Find the argument of `(1)/(1+i)`

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `(-pi)/(4)`

Answer

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

### Step 1: Rationalize the denominator
To simplify \( \frac{1}{1+i} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 1-i \).

\[ \frac{1}{1+i} \cdot \frac{1-i}{1-i} = \frac{1-i}{(1+i)(1-i)} \]

### Step 2: Simplify the denominator
Now, calculate the denominator using the difference of squares:

\[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 3: Write the simplified expression
Now we can write the expression as:

\[ \frac{1-i}{2} = \frac{1}{2} - \frac{i}{2} \]

### Step 4: Identify the coordinates
From the expression \( \frac{1}{2} - \frac{i}{2} \), we can identify the real part \( x = \frac{1}{2} \) and the imaginary part \( y = -\frac{1}{2} \).

### Step 5: Determine the quadrant
Since \( x > 0 \) and \( y < 0 \), the point \( \left( \frac{1}{2}, -\frac{1}{2} \right) \) lies in the fourth quadrant.

### Step 6: Calculate the argument
The argument \( \theta \) of a complex number in the fourth quadrant can be calculated using:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Substituting the values of \( y \) and \( x \):

\[ \theta = \tan^{-1}\left(\frac{-\frac{1}{2}}{\frac{1}{2}}\right) = \tan^{-1}(-1) \]

### Step 7: Find the angle
The angle \( \tan^{-1}(-1) \) corresponds to \( -\frac{\pi}{4} \) (since it is in the fourth quadrant).

### Final Answer
Thus, the argument of \( \frac{1}{1+i} \) is:

\[ \text{Argument} = -\frac{\pi}{4} \]

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