<p>If \(z_1, z_2\) and \(z_3, z_4\) are two pairs of conjugate complex numbers, then find the value of \(\arg(z_1/z_4) + \arg(z_2/z_3)\).</p>
Step-by-Step Solution
Key Concept: When z₁ and z₂ are conjugate pairs, arg(z₁) + arg(z₂) = 0 (mod 2π). Since z₃ = conj(z₄), we have arg(z₃) = -arg(z₄), allowing the arguments to cancel when combined properly.
<p><strong>Step 1:</strong> Let z₁, z₂ be conjugate pairs: z₂ = conj(z₁), so arg(z₂) = -arg(z₁)</p><p><strong>Step 2:</strong> Let z₃, z₄ be conjugate pairs: z₃ = conj(z₄), so arg(z₃) = -arg(z₄)</p><p><strong>Step 3:</strong> Using arg(w₁/w₂) = arg(w₁) - arg(w₂):</p><p>arg(z₁/z₄) = arg(z₁) - arg(z₄)</p><p>arg(z₂/z₃) = arg(z₂) - arg(z₃) = -arg(z₁) - (-arg(z₄)) = -arg(z₁) + arg(z₄)</p><p><strong>Step 4:</strong> Adding both arguments:</p><p>arg(z₁/z₄) + arg(z₂/z₃) = [arg(z₁) - arg(z₄)] + [-arg(z₁) + arg(z₄)] = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0