Examples on Complex Numbers

Comp.145  December 1996  Dr A.J. Mestel

1. Express each of the following complex numbers in the form x + iy where x and y are real numbers.

(a) i^{4} ,

(b) \frac{5}{3+4i} ,

(c) (\frac{\sqrt{2}}{1−i})^{2} .

2. For each part, find all the complex numbers, z satisfying the given relation. (a) z^{6} = 1 ,

(b) z^{4} + 25 = 0 ,

(c) z^{2} + 2iz + 3 = 0 ,

(d) z^{2} = −3 + 4i ,

(e) z^{4} + 6z^{2} + 25 = 0 ,

3. For each part, indicate in a diagram of the complex plane (Argand diagram) the set of complex numbers, z which satisfy the given relation.

(a) z^{6} = 1 ,

(b) |z| = 1 ,

(c) |z − (\frac{1+i}{2})| = 1 ,

(d) |z − 1| = |z − 2| ,

(e) |z − 1| ⩾ |z + 1| ,

(f) |2(1 + i)z − 1| ⩽ 2\sqrt{2} .

4. Demonstrate each of the following equalities.

(a) cos 5θ = cos^{5} θ − 10 cos^{3} θ sin^{2} θ + 5 cos θ sin^{4} θ ,

(b) sin 5θ = 5 cos^{4} θ sin θ − 10 cos^{2} θ sin^{3} θ + sin^{5} θ ,

(c) cos 5θ = 16 cos^{5} θ − 20 cos^{3} θ + 5 cos θ ,

(d) cos 18^{o} ≡ cos \frac{π}{10} = \sqrt{\frac{5+\sqrt{5}}{8}} .

5. For what complex numbers z is the expression z^{5} + z^{3}\overline{z}^{2} + z^{2}\overline{z}^{3} + \overline{z}^{5} a real number?

6. Find all the eigenvalues and eigenvectors of the rotation matrix

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & cos θ & sin θ \\ 0 & − sin θ & cos θ \end{matrix} ) .