# Basic multiplication | Multiplication and division | Arithmetic | Khan Academy

November 21, 2019

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
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,
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
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.