Further Complex Numbers summary

Key learning points

• $[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)$.
• nth roots are equally spaced on a circle in the Argand diagram.
• Loci: $|z-a|=r$ is a circle; $\arg(z-a)=\alpha$ is a half-line.
• Convert to modulus–argument form before raising to powers.

Worked example 1 — Roots

How many distinct nth roots does a non-zero complex number have?

Working

Arguments differ by $\frac{2\pi}{n}$.

Worked example 2 — Key skill 1

Use this method from the Further Complex Numbers sheet: $[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — nth roots are equally spaced on a circle in the Argand

Use this method from the Further Complex Numbers sheet: nth roots are equally spaced on a circle in the Argand diagram.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Loci: is a circle; is a half-line

Use this method from the Further Complex Numbers sheet: Loci: $|z-a|=r$ is a circle; $\arg(z-a)=\alpha$ is a half-line.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Convert to modulus–argument form before raising to powe

Use this method from the Further Complex Numbers sheet: Convert to modulus–argument form before raising to powers.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.