Complex Numbers
Conjugate of Complex Numbers
Grade 11
Question:
<p>The conjugate of a complex number is <span>\(\dfrac{1}{i-1}\)</span>. Then the complex number is</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 a complex number from its conjugate, recognize that if z̄ = 1/(i-1), then z = 1/(−i-1). Simplify by rationalizing the denominator using the conjugate of the denominator.
<p><strong>Step 1:</strong> Given that the conjugate of z is 1/(i-1). Let z = a + bi, then z̄ = a - bi = 1/(i-1).</p><p><strong>Step 2:</strong> To find z, take the conjugate of both sides: z = conjugate of [1/(i-1)] = 1/(−i−1) = 1/(−(i+1)) = −1/(i+1).</p><p><strong>Step 3:</strong> Rationalize by multiplying numerator and denominator by the conjugate (−i+1):<br/>z = −1/(i+1) × (−i+1)/(−i+1) = −(−i+1)/(−i² − i + i + 1) = −(−i+1)/(1+1) = −(−i+1)/2 = (i−1)/2.</p><p><strong>Step 4:</strong> Therefore z = −1/2 + i/2 or (−1+i)/2.</p><p>∴ Answer: C</p>
Correct Answer: C