In mathematics, the additive inverse of an element x, denoted −x, is the element that when added to x, yields the additive identity. This additive identity is often the number 0 (zero), but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number, or its negative.^{[failed verification]} The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.

When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1.

The concept can also be extended to algebraic expressions, which is often used when balancing equations.

The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse:

Conversely, the additive inverse can be thought of as subtraction from zero:

This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.

Given an algebraic structure defined under addition ${\displaystyle (S,+)}$ with an additive identity ${\displaystyle e\in S}$, an element ${\displaystyle x\in S}$ has an additive inverse ${\displaystyle y}$ if and only if ${\displaystyle y\in S}$, ${\displaystyle x+y=e}$, and ${\displaystyle y+x=e}$.