Show that additive inverse of any z=(x, y)∈C is unique and it equals −z=(−x, −y)...

Step 1 
Additive Inverse of Complex Numbers: 
We are aware that zero is the sum of any number and its additive inverse.
To find: Additive Inverse of the Complex Number: 
Lets take the complex numbers ${\displaystyle z=x+iy}$ and its inverse be ${\displaystyle -z=a+ib}$ 
Step 2 
Add the z and -z together to get
${\displaystyle z+(-z)=x+iy+(a+ib)=0}$ 
Combine Like term, we get 
${\displaystyle (x+a)+i(y+b)=0}$ 
Using the zero product property, we can calculate an equivalent as
${\displaystyle a=-x}$ 
${\displaystyle (x+a)=0}$ 
and ${\displaystyle b=-y}$ 
${\displaystyle i(y+b)=0}$ 
${\displaystyle a=-x}$ and ${\displaystyle b=-y}$ 
Thus the additive inverse of the complex numbers z is ${\displaystyle -z=-x+i(-y)}$ 
Step 3 
The multiplicative inverse of a complex number can be found by:
Lets take the complex number ${\displaystyle z=x+iy}$ and its