Let z be a purely imaginary number such that `"lm"(z) le 0`. Then, arg (z) is equal to

Options

A. `pi`
B. `pi//2`
C. 0
D. `-pi//2`

Explanation

To solve the problem, we need to determine the argument of a purely imaginary number \( z \) given that its imaginary part is less than zero. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding Purely Imaginary Numbers**:
A purely imaginary number can be expressed in the form:
\[ z = 0 + i b \]
where \( b \) is a real number. This means that the real part of \( z \) is 0.

2. **Condition on the Imaginary Part**:
The problem states that the imaginary part of \( z \) is less than zero:
\[ b < 0 \]
This indicates that \( z \) lies on the negative imaginary axis of the complex plane.

3. **Visualizing in the Complex Plane**:
In the complex plane, the x-axis represents the real part and the y-axis represents the imaginary part. Since \( z \) is purely imaginary and \( b < 0 \), the point representing \( z \) will be located below the x-axis, specifically on the negative side of the y-axis.

4. **Finding the Argument**:
The argument of a complex number \( z = x + iy \) is defined as the angle \( \theta \) made with the positive x-axis. For purely imaginary numbers:
- If \( b > 0 \), \( \arg(z) = \frac{\pi}{2} \) (positive imaginary axis).
- If \( b < 0 \), \( \arg(z) = -\frac{\pi}{2} \) (negative imaginary axis).

Since we have \( b < 0 \), we can conclude that:
\[ \arg(z) = -\frac{\pi}{2} \]

5. **Final Answer**:
Therefore, the argument of \( z \) is:
\[ \arg(z) = -\frac{\pi}{2} \]