The Opposite of an Opposite
The opposite of an opposite returns you to the original number, expressed as -(-a) = a. For example, -(-11) = 11 because negating a negative number produces its positive counterpart. This rule also applies in subtraction: -5 - (-9) becomes -5 + 9 = 4, since subtracting a negative is the same as adding a positive. Covered in Chapter 7 of Saxon Math Course 2 for 7th grade math, this concept is critical for simplifying expressions with multiple negative signs and for building fluency with integer operations.
Key Concepts
Property Taking the opposite of an opposite brings you back to the original number. $$ ( a) = a$$.
Examples The expression $ ( 11)$ simplifies directly to $11$. To solve $ 5 ( 9)$, we think addition: $( 5) + [ ( 9)] = ( 5) + 9 = 4$. Simplify $[ ( 20)] + ( 12)$ becomes $20 + ( 12) = 8$.
Explanation Think of the minus sign as an 'opposite' command that makes you turn around 180 degrees. If you get two 'opposite' commands in a row, you'll spin around twice and end up facing the same direction you started! So, the opposite of negative five, or $ ( 5)$, just cancels out and brings you right back to 5.
Common Questions
What is the opposite of an opposite in math?
The opposite of an opposite brings you back to the original number. Mathematically, -(-a) = a. For instance, -(-11) = 11. Two negatives cancel each other out, just like making a U-turn twice puts you back in the original direction.
How do you simplify double negatives in math?
Replace the two negative signs with a positive. For -(-9), the answer is simply 9. In expressions like -5 - (-9), rewrite as -5 + 9 = 4. The double negative becomes addition.
Why does subtracting a negative equal adding?
Subtracting means removing, and a negative is already a reduction. Removing a reduction is the same as adding. Think of it as taking away a debt: if someone cancels your 9-dollar debt, you are effectively 9 dollars richer.
How do you simplify expressions with multiple negatives?
Work from the inside out. For [-(-20)] + (-12), first simplify -(-20) = 20, then add: 20 + (-12) = 8. Apply the double-negative rule one pair at a time.
What are common mistakes with double negatives?
Students sometimes forget that two negatives make a positive, writing -(-5) as -5 instead of 5. Another mistake is not recognizing subtraction of a negative as a double negative: 8 - (-3) should be 8 + 3 = 11, not 8 - 3 = 5.
When do students learn about double negatives?
This concept is covered in 7th grade math as part of integer operations. Saxon Math Course 2 teaches it in Chapter 7, helping students handle negative signs confidently in more complex algebraic expressions.