Integers - Seventh Grade
Complete Notes & Formulas
1. Understanding Integers
Definition
Integers are WHOLE NUMBERS that can be
POSITIVE, NEGATIVE, or ZERO
• NO fractions or decimals
• Represented by the symbol Z
Set of Integers
Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}
Three Types of Integers
| Type | Description | Examples |
|---|---|---|
| Positive Integers | Greater than zero | 1, 2, 3, 4, 100, 5000 |
| Negative Integers | Less than zero | −1, −2, −3, −50, −999 |
| Zero | Neither positive nor negative | 0 |
What Are NOT Integers?
✗ Fractions: 1/2, 3/4, 5/8
✗ Decimals: 2.5, −3.14, 0.75
✗ Mixed numbers: 2½, 1¾
2. Integers on Number Lines
Horizontal Number Line
• Positive integers: to the RIGHT of zero
• Negative integers: to the LEFT of zero
• Zero: in the CENTER (origin)
• Numbers INCREASE as you move RIGHT
• Numbers DECREASE as you move LEFT
Vertical Number Line
• Positive integers: ABOVE zero
• Negative integers: BELOW zero
• Zero: in the CENTER
• Numbers INCREASE as you move UP
• Numbers DECREASE as you move DOWN
Real-World Examples
Temperature: 0°C is freezing, +25°C is warm, −10°C is cold
Elevation: 0 ft is sea level, +100 ft is above, −50 ft is below
Money: +$50 is credit, −$30 is debt, $0 is zero balance
Time: +3 hours (future), 0 (now), −2 hours (past)
3. Understanding Absolute Value
Definition
Absolute value is the DISTANCE
of a number from ZERO on the number line
• Always POSITIVE or ZERO
• NEVER negative
• Symbol: | |
Formula
|x| = x, if x ≥ 0
|x| = −x, if x < 0
|0| = 0
Examples
|5| = 5 (5 is already positive)
|−5| = 5 (distance from 0 is 5)
|0| = 0 (zero is zero distance from itself)
|−100| = 100
|47| = 47
Key Point: |5| = |−5| = 5
Remember: Absolute value removes the negative sign and gives the magnitude!
4. Opposite Integers (Additive Inverse)
Definition
Opposite integers (additive inverse) are
two numbers that are the SAME DISTANCE
from zero but on OPPOSITE SIDES
• When added together, they equal ZERO
Formula
a + (−a) = 0
Opposite of a = −a
Examples
| Number | Opposite | Sum |
|---|---|---|
| 7 | −7 | 7 + (−7) = 0 |
| −15 | 15 | −15 + 15 = 0 |
| 0 | 0 | 0 + 0 = 0 |
| 100 | −100 | 100 + (−100) = 0 |
Relationship: Absolute Value & Opposites
Opposites have the SAME absolute value!
|7| = |−7| = 7
|50| = |−50| = 50
5. Quantities That Combine to Zero
Concept
When opposite quantities are combined,
they CANCEL OUT and equal ZERO
Word Problem Examples
Example 1: Temperature
Temperature rises 8°C, then falls 8°C
+8 + (−8) = 0
Net change: 0°C
Example 2: Money
Earn $50, then spend $50
+50 + (−50) = 0
Net change: $0
Example 3: Elevation
Climb 20 feet up, then descend 20 feet
+20 + (−20) = 0
Net change: 0 feet
6. Compare and Order Integers
Comparison Rules
• Positive > Zero: Any positive number is greater than 0
• Zero > Negative: 0 is greater than any negative number
• Positive > Negative: Any positive is greater than any negative
• For negatives: The one CLOSER to zero is GREATER
Inequality Symbols
> greater than
< less than
≥ greater than or equal to
≤ less than or equal to
= equal to
Examples
5 > 2 (5 is greater than 2)
0 > −3 (0 is greater than −3)
−2 > −5 (−2 is closer to 0)
−10 < −1 (−10 is farther from 0)
7 > −100 (positive > negative)
Ordering Integers
Example: Order from least to greatest: 5, −3, 0, −7, 2
Step 1: Identify negatives: −3, −7
Step 2: Order negatives (most negative first): −7, −3
Step 3: Add zero: −7, −3, 0
Step 4: Add positives in order: −7, −3, 0, 2, 5
Answer: −7, −3, 0, 2, 5
7. Integer Inequalities with Absolute Values
Key Concept
When comparing integers with absolute values,
FIRST find the absolute value,
THEN compare the results
Steps
Step 1: Find absolute value of each number
Step 2: Compare the absolute values
Step 3: Use correct inequality symbol
Examples
Example 1: Compare |−8| and |5|
|−8| = 8
|5| = 5
8 > 5
Therefore: |−8| > |5|
Example 2: Compare |−12| and |12|
|−12| = 12
|12| = 12
12 = 12
Therefore: |−12| = |12|
Example 3: Is |−7| < 10 true?
|−7| = 7
7 < 10 ✓
Yes, TRUE
Quick Reference: Integer Formulas & Rules
| Concept | Formula/Rule |
|---|---|
| Integers | Z = {..., −2, −1, 0, 1, 2, ...} |
| Absolute Value | |x| = distance from 0 (always ≥ 0) |
| Opposite/Additive Inverse | a + (−a) = 0 |
| Comparing Integers | Positive > 0 > Negative |
| Comparing Negatives | Closer to 0 is greater |
| Opposites & Absolute Value | |a| = |−a| |
💡 Important Tips to Remember
✓ Integers = whole numbers (positive, negative, or zero)
✓ NO fractions or decimals are integers
✓ Zero is neutral - neither positive nor negative
✓ Absolute value is always positive or zero
✓ Absolute value = distance from zero
✓ Opposites add to zero: a + (−a) = 0
✓ Opposites have same absolute value: |5| = |−5|
✓ On number line: right = larger, left = smaller
✓ For negatives: closer to 0 means greater
✓ −2 > −5 because −2 is closer to zero
🧠 Memory Tricks & Strategies
Integers:
"Integers are whole and complete - no fractions, no decimals to meet!"
Absolute Value:
"Absolute value is the distance you see - from zero, it's always positive or zero, you agree!"
Opposites:
"Opposites attract to make ZERO - add them up, that's the hero!"
Comparing Negatives:
"With negative numbers, here's the key - closer to zero means GREATER, you see!"
Number Line:
"On the number line, it's plain to see - RIGHT is bigger, LEFT is smaller, naturally!"
Zero:
"Zero in the middle stands alone - not positive, not negative, it's in a zone!"
Master Integers! 🔢 ➕ ➖
Remember: Integers include all whole numbers - positive, negative, and zero!