Exercise Set 5

These exercises cover the topic of Complex Numbers.

1. Let \(u=2+3i\) and \(v=5+8i\). Determine the following.

   1. \(u+v\)
   2. \(u-v\)
   3. \(uv\)
   4. \(vu\)
   5. \(\frac{u}{v}\)
   6. \(\frac{v}{u}\)
   7. \(u^2\)
   8. \(v^2\)
   9. \(u^2+v^2\)

2. State the complex conjugates of the following numbers. Also caclulate their modulus.

   1. \(3-4i\)
   2. \(2+2i\)
   3. \(2+15i\)
   4. \(3i\)
   5. \(i\)
   6. \(5\)

3. Let \(z=3-i\). Find \(z^2+7z+13\) in Cartesian form.

4. Find the roots of the following quadratic equations.

   1. \(z^2+2z+26=0\)
   2. \(z^2-2z+3=0\)
   3. \(3z^2-7z+13=0\)

5. Express the following complex numbers in polar form and exponential form.

   1. \(1+i\)
   2. \(-1+i\)
   3. \(\sqrt{3}-i\)
   4. \(z_1=2-3i\)
   5. \(z_2=3+4i\)
   6. \(\frac{z_1-2}{z_2}\)
   7. \(z_1z_2-\frac{z_1-z_2}{z_2}\)
   8. \(w_1=-\sqrt{2}+\sqrt{2}i\)
   9. \(w_2=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\)
   10. \(w_1w_2\)
   11. \(\frac{w_1}{w_2}\)

6. Express the following complex numbers in Cartesian form.

   1. \(2e^{i\frac{\pi}{12}}\)
   2. \(5e^{i 23^\circ}\)
   3. \(2e^{i (-45^\circ)}\)

7. Find the following roots and express them in Cartesian form.

   1. \(1^\frac{1}{3}\)
   2. \(1^\frac{1}{4}\)
   3. The cube roots of \(\sqrt{2}-\sqrt{2}i\)
   4. The square roots of \(3-4i\)

8. Sketch the following complex numbers in the complex plane.

   1. \(2+3i\)
   2. \(-1-i\)
   3. \(-1+i\)
   4. \(-2i\)
   5. \(-3\)

9. What is \(i^i\) in Cartesian form?

10. Find all complex numbers \(z\) such that \(\overline{z}=z^2\).