Basic multiplication

Let's learn to multiply. M U L T I P L Y. And the best way I think to do anything is just to actually do some examples, and then talk through the examples, and try to figure out what they mean. In my first example I have 2 times 3. By now you probably know what 2 plus 3 is. That's equal to 5. And if you need a bit of a review you could think of if I had 2-- I don't know-- 2 magenta-- this color-- cherries. And I wanted to add to it 3 blueberries. How many total pieces of fruit do I now have? And you'd say, oh, 1, 2, 3, 4, 5. Or likewise, if I had our number line, and you probably don't need this review, but it never hurts. Never hurts to reinforce the concept. And it this is 0, 1, 2, 3, 4, 5. If you're sitting 2 to the right of 0 and in general, when we go positive we go to the right. And if you were to add 3 to it, you would move 3 spaces to the right. So if I said, if I just moved over 3 to the right, where do I end up? 1, 2, 3. I end up at 5. So either way, you understand that 2 plus 3 is equal to 5. So what is 2 times 3? An easy way to think about multiplication or timesing something is it's just a simple way of doing addition over and over again. So that you means is, and it's a little tricky. You're not going to add 2 to 3. You're going to add-- and there's actually two ways to think about it. You're going to add 2 to itself three times. Now what does that mean? Well, it means you're going to say 2 plus 2 plus 2. Now where did the 3 go? Well, how many 2's do we have here? Let's see, I have-- this is one 2, I have two 2's, I have three 2's. I'm counting the numbers here the same way that I counted blueberries up here. I had 1, 2, 3 blueberries. I have one, two, three 2's. So this three tells me how many 2's I'm going to have. So what's 2 times 3? Well, I took 2 and I added to itself three times. So 2 plus 2 is 4. 4 plus 2 is equal to 6. Now that's only one way to think about it. The other way we could have thought about this is we could've said, instead of having 2 added to itself three times, we could've added 3 to itself two times. And I know it's maybe becoming a little bit confusing, but the more practice you do it'll make a little sense. So this statement up here, let me rewrite it. 2 times 3. It could also be rewritten as 3 two times. So 3 plus 3. And once again, you're like, where did this 2 go? You know, I had 2 times 3 and whenever you do addition you see I have 2-- oh, I don't know these-- well, I said cherries, but they could be raspberries or anything. And then I had two things, I have three things and the 2 and the 3 never disappear. And I add them together, I get 5. But here I'm saying that 2 times 3 is the same thing as 3 plus 3. Where did the 2 go? 2 in this case, in this scenario, is telling me how many times I'm going to add 3 to itself. But what's interesting is, regardless of which way I interpret 2 times 3, I can interpret it as 2 plus 2 plus 2 or adding 2 to itself three times. I can interpret it that way or I can interpret it as adding 3 to itself two times. But notice, I get the same answer. What's 3 plus 3? That is also equal to 6. And this is probably the first time in mathematics you'll encounter something very neat. Sometimes, regardless of the path you take, as long as you take a correct path you get the same answer. So two people can kind of visualize it-- as long as they're visualizing it correctly, two different problems, but they come up with the name solution. And so you're probably saying, Sal, when is this multiplication thing even useful? And this is where it's useful. Sometimes it simplifies counting. So let's say I have a-- well, let's stick with our fruit analogy. An analogy is just when you kind of use something as-- well, I won't go too much into it. But our fruit example. Let's say I had lemons. Let me draw a bunch of lemons. I'll draw them in rows of 3. So I have 1, 2, 3-- well, I'm not going to count them because that'll give our answer away. I'm just drawing a bunch of lemons. Now, if I said, you tell me how many lemons there are here. And if I did that you would proceed to just count all of the lemons. And it wouldn't take you too long to say, that oh, there's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 lemons I actually already gave you the answer. We know that there are 12 lemons there. But there's an easier way and a faster way to count the number of lemons. Notice: how many lemons are in each row? And a row is kind of the side to side lemons. I think you know what a row is. I don't want to talk down to you. So how many lemons are there in a row? Well, there are 3 lemons in a row. And now let me ask you another question, how many rows are there? Well, this was one row, and this is the second row, this is the third row, and this is the fourth row. So an easy way to count it is say, I have 3 lemons per row and I have 4 of them. So let's say I have 3 lemons per row. I hope I'm not confusing you, but I think you'll enjoy this. And then I have 4 rows. So I have 4 times 3 lemons. And that should be equal to the number of lemons I have-- 12. And just to make that gel with what I just did with the addition, let's think about this. 4 times 3-- literally, when you actually say out the word 4 times 3, I visualize this. I visualize 4 times 3. So 3 four times. 3 plus 3 plus 3. And if we did that we get 3 plus 3 is 6. 6 plus 3 is 9. 9 plus 3 is 12. And we learned up here, this part of the video, we learned that this same multiplication could also be interpreted as 3 times 4. You can switch the order and this is one of the useful and interesting actually, kind of properties of multiplication. But this could also be written as 4 three times. 4 plus 4 plus 4. You add 4 to itself three times. 4 plus 4 is 8. 8 plus 4 is 12. And in the U.S. we always say 4 times 3, but you know, I've met people and a lot of people in my family they kind of learned in the-- I guess, you could call it the English system. And they'll often call this four 3's or three 4's. And that in some ways is a lot more intuitive. It's not intuitive the first time you hear it, but they'll write this multiplication problem or they'll say this multiplication problem and they'll say, what are four 3's? And when they say four 3's, they're literally saying, what are four 3's? So this is one 3, two 3's, three 3's, four 3's. So what are four 3's when you add them up? It's 12. And you could also say, what are three 4's? So let me write this down. Let me do it in a different color. That is four 3's. I mean literally, that's four 3's. If I told you to say, write down for four 3's and add them up, that's what that is. And that is 4 times 3 or 3 four times. And this is-- let me do that in a different color. That is three 4's. And it could also be written as 3 times 4. And they all equal 12. And now you're probably saying, OK, this is nice. It's a cute little trick, Sal, that you've taught me. But it took you less time to count these lemons than to do this problem. And well first of all, that's only right now because you're new to multiplication. But what you'll find is there are times and there are actually many times-- and I don't want to use the word times too much in a video on multiplication-- where each row of lemons, instead of having 3 maybe they have 100 lemons. Maybe there's 100 rows. And then it would take you forever to count all the lemons and that's where multiplication comes really useful. Although, we're not going to learn right now how to multiply 100 times 100. Now, the one thing that I want to get you and this is kind of a trick. I remember my sister just to try to show how much smarter she was than me when I was in kindergarten and she was in third grade, she would say Sal, what is 3 times 1? And I would say, because my brain would say, oh, that's like 3 plus 1. And I would say 3 plus 1 is equal to 4. And so I'd say, 3 times 1? That must be 4 as well. And she would say, no, silly. It's 3. And I was like, how can that be? How can the 3 times some other number still be the same number? And think about what this means. You could view this as three 1's. What are three 1's? That's one 1 plus another one 1 plus another 1. And that's equal to 3. Or you could view this as 3 one time. So what's 3 one time? It's almost silly how easy it is. It's just 3. That's one 3. You could write this as one 3. And that's why anything times 1 or 1 times anything is that anything. So then 3 times 1 is 3. 1 times 3 is 3. And you know, I could say 100 times 1 is equal to 100. I could say that 1 times 39 is equal to 39. And I think you're familiar with numbers this large by now. So that's interesting. Now there's one other really interesting thing about multiplication. And that's when you multiply by 0. And I'll start with the analogy or the example when you add. 3 plus 0 you've hopefully learned is 3. Because I'm adding nothing to the 3. If I have 3 apples and I give you 0 more apples, you still have 3 apples. But what is 3 and maybe I'm just fixated on the number 3 a little bit too much. Let me switch. What is 4 times 0? Well this is saying, 0 four times. So what's 0 plus 0 plus 0 plus 0? Well, that's 0. I have nothing plus nothing plus nothing plus nothing, so I get nothing. Another way to think of it, I could say 4 zero times. So how do I write 4 zero times? Well, I just don't write anything, right? Because if I write anything, if I write 1/4 and I don't have no 4's-- let me write this. This is four 0's, but I could also write zero 4's. And what is zero 4's? I'll just write a big blank here. There, I wrote it. There are no 4's here. So there's just a big blank. And that's another fun thing. So anything times 0 is 0. I could write a huge number, you know, 5,493,692 times 0. What does that equal? That equals 0. And by the way, what's this number times 1? Well, it's that number again. And what's 0 times 17? Once again, that is 0. Anyway, I think I've talked for long enough. See you in the next video.