Let `z in C` be such that `Re(z^(2)) = 0`, then

Options

A. `|Re(z)| + Im(z) = 0`
B. `|Re(z)| = |Im (z)|`
C. `Re(z) + |Im (z) | = 0`
D. `Re(z) = 0 or Im (z) = 0`

Answer

To solve the problem, we need to find the relationship between the real part and the imaginary part of a complex number \( z \) such that \( \text{Re}(z^2) = 0 \).

Let's denote the complex number \( z \) as:
\[ z = x + iy \]
where \( x \) is the real part and \( y \) is the imaginary part of \( z \).

### Step 1: Calculate \( z^2 \)
We first calculate \( z^2 \):
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \]

### Step 2: Find the real part of \( z^2 \)
The real part of \( z^2 \) is given by:
\[ \text{Re}(z^2) = x^2 - y^2 \]

### Step 3: Set the real part equal to zero
According to the problem, we have:
\[ \text{Re}(z^2) = 0 \implies x^2 - y^2 = 0 \]

### Step 4: Solve the equation
From the equation \( x^2 - y^2 = 0 \), we can factor it as:
\[ (x - y)(x + y) = 0 \]
This gives us two cases:
1. \( x - y = 0 \) (i.e., \( x = y \))
2. \( x + y = 0 \) (i.e., \( x = -y \))

### Step 5: Interpret the results
The first case \( x = y \) implies that the real part of \( z \) is equal to the imaginary part of \( z \). The second case \( x = -y \) implies that the real part of \( z \) is the negative of the imaginary part of \( z \).

### Conclusion
Thus, the relationship between the real part and the imaginary part of \( z \) can be summarized as:
- Either \( \text{Re}(z) = \text{Im}(z) \) or \( \text{Re}(z) = -\text{Im}(z) \).