https://file.aiquickdraw.com/mathbot/file/1730569904074i5irs13a.jpg

The image shows a complex number \( Z \) represented in a coordinate system. The coordinates of \( Z \) appear to be in the third quadrant, which is supported by the fact that both the real part (\( x \)-axis) and the imaginary part (\( y \)-axis) are negative.

Here's the breakdown of the reasoning:

1. **Location Analysis**: The complex number \( Z \) has a real part located at \( -\frac{1}{2} \) (to the left of the origin) and an imaginary part at \( -\frac{\sqrt{3}}{2} \) (below the origin).
2. **Quadrant Identification**: Since both parts are negative, \( Z \) is in the **third quadrant**. This aligns with the corrected answer provided below the question in the image.

The angle \( \varphi \), which seems to indicate the argument of \( Z \), further confirms the positioning in the third quadrant, as angles are typically measured clockwise or counterclockwise from the positive real axis.

### Correct Answer
- The complex number \( Z \) is in the **third quadrant** (option **c**).

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### Related Questions
1. What are the characteristics of complex numbers in each quadrant?
2. How is the argument \( \varphi \) of a complex number determined?
3. How do you find the conjugate of a complex number?
4. What is the modulus of a complex number, and how is it calculated?
5. How do changes in the real or imaginary part affect the quadrant of a complex number?

### Tip
When determining the quadrant of a complex number, always check the signs of both the real and imaginary components.

На слици је приказан комплексан број Z. Његов коњуговано комплексан број је у ком квадранту?

Explore how to determine the quadrant of a complex number and its conjugate based on real and imaginary components. This solution helps in identifying the location of complex numbers in coordinate planes using their real and imaginary parts.

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