Understanding Place Value
What is Place Value?
Place value refers to the value of a digit based on its position in a number. Each position in a number has a value that is 10 times the position to its right (in the base-10 system).
For example, in the number 5,274:
- The digit 4 is in the ones place, so its place value is 4 × 1 = 4
- The digit 7 is in the tens place, so its place value is 7 × 10 = 70
- The digit 2 is in the hundreds place, so its place value is 2 × 100 = 200
- The digit 5 is in the thousands place, so its place value is 5 × 1,000 = 5,000
Place Value Chart
| Thousands | Ones | Decimals | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths | ||
| 100,000 | 10,000 | 1,000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 | ||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| The number above is 123,456.789 | ||||||||||
Different Ways to Solve Place Value Problems
1. Expanded Form Method
Express a number by showing the value of each digit.
Example: 3,475
Expanded form: 3,000 + 400 + 70 + 5
= (3 × 1,000) + (4 × 100) + (7 × 10) + (5 × 1)
2. Place Value Disks/Chips Method
Use physical or visual representations to show different place values.
Example: Represent 324 with place value disks
Hundreds
Tens
Ones
3. Base-10 Blocks Method
Use visual blocks representing ones, tens, hundreds, and thousands.
Example: Represent 1,234 with base-10 blocks
4. Place Value Arrow Cards Method
Use cards showing place values that can be combined to create numbers.
Example: Create 358 with arrow cards
5. Word Form Method
Express numbers in words to understand place value.
Example: 2,539
Word form: Two thousand, five hundred thirty-nine
Place Value with Decimals
Place value extends to numbers less than 1 through decimal places.
For example, in the number 42.357:
- The digit 7 is in the thousandths place, so its place value is 7 × 0.001 = 0.007
- The digit 5 is in the hundredths place, so its place value is 5 × 0.01 = 0.05
- The digit 3 is in the tenths place, so its place value is 3 × 0.1 = 0.3
- The digit 2 is in the ones place, so its place value is 2 × 1 = 2
- The digit 4 is in the tens place, so its place value is 4 × 10 = 40
Different Number Systems
1. Base-10 (Decimal)
Our standard system using digits 0-9. Place values are powers of 10 (ones, tens, hundreds, etc.).
2. Base-2 (Binary)
Used in computing. Only uses 0 and 1. Place values are powers of 2 (1, 2, 4, 8, 16, etc.).
Example: The binary number 1011
= 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
= 8 + 0 + 2 + 1 = 11 in decimal
3. Base-16 (Hexadecimal)
Uses digits 0-9 and letters A-F. Place values are powers of 16.
Example: The hexadecimal number 2A
= 2 × 16¹ + 10 × 16⁰ (A represents 10)
= 32 + 10 = 42 in decimal
Common Place Value Problems and Solutions
1. Finding the Value of a Digit
Problem: What is the value of 7 in 5,728?
Solution: 7 is in the hundreds place, so its value is 7 × 100 = 700
2. Comparing Numbers
Problem: Compare 3,542 and 3,524.
Solution: Start comparing from the leftmost digit:
- Thousands place: Both have 3 (equal)
- Hundreds place: Both have 5 (equal)
- Tens place: 4 in 3,542 vs 2 in 3,524 (4 > 2)
- Therefore, 3,542 > 3,524
3. Rounding Numbers
Problem: Round 3,782 to the nearest hundred.
Solution:
- Look at the digit in the tens place (8)
- Since 8 ≥ 5, round up
- 3,782 rounded to the nearest hundred is 3,800
4. Ordering Numbers
Problem: Order 3.45, 3.54, 3.4, and 3.405 from least to greatest.
Solution: Compare place values from left to right:
3.4 < 3.405 < 3.45 < 3.54