Additive inverse

Relation to subtraction

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

a − b = a + (−b).

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

−a = 0 − a.

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.

Formal definition

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 ${\displaystyle y+x=e}$.

Addition is typically only used to refer to a commutative operation, but for some systems of numbers, such as floating point, it might not be associative. When it is associative, so ${\displaystyle (a+b)+c=a+(b+c)}$, the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.

The definition requires closure, that the additive element ${\displaystyle y}$ be found in ${\displaystyle S}$. However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., ${\displaystyle -3}$ for ${\displaystyle 3}$) is not a natural number; it is an integer. Therefore, the natural numbers in set ${\displaystyle S}$ do have additive inverses and their associated inverses are negative numbers.

Further examples

• In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction.
• In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11).
• In a Boolean ring, which has elements ${\displaystyle \{0,1\}}$ addition is often defined as the symmetric difference. So ${\displaystyle 0+0=0}$, ${\displaystyle 0+1=1}$, ${\displaystyle 1+0=1}$, and ${\displaystyle 1+1=0}$. Our additive identity is 0, and both elements are their own additive inverse as ${\displaystyle 0+0=0}$ and ${\displaystyle 1+1=0}$.