COMPLEX NUMBERS
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What classes of complex numbers z satisfy each of the following relations?
a. z* = z
b. z* = -z
c. z*z = z^2
d. |z| = 1
e. arg(z) = pi
f. Re(z) = Im(z)

a. z* = z

HOPE YOU MEAN Z BAR OR CONJUGATE OF Z BY THE SYMBOL Z* …ASSUMING SO …

SO WE GET ….A-BI=A+BI….B=0….SO Z SHALL BE A PURELY REAL NUMBER ….

b. z* = -z

HOPE YOU MEAN Z BAR OR CONJUGATE OF Z BY THE SYMBOL Z* …ASSUMING SO …

A-BI = -A-BI……..A=0…..SO Z SHALL BE A PURELY IMAGINARY NUMBER .

c. z*z = z^2

HOPE YOU MEAN Z BAR OR CONJUGATE OF Z BY THE SYMBOL Z* …ASSUMING SO …

[A+BI][A-BI]=[A+BI]^2……A^2+B^2=A^2+B^2+2A*B*I….SO A*B = 0

THAT IS EITHER A=0 OR B=0 OR BOTH …THAT IS Z SHALL BE EITHER

PURELY IMAGINARY OR PURELY REAL OR ZERO .

d. |z| = 1

X^2+Y^2=1……THAT IS Z SHALL BE OF THE TYPE Z = COS(T) + I*SIN(T)..

OR Z SHALL REPRESENT THE UNIT CIRCLE WITH CENTER AT THE ORIGIN

AND RADIUS = 1 UNIT .

e. arg(z) = pi

ATAN[B/A]=PI …TAN[PI]=B/A=0….B=0…..THAT IS Z SHALL BE PURELY REAL NUMBER .

f. Re(z) = Im(z)

A = B …….THAT IS Z SHALL BE OF THE TYPE …Z=R[1+I] …OR IT SHALL REPRESENT A LINE THROUGH ORIGIN

WITH UNIT SLOPE IN THE ARGAND DIAGRAM