To solve this, let’s proceed by breaking down the expression for $z$ and finding each part step-by-step.

Given: $z = \frac{2 - i}{1 + i}$

1. Simplify $z$:

   To simplify $z$, multiply both the numerator and the denominator by the complex conjugate of the denominator, $1 - i$: $z = \frac{(2 - i)(1 - i)}{(1 + i)(1 - i)}$

   Calculate the denominator: $(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2$

   Calculate the numerator: $(2 - i)(1 - i) = 2 \cdot 1 - 2 \cdot i - i \cdot 1 + i \cdot i = 2 - 2i - i + i^2 = 2 - 3i - 1 = 1 - 3i$

   So, $z = \frac{1 - 3i}{2} = \frac{1}{2} - \frac{3}{2}i$

   Thus, the real part of $z$ is $\text{Re}(z) = \frac{1}{2}$ and the imaginary part is $\text{Im}(z) = -\frac{3}{2}$.

2. Find $z^2$:

   Calculate $z^2$ by squaring $z$: $z^2 = \left( \frac{1}{2} - \frac{3}{2}i \right)^2$

   Expand this expression: $z^2 = \frac{1}{4} - 2 \cdot \frac{1}{2} \cdot \frac{3}{2} i + \left(-\frac{3}{2} i\right)^2$ $= \frac{1}{4} - \frac{3}{2}i + \frac{9}{4}(-1) = \frac{1}{4} - \frac{3}{2}i - \frac{9}{4}$ $= -\frac{8}{4} - \frac{3}{2}i = -2 - \frac{3}{2}i$

   So, $\text{Re}(z^2) = -2$ and $\text{Im}(z^2) = -\frac{3}{2}$.

   Therefore, for part (a): $\text{Re}(z^2) + \text{Im}(z^2) = -2 - \frac{3}{2} = -\frac{7}{2}$

3. Calculate $z + \frac{1}{z}$:

   First, find $\frac{1}{z}$.

   $\frac{1}{z} = \frac{1}{\frac{1}{2} - \frac{3}{2}i}$

   Multiply the numerator and the denominator by the complex conjugate of the denominator: $\frac{1}{z} = \frac{\frac{1}{2} + \frac{3}{2}i}{\left( \frac{1}{2} \right)^2 + \left( \frac{3}{2} \right)^2}$ $= \frac{\frac{1}{2} + \frac{3}{2}i}{\frac{1}{4} + \frac{9}{4}} = \frac{\frac{1}{2} + \frac{3}{2}i}{\frac{10}{4}} = \frac{1 + 3i}{5} = \frac{1}{5} + \frac{3}{5}i$

   Now, compute $z + \frac{1}{z}$: $z + \frac{1}{z} = \left( \frac{1}{2} - \frac{3}{2}i \right) + \left( \frac{1}{5} + \frac{3}{5}i \right)$ $= \frac{1}{2} + \frac{1}{5} - \frac{3}{2}i + \frac{3}{5}i$ Combine real and imaginary parts: $= \left( \frac{1}{2} + \frac{1}{5} \right) + \left( -\frac{3}{2} + \frac{3}{5} \right)i$ $= \frac{5 + 2}{10} + \frac{-15 + 6}{10}i = \frac{7}{10} - \frac{9}{10}i$

   Thus, $\text{Re}\left( z + \frac{1}{z} \right) = \frac{7}{10}$ and $\text{Im}\left( z + \frac{1}{z} \right) = -\frac{9}{10}$.

   So, for part (b): $\text{Re}\left( z + \frac{1}{z} \right) + \text{Im}\left( z + \frac{1}{z} \right) = \frac{7}{10} - \frac{9}{10} = -\frac{2}{10} = -\frac{1}{5}$

Summary of Results:

• (a) $\text{Re}(z^2) + \text{Im}(z^2) = -\frac{7}{2}$
• (b) $\text{Re}\left( z + \frac{1}{z} \right) + \text{Im}\left( z + \frac{1}{z} \right) = -\frac{1}{5}$

Would you like more details on any specific part of the solution, or have any questions?

Further Exploration Questions:

1. How do we find the modulus and argument of a complex number?
2. What is the significance of the real and imaginary parts in complex arithmetic?
3. How does multiplication by the conjugate help simplify complex fractions?
4. What are some applications of complex numbers in physics and engineering?
5. How can complex numbers be represented graphically in the complex plane?

Tip:

When simplifying complex numbers, always try to multiply by the conjugate of the denominator to eliminate any imaginary terms.