Complex Numbers Topic Review summary

Key learning points

• $z=x+iy$; add/multiply in Cartesian form; conjugate $\\bar z=x-iy$.
• $|z|=\\sqrt{x^2+y^2}$; $\\arg z$ from geometry with correct quadrant.
• $z\\bar z=|z|^2$; divide by multiplying by the conjugate.
• Polar / modulus-argument form $r(\\cos\\theta+i\\sin\\theta)$.
• Loci: $|z-a|=r$ circles; $\\arg(z-a)=\\theta$ half-lines.

Worked example 1 — Multiply

$(1+2i)(3-i)$.

Working

Expand and use $i^2=-1$.

Worked example 2 — Modulus

$|3-4i|$.

Working

3-4-5 triangle.

Worked example 3 — Practice point 1

$z=x+iy$; add/multiply in Cartesian form; conjugate $\\\\bar z=x-iy$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

$|z|=\\\\sqrt{x^2+y^2}$; $\\\\arg z$ from geometry with correct quadrant.

Working

Attempt without notes, then check the answers PDF.

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.