>> Hi, this is Julie Harland,
and I'm your Math Gal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
We're going to go
over place value
for whole numbers
in this video,
but before we start let's
think about this.
Before there was a need
to write large numbers,
there were various ways
of writing numerals.
One was this tally system.
For instance,
to write the number 24 we have
this tally you see right here.
Most of you are probably
familiar with this.
This would be five,
another five, another five,
another five,
that's twenty plus four
more strokes.
So this was made
up of these little symbols
that you see here,
and you could easily read
that as 24.
And we still use this
today often.
For instance, if you're trying
to count a few things
up it's a quick way
of tallying a vote,
etc. But there's a problem
if you're trying
to write a really
large number.
So let's say you have this
number, 5,324,
which hopefully you're
familiar with reading
that number.
Well, if you tried to do
that using this tally system
it would just take way
too long.
You couldn't fit it
on a page even.
So this system
that we're used to,
this 5,324,
it actually incorporates
something pretty important
called place value.
So this doesn't mean five plus
three plus two plus four
for instance.
Depending on where the
placement of a digit is,
it represents something.
So this four this is really
four ones,
this is called the ones place.
The two is
in what's called the tenths
place, like two tenths.
The three is
in the hundredths place,
and the five is
in the thousandths place.
So we have five thousand,
three hundred, two tens
and four ones.
Now, you can think
about this as many.
If you had, I wish I had this,
$5,324 you might have five one
thousand dollar bills,
three one hundred dollar
bills, two ten dollar bills
and four ones.
So this really represents five
one thousand dollar bills.
I don't even know
if there is a thousand dollar
bill, but if there were there
would be five of them
because I've never had one
in my hand;
three hundred dollar bills,
and we've got two ten dollar
bills, and we'd have one
one-dollar bill.
So the placement
of these digits,
remember the digits are
from zero to nine is
very important.
What's also important
about place value system is
you might have a zero,
zeros are really important.
Let's say you didn't have any
ten dollar bills.
Then you have
to have this placeholder
of zero to tell you there's no
ten dollar bills.
If you just wrote 534,
you would think
that would be 534,
five one hundreds, three tens
and four ones.
So here's how the place value
system works.
I've gone up to let's say the
millionth place.
So what we have is starting
from the right is the ones,
the tens, the hundreds,
the thousands,
the ten thousands,
the hundred thousands,
the one million.
If I kept going
to the lesser it would be the
ten million,
the hundred million,
etc. So the interesting thing
is in each group there's a
ones, tens and hundreds.
So first in this group we
don't really call it anything
but there's ones, tens,
hundreds, but notice here
there's one thousand,
ten thousand,
one hundred thousand.
And in the millions there will
be the one million,
ten million,
one hundred million,
then we'll go to the billions
and the trillions,
etc. So there's always this
grouping of three
and that is why we have
the comma.
Now, another way
of writing these numbers is
using exponents.
And I'm going
to preview what an exponent
and power is.
An exponent
and a power is a little number
written to the upper right
of a number called the base.
So you see how I have a ten
here, this called a base,
and the little three
to the upper right that's
called the exponent,
and this is what it means.
Ten with that little three
up there means you take the
base ten and you multiply it
by itself three times,
whatever the exponent is.
So this is what it means
when we have whole numbers.
So I'm just going
to show you how it works
with whole numbers
where the base is ten.
So ten with a little three,
ten times ten times ten
or a thousand.
Okay? Ten with a little two we
could read this as ten
to the second power
or ten squared means ten times
ten which is a hundred.
Ten to the sixth power means
ten times ten times ten times
ten times ten times ten.
You're going
to have ten multiplied
by itself six times
which is one million.
So here's the thing.
Any time you have ten
to some number, let's do ten
to the fourth power, well,
look at the pattern up here.
When you had ten
to the third power notice you
had one with three zeros
for the answer.
When you have ten
to the second power you have a
one with two zeroes.
When you have ten
to the sixth power you ended
up one with six zeros.
So the pattern is ten
to the fourth is a one,
the answer ends up being one
with four zeroes.
How about ten
to the first power?
That would be a one
with one zero.
Here's the tricky one
since we're doing
whole numbers.
If you have ten
to the zero power the answer
just ends up being one
with no zeroes.
So if you haven't learned
about exponents before this is
the basic definition of it
when you're taking
at least ten
to some whole number
like zero, one, two, three,
four, five, six,
and there's a lot more to it
that you may learn
in other sections
of arithmetic or pre-algebra.
This number ten is the same
thing as ten
to the first power.
One hundred is ten squared,
ten to the second power,
this will be ten cubed,
ten to the fourth,
ten to the fifth and ten
to the sixth.
So basically this exponent
means for ten cubed ten times
ten times ten
which is a thousand.
Now, see this pattern?
It goes one, two, three, four,
five, six,
the exponents go up.
So this very first one is
actually ten to the zero.
Now, if you haven't had any
algebra yet this will make
more sense as you go along,
but for now I'm just showing
it to you as a pattern.
So these place values they
also could be written
as powers of ten,
and that's useful
because it's kind of hard
to keep writing these
big numbers.
So it might look
like this instead.
If I write the place value
in powers of ten I could sort
of fit more on the page
as you can see.
So you start off with ten
to the zero,
and what that means is just
the number that's a hundred,
ten and one.
Ten to the zero is actually
equal to one.
[ Pause ]
This concept,
ten to the zero equals one,
will be clear once you're
working with the laws
of exponents in algebra.
Alright, so let's look
at this place value system,
and let's say we had the
number, we'll just fill it
in up here, how about 50214,
alright, so in other words
you're looking at this number.
The expanded form is
understanding what each
of these place values is,
what we say is the five is
in the ten
to the fourth place,
so that's five
ten thousandths.
So you could write
that as fifty thousand.
I don't have anything
in the thousandths place
except zero.
So I've got two
in the hundredths place
so this is two hundred.
I have one in the tenths place
so I have one ten
and I have four.
So if you add these numbers
up together it actually is
this exact same number, okay?
So when you put the five here
you have to remember what it
really stands
for is fifty thousand.
When you have the two here it
really stands for two hundred,
the one stands for one ten
which is just ten,
and the units
or the ones place is
just four.
So when we add those numbers
up it really represents the
number 50,214.
These stand
for the same thing.
And this is called expanded
form if you write it
like this,
fifty thousand plus two
hundred plus ten plus four,
we call that the expanded form
of the number.
Alright, so here's one
for you to try.
Write 6,570 in expanded form.
So six is in the thousandths
place so we have six thousand
plus five is
in the hundredths place,
five hundred,
seven is the tens place,
seven tens is seventy,
we don't have ones.
Notice you don't have
to write plus zero, right,
so this is now the expanded
form of 6,570.
Alright, let's just do one
more thing.
So here are the place values
that I wrote out, ones, tens,
hundreds, thousandths,
ten thousandths.
I didn't write out the words,
but I want you to think
about the words now,
and also here is it
in the exponential form
if you want to see that.
And the question is how would
we state the place value
of the four in each
of these numbers?
So in other words four,
depending on its placement,
what does it really stand for?
Alright, try this
on your own first.
See if you can write
out in words what the place
value is.
Okay, so for the first one
what is the four in?
We've got one, ten,
hundredths,
so it's in the
hundredths place.
[ Pause ]
Alright, next one,
so that's in the
hundredths place.
What about this four?
One, ten, a hundred thousand,
or what you could say is,
well, it's in that grouping
with the thousandths
and so it's
in the first remember this
goes one, ten,
a hundred thousand
so it's one thousand.
So this is
in the thousandths place,
alright?
And what about this last one,
where is that four?
It's right in the ones.
Alright, on the next video
what we'll do is learn how
to write the numbers in words
and also how to take a number
in words and write it
as our regular number.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.