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.