Complex Numbers

Answer

Step 1: Recognize that powers of i repeat every 4 terms: i^1=i, i^2=-1, i^3=-i, i^4=1 Step 2: Divide the exponent by 4: 7 ÷ 4 = 1 remainder 3 Step 3: Therefore i^7 = i^3 = -i Answer: -i Key insight: …

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Step 1: For i^1729: 1729 ÷ 4 = 432 remainder 1, so i^1729 = i^1 = i Step 2: For i^1950: 1950 ÷ 4 = 487 remainder 2, so i^1950 = i^2 = -1 Step 3: Add the results: i + (-1) = -1 + i Answer: -1 + i Meth…

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Step 1: Calculate individual terms: i + (-1) + (-i) + 1 + i + (-1) + (-i) + 1 Step 2: Group first 4 terms: (i - 1 - i + 1) = 0 Step 3: Group next 4 terms: (i - 1 - i + 1) = 0 Step 4: Total sum = 0 + 0…

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Step 1: Identify components: Real part = 3, Imaginary part = 4 Step 2: Rectangular form: z = 3 + 4i (already in standard form) Step 3: Check if purely imaginary: A number is purely imaginary if real p…

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Step 1: Expand left side: 2x + ix + y - iy = (2x + y) + i(x - y) Step 2: Equate with 3 + 4i: (2x + y) + i(x - y) = 3 + 4i Step 3: Equate real parts: 2x + y = 3 ... (1) Step 4: Equate imaginary parts: …

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Step 1: Find conjugate: Replace i with -i Step 2: z̄ = 5 + 3i Step 3: Calculate |z|: |5 - 3i| = √(5² + (-3)²) = √(25 + 9) = √34 Step 4: Calculate |z̄|: |5 + 3i| = √(5² + 3²) = √(25 + 9) = √34 Step 5: …

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Step 1: Multiply numerator and denominator by conjugate of denominator Step 2: Conjugate of (5 - 12i) is (5 + 12i) Step 3: [(3 + 4i)(5 + 12i)] / [(5 - 12i)(5 + 12i)] Step 4: Numerator: 15 + 36i + 20i …

Answer

Step 1: Simplify (1+i)/(1-i) by multiplying by (1+i)/(1+i) Step 2: [(1+i)²] / [(1-i)(1+i)] = [1 + 2i + i²] / [1 - i²] Step 3: [1 + 2i - 1] / [1 + 1] = 2i/2 = i Step 4: Find cube: [(1+i)/(1-i)]³ = i³ S…