Whole Numbers and Place Value
Fifth Grade Mathematics - Complete Guide
📊 Place Value System
Understanding Place Value:
Place value is the value of each digit in a number based on its position[web:32].
Place Value Chart (up to Millions):
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Example: 3,456,789 = Three million, four hundred fifty-six thousand, seven hundred eighty-nine[web:26][web:35]
Place Value Formula:
Value of Digit = Digit × Place Value
Example: In 5,678, the value of 5 is \( 5 \times 1000 = 5000 \)[web:32]
🔢 Standard and Expanded Form
Three Forms of Numbers[web:23][web:25]:
- Standard Form: The usual way we write numbers (e.g., 45,678)
- Expanded Form: Numbers written as sum of place values
- Word Form: Numbers written in words
Conversion Formulas:
Formula 1: Standard to Expanded Form
Break each digit × its place value[web:24]
Example:
Standard Form: 456,789
Expanded Form: \( 400{,}000 + 50{,}000 + 6{,}000 + 700 + 80 + 9 \)
Formula 2: Expanded to Standard Form
Add all place values together[web:25]
Example:
Expanded Form: \( 300{,}000 + 20{,}000 + 5{,}000 + 400 + 60 + 7 \)
Standard Form: 325,467
Formula 3: Expanded Product Form
Write as digit × power of 10[web:23]
Example:
Standard Form: 7,842
Expanded Product Form:
\( (7 \times 1000) + (8 \times 100) + (4 \times 10) + (2 \times 1) \)
Formula 4: Exponential Form
Use powers of 10 with exponents[web:23]
Example:
Standard Form: 56,234
Exponential Form: \( (5 \times 10^4) + (6 \times 10^3) + (2 \times 10^2) + (3 \times 10^1) + (4 \times 10^0) \)
🔗 Relationship Between Place Values
Place Value Relationships[web:32]:
Formula 1: Moving LEFT (×10 Rule)
Each place to the LEFT is \( \times 10 \)
- Ones → Tens: \( 1 \times 10 = 10 \)
- Tens → Hundreds: \( 10 \times 10 = 100 \)
- Hundreds → Thousands: \( 100 \times 10 = 1{,}000 \)
- Thousands → Ten Thousands: \( 1{,}000 \times 10 = 10{,}000 \)
- Ten Thousands → Hundred Thousands: \( 10{,}000 \times 10 = 100{,}000 \)
- Hundred Thousands → Millions: \( 100{,}000 \times 10 = 1{,}000{,}000 \)
Formula 2: Moving RIGHT (÷10 Rule)
Each place to the RIGHT is \( \div 10 \)
- Millions → Hundred Thousands: \( 1{,}000{,}000 \div 10 = 100{,}000 \)
- Hundred Thousands → Ten Thousands: \( 100{,}000 \div 10 = 10{,}000 \)
- Ten Thousands → Thousands: \( 10{,}000 \div 10 = 1{,}000 \)
- Thousands → Hundreds: \( 1{,}000 \div 10 = 100 \)
- Hundreds → Tens: \( 100 \div 10 = 10 \)
- Tens → Ones: \( 10 \div 10 = 1 \)
Formula 3: Comparison Formula
Higher Place Value = Lower Place Value × 10
Example: 1 thousand = 10 hundreds = 100 tens = 1,000 ones
✍️ Write Numbers in Words
Rules for Writing Numbers in Words[web:25][web:32]:
Step-by-Step Process:
- Start from the left (highest place value)
- Read the number in each period (millions, thousands, ones)
- Say the period name after each group
- Use commas to separate periods
- Use hyphens for compound numbers (21-99)
Examples:
Standard Form: 456
Word Form: Four hundred fifty-six
Standard Form: 7,834
Word Form: Seven thousand, eight hundred thirty-four
Standard Form: 523,067
Word Form: Five hundred twenty-three thousand, sixty-seven
Standard Form: 2,456,789
Word Form: Two million, four hundred fifty-six thousand, seven hundred eighty-nine
📝 Spelling Number Words (Up to 1 Million)
Key Number Words to Memorize:
| Number | Word | Number | Word |
|---|---|---|---|
| 1 | one | 11 | eleven |
| 2 | two | 12 | twelve |
| 3 | three | 13 | thirteen |
| 4 | four | 14 | fourteen |
| 5 | five | 15 | fifteen |
| 6 | six | 20 | twenty |
| 7 | seven | 30 | thirty |
| 8 | eight | 40 | forty |
| 9 | nine | 50 | fifty |
| 10 | ten | 100 | hundred |
Important Place Value Words:
- 1,000 = thousand
- 10,000 = ten thousand
- 100,000 = hundred thousand
- 1,000,000 = million
🏛️ Roman Numerals (I, V, X, L, C, D, M)
Basic Roman Numeral Symbols[web:27][web:33]:
| Roman Numeral | Value | Roman Numeral | Value |
|---|---|---|---|
| I | 1 | C | 100 |
| V | 5 | D | 500 |
| X | 10 | M | 1,000 |
| L | 50 | — | — |
Roman Numeral Rules & Formulas[web:27][web:30]:
Rule 1: Addition Rule
If smaller or equal value comes AFTER, ADD
Examples:
- VI = \( 5 + 1 = 6 \)
- VII = \( 5 + 1 + 1 = 7 \)
- LXXX = \( 50 + 10 + 10 + 10 = 80 \)
- MCCC = \( 1000 + 100 + 100 + 100 = 1300 \)
Rule 2: Subtraction Rule
If smaller value comes BEFORE larger, SUBTRACT
Examples:
- IV = \( 5 - 1 = 4 \)
- IX = \( 10 - 1 = 9 \)
- XL = \( 50 - 10 = 40 \)
- XC = \( 100 - 10 = 90 \)
- CD = \( 500 - 100 = 400 \)
- CM = \( 1000 - 100 = 900 \)
Rule 3: Repetition Rule
A symbol can repeat up to 3 times maximum[web:27]
Valid Examples:
- III = 3 ✓
- XXX = 30 ✓
- CCC = 300 ✓
- MMM = 3,000 ✓
Invalid: IIII ✗ (use IV instead)
Rule 4: Non-Repeatable Symbols
V, L, D cannot be repeated[web:33]
Use: 10 = X (not VV), 100 = C (not LL), 1000 = M (not DD)
Common Roman Numeral Conversions:
| Number | Roman | Number | Roman | Number | Roman |
|---|---|---|---|---|---|
| 1 | I | 20 | XX | 100 | C |
| 2 | II | 30 | XXX | 200 | CC |
| 3 | III | 40 | XL | 300 | CCC |
| 4 | IV | 50 | L | 400 | CD |
| 5 | V | 60 | LX | 500 | D |
| 6 | VI | 70 | LXX | 600 | DC |
| 7 | VII | 80 | LXXX | 700 | DCC |
| 8 | VIII | 90 | XC | 800 | DCCC |
| 9 | IX | 100 | C | 900 | CM |
| 10 | X | 1,000 | M |
Complex Roman Numeral Examples[web:30]:
MCMXCIV = \( 1000 + (1000-100) + (100-10) + (5-1) = 1000 + 900 + 90 + 4 = 1994 \)
MMXXV = \( 1000 + 1000 + 10 + 10 + 5 = 2025 \)
CDXLVIII = \( (500-100) + (50-10) + 5 + 1 + 1 + 1 = 400 + 40 + 8 = 448 \)
⚡ Quick Reference Formulas
Master Formula Chart:
| Concept | Formula |
|---|---|
| Place Value | Digit × Position Value |
| Standard Form | Regular number format |
| Expanded Form | Sum of (digit × place value) |
| Place Relationship Left | Multiply by 10 |
| Place Relationship Right | Divide by 10 |
| Roman Addition | Smaller after = Add |
| Roman Subtraction | Smaller before = Subtract |
📝 Practice Examples
Worked Examples[web:25]:
Example 1: Complete Conversion
Number: 847,652
Expanded Form: \( 800{,}000 + 40{,}000 + 7{,}000 + 600 + 50 + 2 \)
Product Form: \( (8 \times 100{,}000) + (4 \times 10{,}000) + (7 \times 1{,}000) + (6 \times 100) + (5 \times 10) + (2 \times 1) \)
Word Form: Eight hundred forty-seven thousand, six hundred fifty-two
Example 2: Place Value Relationships
Question: In 456,789, what is the relationship between the two 4s if there's a 4 in the ten thousands place?
Answer: The 4 is in the hundred thousands place = \( 400{,}000 \). If we had a 4 in ten thousands = \( 40{,}000 \), the relationship would be \( 400{,}000 \div 40{,}000 = 10 \) times
Example 3: Roman Numeral Conversion
Question: Convert MCMXLVII to standard form
Solution:
M = 1000
CM = \( 1000 - 100 = 900 \)
XL = \( 50 - 10 = 40 \)
VII = \( 5 + 1 + 1 = 7 \)
Answer: \( 1000 + 900 + 40 + 7 = 1947 \)
🌟 Key Takeaways
- Place value determines the value of each digit in a number[web:32]
- Each place is 10 times the value of the place to its right[web:32]
- Expanded form breaks numbers into place value sums[web:23][web:24]
- Standard form is the regular way we write numbers[web:25]
- Word form writes numbers using words with hyphens for 21-99[web:25]
- Roman numerals use seven symbols: I, V, X, L, C, D, M[web:27][web:33]
- Roman numerals add when smaller values follow, subtract when they come before[web:27][web:30]