.
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.
Solved.