If arg ` z = pi // 4 `

Options

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

Answer

To solve the problem where the argument of the complex number \( z \) is given as \( \arg z = \frac{\pi}{4} \), we will follow these steps:

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

### Step 2: Finding the Relationship Between \( x \) and \( y \)
Taking the tangent of both sides, we have:
\[ \frac{y}{x} = \tan\left(\frac{\pi}{4}\right) \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we can conclude:
\[ \frac{y}{x} = 1 \implies y = x \]

### Step 3: Expressing \( z \) in Terms of \( x \)
Now that we have \( y = x \), we can express \( z \) as:
\[ z = x + iy = x + ix = x(1 + i) \]

### Step 4: Calculating \( z^2 \)
Next, we need to find \( z^2 \):
\[ z^2 = (x(1 + i))^2 = x^2(1 + i)^2 \]
Calculating \( (1 + i)^2 \):
\[ (1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i \]
Thus, we have:
\[ z^2 = x^2 \cdot 2i = 2ix^2 \]

### Step 5: Identifying Real and Imaginary Parts
From \( z^2 = 2ix^2 \), we can see that:
- The real part of \( z^2 \) is \( 0 \) (since there is no real component).
- The imaginary part of \( z^2 \) is \( 2x^2 \).

### Step 6: Conclusion
Since the real part of \( z^2 \) is \( 0 \), we can conclude that:
\[ \text{Real part of } z^2 = 0 \]
Thus, the correct option is:
**Option C: Real value of \( z^2 \) equal to 0.**

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