Additive Inverse
ArithmeticThe 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.
Example
The additive inverse of $6$ is $-6$, because $6 + (-6) = 0$. The additive inverse of $-4$ is $4$, because $-4 + 4 = 0$.
Key Insight
Every number has exactly one additive inverse. They come in pairs that always cancel to zero.
Definition
For any real number $a$, its additive inverse is $-a$, satisfying $a + (-a) = 0$. Additive inverses are used to define subtraction: $a - b = a + (-b)$. The additive inverse of the additive inverse is the original number: $-(-a) = a$.
Example
Additive inverse of $3/4$ is $-3/4$. Of $0$ is $0$. Of $-\pi$ is $\pi$. Subtracting $5$ is the same as adding the additive inverse of $5$: $12 - 5 = 12 + (-5) = 7$.
Key Insight
The additive inverse is what makes subtraction possible on negative numbers. Without it, you could not subtract a larger number from a smaller one and stay in your number system.
Definition
In a group $(G, +)$, the additive inverse of $a$ is the unique element $-a$ satisfying $a + (-a) = 0$. Uniqueness: 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
In the vector space $\mathbb{R}^n$, the additive inverse of $(x_1,\ldots,x_n)$ is $(-x_1,\ldots,-x_n)$. In polynomial rings, the additive inverse of $p(x) = 3x^2 - x + 1$ is $-p(x) = -3x^2 + x - 1$.
Key Insight
The existence of additive inverses (negatives) is the property that distinguishes groups from monoids and rings from semirings. Natural numbers form a semiring (no additive inverses); integers form a ring (additive inverses exist). This extension is the algebraic story of "inventing negative numbers."