Question

Consider the complex number $$ z=\frac{a+i b}{c+i d} $$ where \(a, b, c\) and \(d\) are real. Determine \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\).

Short answer

Question: Determine the real and imaginary parts of the complex number \(\frac{a + i b}{c + i d}\). Answer: The real and imaginary parts of the complex number can be found as follows: \(\operatorname{Re}(z) = \frac{ac + bd}{c^2 + d^2}\) and \(\operatorname{Im}(z) = \frac{bc - ad}{c^2 + d^2}\).

Step-by-step solution

Identify the conjugate of the denominator

The denominator of the given complex number is \(c + id\). Its conjugate is \(c - id\).

Multiply the numerator and denominator by the conjugate of the denominator

To eliminate the imaginary part in the denominator, multiply both the numerator and denominator by \((c - id)\). This gives us: $$ z = \frac{(a + ib)(c - id)}{(c + id)(c - id)} $$

Perform the multiplication and simplify

Distribute to simplify the expression: $$ z = \frac{ac - aid + ibc - i^2 bd}{c^2 - icd + icd - i^2 d^2} $$ Since \(i^2 = -1\) and the imaginary parts in the denominator cancel out, we get: $$ z = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2} $$

Identify the real and imaginary parts of the complex number

From the simplified expression, \(\operatorname{Re}(z) = \frac{ac + bd}{c^2 + d^2}\) and \(\operatorname{Im}(z) = \frac{bc - ad}{c^2 + d^2}\).