Complex Numbers

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A complex number is a number of the form z = a + ib, where a, b ∈ R and i = √(-1) with i² = -1. Example: z = 3 + 4i - Real part: Re(z) = 3 - Imaginary part: Im(z) = 4 Note: The set of complex number…

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Step 1: Recall the pattern of powers of i: i¹ = i, i² = -1, i³ = -i, i⁴ = 1 Step 2: For any positive integer n, divide n by 4: i^n = i^r where r is the remainder when n is divided by 4 Step 3: Divid…

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Condition: Two complex numbers are equal if and only if their corresponding real and imaginary parts are equal. z₁ = z₂ ⟺ a = c and b = d Example: If 3x + i(2y - 1) = 6 + 5i, find x and y. Step 1: …

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Definition: The conjugate of z = a + ib is z̄ = a - ib For z = 5 - 3i: z̄ = 5 - (-3)i = 5 + 3i Verification that z · z̄ = |z|²: Step 1: Calculate z · z̄ z · z̄ = (5 - 3i)(5 + 3i) = 25 + 15i - 15i - …

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Rule: For z₁ = a + ib and z₂ = c + id: z₁ + z₂ = (a + c) + i(b + d) Solution: Step 1: Identify components z₁ = 2 + 3i → a = 2, b = 3 z₂ = -4 + 5i → c = -4, d = 5 Step 2: Add real parts and imaginary…

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Rule: (a + ib)(c + id) = (ac - bd) + i(ad + bc) Step 1: Expand using distributive property z₁ · z₂ = (3 + 2i)(1 - 4i) = 3(1 - 4i) + 2i(1 - 4i) = 3 - 12i + 2i - 8i² Step 2: Substitute i² = -1 = 3 - 1…

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Method: Multiply numerator and denominator by conjugate of denominator Step 1: Write the division z₁/z₂ = (4 + 3i)/(2 - i) Step 2: Find conjugate of denominator Conjugate of (2 - i) = 2 + i Step 3:…

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Step 1: Find the modulus |z| = √(a² + b²) where z = a + bi |z| = √((-3)² + 4²) = √(9 + 16) = √25 = 5 Step 2: Find the argument For z = -3 + 4i, we have a = -3, b = 4 Since a < 0 and b > 0, z lies in …