Every positive number has a partner on the other side of 0. Those partners are called opposites.
Key idea
Opposites are the same distance from 0, on opposite sides: 3 and $-3$. The minus sign can be read "the opposite of", so $-(-3)$ is the opposite of $-3$, which is $3$.
Why it works
Taking the opposite flips a number across 0 to the same distance on the other side. Flip 3 and you land on $-3$. Flip $-3$ and you land back on 3. So $-(-3) = 3$: two flips bring you home. The number 0 does not move when you flip it, so 0 is its own opposite.
Worked example: The opposite of a decimal
What is the opposite of $7.5$?
- $7.5$ is 7.5 units right of 0.
- Its opposite is 7.5 units left of 0.
Answer: $-7.5$
Worked example: Opposite of an opposite
Simplify $-(-12)$.
- $-12$ is the opposite of 12.
- The opposite of $-12$ flips back across 0.
Answer: $12$
Worked example: Three flips
Simplify $-(-(-2))$.
- Inside first: $-(-2) = 2$.
- Then $-(2) = -2$. Three flips end on the negative side.
Answer: $-2$
Common mistake
Thinking that two minus signs "stay negative", so $-(-3) = -3$. Quick check with the picture: each minus sign is one flip across 0. An even number of flips ends where you started.