The problem in "Aufgabe 2" requires converting the complex number $z$ from polar to Cartesian coordinates. The polar coordinates given are:

• $r = 5$ (the modulus)
• $\varphi = -\pi$ (the argument)

To convert from polar form $z = r(\cos \varphi + i \sin \varphi)$ to Cartesian form $z = x + iy$, we use:

$x = r \cos \varphi$ $y = r \sin \varphi$

1. Calculate $x$: $x = 5 \cos(-\pi) = 5 \cdot (-1) = -5$

2. Calculate $y$: $y = 5 \sin(-\pi) = 5 \cdot 0 = 0$

Thus, in Cartesian form, the complex number $z$ is: $z = -5 + 0i \quad \text{or simply} \quad z = -5$

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

Here are some related questions that could deepen understanding:

1. How would the Cartesian form change if $\varphi$ was $\frac{\pi}{2}$?
2. What is the interpretation of negative angles in polar coordinates?
3. How does changing $r$ affect the location of $z$ on the complex plane?
4. Can you convert other polar forms, such as $r = 3$ and $\varphi = \frac{\pi}{4}$, to Cartesian coordinates?
5. What is the significance of the argument $\varphi$ being exactly $-\pi$?

Tip: Remember that angles in polar form can often be expressed in multiple ways, as adding $2\pi$ (or subtracting) gives the same point on the complex plane.