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Let's learn to multiply.

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M U L T I P L Y.

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And the best way I think to do anything is just to actually do

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some examples, and then talk through the examples, and try

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to figure out what they mean.

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In my first example I have 2 times 3.

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By now you probably know what 2 plus 3 is.

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That's equal to 5.

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

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if I had 2-- I don't know-- 2 magenta--

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this color-- cherries.

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And I wanted to add to it 3 blueberries.

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How many total pieces of fruit do I now have?

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And you'd say, oh, 1, 2, 3, 4, 5.

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Or likewise, if I had our number line, and you probably

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don't need this review, but it never hurts.

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Never hurts to reinforce the concept.

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And it this is 0, 1, 2, 3, 4, 5.

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If you're sitting 2 to the right of 0 and in general,

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when we go positive we go to the right.

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And if you were to add 3 to it, you would move

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3 spaces to the right.

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So if I said, if I just moved over 3 to the

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right, where do I end up?

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1, 2, 3.

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I end up at 5.

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So either way, you understand that 2 plus 3 is equal to 5.

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So what is 2 times 3?

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An easy way to think about multiplication or timesing

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something is it's just a simple way of doing addition

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over and over again.

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So that you means is, and it's a little tricky.

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You're not going to add 2 to 3.

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You're going to add-- and there's actually two

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ways to think about it.

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You're going to add 2 to itself three times.

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Now what does that mean?

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Well, it means you're going to say 2 plus 2 plus 2.

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Now where did the 3 go?

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Well, how many 2's do we have here?

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Let's see, I have-- this is one 2, I have two

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2's, I have three 2's.

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I'm counting the numbers here the same way that I counted

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blueberries up here.

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I had 1, 2, 3 blueberries.

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I have one, two, three 2's.

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So this three tells me how many 2's I'm going to have.

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So what's 2 times 3?

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Well, I took 2 and I added to itself three times.

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So 2 plus 2 is 4.

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4 plus 2 is equal to 6.

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Now that's only one way to think about it.

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The other way we could have thought about this is we

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could've said, instead of having 2 added to itself three

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times, we could've added 3 to itself two times.

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And I know it's maybe becoming a little bit confusing, but the

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more practice you do it'll make a little sense.

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So this statement up here, let me rewrite it.

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2 times 4.

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It could also be rewritten as 3 two times.

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So 3 plus 3.

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And once again, you're like, where did this 2 go?

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You know, I had 2 times 3 and whenever you do addition you

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see I have 2-- oh, I don't know these-- well, I said

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cherries, but they could be raspberries or anything.

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And then I had two things, I have three things and the 2

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and the 3 never disappear.

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And I add them together, I get 5.

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But here I'm saying that 2 times 3 is the same

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thing as 3 plus 3.

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Where did the 2 go?

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2 in this case, in this scenario, is telling me

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how many times I'm going to add 3 to itself.

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But what's interesting is, regardless of which way I

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interpret 2 times 3, I can interpret it as 2 plus 2

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plus 2 or adding 2 to itself three times.

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I can interpret it that way or I can interpret it as adding

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3 to itself two times.

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But notice, I get the same answer.

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What's 3 plus 3?

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That is also equal to 6.

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And this is probably the first time in mathematics you'll

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encounter something very neat.

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Sometimes, regardless of the path you take, as long as you

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take a correct path you get the same answer.

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So two people can kind of visualize it-- as long as

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they're visualizing it correctly, two different

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problems, but they come up with the name solution.

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And so you're probably saying, Sal, when is

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this multiplication thing even useful?

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And this is where it's useful.

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Sometimes it simplifies counting.

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So let's say I have a-- well, let's stick with

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our fruit analogy.

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An analogy is just when you kind of use something

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as-- well, I won't go too much into it.

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But our fruit example.

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Let's say I had lemons.

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Let me draw a bunch of lemons.

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I'll draw them in rows of 3.

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So I have 1, 2, 3-- well, I'm not going to count them because

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that'll give our answer away.

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I'm just drawing a bunch of lemons.

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Now, if I said, you tell me how many lemons there are here.

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And if I did that you would proceed to just count

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all of the lemons.

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And it wouldn't take you too long to say, that oh, there's

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 lemons I actually

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already gave you the answer.

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We know that there are 12 lemons there.

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But there's an easier way and a faster way to count

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the number of lemons.

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Notice: how many lemons are in each row?

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And a row is kind of the side to side lemons.

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I think you know what a row is.

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I don't want to talk down to you.

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So how many lemons are there in a row?

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Well, there are 3 lemons in a row.

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And now let me ask you another question, how

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many rows are there?

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Well, this was one row, and this is the second row,

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this is the third row, and this is the fourth row.

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So an easy way to count it is say, I have 3 lemons per

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row and I have 4 of them.

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So let's say I have 3 lemons per row.

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

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And then I have 4 rows.

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So I have 4 times 3 lemons.

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And that should be equal to the number of lemons I have-- 12.

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And just to make that gel with what I just did

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with the addition, let's think about this.

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4 times 3-- literally, when you actually say out the word 4

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times 3, I visualize this.

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I visualize 4 times 3.

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So 3 four times.

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3 plus 3 plus 3.

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And if we did that we get 3 plus 3 is 6.

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6 plus 3 is 9.

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9 plus 3 is 12.

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

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that this same multiplication could also be interpreted

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as 3 times 4.

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You can switch the order and this is one of the useful and

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interesting actually, kind of properties of multiplication.

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But this could also be written as 4 three times.

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4 plus 4 plus 4.

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You add 4 to itself three times.

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4 plus 4 is 8.

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8 plus 4 is 12.

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And in the U.S. we always say 4 times 3, but you know, I've met

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people and a lot of people in my family they kind of learned

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in the-- I guess, you could call it the English system.

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And they'll often call this four 3's or three 4's.

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And that in some ways is a lot more intuitive.

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It's not intuitive the first time you hear it, but they'll

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write this multiplication problem or they'll say this

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multiplication problem and they'll say, what are four 3's?

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And when they say four 3's, they're literally saying,

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what are four 3's?

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So this is one 3, two 3's, three 3's, four 3's.

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So what are four 3's when you add them up?

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It's 12.

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And you could also say, what are three 4's?

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So let me write this down.

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Let me do it in a different color.

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That is four 3's.

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I mean literally, that's four 3's.

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If I told you to say, write down for four 3's and add them

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up, that's what that is.

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And that is 4 times 3 or 3 four times.

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And this is-- let me do that in a different color.

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That is three 4's.

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And it could also be written as 3 times 4.

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And they all equal 12.

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And now you're probably saying, OK, this is nice.

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It's a cute little trick, Sal, that you've taught me.

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But it took you less time to count these lemons

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than to do this problem.

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And well first of all, that's only right now because you're

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new to multiplication.

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But what you'll find is there are times and there are

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actually many times-- and I don't want to use the word

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times too much in a video on multiplication-- where each row

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of lemons, instead of having 3 maybe they have 100 lemons.

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Maybe there's 100 rows.

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And then it would take you forever to count all the lemons

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and that's where multiplication comes really useful.

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Although, we're not going to learn right now how to

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multiply 100 times 100.

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Now, the one thing that I want to get you and this

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is kind of a trick.

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I remember my sister just to try to show how much smarter

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she was than me when I was in kindergarten and she was in

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third grade, she would say Sal, what is 3 times 1?

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And I would say, because my brain would say, oh,

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that's like 3 plus 1.

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And I would say 3 plus 1 is equal to 4.

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And so I'd say, 3 times 1?

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That must be 4 as well.

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And she would say, no, silly.

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It's 3.

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And I was like, how can that be?

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How can the 3 times some other number still

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be the same number?

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And think about what this means.

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You could view this as three 1's.

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What are three 1's?

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That's one 1 plus another one 1 plus another 1.

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And that's equal to 3.

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Or you could view this as 3 one time.

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So what's 3 one time?

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It's almost silly how easy it is.

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It's just 3.

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That's one 3.

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You could write this as one 3.

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And that's why anything times 1 or 1 times

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anything is that anything.

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So then 3 times 1 is 3.

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1 times 3 is 3.

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And you know, I could say 100 times 1 is equal to 100.

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I could say that 1 times 39 is equal to 39.

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And I think you're familiar with numbers this large by now.

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So that's interesting.

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Now there's one other really interesting thing

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about multiplication.

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And that's when you multiply by 0.

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

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3 plus 0 you've hopefully learned is 3.

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Because I'm adding nothing to the 3.

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If I have 3 apples and I give you 0 more apples,

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you still have 3 apples.

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But what is 3 and maybe I'm just fixated on the number

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3 a little bit too much.

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Let me switch.

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What is 4 times 0?

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Well this is saying, 0 four times.

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So what's 0 plus 0 plus 0 plus 0?

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Well, that's 0.

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I have nothing plus nothing plus nothing plus nothing,

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so I get nothing.

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Another way to think of it, I could say 4 zero times.

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So how do I write 4 zero times?

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Well, I just don't write anything, right?

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Because if I write anything, if I write 1/4 and I don't have

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no 4's-- let me write this.

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This is four 0's, but I could also write zero 4's.

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And what is zero 4's?

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I'll just write a big blank here.

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There, I wrote it.

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There are no 4's here.

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So there's just a big blank.

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And that's another fun thing.

256
00:12:48,710 --> 00:12:50,800
So anything times 0 is 0.

257
00:12:50,799 --> 00:13:01,389
I could write a huge number, you know, 5,493,692 times 0.

258
00:13:01,389 --> 00:13:02,559
What does that equal?

259
00:13:02,559 --> 00:13:04,559
That equals 0.

260
00:13:04,559 --> 00:13:06,179
And by the way, what's this number times 1?

261
00:13:06,179 --> 00:13:07,214
Well, it's that number again.

262
00:13:07,215 --> 00:13:12,410
And what's 0 times 17?

263
00:13:12,409 --> 00:13:15,230
Once again, that is 0.

264
00:13:15,230 --> 00:13:17,855
Anyway, I think I've talked for long enough.

265
00:13:17,855 --> 00:13:20,350
See you in the next video.