Integer Operations: Subtraction
Subtracting integers is easier than you think - just add the opposite! This powerful shortcut transforms every subtraction problem into an addition problem you already know how to solve.
Concept
Subtraction of integers follows one golden rule that simplifies everything! **THE GOLDEN RULE OF INTEGER SUBTRACTION:** To subtract an integer, ADD ITS OPPOSITE! a - b = a + (-b) **WHAT IS THE OPPOSITE?** The opposite of a number has the same distance from zero but different sign: - Opposite of 5 is -5 - Opposite of -3 is 3 - Opposite of 0 is 0 **THE TWO-STEP METHOD:** **Step 1: Keep-Change-Change (KCC)** - KEEP the first number - CHANGE subtraction to addition - CHANGE the sign of the second number **Step 2: Use integer addition rules** **EXAMPLES:** Example 1: 8 - 5 - Keep: 8 - Change - to + - Change 5 to -5 - Result: 8 + (-5) = 3 Example 2: 4 - 9 - Keep: 4 - Change - to + - Change 9 to -9 - Result: 4 + (-9) = -5 Example 3: -7 - 3 - Keep: -7 - Change - to + - Change 3 to -3 - Result: -7 + (-3) = -10 Example 4: -6 - (-10) - Keep: -6 - Change - to + - Change -10 to 10 (opposite of -10 is 10!) - Result: -6 + 10 = 4 **IMPORTANT PATTERN:** Subtracting a negative = Adding a positive! - 5 - (-3) = 5 + 3 = 8 - -2 - (-7) = -2 + 7 = 5 Two negatives make a positive in subtraction! **NUMBER LINE VISUALIZATION:** For a - b: - Start at a - If b is positive, move LEFT (subtract means go left) - If b is negative, move RIGHT (subtracting negative = going right) **REAL-WORLD APPLICATIONS:** - Temperature: -3°F - 5°F = -8°F (gets colder) - Elevation: Sea level (0) - underwater depth (-50) = 50 ft difference - Time zones: If it's 3 PM here and 5 hours earlier there → 3 - 5 = -2 (which means 2 PM yesterday... or use 15 - 5 = 10 AM) - Debt: Owing $20 and removing a debt of $30 → -20 - (-30) = $10 (you're $10 ahead!)
Try it
Master integer subtraction through practice! **BASIC SUBTRACTION:** 1. 15 - 8 = ? 2. 7 - 12 = ? 3. 20 - 25 = ? 4. 3 - 15 = ? **SUBTRACTING NEGATIVES:** 5. 10 - (-6) = ? 6. 5 - (-5) = ? 7. -3 - (-8) = ? 8. -12 - (-7) = ? **NEGATIVE MINUS POSITIVE:** 9. -6 - 4 = ? 10. -15 - 9 = ? 11. -2 - 20 = ? **MIXED PRACTICE:** 12. 0 - 8 = ? 13. 0 - (-8) = ? 14. -9 - 0 = ? 15. 14 - (-6) = ? **WORD PROBLEMS:** 16. The temperature was 6°F. It dropped to -4°F. What was the change in temperature? (Hint: Find -4 - 6 or calculate the difference) 17. A hiker starts at elevation 200 feet and descends to -50 feet (below sea level). What is the total change in elevation? 18. Your checking account has $25. You make a payment of $40. What's your new balance? (Express as an integer) **CHALLENGE:** 19. Evaluate: 15 - (-8) - 12 - (-5) (Rewrite each subtraction as addition first!) 20. If a - b = 10 and b = -3, what is a? **THINK DEEPER:** 21. Why does subtracting a negative number give the same result as adding a positive number? Explain using a real-world example.