For binary operation

* : A × A → A

with identity element e

For element a in A,

there is an element b in A

such that

a * b = e = b * a

Then, b is called inverse of a

##
**
Addition
**

+ :
**
R
**
×
**
R
**
→
**
R
**

For element a in A,

there is an element b in A

such that

a * b = e = b * a

Then, b is called inverse of a

Here, e = 0 for addition

So,
*
a * b = e = b * a
*

a + b = 0 = b + a

⇒ b = –a

Since

a + (– a) = 0 = (– a) + a,

So,
**
–a
**
is the inverse of a for addition.

##
**
Multiplication
**

× :
**
R
**
×
**
R
**
→
**
R
**

An element a in
**
R
**
is invertible if,

there is an element b in
**
R
**
such that ,

*
a * b = e = b * a
*

Here, b is the inverse of a

Here, e = 1 for multiplication

So,
*
a * b = e = b * a
*

a × b = 1 = b × a

⇒ b = 1/a

Since

a × 1/a = 1 = 1/a × a

So,
**
1/a
**
is the inverse of a for multiplication.