Complex Numbers
Argument of Complex Numbers
Grade 11
Question:
<p>Let \(z_1 = a + ib\), \(z_2 = a - ib\), \(z_3 = c + id\), \(z_4 = c - id\). Then \(\arg\left(\dfrac{z_1}{z_4}\right) + \arg\left(\dfrac{z_2}{z_3}\right)\) equals:</p>
<p>\(0\)</p>
<p>\(\pi\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(2\pi\)</p>
Step-by-Step Solution
Key Concept: Recognize that z₁ and z₂ are complex conjugates, as are z₃ and z₄. Use the property that arg(z̄) = -arg(z) and arg(z₁/z₂) = arg(z₁) - arg(z₂) to simplify the sum of arguments.
<p><strong>Step 1:</strong> Identify the structure. We have z₁ = a + ib, z₂ = a - ib (conjugate pair), z₃ = c + id, z₄ = c - id (conjugate pair).</p><p><strong>Step 2:</strong> Write the argument sum: arg(z₁/z₄) + arg(z₂/z₃) = arg(z₁) - arg(z₄) + arg(z₂) - arg(z₃)</p><p><strong>Step 3:</strong> Use conjugate properties: arg(z₂) = arg(z̄₁) = -arg(z₁) and arg(z₄) = arg(z̄₃) = -arg(z₃)</p><p><strong>Step 4:</strong> Substitute: = arg(z₁) - (-arg(z₃)) + (-arg(z₁)) - arg(z₃) = arg(z₁) + arg(z₃) - arg(z₁) - arg(z₃) = 0</p><p><strong>Step 5:</strong> Alternatively, compute directly: arg(z₁/z₄) = arg[(a+ib)/(c-id)] and arg(z₂/z₃) = arg[(a-ib)/(c+id)]. Note that z₂/z₃ is the conjugate of z₁/z₄, so their arguments sum to 0.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: A