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

Lorraine Harvey

Lorraine Harvey

Answered

2022-01-16

Show that additive inverse of any z=(x, y)C is unique and it equals z=(x, y)
Show that multiplicative inverse of any non-zero z=(x, y)C is unique and it equals z1=(u, v), where
u=(xx2+y2)
v=yx2+y2

Answer & Explanation

poleglit3

poleglit3

Expert

2022-01-17Added 32 answers

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 z=x+iy and its inverse be z=a+ib 
Step 2 
Add the z and -z together to get
z+(z)=x+iy+(a+ib)=0 
Combine Like term, we get 
(x+a)+i(y+b)=0 
Using the zero product property, we can calculate an equivalent as
a=x 
(x+a)=0 
and b=y 
i(y+b)=0 
a=x and b=y 
Thus the additive inverse of the complex numbers z is z=x+i(y) 
Step 3 
The multiplicative inverse of a complex number can be found by:
Lets take the complex number z=x+iy and its

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