Find real and imaginary parts of the complex number `(a + ib)/(a - ib)`

Answer

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

### Step 1: Rationalize the denominator
To simplify the expression, we will multiply both the numerator and the denominator by the conjugate of the denominator, which is \(a + ib\).

\[ \frac{a + ib}{a - ib} \cdot \frac{a + ib}{a + ib} = \frac{(a + ib)(a + ib)}{(a - ib)(a + ib)} \]

### Step 2: Expand the numerator
Now we will expand the numerator:

\[ (a + ib)(a + ib) = a^2 + 2a(ib) + (ib)(ib) = a^2 + 2abi - b^2 \]

### Step 3: Expand the denominator
Next, we will expand the denominator using the difference of squares:

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

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

\[ \frac{a^2 - b^2 + 2abi}{a^2 + b^2} \]

### Step 5: Separate real and imaginary parts
Now we can separate the real and imaginary parts:

- Real part \(R\):
\[ R = \frac{a^2 - b^2}{a^2 + b^2} \]

- Imaginary part \(I\):
\[ I = \frac{2ab}{a^2 + b^2} \]

### Final Result
Thus, the real part is \(\frac{a^2 - b^2}{a^2 + b^2}\) and the imaginary part is \(\frac{2ab}{a^2 + b^2}\).

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