https://file.aiquickdraw.com/mathbot/file/1729866323677p16es3bn.jpg

The task shown in the image is to match complex numbers with their corresponding arguments (angles in polar form). Each complex number needs to be associated with an angle expressed in radians.

Here is a breakdown of the matching:

1. **Complex number: \( 7 + 7j \)**  
   - This is of the form \( z = x + iy \), where both real and imaginary parts are positive.  
   - The argument corresponds to \( \frac{\pi}{4} \), as it lies in the first quadrant where real and imaginary parts are equal.

2. **Complex number: \( \frac{7}{3} + \frac{7}{\sqrt{3}} j \)**  
   - This is also a complex number in the first quadrant, and the argument is \( \frac{\pi}{6} \) based on the ratio between the real and imaginary parts.

3. **Complex number: \( \frac{7}{\sqrt{3}} + \frac{7}{3} j \)**  
   - This number lies in the first quadrant but with a different ratio of real to imaginary parts, matching \( \frac{\pi}{3} \).

4. **Complex number: \( 7 \)**  
   - This is a purely real number with no imaginary part, so the argument is 0.

5. **Complex number: \( \frac{7}{3} - \frac{7}{\sqrt{3}} j \)**  
   - This complex number lies in the fourth quadrant, and the argument is \( \frac{11\pi}{6} \) or equivalently \( -\frac{\pi}{6} \), which corresponds to \( \frac{7\pi}{6} \) (since the image uses positive angles).

6. **Complex number: \( -7 \)**  
   - This is a purely real number in the negative direction, so the argument is \( \pi \).

7. **Complex number: \( -21 - 7\sqrt{3} j \)**  
   - This lies in the third quadrant (both real and imaginary parts are negative), corresponding to \( \frac{5\pi}{4} \).

8. **Complex number: \( -7 + 7j \)**  
   - This complex number lies in the second quadrant, corresponding to \( \frac{3\pi}{4} \).

Would you like further details on how these arguments are derived, or do you have any questions?

Here are some related questions for practice:
1. How do you calculate the argument of a complex number using arctan?
2. What is the geometric representation of complex numbers in polar form?
3. What is the difference between real and imaginary parts of a complex number?
4. Can a purely imaginary number have a non-zero argument?
5. What is the relationship between polar and rectangular forms of a complex number?

**Tip:** The argument of a complex number in the first quadrant is always between 0 and \( \frac{\pi}{2} \).

Ordnen Sie jeder komplexen Zahl ihr Argument zu! (Assign each complex number to its argument.)

Learn how to match complex numbers with their arguments in polar form. This task covers key mathematical concepts including complex numbers, quadrants, and argument calculation. Suitable for advanced high school or undergraduate students.

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