# Definition:Multiplicative Inverse

## Definition

Let $\strut {A_F, \oplus}$ be a unitary algebra whose unit is $1$ and whose zero is $0$.

Let $a \in A_F$ such that $a \ne 0$.

A multiplicative inverse of $a$ is an element $b \in A_F$ such that:

$a \oplus b = 1 = b \oplus a$

### Field

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $a \in F$ such that $a \ne 0_F$.

Then the inverse element of $a$ with respect to the $\times$ operator is called the multiplicative inverse of $F$.

It is usually denoted $a^{-1}$ or $\dfrac 1 a$.

### Multiplicative Inverse of Number

Let $\Bbb F$ be one of the standard number fields: $\Q$, $\R$, $\C$.

Let $a \in \Bbb F$ be any arbitrary number.

The multiplicative inverse of $a$ is its inverse under addition and can be denoted: $a^{-1}$, $\dfrac 1 a$, $1 / a$, and so on.

$a \times a^{-1} = 1$