Division of complex number

The division of complex numbers is similar to dividing real numbers but involves an additional step of multiplying by the conjugate of the denominator to eliminate the imaginary part in the denominator. Here’s the general process for dividing two complex numbers \(a + bi\) and \(c + di\), where \(i\) is the imaginary unit:

1. **Write the complex numbers:**
\[ \frac{a + bi}{c + di} \]

2. **Multiply the numerator and denominator by the conjugate of the denominator:**
\[ \frac{a + bi}{c + di} \cdot \frac{c – di}{c – di} \]

3. **Perform the multiplication:**
\[ \frac{(a + bi)(c – di)}{(c + di)(c – di)} \]

4. **Simplify the expression:**
\[ \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]

So, the division of two complex numbers \(a + bi\) and \(c + di\) is given by:
\[ \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]

This process ensures that the denominator becomes a real number, eliminating the imaginary part. The final result is in the form \(x + yi\), where \(x\) and \(y\) are real numbers.