SOLUTION: Use De Moivre's Theorem to show that integral powers of (-1 + i)/(√2) are real, and which are imaginary

The complex number  z =   has the modulus 1  and the argument  .


So, in cis-form,  z =  = .


It means that z itself and all integer degrees of z have the modulus 1, i.e. lie on a unit circle 
in complex plane.


According to De Moivre's theorem, the degrees of z are


     = 

     =  =  = -i   (pure imaginary)

     = 

     =  =  = -1    (real number)
    
     = 

     =  =  = i   (pure imaginary)

     = 

     =  =  = 1     (real number)


The degrees of z that follow after  ,  repeat these numbers cyclically


     =   

     =  = -i  (imaginary)

     = 

     =  = -1  (real number)



     =   

     =  =  i  (imaginary)

     = 

     =  =  1  (real number)


So, the pattern is this:    is real     if and only n is of the form  n = 4k  (i.e. n is a multiple of 4), and

                            is pure imaginary if and only n is of the form  n = 4k+2  (i.e. n gives the remainder of 2 when is divided by 4).



ANSWER.    is real if and only if  n == 0  mod 4;

           is pure imaginary if and only if  n == 2  mod 4.