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## Multiplying decimals

# Multiplying decimals: place value

## Video transcript

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.