Understanding Our Number System
5. Patterns in the Place Value System
Learning outcomes
- I can identify patterns in powers of ten.
- I can explain what happens when numbers are multiplied by 10, 100, and 1000.
- I can explain what happens when numbers are divided by 10, 100, and 1000.
- I can recognize repeating patterns in place value charts.
- I can use place value patterns to make predictions about numbers.
Introduction
Our number system is based on the number 10. This is called the base-ten place value system.
Every time we move one place to the left in a number, the value becomes ten times larger. Every time we move one place to the right, the value becomes ten times smaller.
Recognizing these patterns makes it much easier to multiply, divide, estimate, and work with large and small numbers.
The Base-Ten Number System
Our number system uses ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The value of a digit depends on its position.
Each place is worth 10 times the place immediately to its right.
| Place Value | Value |
|---|---|
| Thousands | 1,000 |
| Hundreds | 100 |
| Tens | 10 |
| Ones | 1 |
Notice the pattern:
- 1 × 10 = 10
- 10 × 10 = 100
- 100 × 10 = 1,000
- 1,000 × 10 = 10,000
These are called powers of ten.
Powers of Ten
A power of ten is a number made by multiplying 10 by itself one or more times.
| Power | Value |
|---|---|
| 10⁰ | 1 |
| 10¹ | 10 |
| 10² | 100 |
| 10³ | 1,000 |
| 10⁴ | 10,000 |
| 10⁵ | 100,000 |
Each power of ten is ten times larger than the previous one.
Multiplying by 10, 100, and 1000
When you multiply a whole number by a power of ten, every digit moves to the left.
- ×10 → move one place left.
- ×100 → move two places left.
- ×1000 → move three places left.
The number becomes larger because each digit moves into a place worth ten times as much.
Examples
| Calculation | Answer |
|---|---|
| 6 × 10 | 60 |
| 24 × 10 | 240 |
| 35 × 100 | 3,500 |
| 482 × 100 | 48,200 |
| 7 × 1000 | 7,000 |
| 125 × 1000 | 125,000 |
Notice that the digits move, not simply "zeros are added." Thinking about place value is more important than memorizing shortcuts.
Dividing by 10, 100, and 1000
When dividing by a power of ten, every digit moves to the right.
- ÷10 → move one place right.
- ÷100 → move two places right.
- ÷1000 → move three places right.
The number becomes smaller because each digit moves into a place worth one-tenth as much.
Examples
| Calculation | Answer |
|---|---|
| 80 ÷ 10 | 8 |
| 560 ÷ 10 | 56 |
| 4,300 ÷ 100 | 43 |
| 75,000 ÷ 1000 | 75 |
| 126,000 ÷ 1000 | 126 |
Understanding the Pattern
Look at the number 347.
| Operation | Result |
|---|---|
| 347 | 347 |
| 347 × 10 | 3,470 |
| 347 × 100 | 34,700 |
| 347 × 1000 | 347,000 |
Now divide:
| Operation | Result |
|---|---|
| 347 | 347 |
| 347 ÷ 10 | 34.7 |
| 347 ÷ 100 | 3.47 |
| 347 ÷ 1000 | 0.347 |
Each multiplication moves the digits one place to the left for every factor of 10.
Each division moves the digits one place to the right for every factor of 10.
Patterns in a Place Value Chart
Every place value is related to the next.
| Place | Relationship |
|---|---|
| Ones → Tens | ×10 |
| Tens → Hundreds | ×10 |
| Hundreds → Thousands | ×10 |
| Thousands → Ten Thousands | ×10 |
Moving the opposite direction means dividing by 10 each time.
Making Predictions Using Place Value
Understanding place value patterns helps us predict answers without lengthy calculations.
Example 1
If:
42 × 10 = 420
Then:
42 × 100 = 4,200
Example 2
If:
5,600 ÷ 100 = 56
Then:
5,600 ÷ 1000 = 5.6
Example 3
If a number is multiplied by 100 twice:
35 × 100 × 100
This is the same as:
35 × 10,000
Answer:
350,000
Recognizing these patterns makes calculations much faster.
Common Mistakes
One common mistake is thinking that multiplying by 10 simply means "adding a zero."
This works for many whole numbers, but not for all numbers.
For example:
6.5 × 10 = 65
The digits have moved one place to the left.
Similarly:
650 ÷ 10 = 65
The digits have moved one place to the right.
Thinking in terms of place value works for both whole numbers and decimals.
Real-World Applications
Understanding place value patterns is useful in many situations, including:
- Converting metric units.
- Estimating calculations.
- Working with money.
- Reading population data.
- Scientific measurements.
- Engineering calculations.
- Computer programming.
- Organizing large sets of data.
Worked Examples
Example 1
Calculate:
48 × 100
Answer:
4,800
Example 2
Calculate:
7,500 ÷ 100
Answer:
75
Example 3
Complete the pattern.
8
80
800
8,000
Answer:
Each number is multiplied by 10.
Example 4
What happens to the digits when a number is divided by 100?
Answer:
Each digit moves two places to the right, making the number one hundred times smaller.
Example 5
Predict:
250 × 1000
Answer:
250,000
Did You Know?
The metric system is based on powers of ten. This makes converting between units very simple. For example, 1 kilometre = 1,000 metres, 1 metre = 100 centimetres, and 1 kilogram = 1,000 grams. Because everything is based on powers of ten, metric conversions are much easier than converting between imperial units such as feet, inches, and miles.
Key Terms
| Term | Definition |
|---|---|
| Place Value | The value of a digit determined by its position in a number. |
| Base-Ten System | A number system in which each place value is ten times the value of the place to its right. |
| Power of Ten | A number produced by multiplying 10 by itself one or more times. |
| Multiply | To increase a number by repeated groups or scaling. |
| Divide | To split a number into equal parts or reduce its size by a factor. |
Key Takeaways
- Our number system is based on powers of ten.
- Every place value is ten times greater than the place to its right.
- Multiplying by 10, 100, or 1000 moves every digit to the left by one, two, or three places.
- Dividing by 10, 100, or 1000 moves every digit to the right by one, two, or three places.
- Understanding place value patterns makes calculations faster and helps predict answers.
- Place value patterns are used throughout mathematics, science, engineering, and the metric system.