FAC1004 Tutorial 2 — Complex Numbers

Practice problems on complex numbers, powers, De Moivre's theorem, and roots.

Topics Covered

• Powers of complex numbers in polar form
• De Moivre's theorem applications
• Euler's formula and exponential form
• Finding n-th roots of complex numbers
• Solving polynomial equations with complex roots

Problem Set

1. Powers in Cartesian Form: Given $z_1 = -4 + 4i$ and $z_2 = -2\sqrt{3} + 2i$, find $(z_1)^3$ and $(z_2)^4$

2. Simplification: Simplify expressions involving $(\cos x + i\sin x)$ terms

3. Exponential and Cartesian Form: Express complex numbers in both exponential and Cartesian forms

4. Argument Calculations: Calculate arguments and moduli of complex expressions

5. Euler's Identity Applications: Show that $e^{ix} + e^{-ix} = 2\cos x$ and related identities

6. Multiple Angle Formulas: Prove $\cos(5\theta)$ and $\sin(5\theta)$ formulas

7. Cube Roots: Find cube roots of various complex numbers

8. n-th Roots of Unity: Find tenth roots of 1 and classify by quadrant

9. Polynomial Roots: Find all $z$ for which $z^5 = -32i$ with $\text{Im}(z) > 0$

10. Complex Polynomial Equations: Given a root, find coefficients of polynomial

11. Complex Roots: Find roots of various complex numbers

12. Complex Equations: Solve for $z = x + iy$ in various equations

Related

• FAC1004 - Advanced Mathematics II (Computing) — main course page
• Complex Numbers — concept page
• FAC1004 L01 — Complex Numbers — related lecture
• FAC1004 L02 — Euler's Formula — related lecture
• FAC1004 L5-L6 — Functions of Complex Numbers (n-th Roots) — related lecture

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 2 25-26.pdf