Additive Inverse
The additive inverse of a number is the number that, when added to it, gives zero; it is the opposite or negation of the number.
Formula
a + (-a) = 0
Definition
The additive inverse of a number is its opposite: the number you add to it to get zero. For any real number $a$, the additive inverse is $-a$, satisfying $a + (-a) = 0$, and it is used to define subtraction, $a - b = a + (-b)$; the additive inverse of the additive inverse is the original number, $-(-a)=a$. In a group $(G,+)$, the additive inverse of $a$ is the unique element $-a$ satisfying $a + (-a) = 0$, unique because if $a+b=0$ and $a+c=0$ then $b = b+0 = b+(a+c) = (b+a)+c = 0+c = c$; the map $a \to -a$ is a group automorphism (involution) of $(G,+)$.
Example
The additive inverse of $6$ is $-6$, because $6+(-6)=0$; the additive inverse of $-4$ is $4$. Subtracting $5$ is the same as adding its additive inverse: $12 - 5 = 12 + (-5) = 7$. In $\mathbb{R}^n$, the additive inverse of $(x_1,\ldots,x_n)$ is $(-x_1,\ldots,-x_n)$, and in polynomial rings the additive inverse of $p(x) = 3x^2 - x + 1$ is $-p(x) = -3x^2+x-1$.
Key Insight
Every number has exactly one additive inverse; they come in pairs that always cancel to zero, and this is what makes subtraction possible on negative numbers, since without it you could not subtract a larger number from a smaller one and stay in your number system. The existence of additive inverses (negatives) is precisely the property that distinguishes groups from monoids and rings from semirings: natural numbers form a semiring with no additive inverses, while integers form a ring, an extension that is the algebraic story of "inventing negative numbers."