The complex number \( z = -2i \) can be analyzed as follows:

1. **Standard Form**: The complex number is written in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In this case:
   - Real part: \( a = 0 \) (since there is no real component),
   - Imaginary part: \( b = -2 \).

2. **Modulus**: The modulus (or absolute value) of \( z \), denoted as \( |z| \), is the distance from the origin to the point in the complex plane. It is calculated as:
   \[
   |z| = \sqrt{a^2 + b^2} = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2
   \]

3. **Argument**: The argument \( \arg(z) \) is the angle \( z \) makes with the positive real axis. Since \( z = -2i \) lies on the negative imaginary axis, the argument is:
   \[
   \arg(z) = -\frac{\pi}{2}
   \]
   (or \( 270^\circ \)).

4. **Polar Form**: The polar form of a complex number is given by:
   \[
   z = r(\cos \theta + i \sin \theta)
   \]
   where \( r \) is the modulus and \( \theta \) is the argument. For \( z = -2i \), this becomes:
   \[
   z = 2(\cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2})) = 2(0 - i) = -2i
   \]
   which confirms the polar form matches the original number.

5. **Exponential Form**: Using Euler’s formula, \( z \) can be expressed as:
   \[
   z = re^{i\theta} = 2e^{-i\frac{\pi}{2}}
   \]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What is the general formula for the modulus of any complex number?
2. How would you express a complex number in exponential form?
3. What is the significance of the argument of a complex number?
4. How does the complex plane represent imaginary numbers like \( -2i \)?
5. How can you multiply two complex numbers using their polar forms?

**Tip**: When dealing with complex numbers, always plot them on the complex plane to get a better sense of their modulus and argument.

Explore the complex number z = -2i with detailed steps, including its modulus, argument, polar form, and exponential form. This solution explains key concepts like Euler's formula and the representation of complex numbers in the complex plane. Suitable for Grades 10-12 students.

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