Multiplying and dividing by 10, 100, 1000

• [Instructor] In this video, we're gonna think about what happens when we multiply or divide by 10, 100, or 1,000. Let's just start with an example. Let's say we wanna figure out what 237 times 10 is. Pause this video and see if you can have a go at it.

All right, well, one way to think about it is this is 237 10s, so if I put my place values here, so this is the 1,000s place, this is the 100s place, I'm just not writing it out in words just to save space; sometimes you'll see it written out as the word thousands and hundreds, this is the 10s place, and then this is the 1s place.

Well, if I were to just say seven 10s, I would put the seven over there. If I were to say 37 10s, I would write it like this, because 30 10s is three 100s, and if I were to say 237 10s, I would write it just like that. Because 200 10s is two 1,000s.

And so what just happened? Well, all of my digits shifted one place to the left. So what used to be in the 1s place is now in the 10s place. What used to be in the 10s place is now in the 100s place, and what used to be in the 100s place is now in the 1,000s place, but what's in the 1s place? We have to put something in the 1s place.

Well, we no longer have any 1s in this product, so we would just put a zero there. So 237 times 10 is 2,370. Now, one way to think about it is, when I multiplied by 10, I really just shifted; I put a zero right over here at the end of this number, but it's important to realize that what really happened is I shifted all my digits one place to the left.

But what if we were to go in the other direction? What if we were to take, I don't know, 450 and we were to divide it by 10? Well, you can imagine if multiplying by 10 shifts the digits one place to the left, then dividing by 10 is going to shift the digits one place to the right, and so we have a four in the 100s place, we have a five in the 10s place, and we have nothing in the 1s place.

And so what used to be in the 100s place will now be in the 10s place, so I will put that four right over there, and then what used to be in the 10s place is now going to be in the 1s place, so I will put that five over there, and we have nothing in the 1s place.

If we did, and this is a little bit out of this video's purpose, but you could start to think what might happen if this was not a zero. But we have 450 divided by 10 is 45, and it is indeed 45. Now, another way to think about it, other than the fact that all of the digits shifted one place to the right is that we removed this right zero here when we divided by 10.

Now, let's extend that; let's say that we had 359 times 100. Pause this video and see if you can figure out what that is going to be. Well, there's two ways that you could think about it. You could just think of it in terms of this is 359 hundreds, so let's put some places here.

So let's say this is the 10,000s place, this is the 1,000s place, this is the 100s place, this is the 10s place, and this is the 1s place. So if you were to say nine 100s, you would go right over there, if you were to say 59 hundreds, it would be like that because 50 hundreds are five 1,000s, and if you were to say 359 hundreds, well, 300 hundreds is three 10,000s.

And of course, in this scenario now, you would have no 10s and no 1s. And so what we see happened is all of the digits shifted two places to the left. What was in the 1s place is now in the 100s place. What was in the 10s place is now in the 1,000s place, and what was in the 100s place is now in the 10,000s place, and so this product is 35,900.

Now, another way you could think about it is this is the same thing as 359 times 10 times 10. And we already know that every time you multiply by 10, you shift your digits one place to the left. So if you're going to multiply by 10 twice, you're going to shift all your digits two places to the left, which is exactly what happened here.

So based on that, what do you think—or based on everything we've looked at, what do you think if I were to take 75,000 and I wanted to divide it by 100, what do you think this is going to be? Pause the video and see if you can work through it on your own.

All right, well, we've already seen if you divide by 10, you shift the digits one place to the right, and dividing by 100 is really the same thing as dividing by 10 twice, so we really just need to shift all our digits two places to the right.

So what used to be in the 10,000s place is now going to go to the 100s place. What used to be in the 1,000s place is now going to be in the 10s place, and what used to be in the 100s place is now going to go into the 1s place.

And so what we are going to get is, instead of seven 10,000s, we're going to get seven 100s; instead of five 1,000s, we're going to get five 10s, and then instead of zero 100s, we will now have zero 1s. So this is equal to 750.

Another way to think about it, and we've talked about it before, if you're dividing by 100 and if you have some of these trailing zeros here, well, then, you would remove two of these right-trailing zeros, but it's really important to know why that happened.

Now, you've hopefully seen a pattern, so I'll ask you two more questions. Let's say that someone were to walk up to you on the street and say, "Hey, you, what is 164 times 1,000?" and they were also to ask you, "What is 98,000, I'll just say 98,000 divi--I'll say 198,000 divided by 1,000?" Pause this video and see if you can do this.

All right, you've probably seen the pattern now. Just as multiplying by 100 was multiplying by 10 twice, multiplying by 1,000 is multiplying by 10 three times, so this is equal to 164, and I'll color-code this, times 10 times 10 times 10, so what we're going to do is shift all the digits three places to the left.

So instead of 164, this four is no longer in the 1s place; it is now in the 1,000s place. It is now in the 1,000s place, and that makes sense. If you take four 1s and you multiply it by 1,000, you're going to get four 1,000s.

What was in the 10s place is now in the 10,000s place, and what was in the 100s place is now going to be in the 100,000s place, and so, in order to represent this number, I'll have to put something in the 100s place; we'll have no 100s now, I have to put something in the 10s place; I have no 10s now, and I have to put something in the 1s place.

So it's going to be 164,000, and once again, that's consistent with the pattern we saw. We've added three zeros to the right of this number. So similarly, what's going to happen over here?

Well, we're going to shift all the digits three places to the right, so the one, it's going to go one, two, three places to the right, so what was in the 100,000s place is going to go into the 100s place; what was in the 10,000s place is going to go one, two, three places to the right; it's going to go into the 10s place, and what was in the 1,000s place is going to shift one, two, three digits to the right; it's going to be in the 1s place, or another way to think about it is when we divide by 1,000, especially when we have these three trailing zeros, those are going to go away.

So instead of 100,000 here, this is just going to be 100. Instead of 90,000 here, or nine 10,000s, this is just going to be nine 10s. And instead of eight 1,000s, this is just going to be eight, so it's equal to 198.