1. The problem is to understand the effect of placing a conjugate bar only on the numerator of a complex fraction. 2. Given the original expression: $$\frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)} = \frac{(a_1 a_2 + b_1 b_2) + (b_1 a_2 - a_1 b_2) i}{a_2^2 + b_2^2}$$ 3. Multiplying numerator and denominator by the conjugate of the denominator removes the imaginary part from the denominator, making it a real number: $$a_2^2 + b_2^2$$ 4. Now, if the conjugate bar is only on the numerator, it means we take the conjugate of the numerator only: $$\overline{(a_1 a_2 + b_1 b_2) + (b_1 a_2 - a_1 b_2) i} = (a_1 a_2 + b_1 b_2) - (b_1 a_2 - a_1 b_2) i$$ 5. The denominator remains: $$a_2^2 + b_2^2$$ 6. So the expression with conjugate only on numerator is: $$\frac{(a_1 a_2 + b_1 b_2) - (b_1 a_2 - a_1 b_2) i}{a_2^2 + b_2^2}$$ 7. This is different from taking the conjugate of the entire fraction, which would conjugate numerator and denominator. 8. In summary, placing the conjugate bar only on the numerator changes only the numerator's imaginary part sign, leaving the denominator unchanged. This is important in complex division and rationalizing denominators.