Find the real part of `(1-i)/(1+i)`.

Answer

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

### Step 1: Rationalize the denominator
To eliminate the imaginary part from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\).

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

### Step 2: Simplify the denominator
Now, we calculate the denominator:

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

### Step 3: Simplify the numerator
Next, we calculate the numerator:

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

### Step 4: Combine the results
Now we can combine the results from the numerator and the denominator:

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

### Step 5: Identify the real part
The complex number \(-i\) can be expressed as \(0 - i\), where the real part is \(0\).

Thus, the real part of \(\frac{1-i}{1+i}\) is:

\[ \text{Real part} = 0 \]

### Final Answer
The real part of \(\frac{1-i}{1+i}\) is \(0\).

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