Negative Numbers
Below zero isn't scary — negative numbers model debt, temperature, elevation, and more.
What Are Negative Numbers?
Negative numbers are numbers less than zero. On the number line:
... -5 -4 -3 -2 -1 0 1 2 3 4 5 ...
Numbers get smaller as you go left and larger as you go right. So -5 < -2 < 0 < 3.
Absolute Value
The absolute value of a number is its distance from zero, always positive.
- |5| = 5
- |-5| = 5
- |0| = 0
Think of it as "remove the negative sign."
Adding with Negatives
Same Signs: Add and keep the sign
Example 1: -4 + (-3) = -7 (both negative, add magnitudes, result negative)
Example 2: 5 + 8 = 13 (both positive)
Different Signs: Subtract and keep the sign of the larger absolute value
Example 3: -9 + 4 = -5 (|-9| > |4|, so result is negative, 9-4=5)
Example 4: 7 + (-3) = 4 (|7| > |-3|, so result is positive, 7-3=4)
Subtracting with Negatives
Rule: Subtracting is the same as adding the opposite.
a - b = a + (-b) and a - (-b) = a + b
Example 5: 5 - 8
= 5 + (-8) = -3
Example 6: -3 - (-7)
= -3 + 7 = 4 (subtracting negative = adding positive!)
Example 7: -4 - 6
= -4 + (-6) = -10
Multiplying with Negatives
The sign rules are simple:
- Positive × Positive = Positive: 3 × 4 = 12
- Negative × Positive = Negative: -3 × 4 = -12
- Positive × Negative = Negative: 3 × (-4) = -12
- Negative × Negative = Positive: -3 × (-4) = 12
Easy rule: Same signs → positive. Different signs → negative.
Dividing with Negatives
Same rules as multiplication:
- 12 ÷ 3 = 4 (positive ÷ positive)
- -12 ÷ 3 = -4 (negative ÷ positive)
- 12 ÷ (-3) = -4 (positive ÷ negative)
- -12 ÷ (-3) = 4 (negative ÷ negative)
Comparing Negative Numbers
On the number line, further right = bigger:
- -1 > -5 (negative 1 is closer to zero)
- -100 < -1 (negative 100 is way left)
- Any positive number > any negative number
- 0 > any negative number
Real-World Uses of Negatives
- Temperature: -40°F is brutally cold (fun fact: -40°F = -40°C!)
- Money/Debt: A balance of -$500 means you owe $500
- Elevation: The Dead Sea is at -1,412 feet (below sea level)
- Sports: In golf, -3 means 3 under par (that's good!)
- Stock market: -2.5% means the market dropped 2.5%
Double Negatives
Two negatives make a positive — but ONLY in multiplication/division:
- Multiplication: -(-5) = +5 ✓
- Addition: -5 + (-3) = -8 (NOT positive!) ✗
🎯 Try It Yourself
Test your understanding with these practice problems.
1. What is -7 + 3?
💡 Hint: Start at -7 on the number line, move 3 to the right
2. What is -5 - (-8)?
💡 Hint: Subtracting a negative = adding: -5 + 8
3. What is -6 × 4?
💡 Hint: Negative × positive = negative
4. What is -12 ÷ -3?
💡 Hint: Negative ÷ negative = positive
5. Put in order from least to greatest: 3, -7, 0, -2, 5
💡 Hint: More negative = further left on number line = smaller
⚠️ Common Mistakes
- ✗Thinking -3 is greater than -1. On the number line, -3 is further left, so -3 < -1.
- ✗Confusing 'subtracting a negative' with 'adding a negative'. 5 - (-3) = 5 + 3 = 8. But 5 + (-3) = 5 - 3 = 2.
- ✗Forgetting the sign rules: negative × negative = POSITIVE, not negative.
- ✗Writing -3² as 9 instead of -9. The negative isn't squared unless it's (-3)² = 9.
- ✗Thinking that two negatives always make a positive. Only true for multiplication/division, NOT addition.
🌍 Real Life Example
Temperature Changes
It's -8°F outside in January. The weather forecast says it will warm up 15 degrees by noon, then drop 20 degrees overnight. Noon temperature: -8 + 15 = 7°F. Overnight: 7 - 20 = -13°F. Negative numbers are essential for weather, altitude (Death Valley is -282 feet), bank overdrafts (-$50 balance), and sports scores (golf is 'under par').
💡 Key Takeaway
Negative numbers extend the number line below zero. For addition/subtraction, think of the number line — adding moves right, subtracting moves left. For multiplication/division: same signs = positive, different signs = negative. Subtracting a negative is the same as adding.