Complex Division 4B6Db9

1. **State the problem:** We want to understand what to find first when dividing complex numbers before simplifying. 2. **Key concept:** When dividing complex numbers, the first step is to find the *conjugate* of the denominator. 3. **Why the conjugate?** Multiplying numerator and denominator by the conjugate of the denominator removes the imaginary part from the denominator, making it a real number. 4. **Formula:** If we have $$\frac{a+bi}{c+di}$$, multiply numerator and denominator by the conjugate of the denominator $$c - di$$: $$\frac{a+bi}{c+di} \times \frac{c - di}{c - di} = \frac{(a+bi)(c - di)}{(c+di)(c - di)}$$ 5. **Simplify denominator:** $$(c+di)(c - di) = c^2 - (di)^2 = c^2 - d^2 i^2 = c^2 - d^2(-1) = c^2 + d^2$$ This is a real number. 6. **Simplify numerator:** Use distributive property: $$(a+bi)(c - di) = ac - adi + bci - bdi^2 = ac - adi + bci + bd$$ Since $$i^2 = -1$$. 7. **Final step:** Write the result as a complex number with real and imaginary parts separated, then simplify. **Answer:** The first thing we find before simplifying complex number division is the *conjugate* of the denominator.