Complex Numbers — Topic Review

← Back to Complex Numbers

This review covers all five lessons in the Complex Numbers topic: Cartesian Form, Operations with Complex Numbers, the Argand Diagram and Modulus, Polar Form and De Moivre’s Theorem, and Polynomial Equations with Complex Roots. Questions are exam-style and increase in difficulty.

Mixed Review Questions

1. Fluency

   Q1 — Cartesian Form Basics

   For z = −4 + 7i, state: (a) Re(z)    (b) Im(z)    (c) z̅    (d) z · z̅

2. Fluency

   Q2 — Operations with Complex Numbers

   Let w = 3 + 2i and z = 1 − 4i. Calculate: (a) w + z    (b) wz    (c) w/z (write in the form a + bi)

3. Fluency

   Q3 — Modulus and Argument

   For z = −1 + √3 i, find: (a) |z|    (b) arg(z) (in exact radians, principal value)

4. Fluency

   Q4 — Polar Form

   Write z = 2(cos(π/4) + i sin(π/4)) in Cartesian form a + bi.

5. Fluency

   Q5 — Conjugate Root Theorem

   A polynomial with real coefficients has roots 2 − 3i and −5. State all roots and write the minimal monic polynomial of degree 3.

6. Understanding

   Q6 — De Moivre’s Theorem

   Use De Moivre’s Theorem to find (1 + i)⁸. Express in Cartesian form.

7. Understanding

   Q7 — Argand Diagram Geometry

   On an Argand diagram, the point representing z lies on the circle |z − 2| = 3. Describe this set geometrically and determine whether z = 5 + 0i lies on this circle.

8. Understanding

   Q8 — Finding Complex Roots of a Quadratic

   Find all complex roots of z² − (3+i)z + (2+i) = 0.

9. Understanding

   Q9 — Multiplication in Polar Form

   Let z₁ = 4 cis(π/6) and z₂ = 3 cis(π/3). Find: (a) z₁z₂    (b) z₁/z₂    Give answers in polar and Cartesian form.

10. Understanding

    Q10 — Cube Roots of Unity

    Find all cube roots of 1 (solutions of z³ = 1) and plot them on a sketch of the Argand plane.

11. Understanding

    Q11 — Proving a Polynomial Identity

    Given that z = 1 + i, show that z⁴ = −4.

12. Problem Solving

    Q12 — Solving a Quartic by Factoring

    Solve z⁴ + 13z² + 36 = 0 over ℂ.

13. Problem Solving

    Q13 — Locus on the Argand Plane

    Find the Cartesian equation of the locus of points z = x + iy satisfying arg(z − 1) = π/4.

14. Problem Solving

    Q14 — Gaussian Integer Division

    Express (3 + 11i) / (1 + 2i) in the form a + bi and verify your answer by multiplication.

15. Problem Solving

    Q15 — Fifth Roots of a Complex Number

    Find all fifth roots of −32 (i.e., solve z⁵ = −32) and express in exact polar form cisθ.