The argument of `(1+i)/(1-i)` is (i) 0 (ii) `-(pi)/(2)` (iii) `(pi)/(2)` (iv) `pi`

Options

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

Answer

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

### Step 1: Write the expression
We start with the expression:
\[ z = \frac{1+i}{1-i} \]

### Step 2: Rationalize the denominator
To simplify, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{(1+i)(1+i)}{(1-i)(1+i)} \]

### Step 3: Expand the numerator and denominator
Now, we expand both the numerator and the denominator:
- The numerator:
\[ (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]
- The denominator:
\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 2 \]

### Step 4: Simplify the expression
Now we can simplify \(z\):
\[ z = \frac{2i}{2} = i \]

### Step 5: Find the argument
The argument of a complex number \(z = x + iy\) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
For \(z = i\), we have \(x = 0\) and \(y = 1\):
\[ \text{arg}(i) = \tan^{-1}\left(\frac{1}{0}\right) \]
Since \(\frac{1}{0}\) is undefined, we know that the argument is:
\[ \text{arg}(i) = \frac{\pi}{2} \]

### Conclusion
Thus, the argument of \(\frac{1+i}{1-i}\) is:
\[ \frac{\pi}{2} \]
The correct option is (iii) \(\frac{\pi}{2}\).

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