If ` z = (4)/(1 - i)` then ` bar(z)` is equal to

Options

A. (1 - i)
B. `(1 +i)/(4)`
C. `(4)/(1+i)`
D. `(2)/(1-i)`

Answer

To find the conjugate of the complex number \( z = \frac{4}{1 - i} \), we will follow these steps:

### Step 1: Multiply by the Conjugate
To express \( z \) in standard form, we need to eliminate the imaginary unit from the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( 1 + i \).

\[ z = \frac{4}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{4(1 + i)}{(1 - i)(1 + i)} \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator. The product \( (1 - i)(1 + i) \) can be calculated as follows:

\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 3: Substitute Back
Now, substituting back into our expression for \( z \):

\[ z = \frac{4(1 + i)}{2} = \frac{4}{2}(1 + i) = 2(1 + i) \]

### Step 4: Identify the Conjugate
The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( z = 2(1 + i) \) is:

\[ \bar{z} = 2(1 - i) \]

### Step 5: Final Expression
Thus, we can express \( \bar{z} \) as:

\[ \bar{z} = 2(1 - i) \]

### Conclusion
The conjugate \( \bar{z} \) is equal to \( 2(1 - i) \).

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