Find the real and imaginary parts of each of the following a) 2/3+i + 3/2+i  b) 1/a+bi - 1/a-bi

Let's solve each part step by step.

### Part (a): \( \frac{2}{3+i} + \frac{3}{2+i} \)

To find the real and imaginary parts, we'll first simplify each fraction by multiplying by the conjugate of the denominator.

#### Simplifying \( \frac{2}{3+i} \):
1. Multiply the numerator and the denominator by the conjugate of \( 3+i \), which is \( 3-i \):

\[
\frac{2}{3+i} = \frac{2(3-i)}{(3+i)(3-i)} = \frac{6 - 2i}{9 - i^2} 
\]

2. Since \( i^2 = -1 \):

\[
\frac{6 - 2i}{9 - (-1)} = \frac{6 - 2i}{10} = \frac{6}{10} - \frac{2i}{10} = \frac{3}{5} - \frac{1}{5}i
\]

#### Simplifying \( \frac{3}{2+i} \):
1. Multiply the numerator and the denominator by the conjugate of \( 2+i \), which is \( 2-i \):

\[
\frac{3}{2+i} = \frac{3(2-i)}{(2+i)(2-i)} = \frac{6 - 3i}{4 - i^2}
\]

2. Since \( i^2 = -1 \):

\[
\frac{6 - 3i}{4 - (-1)} = \frac{6 - 3i}{5} = \frac{6}{5} - \frac{3i}{5}
\]

#### Adding the two fractions:

\[
\frac{2}{3+i} + \frac{3}{2+i} = \left(\frac{3}{5} - \frac{1}{5}i\right) + \left(\frac{6}{5} - \frac{3}{5}i\right)
\]

\[
= \left(\frac{3}{5} + \frac{6}{5}\right) + \left(-\frac{1}{5} - \frac{3}{5}\right)i = \frac{9}{5} - \frac{4}{5}i
\]

So, the **real part** is \( \frac{9}{5} \) and the **imaginary part** is \( -\frac{4}{5} \).

### Part (b): \( \frac{1}{a+bi} - \frac{1}{a-bi} \)

1. Multiply each term by the conjugate of its denominator:

\[
\frac{1}{a+bi} = \frac{1(a-bi)}{(a+bi)(a-bi)} = \frac{a-bi}{a^2 + b^2}
\]

\[
\frac{1}{a-bi} = \frac{1(a+bi)}{(a-bi)(a+bi)} = \frac{a+bi}{a^2 + b^2}
\]

2. Subtract the two fractions:

\[
\frac{1}{a+bi} - \frac{1}{a-bi} = \frac{a-bi}{a^2 + b^2} - \frac{a+bi}{a^2 + b^2} = \frac{(a-bi) - (a+bi)}{a^2 + b^2}
\]

3. Simplify the expression:

\[
= \frac{a - bi - a - bi}{a^2 + b^2} = \frac{-2bi}{a^2 + b^2} = \frac{-2b}{a^2 + b^2}i
\]

So, the **real part** is \( 0 \) and the **imaginary part** is \( \frac{-2b}{a^2 + b^2} \).

### Summary:
- **Part (a)**: Real part = \( \frac{9}{5} \), Imaginary part = \( -\frac{4}{5} \)
- **Part (b)**: Real part = \( 0 \), Imaginary part = \( \frac{-2b}{a^2 + b^2} \)

Would you like more details on any of the steps? 

### Follow-up Questions:
1. How do you find the conjugate of a complex number?
2. Why do we multiply by the conjugate in the denominator?
3. Can you explain the significance of \( i^2 = -1 \)?
4. How would the real and imaginary parts change if \( b = 0 \)?
5. What happens if the numerator in part (a) was complex instead of real?
6. How would you represent these results on the complex plane?
7. How do the operations differ when dividing by a complex number?
8. Can these expressions be simplified further?

### Tip:
When simplifying complex fractions, always multiply by the conjugate to remove the imaginary part from the denominator.

Learn how to find the real and imaginary parts of complex numbers with detailed step-by-step solutions. This problem covers key concepts in complex numbers, including finding real and imaginary parts using conjugates. Suitable for advanced high school students.

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