The conjugate of a complex number z is `(2)/(1-i)`. Then Re(z) equals

Options

A. -1
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the real part of the complex number \( z \) given that its conjugate \( \overline{z} \) is \( \frac{2}{1-i} \).

### Step-by-Step Solution:

1. **Identify the conjugate**:
The conjugate of a complex number \( z = x + iy \) is given by \( \overline{z} = x - iy \). In this case, we have \( \overline{z} = \frac{2}{1-i} \).

2. **Rationalize the denominator**:
To simplify \( \frac{2}{1-i} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1 + i \):
\[ \overline{z} = \frac{2(1+i)}{(1-i)(1+i)} \]

3. **Calculate the denominator**:
The denominator can be simplified using the difference of squares:
\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

4. **Substitute back into the equation**:
Now substituting back, we have:
\[ \overline{z} = \frac{2(1+i)}{2} = 1 + i \]

5. **Find the original complex number \( z \)**:
Since \( \overline{z} = 1 + i \), we can find \( z \) by changing the sign of the imaginary part:
\[ z = 1 - i \]

6. **Identify the real part**:
The real part of \( z \) is the coefficient of the real component:
\[ \text{Re}(z) = 1 \]

### Conclusion:
The real part of the complex number \( z \) is \( 1 \).