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Let's solve the three complex expressions and find the real and imaginary parts of each one.

### a) \(\frac{1 + 2i}{i}\)

We can multiply the numerator and the denominator by \(i\) to simplify the expression:
\[
\frac{1 + 2i}{i} \times \frac{i}{i} = \frac{(1 + 2i) \cdot i}{i^2} = \frac{i + 2i^2}{-1}
\]
Since \(i^2 = -1\):
\[
= \frac{i - 2}{-1} = -i + 2
\]
Thus, the real part is 2, and the imaginary part is \(-1\).

### b) \(\frac{1}{i} + \frac{2i}{1 - i}\)

First, simplify \(\frac{1}{i}\):
\[
\frac{1}{i} = -i
\]

Now simplify \(\frac{2i}{1 - i}\) by multiplying the numerator and denominator by the conjugate of the denominator:
\[
\frac{2i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{2i(1 + i)}{(1 - i)(1 + i)} = \frac{2i + 2i^2}{1^2 - i^2} = \frac{2i - 2}{1 + 1} = \frac{2i - 2}{2} = i - 1
\]

So, we have:
\[
-i + (i - 1) = -i + i - 1 = -1
\]
Thus, the real part is \(-1\), and the imaginary part is 0.

### c) \(\frac{1 + 2i}{1 - 2i} - \frac{1 - 2i}{1 + 2i}\)

We need to simplify each fraction separately. Let’s start with the first one:
\[
\frac{1 + 2i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(1 + 2i)^2}{(1 - 2i)(1 + 2i)} = \frac{1^2 + 2 \cdot 1 \cdot 2i + (2i)^2}{1^2 - (2i)^2} = \frac{1 + 4i - 4}{1 + 4} = \frac{-3 + 4i}{5} = -\frac{3}{5} + \frac{4i}{5}
\]

Now, for the second term:
\[
\frac{1 - 2i}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} = \frac{(1 - 2i)^2}{(1 + 2i)(1 - 2i)} = \frac{1^2 - 2 \cdot 1 \cdot 2i + (2i)^2}{1 + 4} = \frac{1 - 4i - 4}{5} = \frac{-3 - 4i}{5} = -\frac{3}{5} - \frac{4i}{5}
\]

Now, subtract the two:
\[
\left(-\frac{3}{5} + \frac{4i}{5}\right) - \left(-\frac{3}{5} - \frac{4i}{5}\right) = -\frac{3}{5} + \frac{4i}{5} + \frac{3}{5} + \frac{4i}{5} = \frac{8i}{5}
\]

Thus, the real part is 0, and the imaginary part is \(\frac{8}{5}\).

---

### Summary:
- **a)** Real part: 2, Imaginary part: \(-1\)
- **b)** Real part: \(-1\), Imaginary part: 0
- **c)** Real part: 0, Imaginary part: \(\frac{8}{5}\)

Would you like more details on any of these steps, or do you have any questions?

### Related Questions:
1. How do you multiply complex numbers with imaginary units?
2. What is the conjugate of a complex number, and why is it useful?
3. How do you divide complex numbers in rectangular form?
4. Can you explain how to simplify complex fractions?
5. How do you convert a complex number to polar form?

### Tip:
When dividing complex numbers, multiplying by the conjugate of the denominator is an essential step to eliminate imaginary components from the denominator.

Encuentra la parte real y la parte imaginaria de los siguientes complejos: a) (1 + 2i) / i b) 1/i + 2i / (1 - i) c) (1 + 2i) / (1 - 2i) - (1 - 2i) / (1 + 2i)

Solve the complex numbers problems, including finding the real and imaginary parts of expressions like (1 + 2i) / i, 1/i + 2i / (1 - i), and (1 + 2i) / (1 - 2i) - (1 - 2i) / (1 + 2i). Learn about the properties of complex numbers and simplification techniques.

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