To determine the polar representations for the complex numbers associated with the red and blue points on the grid, we need to compute the modulus $r$ and the argument $\varphi$ for each point based on their coordinates.

Let's examine the steps in detail:

Step 1: Identify Coordinates

From the grid:

• Red point: Located at $(4, 3)$
• Blue point: Located at $(-3, -4)$

Step 2: Calculate the Modulus $r$

The modulus $r$ for a complex number $z = x + yi$ is given by: $r = \sqrt{x^2 + y^2}$

For the Red Point ($4, 3$):

$r_{\text{rot}} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

For the Blue Point ($-3, -4$):

$r_{\text{blau}} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Step 3: Calculate the Argument $\varphi$

The argument $\varphi$ is the angle that the complex number makes with the positive x-axis, calculated using: $\varphi = \tan^{-1}\left(\frac{y}{x}\right)$

We need to adjust $\varphi$ based on the quadrant of each point to ensure it lies within the range $\frac{\pi}{2} \leq \varphi \leq \frac{3\pi}{2}$.

For the Red Point ($4, 3$) (First Quadrant):

$\varphi_{\text{rot}} = \tan^{-1}\left(\frac{3}{4}\right) \approx 0.64 \text{ radians}$

For the Blue Point ($-3, -4$) (Third Quadrant):

$\varphi_{\text{blau}} = \tan^{-1}\left(\frac{-4}{-3}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 2.21 \text{ radians}$

Summary of Polar Representations

• For $z_{\text{rot}}$:
  • $r_{\text{rot}} = 5$
  • $\varphi_{\text{rot}} \approx 0.64$ radians
• For $z_{\text{blau}}$:
  • $r_{\text{blau}} = 5$
  • $\varphi_{\text{blau}} \approx 2.21$ radians

Would you like further details or have any questions?

Related Questions

1. How can we convert between polar and rectangular forms of complex numbers?
2. What adjustments are necessary if the argument lies outside the desired range?
3. How does the modulus change if a complex number’s coordinates are scaled?
4. Why do arguments differ in sign depending on quadrants?
5. What are practical applications of polar forms in complex numbers?

Tip

When working with polar coordinates, always check which quadrant the complex number is in to apply the correct angle adjustments.