Decimal place value is about understanding the value of each digit in a number, based on its position. It might appear pointless or easy, but it’s really important for more advanced topics like significant figures, rounding, and scientific notation.

We use a base-10 number system. This means each position has a value that’s 10 times greater than the position to its right. E.g in the number 110, the first 1 means 100, and the second 1 means 10. So the 1 to the left gives a value 10 times bigger than the one to the right (100 vs 10).

Understanding this is needed for working with decimals, where digits can represent values less than 1.

TLDR (skip if you are reading the rest of the document)

Decimal place value is all about what each digit is worth based on its position
The decimal point separates the integer part of a number (to the left) from the fractional part (to the right).
Positions to the right of the decimal point represent fractions: tenths (0.1), hundredths (0.01), thousandths (0.001), etc.
Positions to the left of the decimal point represent whole values: ones, tens, hundreds, etc.
To read a decimal number, say the whole number part, then “point”, then either say each digit individually or use the place name
To compare decimal numbers:
- Line up the decimal points
- Compare digits from left to right until you find a difference
- The number with the greater digit at the first difference is larger
Placeholder zeros can be important (0.1 ≠ 0.10) for indicating precision, but zeros at the end of a decimal don’t change the value (0.10 = 0.1).
Precision: If a number is used for a measuring the value of something 0.1 has a lower precision and means a number between 0.05 and 0.15. 0.10 is more precise, and means a number between 0.095 and 0.105.

What You Need to Know

Order positive and negative decimals
Apply the four operations (addition, subtraction, multiplication, and division) to decimals, both positive and negative, and understand place value, especially when working with very large or small numbers
Use decimal quantities where appropriate
Estimate answers and check calculations using approximation and estimation, including answers obtained using technology
Round numbers and measures to an appropriate degree of accuracy (e.g., to a specified number of decimal places or significant figures)

Key Facts

🔑 Key Fact: The decimal point separates whole numbers on the left from fractions on the right of it.

🔑 Key Fact: On the right side of the decimal point, each position you move to the right is 1/10 of the value of the position to its left. The 1 in 0.001 is 1/10 of the value of the 1 in 0.01.

🔑 Key Fact: On the left side of the decimal point, each position you move to the left is 10 times of the value of the position to its right. The 1 in 100 is 10 times the value of the 1 in 10.

🔑 Key Fact: When comparing the size of decimal numbers, compare digits in the same position, starting from the very left, moving right until you find a difference. So comparing 356.0187 and from 346.892 you would start at the left, 3 vs 3 – no difference, then we get 5 vs 4 so 356.0187 is bigger as 5 is greater than 4.

🔑 Key Fact: Adding zeros to the end of a decimal number doesn’t change its value (0.5 = 0.50, 054.2 = 54.2), but zeros between the decimal point and other digits are significant (0.05 ≠ 0.5, 10.54 ≠ 1.54)

Understanding Decimal Place Value

The decimal system is based on powers of 10. Each position represents a specific value:

Position	Thousands	Hundreds	Tens	Ones	.	Tenths	Hundredths	Thous andths
Value	1000	100	10	1	.	0.1	0.01	0.001
Power	10³	10²	10¹	10⁰	.	10⁻¹	10⁻²	10⁻³

Each place value to the right of the decimal point is a negative power of 10:

First decimal place (tenths): $10⁻¹ = \dfrac{1}{10}= 0.1$
Second decimal place (hundredths): $10⁻² = \dfrac{1}{100} = 0.01$
Third decimal place (thousandths): $10⁻³ = \dfrac{1}{1000} = 0.001$
And so on…

Example 1

Let’s break down the number 345.678:

Position	100s	10s	1s	.	0.1s	0.01s	0.001s
Digit	3	4	5	.	6	7	8

So:

3 is in the hundreds place, representing 3 × 100 = 300
4 is in the tens place, representing 4 × 10 = 40
5 is in the ones place, representing 5 × 1 = 5
6 is in the tenths place, representing 6 × 0.1 = 0.6
7 is in the hundredths place, representing 7 × 0.01 = 0.07
8 is in the thousandths place, representing 8 × 0.001 = 0.008

Therefore: 345.678=300+40+5+0.6+0.07+0.008

Reading and Writing Decimal Numbers

When reading decimal numbers aloud, there are two common approaches:

Method 1: Say each digit after the decimal point

For 3.14159:

“Three point one four one five nine”

Method 2: Use the place value name

For 0.25:

“Zero point twenty-five” becomes “Twenty-five hundredths”

For 0.075:

“Zero point zero seven five” becomes “Seventy-five thousandths”

Method 1 is far more common!

Example 2

How do you read 42.056?

Using Method 1: “Forty-two point zero five six”

Using Method 2: “Forty-two and fifty-six thousandths”

Note that we say “and” instead of “point” when using the place value name.

Comparing Decimal Numbers

To compare decimal numbers, follow these steps:

Line up the decimal points
Compare digits from left to right
The first position where the digits differ determines which number is larger
The number with the larger digit at that position is the larger number

Example 3

Compare 0.82 and 0.819

Step 1: Line up the decimal points.

0.82
0.819

Step 2: Compare digits from left to right. First decimal place: Both have 8, so we continue. Second decimal place: Both have 2, so we continue. Third decimal place: 0.82 has no digit (or can be written as 0.820), while 0.819 has 9. Since 9 > 0, we know that 0.819 > 0.82.

Example 4

Compare 1.05 and 1.5

Step 1: Line up the decimal points.

1.05
1.5  or 1.50

Step 2: Compare digits from left to right. Ones place: Both have 1, so we continue. First decimal place: 0 < 5, so 1.05 < 1.5.

Therefore, 1.5 > 1.05.

The Importance of Zeros in Decimals

Zeros in decimal numbers can serve different purposes:

Leading Zeros (Before the First Non-Zero Digit)

These zeros indicate the place value and are significant if after the decimal point.

0.05 ≠ 0.5

0.05 means “five hundredths” (5/100)
0.5 means “five tenths” (5/10 or 1/2)

Before the decimal point they aren’t significant.

0051.6 = 051.6 = 51.6

Trailing Zeros (Zeros After the Last Non-Zero Digit)

If they come after the decimal point, these zeros don’t change the mathematical value but can indicate precision in measurements.

0.50 = 0.5 mathematically, but in measurements:

0.50 indicates precision to the nearest hundredth
0.5 indicates precision to the nearest tenth

When they come before the decimal point, so between the last non-zero digit and the decimal point they do matter. So the zeros in 820 or 5400.2 do change the value. Note 820 can be written as 820.0 to help you see the first zero is after the last non-zero digit and before the decimal point.

Example 5

Understand the difference between 3.04 and 3.4

$3.04 = 3 + 0.04 = 3 + \dfrac{4}{100}$

$3.4 = 3 + 0.4 = 3 + \dfrac{4}{10} = 3 + \dfrac{40}{100}$

So the 4 in 3.4 is a value ten times larger than the 4 in 3.04.

It’s tenths in 3.4 vs hundredths in 3.04

If you write them as fractions with the same denominator:

$3.04 = \dfrac{304}{100}$

$3.4 = \dfrac{340}{100}$

Clearly $\dfrac{340}{100} > \dfrac{304}{100}$, so $3.4 > 3.04$.

Operations with Decimals

Addition and Subtraction

When adding or subtracting decimals, we need to align the decimal points to make sure we’re adding or subtracting the same place values.

Example 6

Calculate 23.45 + 7.8

Step 1: Align the decimal points.

  23.45
+  7.8
------

Step 2: If it helps, add zeros to make the numbers have the same length.

  23.45
+ 07.80
------

Step 3: Add normally, keeping the decimal point in the same position.

  23.45
+ 07.80
------
  31.25

Example 7

Calculate 15.73 – 8.9

Step 1: Align the decimal points.

  15.73
-  8.9
------

Step 2: If it helps, add zeros to make the numbers have the same length.

  15.73
- 08.90
------

Step 3: Subtract normally, keeping the decimal point in the same position.

  15.73
- 08.90
------
   6.83

Multiplication

When multiplying decimals, we don’t need to line up the decimal points. Instead, we multiply like the numbers were whole numbers, then just move the decimal point in the result.

The number of decimal places in the result is the sum of the decimal places in the original numbers.

Example 8

Calculate 2.3 × 1.4

Step 1: Multiply as whole numbers.

   23
×  14
-----
   92
  23
-----
  322

Step 2: Count the total number of decimal places in the original numbers. 2.3 has 1 decimal place 1.4 has 1 decimal place Total: 1 + 1 = 2 decimal places

Step 3: Move the decimal point in the result, counting from the right. 322 → 3.22

So 2.3 × 1.4 = 3.22

Division

Just like in multiplication, when dividing with decimals, we can convert the problem to a problem with whole numbers. Then at the end We move the decimal point in both numbers the same number of places the other way.

Example 9

Calculate 8.4 ÷ 0.2

Step 1: Multiply both numbers by 10 to make them whole numbers. 8.4 ÷ 0.2 = 84 ÷ 2

Step 2: Do the division. 84 ÷ 2 = 42

Therefore 8.4 ÷ 0.2 = 42

Example 10

Calculate 3.75 ÷ 0.25

Step 1: Multiply both numbers by 100 to make them whole numbers. 3.75 ÷ 0.25 = 375 ÷ 25

Step 2: Perform the division. 375 ÷ 25 = 15

Giving us 3.75 ÷ 0.25 = 15

Common Mistakes to Avoid

Ignoring place value alignment when adding or subtracting
- Incorrect: Adding 13.4 + 5.67 vertically without aligning decimal points
```
13.4
5.67
----
```
- Correct: Align decimal points before adding
```
13.4
 5.67
----
```
Misplacing the decimal point in multiplication
- Incorrect: 2.3 × 1.4 = 32.2 (putting decimal point in wrong position)
- Correct: Count total decimal places in original numbers (2) → 2.3 × 1.4 = 3.22
Comparing decimals digit by digit without aligning
- Incorrect: Saying 0.5 < 0.25 because 5 < 25
- Correct: Compare digits in the same place value → 0.5 = 0.50 > 0.25
Misunderstanding the role of zeros
- Incorrect: Thinking 3.02 = 3.2
- Correct: 3.02 (three and two hundredths) ≠ 3.2 (three and two tenths)

Questions

Try these questions to practice your understanding of decimal place value:

Write the value of the bold digit in 45.273.
Order these numbers from smallest to largest: 0.8, 0.08, 0.088, 0.78, 0.780.
Calculate: 5.43 + 2.7
Calculate: 8.1 – 3.95
Calculate: 2.4 × 0.15
Calculate: 4.5 ÷ 0.9
Which is greater: 0.42 or 0.409?
Convert 3.75 to a fraction in its simplest form.
Round 3.456 to: a) 1 decimal place b) 2 decimal places
A rectangle has a length of 3.8 cm and a width of 2.45 cm. Calculate its area.

Solutions

Question 1

Write the value of the underlined digit in 45.273.

The 7 is in the hundredths place. Value = 7 × 0.01 = 0.07

Question 2

Order these numbers from smallest to largest: 0.8, 0.08, 0.088, 0.78, 0.780.

Step 1: Write the numbers with the same number of decimal places. 0.8 = 0.800 0.08 = 0.080 0.088 = 0.088 0.78 = 0.780 0.780 = 0.780

Note we have two numbers that have the same value, 0.78 = 0.780

Step 2: Compare from left to right. 0.080 < 0.088 < 0.780 = 0.800

Therefore, the order from smallest to largest is: 0.08, 0.088, 0.78 = 0.780, 0.8

Question 3

Calculate: 5.43 + 2.7

  5.43
+ 2.70
------
  8.13

Therefore, 5.43 + 2.7 = 8.13

Question 4

Calculate: 8.1 – 3.95

  8.10
- 3.95
------
  4.15

Therefore, 8.1 – 3.95 = 4.15

Question 5

Calculate: 2.4 × 0.15

Step 1: Multiply as whole numbers.

   24
×  15
-----
  120
  24
-----
  360

Step 2: Count decimal places (1 in 2.4, 2 in 0.15, so 3 total). 360 → 0.360 = 0.36

Therefore, 2.4 × 0.15 = 0.36

Question 6

Calculate: 4.5 ÷ 0.9

Step 1: Multiply both by 10 to make them whole numbers. 4.5 ÷ 0.9 = 45 ÷ 9 = 5

Therefore, 4.5 ÷ 0.9 = 5

Question 7

Which is greater: 0.42 or 0.409?

Step 1: Compare digits from left to right. First decimal place: Both have 4, so continue. Second decimal place: Both have 2 and 0, but 2 > 0, so 0.42 > 0.409.

Therefore, 0.42 is greater than 0.409.

Question 8

Convert 3.75 to a fraction in its simplest form.

Step 1: Write as an improper fraction with denominator based on decimal places. $3.75=3751003.75 = \dfrac{375}{100} 3.75=100375$ (2 decimal places → denominator is 100)

Step 2: Simplify. $375100=375÷25100÷25=154\dfrac{375}{100} = \dfrac{375 \div 25}{100 \div 25} = \dfrac{15}{4} 100375=100÷25375÷25=415$

Therefore, $3.75=1543.75 = \dfrac{15}{4} 3.75=415$ in its simplest form.

Question 9

Round 3.456 to: a) 1 decimal place b) 2 decimal places

a) To 1 decimal place: Look at the second decimal place (5). Since 5 ≥ 5, round up: 3.456 ≈ 3.5 (to 1 d.p.)

b) To 2 decimal places: Look at the third decimal place (6). Since 6 > 5, round up: 3.456 ≈ 3.46 (to 2 d.p.)

Question 10

A rectangle has a length of 3.8 cm and a width of 2.45 cm. Calculate its area.

Step 1: Use the formula: Area = length × width Area = 3.8 × 2.45

Step 2: Multiply the decimals.

   380
×  245
------
  1900
 1520
760
------
 93100

Step 3: Place the decimal point (1 + 2 = 3 decimal places from right). 93100 → 9.3100 = 9.31

Therefore, the area of the rectangle is 9.31 cm².

Summary

Decimal place value is determined by the position of a digit relative to the decimal point
Positions to the left of the decimal point represent whole numbers (ones, tens, hundreds, etc.)
Positions to the right of the decimal point represent fractions (tenths, hundredths, thousandths, etc.)
When comparing decimals, compare corresponding place values from left to right
Aligning decimal points is crucial for addition and subtraction
For multiplication, count the total number of decimal places in all the original numbers
For division, convert to an equivalent problem with whole numbers when possible
Understanding place value is essential for accurate calculations, conversions, and real-world applications

Remember, a solid grasp of decimal place value is a vital part of basic maths skills that help with advanced topics like significant figures and scientific notation.