Write the amplitude of `(1)/(i)`

Answer

To find the amplitude (or argument) of the complex number \( \frac{1}{i} \), we can follow these steps:

### Step 1: Simplify \( \frac{1}{i} \)
To simplify \( \frac{1}{i} \), we can multiply the numerator and the denominator by \( i \):
\[ \frac{1}{i} = \frac{1 \cdot i}{i \cdot i} = \frac{i}{-1} = -i \]

### Step 2: Identify the form of the complex number
The complex number \( -i \) can be expressed in the form \( a + bi \), where:
- \( a = 0 \)
- \( b = -1 \)

### Step 3: Determine the quadrant
The point \( (0, -1) \) lies on the negative imaginary axis, which is in the fourth quadrant of the complex plane.

### Step 4: Find the argument
The argument (or amplitude) of a complex number in the fourth quadrant can be calculated as follows:
- The reference angle \( \alpha \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \).
- Since \( a = 0 \) and \( b = -1 \), we have:
\[ \frac{b}{a} = \frac{-1}{0} \]
This indicates that the angle approaches \( \frac{\pi}{2} \) (or \( 90^\circ \)).

### Step 5: Adjust for the quadrant
In the fourth quadrant, the argument is negative. Therefore, the argument of \( -i \) is:
\[ \text{Argument} = -\frac{\pi}{2} \]

### Final Answer
The amplitude of \( \frac{1}{i} \) is:
\[ -\frac{\pi}{2} \]

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