Complex Numbers
Conjugate of Complex Numbers
Grade 11
Question:
<p>Given that the complex number \(\bar{z} = \dfrac{1}{i-1}\), then \(z\) equals:</p>
<p>\(\dfrac{-1}{i+1}\)</p>
<p>\(\dfrac{1}{i+1}\)</p>
<p>\(\dfrac{-1}{i-1}\)</p>
<p>\(\dfrac{1}{i-1}\)</p>
Step-by-Step Solution
Key Concept: To find z, first rationalize the complex fraction by multiplying by the conjugate of the denominator, then take the complex conjugate of the result to get z from the given ̄z.
<p><strong>Step 1:</strong> We are given ̄z = 1/(i-1). Rationalize by multiplying numerator and denominator by the conjugate of (i-1), which is (-i-1):</p><p>̄z = (1)/((i-1)) · ((-i-1))/((-i-1)) = ((-i-1))/((-i-1)(i-1))</p><p><strong>Step 2:</strong> Calculate the denominator: (-i-1)(i-1) = -i² + i - i + 1 = -(-1) + 1 = 1 + 1 = 2</p><p>So: ̄z = (-i-1)/2 = (-1-i)/2</p><p><strong>Step 3:</strong> Since ̄z = (-1-i)/2, find z by taking the complex conjugate:</p><p>z = ̄(̄z) = ̄((-1-i)/2) = (-1+i)/2 = (-1+i)/2</p><p>∴ Answer: z = <strong>(-1+i)/2</strong> or <strong>(-1/2 + i/2)</strong></p>
Correct Answer: A