Key takeaway: Opposite numbers are two numbers that sit the same distance from zero on a number line but on different sides, so their sum is zero, like 6.5 and −6.5.
Grade 66.NS.C.6aNumbers and Operations
Definition of Opposite Numbers
Opposite numbers are mirror images across zero. The numbers 6.5 and −6.5 are each 6.5 units from 0, one to the right and one to the left. Add them and the result is always 0, which is why they are also called additive inverses.
Every number except zero has exactly one partner of this kind, and zero is its own partner. Taking the opposite twice returns the starting number, so the opposite of the opposite of 12 is 12 again. Picture a seesaw balanced at 0: when one end rises by 6.5, the other end drops by exactly 6.5, and the two heights cancel. In money terms, a \$40 deposit and a \$40 withdrawal are partners of this kind.
Formula and notation
A minus sign in front of a number or an expression asks for its opposite:
\[-a \text{ is the opposite of } a \qquad a + (-a) = 0 \qquad -(-a) = a\]
When \(a\) is negative, \(-a\) is positive. A minus sign alone does not tell you the number is negative; read it as “the opposite of”.
Key facts
- Opposite numbers add to zero, and the gap between them is twice their distance from zero.
- Both partners have the same absolute value, because they are equally far from zero.
- Zero is the only number that is its own partner.
- Changing the sign of any nonzero number gives its mirror image across zero.
Opposite Numbers example
A submarine is at \(-215\) feet (215 feet below sea level). Find its opposite, the opposite of \(-18.5\), the opposite of 0, and simplify \(-(-12)\).
-
The mirror image of \(-215\) across zero is \(215\).
-
Flip the sign of \(-18.5\) to get \(18.5\). Zero stays \(0\).
-
\(-(-12)\) means the opposite of \(-12\), which is \(12\).
The results are 215, 18.5, 0 and 12.
Common mistake
Watch out
Dan says the opposite of \(-\frac{3}{5}\) is \(\frac{5}{3}\).
Why it is wrong
He flipped the fraction, which finds the reciprocal. The opposite only changes the sign: \(\frac{3}{5}\). Check by adding: \(-\frac{3}{5} + \frac{3}{5} = 0\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations