Opposite Number
The opposite of a number is the number the same distance from zero on the number line but on the other side; it is the additive inverse.
Formula
\text{opposite of } a = -a; \ a + (-a) = 0
Definition
The opposite of a number is the number on the other side of zero on the number line, the same distance away; adding a number and its opposite always gives zero. Formally, the opposite (additive inverse) of $a$ is $-a$, defined by $a + (-a) = 0$, symmetric to $a$ about zero with the same absolute value, and satisfying the double negative law $-(-a) = a$. In any abelian group $(G, +)$, the additive inverse of $a$ is the unique element $-a$ such that $a + (-a) = 0$ (the identity), its existence one of the group axioms, and in a ring the additive inverse satisfies $(-a)(-b) = ab$, derivable from the distributive law.
Example
The opposite of $5$ is $-5$; the opposite of $-3$ is $3$; the opposite of $0$ is $0$; and $5 + (-5) = 0$. Simplify $-(-(-3))$: $-(-(-3)) = -(3) = -3$, since the notation "$-a$" does not mean $a$ is negative, if $a = -5$ then $-a = -(-5) = 5$. In $\mathbb{Z}/7\mathbb{Z}$, the opposite of $3$ is $4$ (since $3 + 4 = 7 \equiv 0 \pmod 7$), and in the group of $2\times2$ invertible matrices, the additive inverse of matrix $A$ is $-A$.
Key Insight
Opposites always add up to zero, like perfectly balanced weights on either side of a scale. The negative sign means "opposite," and the opposite of a negative is positive. The existence of additive inverses distinguishes groups from monoids: adding inverses to the natural numbers gives the integers, and adding multiplicative inverses (for non-zero elements) gives the rationals, a chain of extensions that is the construction of the number line from scratch.