If z is a purely real complex number such that `"Re"(z) lt 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 real complex number \( z \) where the real part is less than zero.

### Step-by-Step Solution:

1. **Understanding the Complex Number**:
Since \( z \) is a purely real complex number, we can express it as:
\[ z = a + bi \]
where \( b = 0 \). Therefore, we can write:
\[ z = a + 0i = a \]
Here, \( a \) is the real part of \( z \).

2. **Condition on the Real Part**:
We are given that the real part of \( z \) is less than zero:
\[ a < 0 \]
This means \( z \) lies on the negative side of the real axis.

3. **Graphical Representation**:
On the complex plane, the point \( z \) (which is purely real and negative) will be located on the negative real axis. This can be visualized as:
\[ z = -|a| \quad (\text{where } |a| > 0) \]
The point will be to the left of the origin.

4. **Finding the Argument**:
The argument of a complex number \( z \) is defined as the angle \( \theta \) that the line representing the complex number makes with the positive x-axis (real axis). For a purely real number:
- If \( z \) is positive, \( \text{arg}(z) = 0 \).
- If \( z \) is negative (as in our case), \( \text{arg}(z) = \pi \) radians (or 180 degrees).

5. **Conclusion**:
Therefore, since \( z \) is a purely real complex number with \( \text{Re}(z) < 0 \), we conclude that:
\[ \text{arg}(z) = \pi \]

### Final Answer:
\[ \text{arg}(z) = \pi \]