Inverse element

Definition

Let $S$ be a set with a binary operation $\ast$ and an identity element $e$. Let $a, b \in S$. The element $b$ is a left inverse of $a$ when $b \ast a = e$. The element $b$ is a right inverse of $a$ when $a \ast b = e$. The element $b$ is an inverse of $a$ (or two-sided inverse) when it is both a left inverse and a right inverse of $a$. An element that has an inverse is called invertible.

Remarks

For addition on $\mathbb{Z}$, the inverse of $a$ is $-a$. For multiplication on $\mathbb{Q}$, each nonzero $a$ has the inverse $1/a$, and $0$ has no inverse. When $\ast$ is associative, an element has at most one inverse.

Explanations

L1 human

An inverse of an element is a second element that cancels the first one. If you add $5$ and then add $-5$, you come back to $0$, the identity for addition. For multiplication, the inverse of $2$ is $\tfrac{1}{2}$, because $2 \cdot \tfrac{1}{2} = 1$.