Complex Numbers and De Moivre’s Theorem

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A complex number is a number of the form z = a + bi, where: • a and b are real numbers • i = √(-1) is the imaginary unit • a is called the real part [Re(z) = a] • b is called the imaginary part [Im(z)

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Powers of i follow a cyclic pattern: • i¹ = i • i² = -1 • i³ = i² · i = (-1) · i = -i • i⁴ = (i²)² = (-1)² = 1 • i⁵ = i⁴ · i = 1 · i = i • i⁸ = (i⁴)² = 1² = 1 Pattern: i, -1, -i, 1 (repeats every 4 p

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For z = 3 - 5i, the conjugate z̄ = 3 + 5i General Rule: The conjugate of z = a + bi is z̄ = a - bi • Change the sign of the imaginary part only • Real part remains unchanged Properties of Conjugates

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For z = 4 + 3i: |z| = √(a² + b²) = √(4² + 3²) = √(16 + 9) = √25 = 5 General Formula: |z| = |a + bi| = √(a² + b²) Geometric Interpretation: • Modulus represents the distance from origin to point (a,b

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Two complex numbers are equal if and only if their real parts and imaginary parts are respectively equal. General Rule: a + bi = c + di ⟺ a = c and b = d Solving 2x + 3yi = 6 + 9i: Step 1: Compare r

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Solution: (3 + 2i) + (-1 + 4i) = (3 + (-1)) + (2 + 4)i = 2 + 6i General Rule for Addition: (a + bi) + (c + di) = (a + c) + (b + d)i • Add real parts separately • Add imaginary parts separately Prope

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Solution: (5 + 3i) - (2 - i) = (5 - 2) + (3 - (-1))i = 3 + 4i General Rule for Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i Geometric Interpretation: • Complex numbers are points in Argand

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Step 1: Apply distributive property (2 + 3i)(1 - 2i) = 2(1 - 2i) + 3i(1 - 2i) Step 2: Expand each term = 2 - 4i + 3i - 6i² Step 3: Substitute i² = -1 = 2 - 4i + 3i - 6(-1) = 2 - 4i + 3i + 6 Step 4: