IB Maths AA Analysis and Approaches How to Solve: IB AA HL – Complex Numbers (De Moivre, Roots of Unity, Argand Diagram Proofs)

How to Solve: IB AA HL – Complex Numbers (De Moivre, Roots of Unity, Argand Diagram Proofs)

Complete Guide

Introduction

Mastering De Moivre’s Theorem and roots of unity doesn’t just earn you 7–9 marks on your IB AA HL exam—it unlocks quantum mechanics wave functions, AC circuit analysis, and even fractal geometry in university. One question on this topic can decide whether you hit a 6 or a 7.

WHAT YOU NEED TO KNOW FIRST

1. Polar form of complex numbers: ( z = r(\cos \theta + i \sin \theta) )
2. Euler’s formula: ( e^{i\theta} = \cos \theta + i \sin \theta ) (given on exam sheet)
3. Basic trigonometric identities (double-angle, sum-to-product)

KEY TERMS & FORMULAS

1. De Moivre’s Theorem

Formula: ( z^n = [r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta) ) Variables: - ( z ) = complex number - ( r ) = modulus (magnitude) - ( \theta ) = argument (angle) - ( n ) = integer power

MEMORISE THIS – Not on the formula sheet!

2. Roots of Unity

Formula: The ( n )-th roots of unity are given by: ( \omega_k = e^{2\pi i k / n} = \cos \left( \frac{2\pi k}{n} \right) + i \sin \left( \frac{2\pi k}{n} \right) ) for ( k = 0, 1, 2, \dots, n-1 ).

MEMORISE THIS – The pattern, not the exact formula.

3. Argand Diagram Proofs

Key Idea: - Geometric interpretation: Multiplying by ( e^{i\theta} ) rotates a complex number by ( \theta ) radians. - Sum of roots of unity: Always zero for ( n \geq 2 ).

Given on exam sheet: Basic complex number properties.

STEP-BY-STEP METHOD

Step 1: Convert to Polar Form

• Find the modulus ( r = |z| = \sqrt{a^2 + b^2} ).
• Find the argument ( \theta = \tan^{-1}(b/a) ) (adjust for quadrant).
• Write ( z = r(\cos \theta + i \sin \theta) ).

Step 2: Apply De Moivre’s Theorem

• Raise ( r ) to the power ( n ).
• Multiply ( \theta ) by ( n ).
• Write the result in polar form.

Step 3: Convert Back to Cartesian (if needed)

• Expand ( r^n (\cos n\theta + i \sin n\theta) ) using trigonometric identities.

Step 4: For Roots of Unity

• Use the formula ( \omega_k = e^{2\pi i k / n} ).
• Plot the roots on the unit circle (equally spaced).
• Sum the roots: ( 1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0 ).

Step 5: Argand Diagram Proofs

• Use geometric properties (rotation, reflection).
• Show symmetry (e.g., roots of unity form a regular polygon).

WORKED EXAMPLES

Example 1 – Basic (De Moivre’s Theorem)

Question: Find ( (1 + i)^5 ) in Cartesian form.

Step 1: Convert to polar form. - ( r = \sqrt{1^2 + 1^2} = \sqrt{2} ) - ( \theta = \tan^{-1}(1/1) = \pi/4 ) - ( z = \sqrt{2} (\cos \pi/4 + i \sin \pi/4) )

Step 2: Apply De Moivre’s Theorem. - ( z^5 = (\sqrt{2})^5 (\cos 5\pi/4 + i \sin 5\pi/4) ) - ( = 4\sqrt{2} (\cos 5\pi/4 + i \sin 5\pi/4) )

Step 3: Convert back to Cartesian. - ( \cos 5\pi/4 = -\sqrt{2}/2 ), ( \sin 5\pi/4 = -\sqrt{2}/2 ) - ( z^5 = 4\sqrt{2} (-\sqrt{2}/2 - i \sqrt{2}/2) = -4 - 4i )

What we did and why: We used De Moivre’s Theorem to raise a complex number to a power efficiently, avoiding binomial expansion.

Example 2 – Medium (Roots of Unity)

Question: Find all 4th roots of unity and plot them on an Argand diagram.

Step 1: Use the roots of unity formula. - ( \omega_k = e^{2\pi i k / 4} ) for ( k = 0, 1, 2, 3 ).

Step 2: Calculate each root. - ( \omega_0 = e^0 = 1 ) - ( \omega_1 = e^{i\pi/2} = i ) - ( \omega_2 = e^{i\pi} = -1 ) - ( \omega_3 = e^{3i\pi/2} = -i )

Step 3: Plot on the unit circle. - Points at ( 1, i, -1, -i ) (square shape).

What we did and why: We used the roots of unity formula to find all solutions to ( z^4 = 1 ) and verified their geometric symmetry.

Example 3 – Exam-Style (Argand Diagram Proof)

Question: Prove that the sum of the 5th roots of unity is zero.

Step 1: Write the roots. - ( \omega_k = e^{2\pi i k / 5} ) for ( k = 0, 1, 2, 3, 4 ).

Step 2: Sum the roots. - ( S = 1 + \omega + \omega^2 + \omega^3 + \omega^4 ).

Step 3: Use the geometric series formula. - ( S = \frac{1 - \omega^5}{1 - \omega} ). - Since ( \omega^5 = 1 ), ( S = \frac{1 - 1}{1 - \omega} = 0 ).

Step 4: Argand diagram interpretation. - The roots form a regular pentagon, and their vector sum is zero.

What we did and why: We combined algebraic and geometric reasoning to prove a key property of roots of unity.

COMMON MISTAKES

1. MISTAKE: Forgetting to adjust the argument for quadrant. WHY IT HAPPENS: Using ( \tan^{-1} ) without checking signs of ( a ) and ( b ). CORRECT APPROACH: Always sketch the complex number first.

2. MISTAKE: Misapplying De Moivre’s Theorem to non-integer powers. WHY IT HAPPENS: Confusing ( z^n ) with ( z^{1/n} ). CORRECT APPROACH: De Moivre’s only works for integer ( n ).

3. MISTAKE: Incorrectly calculating roots of unity. WHY IT HAPPENS: Forgetting ( k ) runs from ( 0 ) to ( n-1 ). CORRECT APPROACH: List all ( n ) roots explicitly.

4. MISTAKE: Not converting back to Cartesian form. WHY IT HAPPENS: Leaving answers in polar form when Cartesian is required. CORRECT APPROACH: Always check the question’s required form.

5. MISTAKE: Ignoring the geometric interpretation. WHY IT HAPPENS: Treating complex numbers as purely algebraic. CORRECT APPROACH: Sketch the Argand diagram for every problem.

EXAM TRAPS

1. TRAP: Questions asking for "all possible values" but hiding ( n )-th roots. HOW TO SPOT IT: Phrases like "solve ( z^3 = 8 )" or "find all cube roots." HOW TO AVOID IT: Always find all ( n ) roots, not just the principal root.

2. TRAP: Mixing degrees and radians. HOW TO SPOT IT: Angles given in degrees but formulas require radians. HOW TO AVOID IT: Convert all angles to radians before applying formulas.

3. TRAP: Proofs requiring geometric reasoning. HOW TO SPOT IT: Questions like "show that the roots lie on a circle" or "prove the sum is zero." HOW TO AVOID IT: Draw the Argand diagram and use symmetry.

1-MINUTE RECAP

"Listen up—this is your 60-second crash course for acing complex numbers on the IB exam. First, polar form is your best friend: convert ( a + bi ) to ( r(\cos \theta + i \sin \theta) ). Then, De Moivre’s Theorem lets you raise it to any power in one step—just multiply the angle and raise the modulus. For roots of unity, remember they’re equally spaced on the unit circle, and their sum is always zero. Sketch the Argand diagram for every problem—it’s not just for show, it’s your secret weapon. And watch out for exam traps: always find all roots, never mix degrees and radians, and don’t skip the geometric proof. You’ve got this—go smash that exam!