Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
Key
On screen content
Narration – voice-over
Web links
Activity – Under the Activities heading of the online program
Quiz – Under the Activities heading of the online program
The Place Value System for Whole Numbers
The Decimal System for whole numbers is based on Place Values, which increase 10 times as each
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place moves to the left and is 10 less each time a place moves to the right, as seen below.
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As we saw in the previous topic, our number system – The Decimal System – is based on the number
10 using the 10 Hindu-Arabic digit numerals. We also saw that the value of any whole number digit
numeral is determined by the PLACE that it is in as you can see in the chart. It is also the case with
numbers less than 1 (which are called fractional numbers or decimal numbers) that their value is
determined by the PLACE that they are in. Place value increases by 10 as we move left, and decreases
1
by 10 as we move right.
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There is no end to the number of PLACES there are moving to the left or to the right.
Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
The Place Value System for Decimal Numbers Less than One
What happens to place value for decimal numbers?
Exactly the same system applies for numbers which are less than one (Decimal Numbers).
The decimal system for decimal numbers is just an extension of the same system for whole numbers,
based on Place Values. You can see that the decimal point separates our whole numbers to the left,
from our decimal numbers to the right.
Looking at the whole numbers on our chart and starting with 1 we can see this carries a place value
of just one. We can see that the next column is the tens place which is ten times bigger than the ones
place and so on.
Similarly, every time a place moves to the right, the value of any digit in that place decreases by 10 or
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is 10 th the value of the previous column, as you can see in the chart.
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Let’s now look at the decimal numbers or numbers less than one. We will see that they also decrease
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by 10 as we move to the right.
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Starting with the first decimal place column and moving to the right, 1/10 is ten times smaller than 1
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whole. 100 is ten times smaller than 10 . And 1 000 is ten times smaller than 100 .
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The main thing to remember here is that the decimal point separates the whole numbers from the
decimal or fractional numbers, which are less than 1.
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Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
Place Value
Let’s now look at how the numeral 8 can change its value by being in a different “PLACE” in our
Decimal Place Value System.
Consider the last row in the table above (highlighted in green), which shows the different values of an
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8 in three different places (80 , 10 and 1 000 )
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An 8 digit in the 2nd Place has a value of 80 (8 tens).
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However, an 8 in the 1st Decimal Place gives a value of 10 (8 tenths).
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An 8 in the 3rd Decimal Place gives a value of 1 000 (8 thousandths).
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Together these three “8s” give a number with a total value of 80.808 (eighty point eight zero
eight)
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Go to the following link to see another explanation of this:
Decimals: Comparing place values [link to: https://www.khanacademy.org/math/pre-algebra/decimals-prealg/decimal-place-value-pre-alg/v/comparing-place-values-in-decimals]
Consider the last row in the table, highlighted in green. This shows the different values of 8 in three
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different places (80 , 10 and 1 000 )
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An 8 digit in the 2nd place has a value of 80 (8 tens). However an 8 in the 1st decimal place gives a value
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of 8 tenths or 10 . And an 8 in the 3rd decimal place gives a value of 8 thousandths or alternatively 1 000 .
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Together these three "8s" give a number with a total value of 80.808 (eighty point eight zero eight).
Please notice that in Australia (and some other countries) when writing large numbers, the number is
split into groups of three starting from the right hand side. In Australia we show this separation by
using a space. In other countries they may use a comma to separate the groups of three numbers, as
you will see in the YouTube video.
Click on the Decimals: Comparing place values link to see another way of understanding place value.
Page 3 of 7
Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
Using the Decimal System to Create Numbers
The value (or size) of the number in the example in the bottom row of the table is written as:
9 573.842 (nine thousand, five hundred and seventy three point eight four two).
Decimal numbers that contain no whole numbers always start with a zero. E.g., 0.4 (zero point four)
or 0.572 (zero point five seven two).
Go to Activity One (under Activities in the right-hand side of the screen) and have a go at
creating some numbers yourself, remembering to separate the numbers into groups of three.
Looking at the example in the bottom row of the table, we have created a number with a value of
9 573.842. We can see that this number consists of a 2 in the third decimal place or one thousandth
place giving a value of just two thousandths; when a 2 is placed in the third decimal place or one
thousandths place it gives a value of 2 thousandths; a 4 in the second place or one hundredths place
giving a value of just four hundredths; when a 4 is put in this 2nd it no longer has a value of 4 but has a
value of 4 lots of one hundredth which is 4 hundredths; an 8 in the first decimal place or tenths place
gives a value of 8 tenths; when an 8 is put in this 1st decimal place it no longer has a value of 8 but has
a value of 8 hundredths.
As we saw in the previous topic about whole numbers the only place that has the same face and
place value is the 1st place; it looks like a 3 and has a value of 3. And with the other whole numbers
the 7 is really 70, the 5 is really 500 and the 9 is really 9 000.
Remember that when writing decimal numbers with no whole numbers present, you always start the
number with a zero. In the example of 0.4, which means four tenths, it is simply written as 0.4; the
zero in front of the four means that there are no whole numbers present in this number. Similarly the
in the next example, 0.572, the zero also shows that the number contains no whole numbers and is
just made up of 5 tenths, 7 hundredths and 2 thousandths.
Have a go to Activity One. Remember that all numbers (both whole and decimal numbers) need to
be separated into groups of three using a space between each group of three as you have been
shown.
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Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
Expanded Notation
“Expanded Notation” is simply a way of writing a number by adding together the value of each digit:
You can see how fractions can
be included in expanded
notation in the second
example.
Example 1: 7 392 = 7 000 + 300 + 90 + 2
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Example 2: 436.58 = 400 + 30 + 6 + 10 + 100
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These values can then be further expanded to include the place value of each digit:
Example 2: 436.58 = 400 + 30 + 6 +
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= (4 x 100) + (3 x 10) + (6 x 1) + (5 x 10 ) + (8 x 100 )
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The arrows show you which expanded
notation relates to the number. 400 is
simply equal to 4 x 100.
Go to the following link to see another explanation of this:
Decimals: expanding out place value
(https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-place-value-prealg/v/expanding-out-a-decimal-by-place-value)
By analysing numbers this way we can get a good understanding of place value and an appreciation of
how place value works using decimal numbers in our Decimal Number System.
Have a go at Activity Two to practise analysing some bigger numbers according to their place value.
Expanded Notation is simply a way of writing a number by adding together the value of each digit. In
the example of 7 392, you can see that ‘7’ is not just a seven, but because of the place it is in (it is in
the fourth place), it is 7 000, and the ‘3’ is not just three, but because of the place it is in (it is in the
third place), it is 300; and similarly, the ‘9’ is not just a nine, but in fact is 90 because it is in the
second place; and the ‘2’ is just two because it is in the first place, the ones place.
Look at the second example (four hundred and thirty six point five eight), which contains decimal
numbers. Similarly, the ‘4’ is not just a four, but is four hundreds because of the place it is in. The
three is really three tens (30) and the 6 is just 6 (six ones). Looking at the numbers to the right of the
decimal point, the 5 stands for 5 tenths and the 8 is really 8 hundredths.
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Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
In the second stage of expanding the decimal number in Example 2, it is a simple matter of
multiplying each digit by its place value. 400 can then be written as 4 lots of 100. 30 can then be
written as 3 lots of 10. The 6 can be written as 6 lots of 1 as it is in the ones place. After the decimal
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point, the 5 can be written as (5 x 10 ), and the 8 can be written as (8 x 100 ).
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Go to the Decimals: expanding out place value link to see another way of looking at expanded
notation and how it relates to place value.
By analysing numbers this way we can get a good understanding of place value and an appreciation
of how place value works in our Decimal Number System.
Have a go at Activity Two to practise analysing and expanding some decimal numbers according to
their place value.
Reading Decimal Numbers
Consider this number:
The problem?
537 894 230.814 926
It may be difficult to read because the place values of the first three numbers are
not apparent by just looking at it.
The solution.
It is easier to read if we identify each of the groups of three digits:
Have a go at Activity Three to practise reading and writing some of these larger decimal numbers.
With the large amounts of places involved in some of these bigger numbers, the difficulty in reading
the numbers is establishing what the place values of the first three digits are.
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Better Math – Topic 1: Understanding Numbers
Topic 1.2 – The Decimal Number System and Place Value (Decimal Numbers)
An easy way to remember to read and write numbers with decimal places is to look at the decimal
point, and starting from the right hand side, going from the smaller place values to the bigger place
values.
The first group of three to the left of the decimal point is always the hundreds, tens and ones; the
next group is always the thousands, then millions and so on.
Once we have established what the place value of the extreme left hand group of three digits is
(millions in this case), then we can read the numbers normally from left to right which in this case is
five hundred and thirty seven million, eight hundred and ninety four thousand, two hundred and
thirty point eight one four nine two six.
Have a go at Activity Three.
Quiz
Now that you have completed Topic 1.2 about how our decimal number system works, test
your understanding of the Topic by completing the Quiz (Under Activities in the right-hand-side of
the screen).
Are you satisfied with your understanding of this topic? If not, we suggest that you go through this
topic again and then have another go at the quiz.
If you’re happy, you should now move onto Topic 1.3 which is still about understanding our number
system.
After you have worked through this topic, have a go at the Quiz (under the Activities heading) to test
your understanding of the concepts covered. The correct answers have also been provided. If you
did not score well, go through the topic again and have another go at the Quiz. If you are happy with
your score, move onto the next topic, which is still about understanding our number system.
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