amazon.co.ukComplex Numbers
A-Level Further Mathematics revision: Complex Numbers. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Cartesian form: z = a + bi; Polar form: z = r(cos θ + i sin θ) = re^{iθ}
Key Fact: Modulus |z| = sqrt(a^2+b^2); Argument arg(z) = arctan(b/a) (adjust for quadrant)
Key Fact: Conjugate: z̄ = a - bi = re^{-iθ}; Properties: z + z̄ = 2a, z z̄ = |z|^2
Key Fact: de Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ; (re^{iθ})ⁿ = rⁿe^{inθ}
Key Fact: n-th roots: r^{1/n} e^{i(θ+2kpi)/n} for k = 0,1,...,n-1 (n distinct roots)
Key Fact: Complex roots of real polynomials occur in conjugate pairs
Key Fact: Loci: |z - a| = r (circle), arg(z - a) = α (half-line), |z - a| = |z - b| (perp bisector)
Key Fact: Euler's formula: e^{iθ} = cos θ + i sin θ; cos θ = ½(e^{iθ}+e^{-iθ}), sin θ = (e^{iθ}-e^{-iθ})/(2i)
🎯 Learning Objectives
- Perform arithmetic with complex numbers in cartesian and polar form
- Represent complex numbers on the Argand diagram
- Find modulus and argument of complex numbers
- Apply de Moivre's theorem for powers and roots
- Solve polynomial equations with complex roots
- Understand loci on the Argand diagram (circles, lines, half-lines)
- Use complex numbers to solve geometric problems
💡 Worked Example
Exam-Style Question
Question: Solve z^3 = -8. Plot the roots on an Argand diagram and show they form an equilateral triangle
Model Answer:
z^3 = 8e^{ipi} -> z = 2e^{i(pi+2kpi)/3} for k=0,1,2. Roots: 2e^{ipi/3} = 1+isqrt3, 2e^{ipi} = -2, 2e^{i5pi/3} = 1-isqrt3. Distance between any two = sqrt((1-(-2))^2+(sqrt3-0)^2) = sqrt(9+3) = 2sqrt3 -- all equal, equilateral triangle
❓ Practice Questions
Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.
Questions:
- Express (1+isqrt3)⁶ in polar form✗ answer coming soon
- Find all solutions to z⁴ = -16✗ answer coming soon
- Solve z^2 + 4z + 13 = 0, plot roots✗ answer coming soon
- Show that |z-3| = 2|z+3| is a circle, find its centre and radius✗ answer coming soon
- If z = cos θ + i sin θ, express cos 3θ in terms of cos θ✗ answer coming soon
🎬 Video Resources
- Complex Numbers - Overview -- YouTube Search
- Complex Numbers - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Complex Numbers -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Complex Numbers -- Free resources, slides, and interactive questions
- BBC Bitesize - Complex Numbers -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of complex numbers with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Solve z^3 = -8. Plot the roots on an Argand diagram and show they form an equilateral triangle. Solution: z^3 = 8e^{ipi} -> z = 2e^{i(pi+2kpi)/3} for k=0,1,2. Roots: 2e^{ipi/3} = 1+isqrt3, 2e^{ipi} = -2, 2e^{i5pi/3} = 1-isqrt3. Distance between any two = sqrt((1-(-2))^2+(sqrt3-0)^2) = sqrt(9+3) = 2sqrt3 -- all equal, equilateral triangle
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Express (1+isqrt3)⁶ in polar form
- Find all solutions to z⁴ = -16
- Solve z^2 + 4z + 13 = 0, plot roots
- Show that |z-3| = 2|z+3| is a circle, find its centre and radius
- If z = cos θ + i sin θ, express cos 3θ in terms of cos θ
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.