Subtracting a number from itself always equals zero.
This rule works for both positive and negative numbers.
Subtracting a negative number is the same as adding.
💡 Main Idea
When you subtract a number from itself, the result is always zero. Since this must also be true for negative numbers, subtracting a negative must turn into addition in order to keep the equation balanced.
✏️ Worked Examples — Why Subtracting a Negative Becomes Addition
Example 1 — A Number Minus Itself
Evaluate:
\(\displaystyle 9-9\)
Step 1: Recognize that the same quantity is being removed from itself.
\(\displaystyle 9-9=0\)
Step 2: Apply the additive-inverse idea.
Positive 9 and negative 9 are opposites, so together they form a zero pair.
\(\displaystyle 9+(-9)=0\)
Final Answer:
\(\boxed{0}\)
Example 2 — A Negative Number Minus Itself
Evaluate:
\(\displaystyle -9-(-9)\)
Step 1: A number subtracted from itself must equal zero.
\(\displaystyle -9-(-9)=0\)
Step 2: Rewrite subtraction as adding the opposite.
The opposite of negative 9 is positive 9.
\(\displaystyle -9-(-9)=-9+9\)
Step 3: Combine the opposites.
\(\displaystyle -9+9=0\)
This demonstrates why subtracting a negative is equivalent to adding a positive.
Final Answer:
\(\boxed{0}\)
Example 3 — Apply the Rule
Evaluate:
\(\displaystyle 7-(-4)\)
Step 1: Identify that a negative number is being subtracted.
Subtracting negative 4 means adding its opposite, positive 4.
\(\displaystyle 7-(-4)=7+4\)
Step 2: Add the positive integers.
\(\displaystyle 7+4=11\)
Interpretation: Removing a negative amount increases the value.
Final Answer:
\(\boxed{11}\)
🧠 Math Vocabulary
Subtract: To take away one number from another.
Opposite: A number the same distance from zero but on the other side.
Zero: The result when equal numbers cancel each other out.
Additive Inverse: Two numbers that add to zero.
Balance: Keeping an equation mathematically true.
📚 What You Should Already Know
Students should know how to add integers and understand that opposite numbers cancel each other out to make zero.
🚀 What Comes Next
Students will apply the rule for subtracting negatives when solving multi-step integer expressions and real-world problems involving temperature changes, elevation, and gains and losses.
