Complex Numbers
Conjugate and Argument
Grade 11
Question:
<p>If <span>\(z_1, z_2\)</span> and <span>\(z_3, z_4\)</span> are 2 pairs of complex conjugate numbers, then <span>\(\arg\left(\dfrac{z_1}{z_4}\right)+\arg\left(\dfrac{z_2}{z_3}\right)\)</span> equals</p>
<p>0</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{3\pi}{2}\)</p>
<p>\(\pi\)</p>
Step-by-Step Solution
Key Concept: If z₁, z₂ are conjugates and z₃, z₄ are conjugates, then z₂ = z̄₁ and z₄ = z̄₃. Use the property that arg(z̄) = -arg(z) and arg(z/w) = arg(z) - arg(w) to simplify the sum of arguments.
<p><strong>Step 1:</strong> Set up the conjugate relationships. Since z₁, z₂ are conjugate pairs: z₂ = z̄₁. Since z₃, z₄ are conjugate pairs: z₄ = z̄₃.</p><p><strong>Step 2:</strong> Substitute into the expression:</p><p>arg(z₁/z₄) + arg(z₂/z₃) = arg(z₁/z̄₃) + arg(z̄₁/z₃)</p><p><strong>Step 3:</strong> Apply arg(z/w) = arg(z) - arg(w) and arg(z̄) = -arg(z):</p><p>arg(z₁/z̄₃) = arg(z₁) - arg(z̄₃) = arg(z₁) - (-arg(z₃)) = arg(z₁) + arg(z₃)</p><p>arg(z̄₁/z₃) = arg(z̄₁) - arg(z₃) = -arg(z₁) - arg(z₃)</p><p><strong>Step 4:</strong> Add the results:</p><p>[arg(z₁) + arg(z₃)] + [-arg(z₁) - arg(z₃)] = 0</p><p>∴ Answer: <strong>0</strong> (or <strong>2nπ</strong> for integer n)</p>
Correct Answer: A