Complex Numbers

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The imaginary unit is denoted by 'i' where i = √(-1) and i² = -1. Key Properties: 1) i × 0 = 0 2) If a ∈ R, then √(-a²) = i√(a²) = ±ia 3) If a, b ∈ R, and ai = bi then a = b Powers of i follow a cyc…

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A complex number is of the form z = a + ib where a, b ∈ R and i = √(-1). For z = 3 + 4i: • Real part: Re(z) = 3 • Imaginary part: Im(z) = 4 Note: The imaginary part is the coefficient of i (not incl…

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The conjugate of z = a + ib is z̄ = a - ib For z = -2 + 7i: Conjugate z̄ = -2 - 7i Key Properties of conjugates: 1) (z̄)̄ = z 2) If z = z̄, then z is real 3) If z = -z̄, then z is pure imaginary 4) …

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Step 1: Group real and imaginary parts separately (3 + 2i) + (5 - 4i) + (-1 + 6i) Step 2: Add real parts Real parts: 3 + 5 + (-1) = 7 Step 3: Add imaginary parts Imaginary parts: 2 + (-4) + 6 = 4 S…

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Step 1: Use FOIL method or distributive property (2 + 3i)(4 - 5i) = 2(4 - 5i) + 3i(4 - 5i) Step 2: Distribute each term = 2×4 + 2×(-5i) + 3i×4 + 3i×(-5i) = 8 - 10i + 12i - 15i² Step 3: Substitute i²…

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Step 1: Multiply numerator and denominator by conjugate of denominator (3 + 4i)/(1 + 2i) × (1 - 2i)/(1 - 2i) Step 2: Calculate numerator (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i² = 3 - 2i + 8 = 11 - 2i S…

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Step 1: Use the pattern that powers of i repeat every 4 terms i¹ = i, i² = -1, i³ = -i, i⁴ = 1, i⁵ = i, ... Step 2: Divide the exponent by 4 45 ÷ 4 = 11 remainder 1 So 45 = 4×11 + 1 Step 3: Use the …

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Step 1: Apply equality condition for complex numbers Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal. Step 2: Compare real and imaginary pa…