SOLUTION: Find all complex numbers z such that z^2=z with line on top or complex conjugate

It is better and much easier to analyse and to solve this problem in polar trigonometric form.


If z^2 = z complex cojugate,  then, firstly, the modulus of z is equal to 1.

In other words, z lies on the unit circle in a coordinate plane.


Next, if the argument of z is polar angle ,  then the polar angle of z^2 is  ,

while the polar angle of (z conjugate) is    or  .


So, we get the equation for the polar angle


    Case 1.   = -,  which implies   = 0,  or   = 0.   Then the solution is z = 1.


OR


    Case 2.   = ,  which implies   = .    Hence,   = .


ANSWER.  There are TWO solutions.  One solution is z = 1.

          The other solution is  z =  = .