Skip to content

Complex Numbers Practice

Intuition

Mathematics is the language of patterns and logic — a tool for describing relationships and solving problems.

A-Level Complex Numbers — Interactive Practice

8 auto-graded practice problems covering Argand diagrams, modulus-argument form, De Moivre’s theorem, and roots of equations.

Argand Diagrams

Modulus-Argument Form

De Moivre’s Theorem

Roots of Equations

Cross-References

Common Mistakes

Confusing modulus with argument when converting to polar form: The modulus r is always non-negative (r = |z| = √(a² + b²)), but the argument θ depends on the quadrant. Students often compute arctan(b/a) and forget to adjust for the correct quadrant — if the complex number is in the second quadrant, add π to the arctan result. Using the two-argument arctan (atan2) avoids this.

Applying De Moivre’s theorem to the wrong form: De Moivre’s theorem (r cis θ)ⁿ = rⁿ cis(nθ) only works in modulus-argument form. Attempting to apply it to rectangular form a + bi directly gives incorrect answers. Always convert to polar form first, apply the theorem, then convert back if needed.

Forgetting that complex roots come in conjugate pairs for real polynomials: If a polynomial has real coefficients and z is a complex root, then z̄ (its conjugate) must also be a root. This is sometimes called the Complex Conjugate Root Theorem. Forgetting this leads to incomplete solutions when finding roots of quadratic or higher-degree equations with complex roots.