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# Comparing decimal place values

Video transcript

We have the number
7,346,521.032. And what I want
to think about is if I look at the same digit
in two different places, in particular, I'm going
to look at the digit 3 here and the digit 3 here, how much
more value does this left 3 represent than this right 3? In order to think
about that, we have to think about place value. So let's write down
all the place values. So this right over here,
this is the ones place. Now, we could move to the right. And as we move to the
right in place values, each place represents 1/10
of the place before it. Or you could divide by 10 as
we're moving to the right. So this is the ones place. This is divide by 10. This is the 1/10 place,
or the tenths place. Divide by 10 again, this
is the hundredths place. Divide by 10 again, this
is the thousandths place. And that "s" I'm just saying
to be plural-- hundredths, thousandths. Now, if we go to the
left, now each place represents a factor of 10 more. So if this is ones, multiply
by 10, this is the tens place. This is the hundreds place. This is the thousands place. This is the ten thousands place. I'm going to have to write
a little bit smaller. This is the hundred
thousands place. And then the 7 is in
the millions place. So what does this number,
what does this 3 represent? Well, it's in the
hundred thousands place. It literally represents
3 hundred thousands, or you could say 300,000,
3 followed by five zeroes. Now, what does this 3 represent? It's in the hundredths place. It literally represents
3 hundredths. It represents 3
times 1/100, which is the same thing as 3, which
is equal to 3 over-- let me do the 3 in
that purple color. Which is the same
thing as 3/100, which is the same thing as 0.03. These are all
equivalent statements. Now let's try to answer
our original question. How much larger is this
3 than that 3 there? Well, one way to think about
it is how much would you have to multiply this 3 by
to get to this 3 over here? Well, one way to think
about is to look directly at place value. So we got to multiply by 10. Every time we
multiply by 10, that's equivalent to thinking about
shifting it to one place to the left. So we would have to multiply by
10 one, two, three, four, five, six, seven times. So multiplying by
10 seven times. Let me write this down. So this multiplied by 10 seven
times should be equal to this. Let me rewrite this. 300,000 should be
equal to 3/100-- let me write it the same way. 3/100 multiplied by 10 seven
times, so times 10 times 10 times 10 times 10
times 10-- let's see, that's five times--
times 10 times 10. Now, multiplying
by 10 seven times is the same thing as multiplying
by 1 followed by seven zeroes. Every time you
multiply by 10, you're going to get another zero here. So this is the same
thing as 3/100 times 1 followed by one, two, three,
four, five, six, seven zeroes. So this is literally
3/100 times 10 million. So let's see if this
actually is the case. Does this actually
equal 300,000? Well, if you divide 10 million
by 100, dividing 10 million by 100, or I guess you'd
say in the numerator, you have 10 million
and the denominator you have 100, if you were to
just multiply it like this, if you view this as 3 over
100 times 10 million over 1. Well, you divide the
numerator by 100, you're going to get rid
of two of these zeroes. Divide the denominator
by 100, you're going to get rid
of this 100 here. And so you're going to be
left with 3 times-- now we got to be careful
with the commas here, because since
I removed two zeroes, the commas are going
to be different. It's going to be
3 times-- we put our commas in the right
place, so just like that. So this simplified
to 3 times 100,000, which is indeed 300,000. So it did work out. Shifting the 3 one, two,
three, four, five, six, seven decimal places
makes that 3 worth 1 followed by seven zeroes
more, or it essentially makes that 3 worth
10 million more. So this 3 represents 10 million
times the value of this 3. Let me write down the numbers. So this 3 is 10 million
times the value of that 3.