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.