PLACE VALUE
This tries to explain how the number system we normally use
is different from other number systems.
Iʼll start with the Roman Numeral system. This is a very old
number system that still is in use today. Youʼll see it used as the
date of a movie, numbering pages, or on a clock.
I, V, X, L, C, D, M are the symbols used over and over.
The number system we usually use has these symbols:
0,1, 2, 3, 4, 5, 6, 7, 8, 9
Our number system also uses these with place value, but
the Roman system does not. It has its own way of working.
The Roman numeral for this amount is III (one and one and one).
In our place value system it is 3.
Here we have IV (one less than V or 5). We just call it 4.
The Roman Numeral system adds and subtracts symbols by a
set of rules.
A symbol to the right of the same symbol is added. If it is to the
left of the next biggest, it is subtracted.
I, X, C, M are the only symbols that can be added or subtracted.
They can be added 3 times, and except for M, subtracted once.
I
V, L, D can be used once only, but can be added to up to three
times, and subtracted from once by a symbol smaller in value.
(value means the amount it is counting)
We can have IV, IX, but not IL or IC.
The only way to make one less than L (50) or C (100) is to write
XLIX and XCIX.
XCIX is ten less than a hundred, plus one less than ten.
Itʼs a very different system from ours.
One thousand is M (1000), but 999 is CMXCIX.
A movie made is 1999 would have its date as MCMXCIX.
You will also notice there is no zero (0).
0 is very important to our place value number system.
It means nothing, but it also shows something.
If you have $0 you have no money!
But if you have $1 you have something.
And if you have $10 you have ten times as much!
$100 means one hundred times as much.
Itʼs all shown by the 0ʼs, and place value.
Our number system is a place value system, based on ten.
In this system ten things form one group.
Thatʼs its simple rule.
II
IS ONE SINGLE THING (1).
IS ONE GROUP OF TEN THINGS (10).
In “10” the 0 sits to the right of the 1, the place where up to 9
single things can be counted, the oneʼs place. It is always the
last to the right, often with a period to the right of it
Once we have ten things they are thought of as one group.
That one group is now counted as a single thing.
The “1” in 10 is the first to the left of the oneʼs place.
It is in the tenʼs place.
Every place to the left of the oneʼs place will count groups of
10.
1 means one single thing but only in the oneʼs place.
10 means one group of ten and no single things.
Zero is used to show a place is there. 0 puts the 1 into the
tenʼs place.
0 has a second job, as a place holder. It shows places that
are there, but have nothing in them.
If you have $0 you have no money!
But if you have $1 you have something.
And if you have $10 you have ten times as much as $1!
$100 means one hundred times as much as $1.
In $100 there are two empty places. The two 0ʼs hold them open,
and show what the 1 is counting.
III
Above you see ten groups of ten.
THE VERY SIMPLE RULE FOR OUR NUMBER SYSTEM IS
TEN THINGS MAKE ONE GROUP.
THEN THIS IS ONE GROUP.
WE CAN COUNT GROUPS OF GROUPS.
The oneʼs place is the first to the right.
From it we move to the left to the tenʼs place.
The oneʼs place shows an amount up to 9 of single things.
The tenʼs place shows an amount up to 9 of groups of ten.
Now we have ten groups of ten,
or one group of a hundred. (100).
To write this number we use 0 to show there are no ones,
and no tens, and to push the 1 into the third place.
IV
100 means, reading as we do from left to right,
one group of ten groups of ten things,
and no single groups of ten things,
and no single things.
The “1” is said to be in the hundredʼs place.
A “0” is in the tenʼs place,
and another “0” is in the oneʼs place.
To understand the value of the “1” we have to first count left from
the oneʼs place.
The “1” is in the third place from the right, the hundredʼs place.
Thatʼs what gives it the value of one hundred.
In 111 each “1” has a different value, shown by its place.
The first “1” on the left is worth 100.
The middle “1” is worth 10.
The last “1” on the right is just 1.
Our PLACE VALUE NUMBER SYSTEM has this one simple rule:
ten things make one group, and those ten things can be groups.
Any of our 0,1, 2, 3, 4, 5, 6, 7, 8, 9 numerals can sit in any place,
used as many times as needed,
and the number of places is unlimited.
We can have 620,351,289,857 and this is
something Roman Numerals canʼt do.
To understand it we look to the right side where the 7 is in the
oneʼs place.Then we move to the left from it.
V
620,351,289,857
Each numeral sits in a place that shows its value
(how much it is counting).
How do we quickly tell the value? What is that 6 meaning?
Notice the large number is broken by commas into sets of 3
numerals, or digits.
This could be called a convention, not a rule. It is something
done by a set of people, not everyone and not all the time.
It is done to make the whole number easier to read.
The first three digits on the right are, from right to left,
in the oneʼs place, the tenʼs place, and the hundredʼs place.
Every set of three digits is like the first three, BUT in the next
three the 9 is counting groups of ten hundreds, or thousands.
The 8 to its left is counting groups of ten thousands,
and the 2 to its left is counting one hundred thousands.
(The sets of three digits are always hundreds, tens, ones,
but what of?.)
So the second set of three from the right are counting thousands.
The 1 above is counting a group of ten hundred thousands,
or a million.
To the left of it is fifty million, and to the left of that is
three hundred million.
That 0 is sitting in the empty place for counting groups of ten
hundred million, or a thousand million, or a billion.
The number is read from left to right: six hundred and twenty
billion, three hundred and fifty-one million, two hundred and
eighty-nine thousand, eight hundred and fifty-seven.
VI
620,351,289,857 is a very big number!
A millionaire has at least ten hundred thousand dollars.
A billionaire has at least ten hundred million dollars.
The next set of three to the left takes us to the trillions.
One countryʼs national debt is now measured in this term, at least
ten hundred billions that is a thousand billions.
Thatʼs a lot of money! Yet some adults confuse trillions, billions,
and millions when politicians use these words.
BIG, BIG, BIG MISTAKE.
A TRILLION IS 1,000,000,000,000.
A MILLION IS 1,000,000.
A TRILLION IS A MILLION TIMES AS BIG AS A MILLION.
An understanding of place value is always very, very important.
Errors in place value can be very big.
Young children have to completely learn the 0 to 9 numerals, and
then learn to use them in place value when they get to 10.
They make errors if they donʼt understand what the 0 to 9 are
counting, in the place the 0 to 9 are sitting.
Go back to the big number at the top of this page.
Six may be less than seven, but the place they are sitting in
makes the 6 worth tremendously more than the 7.
A common mistake is when given the question, 80 - 7, some
children want to do 8 - 7 because you canʼt do 0 - 7 at that age.
A child has to learn to use place value to take a ten from the 8,
make the first question into 10 - 7 ones to get 7 tens and 3 ones,
or 73.
VII
This is why teacherʼs want numbers to be written in neat clear
vertical columns when doing certain questions at any age.
80
-7
73
The place value of each digit should be clear.
8000
-7
Imagine doing 8 - 7 for this. It looks simple, but it takes a real
understanding of place value to get the answer.
The secret lies in knowing that 1 to the left makes 10 to the right.
Take 1 from the 8, leaving 7 thousand.
The 1 thousand becomes 10 hundred to the right.
Take 1 hundred away from this 10 hundred to leave 9 hundred.
That 1 hundred becomes 10 tens to the right.
Take 1 ten away again and leave 9 tens.
That 1 ten becomes 10 ones to the right.
10 ones take away 7 ones leaves 3 ones!
8000
-7
7993
Adding, Subtracting, Multiplying, Dividing.
It all gets more complicated as a child gets older.
The secret lies in knowing:
1 10
where the 1 is in a place to the left,
the 10 is in a place to the right.
VIII
Place value is not a natural thing for children, it has to be taught.
By now you should see a child has to have a sense of order,
a complete grasp of left and right,
and a knowledge of what 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 all mean.
In my primary classes we had ways to keep track of the days we
were at school.
Several of these required numerals to be put in places.
For example, 3 hooks screwed into a board.
On the first day we hung a 1 card over the 0 card already on the
hook farthest right.
The next day we hung a 2 on top of the 1.
This went on until the tenth day.
Thatʼs when we started place value.
We also placed 1 block on the chalkboard each day.
On the tenth day, masking tape went around those 10 blocks.
We had one group of blocks, and no singles blocks.
We learned that we could put a new 1 card on the middle hook,
beside the 0-9 cards to the right.
But what to do with the 0-9 cards there already?
We took them off and reversed their order
so that the 0 card was in front.
Now the children were looking at 1 group of 10 and 0 ones - 10.
The next nine days filled the oneʼs hook up to 9 again - 19.
Then the next day we had to use place value again to get 20.
Eventually we got to 99. We could start on the third hook next.
100 was a big day.
But I think most important was when we then got to 111.
We called this Place Value Day.
We had three 1ʼs, each with a different value!
IX
There is a pattern in the use of our number system.
HUNDREDS
TENS
ONES.
This is used over and over to name very large amounts.
What about very small amounts?
OUR PLACE VALUE SYSTEM ALSO CAN DO THIS.
Iʼll use money as an example.
Take $1.35
That 1 is still in the oneʼs place, counting one dollar.
The three to the right of it, past the period,
counts tenths of a dollar.
The five to the 3ʼs right counts hundredths of a dollar.
In $1.35 there is one single dollar, three tenths of a dollar,
and five hundredths of a dollar.
The pattern above to the left of the period is now similar, but in
reverse, with the ones sitting at the center:
hundreds tens ones, hundreds tens ones, hundreds tens ones, hundreds tens ones,
hundreds tens ones, hundreds tens ones, hundreds tens ones, hundreds tens ones,
hundreds tens ones, hundreds tens
ten-thousandths
ones .
tenths
hundredths
hundred-thousandths
thousandths
millionths
Our place value system, based on 10, is not the only
counting system, but it is the one we mainly use.
X