To convert the complex number $0 - 2i$ into polar form, we need to represent it in terms of its magnitude (modulus) and argument (angle). The polar form of a complex number $z$ is given by:

$z = r (\cos \theta + i \sin \theta)$

where:

• $r$ is the modulus (magnitude) of the complex number
• $\theta$ is the argument (angle) of the complex number

Step 1: Find the modulus $r$

The modulus of a complex number $a + bi$ is given by:

$r = \sqrt{a^2 + b^2}$

For the complex number $0 - 2i$, the real part $a = 0$ and the imaginary part $b = -2$. Therefore:

$r = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2$

Step 2: Find the argument $\theta$

The argument $\theta$ is the angle the complex number makes with the positive real axis in the complex plane. It is given by:

$\theta = \tan^{-1}\left( \frac{b}{a} \right)$

For $a = 0$ and $b = -2$, the complex number lies on the negative imaginary axis, which corresponds to an angle of $\theta = -\frac{\pi}{2}$ radians.

Step 3: Write the polar form

Now we can write the complex number in polar form:

$z = r (\cos \theta + i \sin \theta)$

Substituting $r = 2$ and $\theta = -\frac{\pi}{2}$:

$z = 2 \left( \cos\left( -\frac{\pi}{2} \right) + i \sin\left( -\frac{\pi}{2} \right) \right)$

Thus, the polar form of $0 - 2i$ is:

$z = 2 \left( \cos\left( -\frac{\pi}{2} \right) + i \sin\left( -\frac{\pi}{2} \right) \right)$

Alternatively, you can express it as:

$z = 2 \text{cis}\left( -\frac{\pi}{2} \right)$

where "cis" is shorthand for $\cos \theta + i \sin \theta$.

Final Answer:

The polar form of $0 - 2i$ is:

$2 \text{cis} \left( -\frac{\pi}{2} \right)$ or $2 \left( \cos\left( -\frac{\pi}{2} \right) + i \sin\left( -\frac{\pi}{2} \right) \right)$

Would you like further clarification or have any questions on this? Here's some related material you may find interesting:

1. How do you convert from polar form to rectangular form for complex numbers?
2. What are the properties of complex numbers in polar form?
3. How do you multiply two complex numbers in polar form?
4. How do you divide two complex numbers in polar form?
5. What is the significance of the modulus and argument in complex numbers?

Tip: When the real part of a complex number is zero, the argument is simply $\pm \frac{\pi}{2}$, depending on whether the number is above or below the real axis.