Multiplying decimals: place value

Let's see if we can calculate 2.91 times 3.2. And I encourage you to pause this video and try it out on your own. So the way I'm going to think about it is 2.91 is the same thing as 291 divided by 10. Or not divided by 10, divided by 100. And we know that if you divide something by 100, you are going to move the decimal place two places to the left-- one, two. And you would end up at 2.91. It also make sense, if I take 2, and I multiply it by 100, I'd get 200. Or if I take 200 and divided by 100, I would get 2. So it makes sense that 2.91 is the same thing as 291 divided by 100. Similarlarly-- I can never say that word-- 3.2 can be rewritten. It's the same thing as 32 divided by 10. Now, why is all of this interesting? Well, I could rewrite 2.91 times 3.2 as being the same thing as. Instead of 2.91, I can write 291 divided by 100. And then times-- instead of writing 3.2, I could write 32 divided by 10. And this can be rewritten as-- this is going to be equal to 291 times 32 divided by 100. I'm just reordering this-- divided by 100, divided by 10. Or I could rewrite this. This is equal to 291 times 32. If I divide by 100 and then I divide by 10 again, I'm essentially dividing by 1,000. So this part right over here, I could rewrite as dividing by 1,000. Now, why is this interesting? Well, I already know how to multiply 291 times 32. And then we know how to move the decimal so that when we divide by 1,000. So let's calculate 291 times 32. Let me write it right over here. 291 times 32. Notice, I've just essentially rewritten this without the decimals. So this right over here-- but of course, these are different quantities than this one is right over here. To go from this product to this product, I have to divide by 1,000. But let's just think about this. We already know how to compute this type of thing. 2 times 1 is 2. 2 times 9 is 18. Carry the 1. 2 times 2 is 4, plus 1 is 5. And now we can think about the 3. 3 times 1-- oh, let me throw a 0 here. Because this isn't a 3. This is now a 30. So this is in the tens place. So that's why I put a 0 there. So 30 times 1 is 30. That's why we say 3 times 1 is 3, but notice, it's in the tens place right now. And then 3 times 9 is 27. Carry the 2. 3 times 2 is 6, plus 2 is 8. And now we can add. And we would get 2. 8 plus 3 is 11. 6 plus 3 is 13. And then you get 9. So you get 9,312. So this is going to be equal to 9,312 divided by 1,000. And what's this going to be equal to? Well, if we start with 9,312-- and let me throw a decimal there. Dividing by 1,000 is equivalent to moving the decimal over three places to the left. So you divide by 10, divide by 100, divide by 1,000. So that's going to be 9.312. So if you divide by 1,000, you will get to 9.312. Let me write the decimal in purple. Now, there's something very interesting here. In our original, when we wrote the expression, we had one, two, three total numbers behind the decimal. And then over here, we have one, two, three total numbers to the right of the decimal. Why is this? Well, let's think about it. We re-expressed this as 291 divided by 100. And this is 32 divided by 10. Dividing by 100 and dividing by 10-- this essentially accounts for these three decimal places. So we essentially get rid of those decimal places. But then we have to reintroduce those three decimal places by dividing. Or we have to shift-- here we shift the decimal an aggregate to the right three times-- one, two, and then three. Now, in order to make sure we get the right product, we've got to shift it back to the left. So we're shifting it one, two, three. So we went from this to this. For the whole product, it was like multiplying. We essentially, to go from here to here, we multiplied by 100. To go from here to here, we multiplied by 10. So in aggregate, we multiplied by 1,000, if you think about both of these. And so now we have to divide by 1,000 to get the right value. So that's why three spaces to the right of the decimal here, three digits to the right of the decimal here.