91Ó°ÊÓ

Show that \(\mathrm{e}^{-\mathrm{i} \theta}\) is the conjugate of \(\mathrm{e}^{\mathrm{i} \theta}\).

Find the fourth roots of unity. Express them in polar exponential and Cartesian form, and plot them in the complex plane.

Find the cube roots of \(8 \mathrm{i}\). Express them in polar exponential and Cartesian form, and plot them in the complex plane.

Interpret geometrically the sets of points in the complex plane satisfying: (i) \(|z-\mathrm{i}|=|z-1|\) (ii) \(|z+1|=2\) (iii) \(\operatorname{Re}(z)=-3\), (iv) \(\operatorname{Im}(z) \leq 1\)

Show that if \(|z|=1\) and \(\operatorname{Re}(z)=-\frac{1}{2}\) then \(z\) is a cube root of unity.

Find the primitive 8 th roots of unity.