Complex Numbers

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Complex Numbers — Key Facts for BUET Definition: z = x + iy where x, y ∈ ℝ and i = √−1; set of complex numbers denoted ℂ Real part: Re(z) = x; Imaginary part: Im(z) = y Modulus: |z| = √(x² + y²); Argument: arg(z) = θ where tan θ = y/x (with quadrant check) Conjugate: z̄ = x − iy; property: z·z̄ = |z|² ⚡ Exam tip: For BUET, complex numbers combine with algebra and geometry — polar form and de Moivre’s theorem are essential!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Complex Numbers — BUET Study Guide

Polar form: z = r(cos θ + i sin θ) = r cis θ where r = |z|, θ = arg(z) Euler’s formula: e^{iθ} = cos θ + i sin θ, so z = re^{iθ}

Algebraic operations:

• Addition: z₁ + z₂ = (x₁+x₂) + i(y₁+y₂)
• Multiplication: z₁·z₂ = (x₁x₂ − y₁y₂) + i(x₁y₂ + y₁x₂)
• Division: z₁/z₂ = (z₁·z₂̄)/|z₂|²

Modulus properties:

• |z₁z₂| = |z₁||z₂|
• |z₁/z₂| = |z₁|/|z₂|
• |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality)
• ||z₁| − |z₂|| ≤ |z₁ − z₂|

Conjugate properties:

• z + z̄ = 2Re(z)
• z − z̄ = 2i Im(z)
• z·z̄ = |z|²
• (z₁ + z₂)̄ = z̄₁ + z̄₂
• (z₁z₂)̄ = z̄₁·z̄₂

De Moivre’s theorem: [cos θ + i sin θ]^n = cos(nθ) + i sin(nθ) = cis(nθ) This extends to any complex number in polar form

nth roots of unity: Solutions to z^n = 1: z_k = e^{2πik/n} for k = 0, 1, …, n−1 Sum of all nth roots of unity = 0

Cube roots of unity: Solutions to z³ = 1: 1, ω, ω² where ω = e^{2πi/3} = −½ + i√3/2 Property: 1 + ω + ω² = 0; ω³ = 1; ω² = ω̄

Argument properties:

• arg(z₁z₂) = arg(z₁) + arg(z₂)
• arg(z₁/z₂) = arg(z₁) − arg(z₂)
• arg(z^n) = n arg(z)

Important identities:

• |z₁ + z₂|² = |z₁|² + |z₂|² + 2 Re(z₁z₂̄)
• |z₁ − z₂|² = |z₁|² + |z₂|² − 2 Re(z₁z₂̄)

Geometric interpretation:

• |z − z₀| = r: circle with centre z₀, radius r

• |z − z₁| = |z − z₂|: perpendicular bisector of segment joining z₁ and z₂

• Key formula: z·z̄ = |z|²; [r cis θ]^n = r^n cis nθ; 1 + ω + ω² = 0 for cube roots of unity

• Common trap: arg(z) is NOT simply tan⁻¹(y/x) — always check the quadrant: Q1: tan⁻¹(y/x); Q2: π + tan⁻¹(y/x); Q3: tan⁻¹(y/x) − π; Q4: tan⁻¹(y/x)

• Exam weight: 1–2 questions per exam (4–8 marks); frequently combined with other topics

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Complex Numbers — Comprehensive BUET Notes

Argand plane: Every complex number z = x + iy corresponds to point (x, y) Distance from origin = |z|; direction from positive x-axis = arg(z)

General nth roots: For z^n = w, solutions are z = r^{1/n} cis((θ + 2πk)/n) for k = 0, 1, …, n−1 where w = r cis θ

Argument conventions: Principal argument Arg(z) is in (−π, π] For positive real axis reference, measure counterclockwise

Locus problems in complex form:

• Circle: |z − a| = r
• Perpendicular bisector: |z − z₁| = |z − z₂|
• Ellipse: |z − z₁| + |z − z₂| = constant (constant > |z₁ − z₂|)
• Half-line from a at angle θ: arg(z − a) = θ

Rotation in Argand plane: To rotate z by angle θ about origin: z’ = z·e^{iθ} To rotate about point α: z’ = α + (z − α)·e^{iθ}

De Moivre’s theorem applications:

• Express cos 5θ in terms of cos θ
• Express sin 5θ in terms of sin θ
• Find product of roots of unity

Solving equations: For z² = −1: z = ±i For z² = a (a > 0): z = ±√a For z² = a (a < 0): z = ±i√|a|

Polynomial equations with complex roots: If a + ib is a root of polynomial with real coefficients, then a − ib is also a root

Exponential form applications: e^{iθ} = cos θ + i sin θ Useful for evaluating expressions like (1 + i)^8

Matrix representation: Complex number x + iy corresponds to matrix [[x, −y], [y, x]]

Properties of modulus:

• |z| = 0 only if z = 0
• |z| = |z̄|
• |z₁ + z₂|² = |z₁|² + |z₂|² + 2 Re(z₁z₂̄)
• This is derived from (z₁ + z₂)(z̄₁ + z̄₂)

Cauchy-Schwarz inequality: |z₁·z₂ + z̄₁·z̄₂| ≤ 2|z₁||z₂|

Triangle inequality applications:

• |z₁| − |z₂| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|
• |z₁ − z₂| ≥ ||z₁| − |z₂||

Geometric transformations:

• z → z + a: translation by vector a
• z → kz: dilation by factor k from origin
• z → e^{iθ}z: rotation by θ about origin
• z → z̄: reflection across real axis

Sum of powers of nth roots: For n > 1: 1^p + ω^p + ω^{2p} + … + ω^{(n−1)p} = 0 unless n|p where ω is primitive nth root of unity

Cubic equations and complex roots: If coefficients are real and discriminant < 0, roots include complex conjugate pair

• Remember: z·z̄ = |z|²; [r cis θ]^n = r^n cis nθ; arg(z₁z₂) = arg(z₁) + arg(z₂); rotation: multiply by e^{iθ}
• Previous years: “Find modulus and argument of (1 + i)” [2023 BUET]; “Find cube roots of unity and verify sum = 0” [2024 BUET]; “Express (1 + i)⁸ in form a + ib” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Mathematics

• Calculus (Differentiation + Integration) — highest weight (20–30 marks)
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Coordinate Geometry (Circle, Conics) — high weight
• Trigonometry — medium-high weight
• Complex Numbers — medium weight

📝 Previous Year Question Patterns

• Complex Numbers: 1–2 questions per exam, 4–8 marks
• Common patterns: polar form conversion, de Moivre’s theorem, nth roots, geometry in Argand plane
• Weight: medium — important as foundation

💡 Pro Tips

• For converting to polar form, find r = √(x²+y²) and θ = tan⁻¹(y/x) adjusted for quadrant
• De Moivre’s theorem is essential for (1 + i)^n type problems — note that 1 + i = √2 cis 45°
• For cube roots of unity, remember 1 + ω + ω² = 0 and ω³ = 1
• For locus problems, set z = x + iy and convert to real equation
• Rotation in complex plane: multiply by e^{iθ} for rotation about origin

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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