Directed Numbers
Subtracting Directed Numbers
⭐ The Golden Rule of Subtraction
Subtraction = adding the opposite!
\(a - b = a + (-b)\)
💡 In short: instead of subtracting a number, add its opposite!
This turns every subtraction into an addition — which we already know!
✏️ Basic Examples
Example 1: \(8 - 5 = ?\)
Convert to addition: \(8 - 5 = 8 + (-5)\)
Different signs → subtract: \(8 - 5 = 3\)
Larger is positive → result is positive
Answer: 3
Example 2: \(5 - 8 = ?\)
Convert to addition: \(5 - 8 = 5 + (-8)\)
Different signs → subtract: \(8 - 5 = 3\)
Larger is negative → result is negative
Answer: −3
➖➖ Subtracting a Negative Number (Surprise!)
Minus of a minus = plus!
\(a - (-b) = a + b\)
✏️ Example 1: \(7 - (-3) = ?\)
The opposite of (−3) is 3
\(7 - (-3) = 7 + 3 = 10\)
Answer: 10
✏️ Example 2: \((-5) - (-8) = ?\)
The opposite of (−8) is 8
\((-5) - (-8) = (-5) + 8\)
Different signs → subtract: \(8 - 5 = 3\), larger is positive
Answer: 3
✏️ Example 3: \((-2) - (-10) = ?\)
\((-2) - (-10) = (-2) + 10 = 8\)
Answer: 8
➖ Subtracting a Positive from a Negative
✏️ Example 1: \((-7) - 3 = ?\)
Convert to addition: \((-7) - 3 = (-7) + (-3)\)
Both negative → add, result is negative
\((-7) + (-3) = -10\)
Answer: −10
✏️ Example 2: \((-4) - 6 = ?\)
\((-4) - 6 = (-4) + (-6) = -10\)
Answer: −10
🔄 Quick Conversion Table
| What you see | Convert to… | Example |
|---|---|---|
| + + | + | \(5 + (+3) = 5 + 3 = 8\) |
| + - | - | \(5 + (-3) = 5 - 3 = 2\) |
| - + | - | \(5 - (+3) = 5 - 3 = 2\) |
| - - | + | \(5 - (-3) = 5 + 3 = 8\) |
💡 Rule of thumb:
Same signs → plus (+) | Different signs → minus (−)
📝 Practice
Solve:
\(10 - 15 = ?\)
= 10 + (-15) = -5
\((-6) - (-6) = ?\)
= (-6) + 6 = 0
\((-3) - 7 = ?\)
= (-3) + (-7) = -10
\(4 - (-9) = ?\)
= 4 + 9 = 13
📝 Summary
Subtraction = add the opposite
\(a - b = a + (-b)\)
Minus minus = plus!