Inverse element
For addition of numbers, for every number $a$, there is another number $-a$ such that adding $a + (-a)$ is zero. We call $a$ the additive inverse $a$. For example, the additive inverse of $5$ is $-5$.
The multiplicative inverse (or reciprocal) of a nonzero number $a$ is the number $\frac{1}{a}$ that you multiply by $a$ to get $1$:
\[a \times \frac{1}{a} = 1.\]For example, the multiplicative inverse of $5$ is $\frac{1}{5}$. The number $0$ has no multiplicative inverse, because $0$ times every number is $0$.
Consider a way of combining two things, such as addition of numbers, that has an element that changes nothing, such as $0$ for addition. An inverse of an element $a$ is an element that undoes $a$: combining $a$ with the inverse gives the element that changes nothing. For example, the additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.
Consider a set $S$ with a binary operation $\ast$ and an identity element $e$. An inverse of an element $a \in S$ is an element $b \in S$ such that
\[a \ast b = b \ast a = e.\]Properties
- If $\ast$ is associative, then each element has at most one inverse, which is written $a^{-1}$ (or $-a$ when the operation is addition).