Basic Multiplication

Introduction to 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 two times three.

By now you probably know what two plus three is.

Two plus three.

That's equal to five.

And if you need a bit of a review you could think of

if I had two-- I don't know-- two magenta--

this color-- cherries.

And I wanted to add to it three blueberries.

How many total pieces of fruit do I now have?

And you'd say, oh, one, two, three, four, five.

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 zero, one, two, three, four, five.

If you're sitting two to the right of zero

and in general when we go positive we go to the right.

And if you were to add three to it,

you would move three spaces to the right.

So if I said, if I just moved over three to the right,

where do I end up?

One, two, three.

I end up at five.

So either way, you understand that two plus three is equal to five.

So what is two times three?

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 two to three.

You're going to add--

and there's actually two ways to think about it.

You're going to add two to itself three times.

Now what does that mean?

Well, it means you're going to say two plus two plus two.

Now where did the three go?

Well, how many twos do we have here?

Let's see, I have-- this is one two, I have two twos,

I have three twos.

I'm counting the numbers here

the same way that I counted blueberries up here.

I had one, two, three blueberries.

I have one, two, three twos.

So this three tells me how many twos I'm going to have.

So what's two times three?

Well, I took two and I added it to itself three times.

So two plus two is four.

Four plus two is equal to six.

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 two added to itself three times,

we could have added three 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.

Two times three.

It could also be rewritten as three two times.

So three plus three.

And once again, you're like, where did this two go?

You know, I had two times three

and whenever you do addition, you see I have two-- oh, I don't know these--

well, I said cherries, but they could be raspberries or anything.

And then I have two things, I have three things

and the two and the three never disappear.

And I add them together, I get five.

But here I'm saying that two times three

is the same thing as three plus three.

Where did the two go?

Two in this case, in this scenario,

is telling me how many times I'm going to add three to itself.

But what's interesting is, regardless of which way I interpret two times three,

I can interpret it as two plus two plus two,

or adding two to itself three times.

I can interpret it that way or I can interpret it

as adding three to itself two times.

But notice, I get the same answer.

What's three plus three?

That is also equal to six.

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 same 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 three.

So I have one, two, three-- 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 one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve lemons.

I actually already gave you the answer.

We know that there are twelve 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 three 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 three lemons per row

and I have four of them.

So let's say I have three lemons per row.

I hope I'm not confusing you, but I think you'll enjoy this.

And then I have four rows.

So I have four times three lemons.

Four times three lemons.

And that should be equal to the number of lemons I have-- twelve.

And just to make that gel with what I just did with the addition,

let's think about this.

Four times three, literally when you--

and you know, when you actually say the words four times three,

I visualize this.

I visualize four times three.

So three four times.

Three, plus three, plus three, plus three.

And if we did that we get:

Three plus three is six.

Six plus three is nine.

Nine plus three is twelve.

And we learned, up here, in this part of the video,

We learned that this same multiplication

could also be interpreted

as three times four.

You can switch the order.

And this one of the useful

and interesting, actually, kind of properties of multiplication.

But this could also be written as four three times.

Four, plus four, plus four.

You add four to itself three times.

Four plus four is eight.

Eight plus four is twelve.

And in the U.S. we always say four times three,

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 threes, or three fours.

And that in someways 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.

They'll say, what are four threes?

And when they say four threes,

They're literally saying, what are four threes?

So this is one three, two threes, three threes, four threes.

So what are four threes when you add them up?

It's twelve.

And you could also say, what are three fours?

So let me write this down.

Let me do it in a different color.

That is four threes.

I mean literally, that's four threes.

If I told you, say, write down four threes and add them up,

that's what that is.

And that is four times three.

Or three four times.

And this is-- let me do it in a different color,

that is three fours.

And it could also be written as three times four.

And they all equal twelve.

And now you're probably saying,

okay, 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 you know, 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 that there are times,

and there are actually many times--

I don't want to use the word times too much in a video on multiplication--

where each row of lemons,

instead of having three,

maybe they have one hundred lemons!

Maybe there's one hundred rows!

And it'll take you forever to count all the lemons,

and that's where multiplication comes in useful,

although we're not going to learn right now how to multiply one hundred times one hundred.

Now the one thing that I want to give 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 three times one?"

And I would say, because my brain would say,

Oh! That's like three plus one,

and I would say three plus one is equal to four.

And so I'd say,

Oh! You know, three times one, that must be four as well.

And she'd say,"No, silly! It's three!"

And I was like, how can that be?

How can, you know, three times some other number still be the same number?

And think about what this means.

You could view this as three ones.

And what are three ones?

That's one one, plus another one, plus another one.

That's equal to three.

Or you could do this as three one time.

So what's three one time?

It's almost silly how easy it is!

It's just three.

That's one three.

You could write this as one three.

And that's why anything times one,

or one times anything,

is that anything!

So then, three times one is three.

One times three is three.

And you know, I could say, one hundred times one

is equal to one hundred.

I could say that one times thirty-nine

is equal to thirty-nine.

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 zero.

And I'll start with the analogy, or the example, of when you add.

Three plus zero, you've hopefully learned,

is three.

Because I'm adding nothing to the three.

If you have three apples,

and I give you zero more apples,

you still have three apples.

But what is three--

and maybe I'm just fixated on the number three a little too much--

well, so let me switch--

What is four times zero?

Well this is saying zero four times.

So what's zero, plus zero, plus zero, plus zero?

Well, that's zero!

Right? I have nothing, plus nothing, plus nothing, plus nothing.

So I get nothing!

Another way to think of it,

I could say, four zero times.

So how do I write four zero times?

Well I just don't write anything, right?

Because if I write anything,

if I write one four, then I don't have "no fours".

So this is saying--

so this is four--

let me write this--

this is four zeros.

But I could also write zero fours.

And what are zero fours?

Well, I just write a big blank here.

There, I wrote it!

There are no fours here!

So it's just a big blank.

And that's another fun thing.

So, anything times zero is zero!

I could write a huge number.

You know, five million four hundred ninety-three thousand six hundred ninety-two

times zero.

What does that equal?

That equals zero.

And by the way,

what's this number times one?

Well it's that number again.

What's zero times seventeen?

Once again, that is zero.

Anyway, I think I've talked for long enough.

See you in the next video!