If `z ne 0` be a complex number and `"arg"(z)=pi//4`, then

Options

A. `"Re"(z)="Im"(z)` only
B. `Re(z) = Im(z) gt 0`
C. `Re(z^(2))=Im(z^(2))`
D. none of these

Answer

To solve the problem, we start with the given information about the complex number \( z \).

### Step 1: Understand the argument of the complex number
The argument of a complex number \( z = x + iy \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
Given that \( \text{arg}(z) = \frac{\pi}{4} \), we can set up the equation:
\[ \tan^{-1}\left(\frac{y}{x}\right) = \frac{\pi}{4} \]

### Step 2: Solve for the ratio of the imaginary part to the real part
From the tangent function, we know:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we have:
\[ \frac{y}{x} = 1 \]
This implies:
\[ y = x \]

### Step 3: Analyze the implications of \( y = x \)
Since \( y = x \), we can express \( z \) as:
\[ z = x + ix \]
This means both the real part and the imaginary part of \( z \) are equal.

### Step 4: Determine the conditions for \( x \) and \( y \)
Since \( z \neq 0 \), we know that \( x \) cannot be zero. Additionally, for the argument to be \( \frac{\pi}{4} \), both \( x \) and \( y \) must be positive (as the angle \( \frac{\pi}{4} \) corresponds to the first quadrant in the complex plane).

Thus, we conclude:
\[ x > 0 \quad \text{and} \quad y > 0 \]

### Conclusion
The real part of \( z \) is equal to the imaginary part of \( z \), and both are greater than 0. Therefore, the correct answer is that the real part of \( z \) is equal to the imaginary part of \( z \), and both are greater than 0.

### Final Answer
The correct option is: "The real part of \( z \) is equal to the imaginary part of \( z \) and both are greater than 0."

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