Find real and imaginary parts of `(1-i)/(1+i)`.

Answer

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

### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\).

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

### Step 2: Simplify the Denominator
Now we simplify the denominator using the formula \(a^2 - b^2\):

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

### Step 3: Simplify the Numerator
Next, we simplify the numerator:

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

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

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

### Step 5: Identify Real and Imaginary Parts
The complex number \(-i\) can be expressed in the standard form \(x + iy\):

\[ z = 0 - 1i \]

From this, we can identify:

- Real part \(x = 0\)
- Imaginary part \(y = -1\)

### Final Answer
Thus, the real and imaginary parts of \(\frac{1-i}{1+i}\) are:
- Real part: \(0\)
- Imaginary part: \(-1\)

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