Complex Numbers

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Step 1: Write -36 = 36 × (-1) Step 2: √(-36) = √36 × √(-1) = 6 × i Answer: 6i Key concept: √(-a) = i√a for positive real a…

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Step 1: Divide 47 by 4 → 47 = 4(11) + 3 Step 2: The remainder is 3, so i^47 = i^3 Step 3: i^3 = i^2 × i = (-1) × i = -i Answer: -i Pattern: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 (repeats every 4 power…

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Step 1: Convert each power - i^10 = (i^2)^5 = (-1)^5 = -1 - i^20 = (i^2)^10 = (-1)^10 = 1 - i^30 = (i^2)^15 = (-1)^15 = -1 Step 2: Add all terms 1 + (-1) + 1 + (-1) = 0 Answer: 0…

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The conjugate of a complex number a + bi is a - bi For z = 3 - 4i: Change the sign of the imaginary part z̄ = 3 + 4i Note: To find conjugate, only change the sign in front of i…

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Step 1: Expand (2 + i)^2 (2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i Step 2: Find conjugate of 3 + 4i z̄ = 3 - 4i Answer: 3 - 4i Alternative approach: z = (2 + i)^2, so z̄ = (2̄ + ī)^2 = (2 - i)…

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Formula: For z = a + bi, |z| = √(a^2 + b^2) For z = 5 + 12i: Step 1: a = 5, b = 12 Step 2: |z| = √(5^2 + 12^2) = √(25 + 144) = √169 = 13 Answer: |z| = 13 The modulus represents the distance from or…

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Given z = 3 - 4i Step 1: Calculate |z| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5 Step 2: -z = -3 + 4i, so |-z| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5 Step 3: z̄ = 3 + 4i, so |z̄| = √(3^2 + 4^2) = √(…

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Method 1: Using property |z₁z₂| = |z₁||z₂| Step 1: |1 + 2i| = √(1 + 4) = √5 Step 2: |3 - i| = √(9 + 1) = √10 Step 3: |(1 + 2i)(3 - i)| = √5 × √10 = √50 = 5√2 Method 2: Multiply first, then find modul…