Algebra of Complex Numbers

• Addition, subtraction, multiplication, and division of complex numbers in standard form.
• The conjugate z̄ = a − ib and its role in rationalising the denominator.
• De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ).
• Cube roots of unity ω with the key identity 1 + ω + ω² = 0.
• nth roots of a complex number using polar form.

Level 1 — Four operations and conjugate

Addition/Subtraction: (a + ib) ± (c + id) = (a ± c) + i(b ± d). Treat i like a variable.

Multiplication: (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac − bd) + i(ad + bc). Use i² = −1 to simplify.

Conjugate: z̄ = a − ib. Key properties: z + z̄ = 2a (real), z − z̄ = 2ib (imaginary), z·z̄ = a² + b² = |z|².

Division: Multiply numerator and denominator by conjugate of denominator: (a + ib)/(c + id) = (a + ib)(c − id)/((c + id)(c − id)) = [(ac + bd) + i(bc − ad)]/(c² + d²).

Level 2 — De Moivre's theorem, cube roots of unity, nth roots

De Moivre's theorem: For integer n, (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). In Euler form: (e^(iθ))ⁿ = e^(inθ). Applies for any real n (fractional too), giving multiple values when n = p/q.

Applications of De Moivre: Expand (cosθ + isinθ)³ to get cos3θ = 4cos³θ − 3cosθ (triple angle formula). Similarly sin3θ = 3sinθ − 4sin³θ.

Cube roots of unity: Solve z³ = 1. Roots are 1, ω, ω² where ω = e^(2πi/3) = −1/2 + i√3/2.

• 1 + ω + ω² = 0 (sum of all three cube roots = 0)
• ω³ = 1 (defining property)
• ω̄ = ω² (conjugates of each other)
• For any a, b: a³ + b³ = (a + b)(a + ωb)(a + ω²b)

nth roots of z = r·e^(iθ): The n distinct nth roots are: zₖ = r^(1/n) · e^(i(θ + 2kπ)/n), k = 0, 1, 2, …, n−1. The roots are equally spaced on a circle of radius r^(1/n) in the Argand plane, separated by angle 2π/n.

Key identities using conjugate:

• Re(z) = (z + z̄)/2
• Im(z) = (z − z̄)/(2i)
• |z|² = z·z̄

NCERT spotlight

For JEE, the identity 1 + ω + ω² = 0 is used to evaluate sums like ωⁿ + ω²ⁿ (equals −1 if n not divisible by 3, equals 2 if divisible by 3). Division: always multiply by conjugate of denominator to get real denominator. Memorise: z·z̄ = |z|² is the fastest way to compute |z|² without square roots.

Compute (2 + 3i)/(1 − 2i) and express in a + ib form.

Step 1 — Conjugate of denominator: (1 − 2i)̄ = (1 + 2i).
Step 2 — Multiply numerator and denominator:
         Numerator: (2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i² = 2 + 7i − 6 = −4 + 7i.
         Denominator: (1 − 2i)(1 + 2i) = 1 + 4 = 5.
Step 3 — Result: (−4 + 7i)/5 = −4/5 + (7/5)i.
Step 4 — Check: Re = −4/5, Im = 7/5.

If ω is a complex cube root of unity, evaluate ω¹⁰⁰ + ω²⁰⁰.

Step 1 — Find remainders: 100 = 33×3 + 1, so ω¹⁰⁰ = (ω³)³³ · ω¹ = 1·ω = ω.
Step 2 — Similarly: 200 = 66×3 + 2, so ω²⁰⁰ = ω².
Step 3 — Sum: ω + ω² = −1 (from 1 + ω + ω² = 0).
Step 4 — Answer: ω¹⁰⁰ + ω²⁰⁰ = −1.

1. Compute (1 + i)⁴ using De Moivre or direct expansion.
2. Find the conjugate of (3 − 4i)/(2 + i).
3. Evaluate (cos(π/5) + i sin(π/5))¹⁰.
4. If 1 + ω + ω² = 0, find the value of (1 + ω)³ − ω³.
5. Stretch: Find all cube roots of −8 using the polar form method and locate them on the Argand plane.

Open the Practice tab for graded questions on Complex Numbers — Algebra.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.