Decimal numbers give us a convenient way to describe quantities that include whole numbers and parts of a whole.
You see decimals in measurements, prices, distances, weights, temperatures, sports results and many other everyday situations.
The key to understanding decimals is place value.
Once you understand what each position means, decimals become much easier to read, compare and use.
What is a decimal number?
A decimal number uses the base ten place value system to represent whole numbers and parts of a whole.
For example:
2.5
represents two wholes and five tenths.
The number:
0.5
represents five tenths and no whole units.
The decimal point
The decimal point separates the whole number places from the places representing parts smaller than one whole.
Consider:
34.27
To the left of the decimal point are the whole number places.
To the right are decimal places.
In this example:
- 3 is in the tens place
- 4 is in the ones place
- 2 is in the tenths place
- 7 is in the hundredths place
Place value continues in both directions
Whole number place value may already be familiar.
Moving left from the ones place gives:
- tens
- hundreds
- thousands
Moving right from the decimal point gives:
- tenths
- hundredths
- thousandths
Each move one place to the right represents a value one tenth as large as the previous place.
Tenths
Imagine one whole divided into ten equal parts.
Each part is one tenth.
One tenth can be written as:
1/10
or:
0.1
Seven tenths can therefore be written as:
7/10 = 0.7
Hundredths
If one whole is divided into 100 equal parts, each part is one hundredth.
One hundredth is:
1/100 = 0.01
Twenty five hundredths is:
25/100 = 0.25
Thousandths
One thousandth means one of 1,000 equal parts of a whole.
It can be written as:
1/1000 = 0.001
For example:
0.347
contains:
- 3 tenths
- 4 hundredths
- 7 thousandths
The position of a digit changes its value
Look at the digit 5 in these numbers:
5
0.5
0.05
0.005
The digit is the same, but its value changes because its position changes.
They represent:
- 5 ones
- 5 tenths
- 5 hundredths
- 5 thousandths
This is why place value matters.
Zero can hold a place
Zero is important in decimal notation.
Compare:
0.5
with:
0.05
The first number is five tenths.
The second is five hundredths.
The zero in the tenths position of 0.05 shows that there are no tenths.
Decimal numbers can be greater than one
Decimals are not limited to numbers between zero and one.
For example:
3.6
means:
3 wholes + 6 tenths
And:
12.47
means 12 wholes plus 47 hundredths.
How to read a decimal
The number:
4.72
can be read aloud as:
four point seven two
Thinking about place value, it can also be described as:
four and seventy two hundredths
Both ways can help, but the second description makes the place value especially clear.
Expanded form
Expanded form shows the value contributed by each digit.
For example:
3.47
can be written as:
3 + 0.4 + 0.07
Or using fractions:
3 + 4/10 + 7/100
This shows that a decimal is built from place values.
Decimals are numbers on a number line
Decimals have exact positions on a number line.
For example:
0 0.25 0.5 0.75 1
The number 0.5 is halfway between zero and one.
The number 0.25 is halfway between zero and 0.5.
Thinking about position on a number line can help when comparing decimal numbers.
Decimals and fractions can represent the same number
Many decimals can also be written as fractions.
For example:
0.1 = 1/10
0.5 = 5/10 = 1/2
0.25 = 25/100 = 1/4
0.75 = 75/100 = 3/4
Decimals and fractions are different ways of representing the same quantities.
Convert tenths to decimals
A fraction with denominator 10 can be written directly using the tenths place.
For example:
3/10 = 0.3
and:
9/10 = 0.9
Convert hundredths to decimals
A fraction with denominator 100 can be written using two decimal places.
For example:
7/100 = 0.07
Notice the zero in the tenths place.
Another example is:
42/100 = 0.42
Equivalent decimal forms
Adding a zero at the end of a decimal does not change its value.
For example:
0.5 = 0.50
Five tenths and fifty hundredths describe the same number.
Likewise:
0.7 = 0.70 = 0.700
The extra zeros can sometimes help when comparing numbers with different numbers of decimal places.
Do not assume more digits means a larger decimal
With whole positive numbers, more digits often indicate a larger number.
That rule cannot simply be copied to decimals.
Consider:
0.8 and 0.75
You might be tempted to think 75 makes the second number larger.
Instead, compare place values.
Write 0.8 as:
0.80
Now compare:
0.80 and 0.75
Eighty hundredths is greater than seventy five hundredths.
Therefore:
0.8 is greater than 0.75
Compare whole number parts first
Consider:
3.4 and 2.95
The first number has three whole units.
The second has only two whole units.
Therefore:
3.4 is greater than 2.95
You do not need to compare the decimal places once the whole number parts are different.
Compare tenths next
Now compare:
2.7 and 2.4
The whole number parts are both 2.
Compare the tenths.
Seven tenths is greater than four tenths.
So:
2.7 is greater than 2.4
Compare hundredths when necessary
Consider:
1.36 and 1.32
The whole number parts are equal.
The tenths are also equal.
Compare the hundredths.
Six hundredths is greater than two hundredths.
Therefore:
1.36 is greater than 1.32
Compare decimals with different lengths
Compare:
0.62 and 0.607
You can write 0.62 as:
0.620
Now compare:
0.620 and 0.607
The tenths are equal.
At the hundredths place, 2 is greater than 0.
Therefore:
0.62 is greater than 0.607
Decimals in money
Decimal notation is commonly used when recording money.
For example, a price might be written as:
12.50
The decimal places allow the main currency unit and smaller parts of that unit to be recorded together.
The exact number of smaller units depends on the currency being used.
Decimals in measurement
Metric measurements often use decimals.
For example, a length might be:
1.25 metres
A mass might be:
2.5 kilograms
A distance might be:
3.75 kilometres
Decimals make it possible to express quantities between whole units conveniently.
Everyday example: distance
Suppose one walking route is 2.4 kilometres long and another is 2.75 kilometres long.
Write the first distance as:
2.40 kilometres
Now compare:
2.40 and 2.75
The second route is longer because 2.75 is greater than 2.40.
Everyday example: measurement
Suppose two objects measure:
1.35 metres
and:
1.5 metres
Write 1.5 as:
1.50
Now compare:
1.35 and 1.50
The second measurement is greater.
Multiplying by 10 changes place value
Consider:
0.7 × 10 = 7
The digit 7 has changed from representing seven tenths to representing seven ones.
Each digit has a value ten times as large.
Another example is:
2.35 × 10 = 23.5
Thinking about place value is more useful than imagining that the decimal point itself moves.
Dividing by 10 changes place value
Consider:
8 divided by 10 = 0.8
The 8 changes from eight ones to eight tenths.
Likewise:
35 divided by 10 = 3.5
Each digit represents one tenth of its previous value.
Multiplying and dividing by 100
Multiplying by 100 changes each digit to a place worth 100 times as much.
For example:
0.24 × 100 = 24
Dividing by 100 has the opposite effect.
24 divided by 100 = 0.24
This connection helps explain why hundredths are related to division by 100.
Rounding a decimal to the nearest whole number
Rounding gives an approximate value that can be easier to use.
Consider:
6.3
The number is between 6 and 7.
It is closer to 6.
So 6.3 rounds to:
6
Now consider:
6.8
It is closer to 7.
So 6.8 rounds to:
7
Use the number line to understand rounding
Suppose you are rounding 4.6 to the nearest whole number.
The neighbouring whole numbers are 4 and 5.
The halfway point is:
4.5
Since 4.6 is above the halfway point, it is closer to 5.
Therefore:
4.6 rounds to 5
Rounding to one decimal place
Suppose you want to round:
3.47
to one decimal place.
The tenths digit is 4.
Look at the hundredths digit, which is 7.
The value is closer to 3.5 than 3.4.
Therefore:
3.47 rounds to 3.5
Rounding is not the same as an exact value
If a measurement of 3.47 metres is rounded to 3.5 metres, the original measurement has not changed.
The rounded number is an approximation.
Keep the exact value when the context requires exact information.
Use estimation to check decimal answers
Suppose someone claims:
2.8 + 3.1 = 15.9
You do not need a detailed calculation to see a problem.
2.8 is close to 3.
3.1 is also close to 3.
The total should therefore be close to 6, not close to 16.
Estimation is a useful way to detect mistakes.
Adding zeros can help organise decimal calculations
Consider the numbers:
2.5 and 1.37
You can write:
2.50 and 1.37
This makes the tenths and hundredths positions easier to see.
The value of 2.5 has not changed.
You will use this place value idea in the Practical Mathematics lesson on adding and subtracting decimal numbers.
Practice 1: Identify place value
In:
4.63
what value does the digit 6 represent?
Answer
The 6 is in the tenths place.
It represents:
6/10 = 0.6
Practice 2: Identify a hundredth
In:
2.57
which digit is in the hundredths place?
Answer
The digit 7.
Practice 3: Write a fraction as a decimal
Write:
6/10
as a decimal.
Answer
0.6
Practice 4: Write hundredths as a decimal
Write:
9/100
as a decimal.
Answer
0.09
The zero shows that there are no tenths.
Practice 5: Compare two decimals
Which is greater?
0.7 or 0.65
Answer
Write 0.7 as 0.70.
Then compare:
0.70 and 0.65
Therefore:
0.7 is greater than 0.65
Practice 6: Compare different decimal lengths
Which is greater?
1.08 or 1.8
Answer
Write 1.8 as:
1.80
Then:
1.80 is greater than 1.08
Practice 7: Find an equivalent decimal
Complete:
0.4 = 0.__
using an equivalent form with two decimal places.
Answer
0.4 = 0.40
Practice 8: Multiply by 10
Calculate:
0.36 × 10
Answer
3.6
Each digit now has a value ten times as large.
Practice 9: Divide by 100
Calculate:
45 divided by 100
Answer
0.45
Practice 10: Round to the nearest whole number
Round:
8.7
to the nearest whole number.
Answer
9
8.7 is closer to 9 than to 8.
Practice 11: Round to one decimal place
Round:
5.26
to one decimal place.
Answer
5.3
Practice 12: Check reasonableness
A calculation gives:
4.9 + 5.2 = 30.1
Is this reasonable?
Answer
No.
Both numbers are close to 5, so their sum should be close to 10.
A result near 30 cannot be reasonable.
Common decimal mistakes
- assuming a decimal with more digits must be larger
- ignoring the place occupied by a zero
- thinking 0.05 means five tenths instead of five hundredths
- forgetting that 0.5 and 0.50 have the same value
- comparing digits without first checking their place values
- treating the decimal point as if it creates a completely different number system
- forgetting that rounding produces an approximate value
- accepting a calculation without checking whether the answer is reasonable
A decimal number checklist
When working with decimal numbers, ask:
- What does each digit represent?
- Which digit is in the tenths place?
- Which digit is in the hundredths place?
- Do any zeros act as important place holders?
- Can I write an equivalent form such as 0.5 = 0.50?
- Can a fraction or number line help me understand the value?
- If I am comparing decimals, have I compared corresponding place values?
- If I rounded the number, do I remember that the result is approximate?
- Does my answer make sense compared with the original quantities?
The main idea
Decimal numbers extend the base ten place value system to represent parts smaller than one whole.
The first places to the right of the decimal point are tenths, hundredths and thousandths.
A digit has meaning because of both the digit itself and the place where it appears.
Decimals can be represented using fractions and number lines, and equivalent decimal forms such as 0.5 and 0.50 have the same value.
Understanding place value makes it easier to read, compare, round and calculate with decimal numbers.
Continue learning
Review Practical Number Sense for Everyday Mathematics to strengthen place value, estimation and reasonableness checking.
Review Fractions Explained with Everyday Examples to strengthen the connection between fractions and decimals.
Continue with Percentages and Where We Use Them to see how decimals, fractions and percentages can represent the same proportions.
You will use these decimal place value ideas again in the Practical Mathematics lesson on adding and subtracting decimal numbers.