The conjugate of a complex number is `1/(i-1)` . Then the complex number is

Options

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

Explanation

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

### Step 1: Write down the conjugate
Given:
\[ \overline{z} = \frac{1}{i - 1} \]

### Step 2: Find the complex number \( z \)
The relationship between a complex number and its conjugate is:
\[ z = \overline{z}^* \]
where \( \overline{z}^* \) denotes the conjugate of \( \overline{z} \).

### Step 3: Calculate the conjugate of \( \overline{z} \)
To find \( z \), we first need to compute the conjugate of \( \overline{z} \):
\[ z = \overline{\left(\frac{1}{i - 1}\right)} \]

### Step 4: Simplify \( \overline{z} \)
To find the conjugate of \( \frac{1}{i - 1} \), we first simplify \( \frac{1}{i - 1} \):
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{i - 1} \cdot \frac{-1 - i}{-1 - i} = \frac{-1 - i}{(-1)^2 + (1)^2} = \frac{-1 - i}{1 + 1} = \frac{-1 - i}{2} \]
Thus,
\[ \overline{z} = \frac{-1 - i}{2} \]

### Step 5: Find the conjugate of \( \overline{z} \)
Now, we find the conjugate of \( \overline{z} \):
\[ z = \overline{\left(\frac{-1 - i}{2}\right)} = \frac{-1 + i}{2} \]

### Conclusion
The complex number \( z \) is:
\[ z = \frac{-1 + i}{2} \]

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