Rational Numbers - Seventh Grade
Complete Notes & Formulas
1. Converting Fractions to Decimals
Method: Division
Divide the NUMERATOR by the DENOMINATOR
Step 1: Set up division: numerator ÷ denominator
Step 2: Perform long division
Step 3: Continue until remainder is 0 or pattern repeats
a/b = a ÷ b
Example 1: Convert 3/4 to decimal
3 ÷ 4
0.75
4 ) 3.00
2 8
---
20
20
---
0Answer: 3/4 = 0.75
Example 2: Convert 2 1/5 to decimal
Method 1: Convert to improper fraction first
2 1/5 = 11/5
11 ÷ 5 = 2.2
Method 2: Convert fraction part only
1/5 = 1 ÷ 5 = 0.2
Add whole number: 2 + 0.2 = 2.2
Answer: 2 1/5 = 2.2
Types of Decimals
| Type | Description | Example |
|---|---|---|
| Terminating | Decimal ends | 3/4 = 0.75 |
| Repeating | Decimal repeats pattern | 1/3 = 0.333... = 0.3̄ |
2. Converting Decimals to Fractions
Steps for Conversion
Step 1: Write decimal over 1 (make it a fraction)
Step 2: Multiply by 10^n (n = number of decimal places)
Step 3: Simplify to lowest terms
Place Value Method
Tenths: 0.7 = 7/10
Hundredths: 0.25 = 25/100
Thousandths: 0.125 = 125/1000
Example 1: Convert 0.6 to fraction
Step 1: 0.6 = 0.6/1
Step 2: 1 decimal place, multiply by 10
(0.6 × 10)/(1 × 10) = 6/10
Step 3: Simplify: 6/10 = 3/5 (divide by 2)
Answer: 0.6 = 3/5
Example 2: Convert 2.75 to mixed number
Whole number part: 2
Decimal part: 0.75
Convert 0.75: 75/100 = 3/4
Combine: 2 3/4
Answer: 2.75 = 2 3/4
3. What are Rational Numbers?
Definition
A rational number can be expressed as
a FRACTION p/q, where:
• p and q are INTEGERS
• q ≠ 0 (denominator cannot be zero)
Rational Number = p/q
where p, q ∈ integers and q ≠ 0
Examples of Rational Numbers
Fractions: 1/2, 3/4, 5/8, 7/9
Integers: 5 = 5/1, -3 = -3/1, 0 = 0/1
Terminating Decimals: 0.5 = 1/2, 0.75 = 3/4
Repeating Decimals: 0.333... = 1/3
Mixed Numbers: 2 1/2 = 5/2
What are NOT Rational Numbers?
Irrational Numbers (cannot be written as p/q):
• π (pi) = 3.14159...
• √2 = 1.414213...
• √3, √5, √7 (non-perfect square roots)
• e = 2.71828...
4. Classifying Rational Numbers
Number System Hierarchy
Real Numbers
├─ Rational Numbers
├─ Integers
├─ Whole Numbers (0, 1, 2, 3, ...)
└─ Natural Numbers (1, 2, 3, ...)
└─ Negative Integers (-1, -2, -3, ...)
└─ Fractions (1/2, 3/4, etc.)
└─ Irrational Numbers (π, √2, etc.)
Classification Chart
| Type | Definition | Examples |
|---|---|---|
| Natural Numbers | Counting numbers | 1, 2, 3, 4, 5... |
| Whole Numbers | Natural numbers + 0 | 0, 1, 2, 3, 4... |
| Integers | Whole numbers + negatives | ..., -2, -1, 0, 1, 2... |
| Rational Numbers | Can be written as p/q | 1/2, 0.75, -3, 2.5 |
| Irrational Numbers | Cannot be written as p/q | π, √2, √3 |
5. Absolute Value of Rational Numbers
Definition
Absolute value is the DISTANCE from zero
on a number line (always positive or zero)
• Removes the negative sign
• Symbol: | |
|a| = a if a ≥ 0
|a| = -a if a < 0
Examples
|5| = 5
|-5| = 5
|3.7| = 3.7
|-3.7| = 3.7
|2/3| = 2/3
|-2/3| = 2/3
|0| = 0
6. Comparing and Ordering Rational Numbers
Steps to Compare
Step 1: Convert all numbers to SAME FORM (all decimals or all fractions)
Step 2: If fractions, find LCD and convert
Step 3: Compare the values
Step 4: Order from least to greatest or greatest to least
Comparison Symbols
> greater than
< less than
= equal to
≥ greater than or equal to
≤ less than or equal to
Example: Order -2.5, 1/2, -1, 0.75, -3/4
Step 1: Convert all to decimals
-2.5 = -2.5
1/2 = 0.5
-1 = -1
0.75 = 0.75
-3/4 = -0.75
Step 2: Order from least to greatest
-2.5 < -1 < -0.75 < 0.5 < 0.75
Answer: -2.5, -1, -3/4, 1/2, 0.75
7. Signs of Rational Numbers
Rules for Operations
Multiplication & Division:
• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative
Addition:
• Same signs: Add and keep sign
• Different signs: Subtract and use sign of larger absolute value
Quick Reference: Rational Numbers
| Concept | Formula/Rule |
|---|---|
| Fraction to Decimal | Divide numerator by denominator |
| Decimal to Fraction | Use place value, simplify |
| Rational Number | p/q where p, q are integers, q ≠ 0 |
| Absolute Value | Distance from zero (always ≥ 0) |
| Comparing | Convert to same form, then compare |
💡 Important Tips to Remember
✓ Fraction to decimal: Divide numerator by denominator
✓ Decimal to fraction: Use place value, simplify to lowest terms
✓ All integers are rational numbers (can be written as n/1)
✓ Terminating decimals are rational numbers
✓ Repeating decimals are rational numbers
✓ Non-repeating, non-terminating decimals are irrational
✓ Absolute value is always positive or zero
✓ When comparing: Convert to same form first
✓ Number line: Numbers increase as you move right
✓ Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
🧠 Memory Tricks & Strategies
Fraction to Decimal:
"Top divided by bottom, that's the way - turns fractions to decimals every day!"
Decimal to Fraction:
"Say the place, write the face - decimal to fraction, that's the case!"
Rational Numbers:
"If you can write it as a fraction neat - rational number is complete!"
Absolute Value:
"Distance from zero, always bright - absolute value is never negative, that's right!"
Number Sets:
"Natural, Whole, Integer, Rational - each set grows, that'sational!"
Comparing:
"Same form before you compare - decimals or fractions, handle with care!"
Master Rational Numbers! 🔢 ✨
Remember: Rational numbers can be expressed as fractions!