---
tag: 0BRHKKP333
kind: definition
name: Inverse element
msc: [08A02]
defines: 0ZNZH0EXFX
uses: [0580ZJSA3Q, 0RQPZZ44T4]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_element
    locator: Definitions and basic properties
---

## Definition

Let $S$ be a set with a [[binary operation|0M418FE7BB]] $\ast$ and an [[identity element|07WYQPB8E3]] $e$.
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|0K8F7PVDMX]], 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 $2 \cdot \tfrac{1}{2} = 1$.