Write the argument of `-i`.

Answer

To find the argument of the complex number \(-i\), we can follow these steps:

### Step 1: Identify the complex number in standard form
The complex number can be expressed as:
\[ z = -i = 0 - i \]
Here, we identify \(a = 0\) and \(b = -1\), where \(z = a + bi\).

### Step 2: Determine the position in the complex plane
The point \((a, b)\) corresponds to the coordinates \((0, -1)\) in the complex plane. This point lies on the negative imaginary axis, which is in the fourth quadrant.

### Step 3: Use the formula for the argument in the fourth quadrant
In the fourth quadrant, the argument \(\theta\) of a complex number is given by:
\[ \theta = -\alpha \]
where \(\alpha\) is the angle formed with the positive real axis.

### Step 4: Calculate \(\alpha\) using the tangent function
We can find \(\alpha\) using the tangent function:
\[ \tan(\alpha) = \frac{|b|}{|a|} = \frac{|-1|}{|0|} = \frac{1}{0} \]
Since division by zero leads to infinity, we conclude that:
\[ \tan(\alpha) = \infty \]
This implies that \(\alpha\) corresponds to \(\frac{\pi}{2}\) radians.

### Step 5: Calculate the argument
Now, substituting \(\alpha\) back into the formula for the argument in the fourth quadrant:
\[ \theta = -\alpha = -\frac{\pi}{2} \]

### Final Result
Thus, the argument of \(-i\) is:
\[ \text{arg}(-i) = -\frac{\pi}{2} \]

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