JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
COMPLEX NUMBERS
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JEE MATH APEX
1. Definition & Representation
📌 Complex Number Forms
• Standard form: z = a + bi (where a, b ∈ R, i² = −1)
• Real part: Re(z) = a
• Imaginary part: Im(z) = b
• Conjugate: z̄ = a − bi
• Modulus: |z| = √(a² + b²)
• Argument: arg(z) = θ = tan⁻¹(b/a)
• Real part: Re(z) = a
• Imaginary part: Im(z) = b
• Conjugate: z̄ = a − bi
• Modulus: |z| = √(a² + b²)
• Argument: arg(z) = θ = tan⁻¹(b/a)
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2. Algebra of Complex Numbers
📌 Addition, Subtraction, Multiplication, Division
• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
• Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
• Division: (a + bi)/(c + di) = [(a + bi)(c − di)] / (c² + d²)
• i² = −1, i³ = −i, i⁴ = 1
• Powers of i: i4n = 1, i4n+1 = i, i4n+2 = −1, i4n+3 = −i
• Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
• Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
• Division: (a + bi)/(c + di) = [(a + bi)(c − di)] / (c² + d²)
• i² = −1, i³ = −i, i⁴ = 1
• Powers of i: i4n = 1, i4n+1 = i, i4n+2 = −1, i4n+3 = −i
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3. Conjugate Properties
📌 Properties of Conjugate (z̄)
• (z̄)̄ = z
• z + z̄ = 2Re(z)
• z − z̄ = 2i·Im(z)
• (z₁ + z₂)̄ = z̄₁ + z̄₂
• (z₁ · z₂)̄ = z̄₁ · z̄₂
• (z₁/z₂)̄ = z̄₁/z̄₂ (z₂ ≠ 0)
• z · z̄ = |z|²
• (z̄)n = (zn)̄
• z + z̄ = 2Re(z)
• z − z̄ = 2i·Im(z)
• (z₁ + z₂)̄ = z̄₁ + z̄₂
• (z₁ · z₂)̄ = z̄₁ · z̄₂
• (z₁/z₂)̄ = z̄₁/z̄₂ (z₂ ≠ 0)
• z · z̄ = |z|²
• (z̄)n = (zn)̄
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4. Modulus Properties
📌 Properties of Modulus (|z|)
• |z| ≥ 0 and |z| = 0 ⇔ z = 0
• |z| = |z̄| = |−z|
• |z₁ · z₂| = |z₁| · |z₂|
• |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0)
• |zn| = |z|n
• |z₁ + z₂| ≤ |z₁| + |z₂| (Triangle Inequality)
• |z₁ − z₂| ≥ ||z₁| − |z₂||
• |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁z̄₂)
• |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²)
• |z| = |z̄| = |−z|
• |z₁ · z₂| = |z₁| · |z₂|
• |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0)
• |zn| = |z|n
• |z₁ + z₂| ≤ |z₁| + |z₂| (Triangle Inequality)
• |z₁ − z₂| ≥ ||z₁| − |z₂||
• |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁z̄₂)
• |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²)
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5. Argument Properties
📌 Properties of Argument
• arg(z₁ · z₂) = arg(z₁) + arg(z₂)
• arg(z₁/z₂) = arg(z₁) − arg(z₂)
• arg(z̄) = −arg(z)
• arg(zn) = n·arg(z)
• arg(−z) = arg(z) + π (if arg(z) < π)
• Principal argument: −π < Arg(z) ≤ π
• arg(z₁/z₂) = arg(z₁) − arg(z₂)
• arg(z̄) = −arg(z)
• arg(zn) = n·arg(z)
• arg(−z) = arg(z) + π (if arg(z) < π)
• Principal argument: −π < Arg(z) ≤ π
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6. Polar & Euler Form
📌 Polar Form, Euler Formula & Exponential Form
• Polar form: z = r(cos θ + i sin θ) where r = |z|, θ = arg(z)
• Euler’s formula: eiθ = cos θ + i sin θ
• Exponential form: z = r·eiθ
• cos θ = (eiθ + e−iθ)/2
• sin θ = (eiθ − e−iθ)/2i
• Euler’s formula: eiθ = cos θ + i sin θ
• Exponential form: z = r·eiθ
• cos θ = (eiθ + e−iθ)/2
• sin θ = (eiθ − e−iθ)/2i
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7. De Moivre’s Theorem
📌 De Moivre’s Theorem & Applications
• (cos θ + i sin θ)n = cos(nθ) + i sin(nθ)
• zn = [r(cos θ + i sin θ)]n = rn[cos(nθ) + i sin(nθ)]
• For negative n: (cos θ + i sin θ)−n = cos(nθ) − i sin(nθ)
• (cos θ − i sin θ)n = cos(nθ) − i sin(nθ)
• 1/z = (1/r)·e−iθ
• zn = [r(cos θ + i sin θ)]n = rn[cos(nθ) + i sin(nθ)]
• For negative n: (cos θ + i sin θ)−n = cos(nθ) − i sin(nθ)
• (cos θ − i sin θ)n = cos(nθ) − i sin(nθ)
• 1/z = (1/r)·e−iθ
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8. Roots of Complex Numbers
📌 nth Roots of Unity & Complex Numbers
• nth roots of unity: ωk = cos(2πk/n) + i sin(2πk/n), k = 0, 1, …, n−1
• Cube roots of unity: 1, ω, ω² where ω = (−1 + i√3)/2
• ω³ = 1, 1 + ω + ω² = 0
• ω² = ω̄
• Sum of nth roots of unity = 0
• Product of nth roots of unity = (−1)n−1
• Cube roots of unity: 1, ω, ω² where ω = (−1 + i√3)/2
• ω³ = 1, 1 + ω + ω² = 0
• ω² = ω̄
• Sum of nth roots of unity = 0
• Product of nth roots of unity = (−1)n−1
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9. Geometry in Complex Plane
📌 Distance, Section Formula & Geometric Shapes
• Distance between z₁ and z₂: |z₁ − z₂|
• Section formula: z = (mz₂ + nz₁)/(m + n)
• Midpoint: z = (z₁ + z₂)/2
• Circle: |z − z₀| = r (center z₀, radius r)
• Ellipse: |z − z₁| + |z − z₂| = 2a
• Hyperbola: |z − z₁| − |z − z₂| = 2a
• Line: z = z₁ + t(z₂ − z₁)
• Section formula: z = (mz₂ + nz₁)/(m + n)
• Midpoint: z = (z₁ + z₂)/2
• Circle: |z − z₀| = r (center z₀, radius r)
• Ellipse: |z − z₁| + |z − z₂| = 2a
• Hyperbola: |z − z₁| − |z − z₂| = 2a
• Line: z = z₁ + t(z₂ − z₁)
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10. Rotation in Complex Plane
📌 Rotation Theorem
• If z rotates by angle θ about origin: z’ = z·eiθ
• Rotation about point z₀: (z’ − z₀) = (z − z₀)·eiθ
• Rotation of z₁ about z₂ by θ: z₁’ = z₂ + (z₁ − z₂)eiθ
• If z₁, z₂, z₃ are collinear: (z₁ − z₃)/(z₂ − z₃) is real
• If z₁z₂ ⟂ z₃z₄: (z₁ − z₂)/(z₃ − z₄) is purely imaginary
• Rotation about point z₀: (z’ − z₀) = (z − z₀)·eiθ
• Rotation of z₁ about z₂ by θ: z₁’ = z₂ + (z₁ − z₂)eiθ
• If z₁, z₂, z₃ are collinear: (z₁ − z₃)/(z₂ − z₃) is real
• If z₁z₂ ⟂ z₃z₄: (z₁ − z₂)/(z₃ − z₄) is purely imaginary
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11. Logarithm of Complex Numbers
📌 Complex Logarithm
• log z = log|z| + i·arg(z)
• Log z = log|z| + i·Arg(z) (Principal value)
• log(i) = iπ/2
• log(−1) = iπ
• ii = e−π/2
• Log z = log|z| + i·Arg(z) (Principal value)
• log(i) = iπ/2
• log(−1) = iπ
• ii = e−π/2
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12. Trigonometric Series (Advanced)
📌 Sum of Series Using Complex Numbers
• cos θ + cos 2θ + … + cos nθ = [sin(nθ/2)/sin(θ/2)] · cos[(n+1)θ/2]
• sin θ + sin 2θ + … + sin nθ = [sin(nθ/2)/sin(θ/2)] · sin[(n+1)θ/2]
• C + iS = (eiθ + e2iθ + … + eniθ)
• Sum of GP with complex ratio: Sn = a(1 − rn)/(1 − r)
• sin θ + sin 2θ + … + sin nθ = [sin(nθ/2)/sin(θ/2)] · sin[(n+1)θ/2]
• C + iS = (eiθ + e2iθ + … + eniθ)
• Sum of GP with complex ratio: Sn = a(1 − rn)/(1 − r)
✅ COMPLEX NUMBERS ✅
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