If the conjugate of a complex numbers is `1/(i-1)`, where `i=sqrt(-1)`. Then, the complex number is

Options

A. `(-1)/(i-1)`
B. `1/(i+1)`
C. `(-1)/(i+1)`
D. `1/(i-1)`

Answer

To find the complex number \( z \) given that its conjugate is \( \frac{1}{i - 1} \), we can follow these steps:

### Step 1: Write down the conjugate
Given that the conjugate of the complex number \( z \) is:
\[ \overline{z} = \frac{1}{i - 1} \]

### Step 2: Simplify the expression
To simplify \( \frac{1}{i - 1} \), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( i + 1 \):
\[ \overline{z} = \frac{1 \cdot (i + 1)}{(i - 1)(i + 1)} = \frac{i + 1}{i^2 - 1^2} \]
Since \( i^2 = -1 \), we have:
\[ \overline{z} = \frac{i + 1}{-1 - 1} = \frac{i + 1}{-2} = -\frac{1}{2} - \frac{i}{2} \]

### Step 3: Find \( z \)
We know that if \( \overline{z} = a - bi \), then \( z = a + bi \). Therefore, we can write:
\[ z = -\frac{1}{2} + \frac{i}{2} \]

### Step 4: Final answer
Thus, the complex number \( z \) is:
\[ z = -\frac{1}{2} + \frac{i}{2} \]