To divide the complex numbers $\frac{2 + i}{2 - i}$, we use the method of multiplying by the conjugate of the denominator. The conjugate of $2 - i$ is $2 + i$. Here's the step-by-step process:

1. Multiply the numerator and the denominator by the conjugate of the denominator: $\frac{2 + i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{(2 + i) \times (2 + i)}{(2 - i) \times (2 + i)}$

2. Expand the numerator and the denominator:

   • Numerator: $(2 + i) \times (2 + i) = 2^2 + 2 \times 2 \times i + i^2 = 4 + 4i + i^2$ Since $i^2 = -1$: $4 + 4i + (-1) = 3 + 4i$

   • Denominator: $(2 - i) \times (2 + i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5$

3. Combine the results: $\frac{3 + 4i}{5}$

4. Separate into real and imaginary parts: $\frac{3 + 4i}{5} = \frac{3}{5} + \frac{4i}{5}$

So, the final result is: $\frac{2 + i}{2 - i} = \frac{3}{5} + \frac{4i}{5}$

Would you like more details or have any questions about this process?

Here are 5 related questions to expand your understanding:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate when dividing complex numbers?
3. Can you explain why $i^2 = -1$ and how it affects complex number multiplication?
4. How do you add and subtract complex numbers?
5. What is the geometric interpretation of complex number division?

Tip: When dividing complex numbers, multiplying by the conjugate helps eliminate the imaginary part in the denominator, simplifying the expression to a standard form.