if z=3 -2i, then verify that
(i) ` z + barz = 2Rez`
(ii) ` z - barz = 2ilm z `

Answer

To verify the given statements for the complex number \( z = 3 - 2i \), we will follow these steps:

### Step 1: Find the conjugate of \( z \)
The complex conjugate of a complex number \( z = x + iy \) is given by \( \bar{z} = x - iy \).

For our case:
\[ z = 3 - 2i \implies \bar{z} = 3 + 2i \]

### Step 2: Calculate \( z + \bar{z} \)
Now, we will calculate \( z + \bar{z} \):
\[ z + \bar{z} = (3 - 2i) + (3 + 2i) \]
Combine the real and imaginary parts:
\[ z + \bar{z} = 3 + 3 + (-2i + 2i) = 6 + 0i = 6 \]

### Step 3: Calculate \( 2 \text{Re}(z) \)
The real part of \( z \) is:
\[ \text{Re}(z) = 3 \]
Now, calculate \( 2 \text{Re}(z) \):
\[ 2 \text{Re}(z) = 2 \times 3 = 6 \]

### Step 4: Verify \( z + \bar{z} = 2 \text{Re}(z) \)
From our calculations:
\[ z + \bar{z} = 6 \quad \text{and} \quad 2 \text{Re}(z) = 6 \]
Thus, we have:
\[ z + \bar{z} = 2 \text{Re}(z) \quad \text{(Verified)} \]

### Step 5: Calculate \( z - \bar{z} \)
Now, we will calculate \( z - \bar{z} \):
\[ z - \bar{z} = (3 - 2i) - (3 + 2i) \]
Combine the real and imaginary parts:
\[ z - \bar{z} = 3 - 3 + (-2i - 2i) = 0 - 4i = -4i \]

### Step 6: Calculate \( 2i \text{Im}(z) \)
The imaginary part of \( z \) is:
\[ \text{Im}(z) = -2 \]
Now, calculate \( 2i \text{Im}(z) \):
\[ 2i \text{Im}(z) = 2i \times (-2) = -4i \]

### Step 7: Verify \( z - \bar{z} = 2i \text{Im}(z) \)
From our calculations:
\[ z - \bar{z} = -4i \quad \text{and} \quad 2i \text{Im}(z) = -4i \]
Thus, we have:
\[ z - \bar{z} = 2i \text{Im}(z) \quad \text{(Verified)} \]

### Conclusion
Both statements have been verified:
1. \( z + \bar{z} = 2 \text{Re}(z) \)
2. \( z - \bar{z} = 2i \text{Im}(z) \)