Given that IzI=1 ,Compute all possible values of I1+Z I^2 + I1-ZI^2

Question

Hi I am doing this course on complex numbers and the material I am using to study for it is not that helpful.Can you please recommend a source .Thank you very much.

 

Given that IzI=1 ,Compute all possible values of I1+Z I^2 + I1-ZI^2

Answer 1

Let z = a+bi

 

Since lzl = 1, a²+b²=1.

 

l 1+z l² + l 1-z l² = l (1+a) + bi l + l (1-a) - bi l

 

                         = (1+a)² + b² + (1-a)² + (-b)²

 

                         = 1+2a+a²+1-2a+a²+2b²

 

                         = 2(a²+b²+1)

 

                         = 4 (since, by assumption, a² + b² = 1)

 

Answer 2

If |z| = 1 then

z = 1 or -1

 

|1+1|² + |1-1|² = 4

 

|1-1|² + |1-(-1)|² = 4