Rational Numbers - Sixth Grade
Complete Notes & Formulas
1. What are Rational Numbers?
Definition
A rational number is any number that can be written as
p/q
where p and q are INTEGERS and q ≠ 0
Set Notation
ℚ = Set of all Rational Numbers
Examples of Rational Numbers
• Integers: 5 (= 5/1), −3 (= −3/1), 0 (= 0/1)
• Fractions: 2/3, −5/7, 11/4
• Terminating Decimals: 0.75 (= 3/4), −2.5 (= −5/2)
• Repeating Decimals: 0.333... (= 1/3), 0.666... (= 2/3)
• Mixed Numbers: 2½ (= 5/2), −3¾ (= −15/4)
Remember: ALL integers, fractions, and decimals (terminating/repeating) are rational numbers!
2. Classifying Numbers
Number Sets Hierarchy
Natural Numbers (ℕ) ⊂ Whole Numbers (W) ⊂ Integers (ℤ) ⊂ Rational Numbers (ℚ)
| Number Set | Definition | Examples |
|---|---|---|
| Natural (ℕ) | Counting numbers | 1, 2, 3, 4, 5, ... |
| Whole (W) | Natural + 0 | 0, 1, 2, 3, 4, ... |
| Integers (ℤ) | Whole + negatives | ..., −2, −1, 0, 1, 2, ... |
| Rational (ℚ) | p/q where q ≠ 0 | 1/2, −3, 0.75, 2⅓ |
Classification Examples
Example 1: Classify 5
✓ Natural (counting number)
✓ Whole (natural + 0)
✓ Integer (whole + negatives)
✓ Rational (5 = 5/1)
Example 2: Classify −3/4
✗ Not Natural
✗ Not Whole
✗ Not Integer
✓ Rational (fraction form)
3. Rational Numbers on Number Lines
Key Points
• Every rational number has a specific location on the number line
• Numbers increase from LEFT to RIGHT
• Positive numbers are to the RIGHT of 0
• Negative numbers are to the LEFT of 0
Plotting Fractions
Example: Plot 3/4 on a number line
Step 1: Divide the space between 0 and 1 into 4 equal parts
Step 2: Count 3 parts from 0
Step 3: Mark the point at 3/4
4. Comparing Rational Numbers
Methods to Compare
Method 1: Convert all to decimals and compare
Method 2: Find common denominator (for fractions)
Method 3: Use number line position
Comparison Symbols
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | 3/4 > 1/2 |
| < | Less than | −2/3 < 1/3 |
| = | Equal to | 1/2 = 2/4 |
Example: Compare 2/3 and 3/4
Method 1: Convert to decimals
2/3 = 0.666...
3/4 = 0.75
0.666... < 0.75
Answer: 2/3 < 3/4
Rules for Comparing
• Positive > 0 > Negative (always)
• For positive numbers: larger value = greater
• For negative numbers: closer to 0 = greater
5. Ordering Rational Numbers
Steps to Order
Step 1: Convert all numbers to same form (decimals or fractions)
Step 2: Compare values
Step 3: Arrange from least to greatest (or vice versa)
Example: Order from Least to Greatest
Problem: Order: 1/2, −0.75, 3/4, −1, 0.2
Step 1: Convert to decimals
1/2 = 0.5
−0.75 = −0.75
3/4 = 0.75
−1 = −1.0
0.2 = 0.2
Step 2: Order
−1.0 < −0.75 < 0.2 < 0.5 < 0.75
Answer: −1, −0.75, 0.2, 1/2, 3/4
6. Rational Numbers: Equal or Not Equal
Testing Equality
Method 1: Convert both to decimals and compare
Method 2: Cross-multiply fractions: a/b = c/d if a×d = b×c
Method 3: Simplify both fractions to lowest terms
Example: Are 2/3 and 4/6 equal?
Method 1: Decimals
2/3 = 0.666...
4/6 = 0.666...
Method 2: Cross-multiply
2 × 6 = 12
3 × 4 = 12
12 = 12 ✓
Answer: YES, they are equal!
7. Opposites of Rational Numbers
Definition
Opposites are the SAME distance from 0
but on OPPOSITE sides of the number line
Formula
Opposite of a = −a
Opposite of −a = a
Examples
| Number | Opposite |
|---|---|
| 2/3 | −2/3 |
| −4.5 | 4.5 |
| 0 | 0 |
| 3¾ | −3¾ |
8. Absolute Value of Rational Numbers
Definition
Absolute value = DISTANCE from 0
Always POSITIVE or ZERO
Notation
|a| = absolute value of a
Examples
|3/4| = 3/4
|−2.5| = 2.5
|0| = 0
|−7/8| = 7/8
|4.75| = 4.75
9. Rational Numbers: Find the Sign
Rules for Operations
Addition/Subtraction:
• Same signs: Keep the sign
• Different signs: Use sign of larger absolute value
Multiplication/Division:
• Same signs: Result is POSITIVE
• Different signs: Result is NEGATIVE
Examples
• 2/3 × (−4/5) = negative (different signs)
• −1.5 ÷ (−3) = positive (same signs)
• 3/4 + 1/2 = positive (both positive)
• −5/6 − 2/3 = negative (both negative)
10. Word Problems
Example 1: Comparing Heights
Problem: Sarah is 5¾ feet tall, and Tom is 5.6 feet tall. Who is taller?
Convert to decimals:
5¾ = 5.75 feet
5.6 = 5.60 feet
5.75 > 5.60
Answer: Sarah is taller
Example 2: Temperature
Problem: Order these temperatures from coldest to warmest: 2.5°C, −3/4°C, 0°C, −1.2°C
Convert: −3/4 = −0.75°C
Order: −1.2 < −0.75 < 0 < 2.5
Answer: −1.2°C, −3/4°C, 0°C, 2.5°C
Quick Reference: Rational Numbers
| Concept | Key Point |
|---|---|
| Definition | p/q where q ≠ 0 |
| Includes | Integers, fractions, terminating/repeating decimals |
| Opposite | Same distance from 0, opposite side |
| Absolute Value | Distance from 0 (always positive) |
| Comparing | Convert to same form (decimals or fractions) |
💡 Important Tips to Remember
✓ Rational = can be written as p/q where q ≠ 0
✓ All integers ARE rational (can write as n/1)
✓ Decimals: Terminating and repeating are rational
✓ To compare: Convert to same form (usually decimals)
✓ On number line: Right = greater, Left = smaller
✓ Opposite: Change the sign (positive ↔ negative)
✓ Absolute value: Always positive (distance from 0)
✓ Equivalent fractions: Use cross-multiplication to check
✓ Natural ⊂ Whole ⊂ Integer ⊂ Rational
✓ Practice converting between fractions and decimals
🧠 Memory Tricks & Strategies
Rational Numbers:
"Rational is a ratio, like a fraction you know!"
Number Sets:
"Natural, Whole, Integers grow - Rationals include them all, you know!"
Comparing:
"Decimals make it easy to see, which number is greater - 1, 2, 3!"
Opposites:
"Flip the sign, you'll be fine - opposite found every time!"
Absolute Value:
"Distance from zero, never negative - always positive, quite definitive!"
Number Line:
"Right is more, left is less - number line helps you pass the test!"
Master Rational Numbers! 🔢 📊 🎯
Remember: If it can be a fraction, it's rational!