Integer Operations

Understanding Integers

Integers are whole numbers and their opposites: ..., -3, -2, -1, 0, 1, 2, 3, ...

They include:

• Positive integers: 1, 2, 3, 4...
• Zero: 0
• Negative integers: -1, -2, -3, -4...

Adding Integers

Same signs → Add and keep the sign

Example: -5 + (-3)

• Both negative
• Add: 5 + 3 = 8
• Keep negative sign
• Answer: -8

Example: 7 + 4

• Both positive
• Add: 7 + 4 = 11
• Answer: 11

Different signs → Subtract and use the sign of the larger number

Example: -8 + 5

• Different signs
• Subtract: 8 - 5 = 3
• 8 is larger and negative
• Answer: -3

Example: 9 + (-2)

• Different signs
• Subtract: 9 - 2 = 7
• 9 is larger and positive
• Answer: 7

Adding Using a Number Line

Start at the first number, move right for positive, left for negative.

Example: -2 + 5

        start        end
          ↓           ↓
... -3 -2 -1  0  1  2  3  4 ...
       →  →  →  →  →
     (move right 5)

Answer: 3

Subtracting Integers

Key rule: Subtracting is the same as adding the opposite!

Change subtraction to addition:

• a - b = a + (-b)

Example: 7 - 10

• Change to: 7 + (-10)
• Different signs, subtract: 10 - 7 = 3
• Larger number (10) is negative
• Answer: -3

Example: -4 - 6

• Change to: -4 + (-6)
• Same signs (both negative), add
• Answer: -10

Example: -3 - (-5)

• Change to: -3 + 5
• Different signs, subtract: 5 - 3 = 2
• Larger number (5) is positive
• Answer: 2

Multiplying Integers

Signs determine the answer:

Memory trick:

• "Friends with friends = friends" (positive × positive = positive)
• "Enemies with enemies = friends" (negative × negative = positive)
• "Friends with enemies = enemies" (positive × negative = negative)

Examples:

• 5 × 8 = 40 (both positive → positive)
• (-5) × (-8) = 40 (both negative → positive)
• 5 × (-8) = -40 (different → negative)
• (-5) × 8 = -40 (different → negative)

Dividing Integers

Same rules as multiplication!

Examples:

• 20 ÷ 4 = 5 (same → positive)
• (-20) ÷ (-4) = 5 (same → positive)
• 20 ÷ (-4) = -5 (different → negative)
• (-20) ÷ 4 = -5 (different → negative)

Order of Operations with Integers

PEMDAS still applies!

Example: -3 + 5 × (-2)

1. Multiply first: 5 × (-2) = -10
2. Then add: -3 + (-10) = -13 Answer: -13

Example: (-4 + 8) ÷ (-2)

1. Parentheses: -4 + 8 = 4
2. Divide: 4 ÷ (-2) = -2 Answer: -2

Real-World Applications

Temperature: -5°F increased by 12° → -5 + 12 = 7°F

Money: Owe $50 (−50), pay back $30 → -50 + 30 = -20 (still owe $20)

Elevation: 200 ft below sea level (−200), go up 350 ft → -200 + 350 = 150 ft above sea level

Time zones: 3 hours behind (−3), add 5 hours → -3 + 5 = 2 hours ahead

Practice