whole number, but, remember, a whole number is a non-negative number that doesn't need to be represented as the ratio of two integers. And this one, even though we did represent it as the ratio of two integers, it doesn't need to be represented as the ratio of two integers. You could have represent this as just two. So, that's going to be a whole number. 14 over seven, which is the same thing as two, that is a whole number. Now, two-pi. Now pi is an irrational... Pi is an irrational number. So if we just take a multiple of pi, if we just take a integer multiple of pi, like that, this is also going to be an irrational number. If you looked at its decimal representation, it will never repeat. So that's two-pi, right over there. Now what about... Let me do that same, since I've been consistent, relatively consistent, with the colors. So, this is two-pi right over there. Now, what about the negative square root of 25. Well, 25's a perfect square. Square root of that's just gonna be five. So, this thing is going to be, this thing is going to be equivalent to negative five. So, this is just another representation of this, right over here. So, it is an integer. It's not a whole number because it's negative, but it's an integer. Negative square root of 25. These two things are actually... These two things are actually the same number, just different ways of representing them. And then you have, let's see, you have the square root of nine over... The square root of nine over seven. Well, what's the principal root of nine? This thing is gonna be the same thing, this thing is the same... Let me do this in a different color. This is the same thing as, square root of nine is three, it's the principal root of nine, so it's three-sevenths. So, this is a ratio of two integers. This is a rational number. Square root of nine over seven is the same thing as three-sevenths. Now, let me just give you one more just for the road. What about pi over pi? What is that going to be? Well, pi divided by pi is going to be equal to one. So, this is actually a whole number. So I could write pi over pi, right over there. That's just a very fancy way of saying one.

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Classifying numbers | Algebra (video) | Khan Academy

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Voiceover] So we have a bunch of numbers listed up here, and my goal, in this video, is to see if we can classify them into different types of number categories, and let me draw the categories.

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Classify numbers | Algebra (practice) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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You already know that we can view multiplication as repeated addition. So, if we had 2 times 3 (2 × 3), we could literally view this as 3 2's being added together. So it could be 2 + 2 + 2. Notice this is [COUNTING: 1, 2] 3 2's. And when you add those 2's together, you get 6. What we're going to introduce you to in this video is the idea of repeated multiplication - a new operation that really can be viewed as repeated multiplication. And that's the operation of taking an 'exponent.' And it sounds very fancy. But we'll see with a few examples that it's not too bad. So now, let's take the idea of 2 to the 3rd power (2^3) - which is how we would say this. (So let me write this down in the appropriate colors.) So 2 to the 3rd power. (2^3.) So you might be tempted to say, "Hey, maybe this is 2 × 3, which would be 6." But remember, I just said this is repeated multiplication. So if I have 2 to the 3rd power, (2^3), this literally means multiplying 3 2's together. So this would be equal to, not 2 + 2 + 2, but 2 × ... (And I'll use a little dot to signify multiplication.) ... 2 × 2 × 2. Well, what's 2 × 2 × 2? Well that is equal to 8. (2 × 2 × 2 = 8.) So 2 to the 3rd power is equal to 8. (2^3 = 8.) Let's try a few more examples here. What is 3 to the 2nd power (3^2) going to be equal to? And I'll let you think about that for a second. I encourage you to pause the video. So let's think it through. This literally means multiplying 2 3's. So let's multiply 3 - (Let me do that in yellow.) Let's multiply 3 × 3. So this is going to be equal to 9. Let's do a few more examples. What is, say, 5 to the - let's say - 5 to the 4th power (5^4)? And what you'll see here is this number is going to get large very, very, very fast. So 5 to the 4th power (5^4) is going to be equal to multiplying 4 5's together. So 5^4 = 5 × 5 × 5 × 5. Notice, we have [COUNTING: 1, 2, 3] 4 5's. And we are multiplying them. We are not adding them. This is not 5 × 4. This is not 20. This is 5 × 5 × 5 × 5. So what is this going to be? Well 5 × 5 is 25. (5 × 5 = 25.) 25 × 5 is 125. (25 × 5 = 125.) 125 × 5 is 625. (125 × 5 = 625.)

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Intro to exponents (video) | Exponents | Khan Academy

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You already know that we can view multiplication as repeated addition. So, if we had 2 times 3 (2 × 3), we could literally view this as 3 2's being added together. So it could be 2 + 2 + 2. Notice this is [COUNTING: 1, 2] 3 2's.

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- [Instructor] In this video we are going to introduce ourselves to the idea of permutations, which is a fancy word for a pretty straight forward concept, which is what are the number of ways that we can arrange things? How many different possibilities are there? And to make that a little bit tangible, let's have an example with say a sofa. My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there's several ways to approach this. One way is just try to think through all of the possibilities. You could do it systematically. You could say alright, if I have person A in seat number one, then I could have person B in seat number two, and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C. So it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one, and then swap A and C. So C and then A. And then if I put C in seat number one, well I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations and you can see that there are one, two, three, four, five, six. Now this wasn't too bad. And in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well the way that you would do it, and this is going to be a technique that you can use for really any number of people and any number of seats is to really just build off of what we just did here. What we did here is we started with seat number one and we said alright, how many different possibilities are, how many different people could sit in seat number one assuming no one has sat down before. Well, three different people could sit in seat number one. You could see it right over here. This is where A is sitting in seat number one, this is where B is sitting in seat number one, and this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is a tongue-twister, there's two different possibilities for seat number two. And so, you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways of getting that, and so there's two people left who could sit in the second seat and we saw that right over here, where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial, which literally just means hey, start with that number, and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial, which is going to be equal to six, which is exactly what we got here. And to appreciate the power of this, let's extend our example. Let's say that we have five seats. One, two, three, four, five. And we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say well that's going to be five factorial, which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. If no one's sat down, there's five different possibilities for seat number one. And then for each of those possibilities, there's four people who could sit in seat number two. And then for each of those 20 possibilities in seat numbers one and two, well there's gonna be three people who could sit in seat number three. And for each of these 60 possibilities, there's two people who can sit in seat number four. And then once you know who's in the first four seats, you know who has to sit in that fifth seat. And that's where we got that 120 from.

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Factorial and counting seat arrangements (video) | Khan Academy

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Instructor] In this video we are going to introduce ourselves to the idea of permutations, which is a fancy word for a pretty straight forward concept, which is what are the number of ways that we can arrange things? How many different possibilities are there?

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- [Voiceover] We know that if we had five people, let's say person A, person B, person C, person D, and person E, and we wanted to put them in five different, let's say, positions, or chairs, so position one, position two, position three, position four, and position five. If we wanted to count the number of scenarios, or we could say the number of permutations of putting these five people in these five chairs, we could say, well, we have five different ... If we seated people in order, which we might as well do, we could say, look, five different people could sit in chair one. So, for each of those scenarios, four different people could sit in chair two. Now, for each of these scenarios, now, so we have 20 scenarios. Five times four, we have 20 scenarios where we've seated seat one and seat two, how many people could we now seat in seat three for each of those 20 scenarios? Well, three people haven't sat down yet, so there's three possibilities there. So now, there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yet, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats. For each of those, how many possibilities are there for the fifth seat? Well, one. For each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. So the number of permutations, the number of ... Let me write this down. The number of permutations, permutations, of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which, of course, is equal to, let's see, 20 times six, which is equal to 120. We have already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. Let's say that we only have three chairs. We have chair one, we have chair two, and we have chair three. How many ways can you have five people, where only three of them are going to sit down in these three chairs, and we care which chair they sit in? I encourage you to pause the video and think about it. I am assuming you have had your go at it. Let's use the same logic. If we seat them in order, and we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we had five different people, five different people could sit in seat one. For each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? In each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we've seated seats one and seat two. For each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three, so that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now this, and my brain, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, when we just did five different people in five different chairs, and we cared which seat they sit in, we had this five factorial. Factorial is kind of neat little operation there. How can I relate factorial to what we did just now? It looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one. So one way to think about what we just did, is we just did five times four times three times two times one, but of course we actually didn't do the two times one, so you could take that and you could divide by two times one. If you did that, this two times one would cancel with that two times one and you'd be left with five times four times three. The whole reason I'm writing this way is that now I can write it in terms of factorial. I could write this as five factorial, five factorial, over two factorial, over two factorial. But then you might have the question where did this two come from? I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three, I kept going until I had that many seats, and then I didn't do the remainder. So the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs, so I was trying to put five things in three places. Five minus three, that gave me two left over. So I could write it like this. I could write it as five ... Let me use the same colors. I could write it as five factorial over, over five minus three, which of course is two, five minus three factorial. Another way of thinking about it, if we wanted to generalize, is if you're trying to figure out the number of permutations and there's a bunch of notations for writing this, if you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations you could put n people in r seats, and there's other notations as well, well, this is just going to be n factorial over n minus r factorial. Here n was five, r was three. Five minus three is two. Now, you'll see this in a probability or a statistics class, and people might memorize this thing. It seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not some type of voodoo magic. But I will tell that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going, wait, does this formula apply there? What's n? What's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything. You don't feel like you're just memorizing without understanding. You're just using your deductive reasoning, your logic. That's especially valuable because as we'll see, not every scenario's going fit so cleanly into what we did. There might be some tweaks on this, where like maybe only person B likes sitting in one of the chairs, or who knows what it might be? Then your formula is going to be useless. So I like reasoning through it like this, but I just showed you this so that you could connect to it a formula that you might see in a lecture or in a class.

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Permutation formula (video) | Permutations | Khan Academy

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Voiceover] We know that if we had five people, let's say person A, person B, person C, person D, and person E, and we wanted to put them in five different, let's say, positions, or chairs, so position one, position two, position three, position four, and position five.