Practice

1. Convert each number to polar form and evaluate:
   
2. Find the cube roots of:
   (a) 125 j

To convert numbers to polar form, you first find the magnitude and angle (argument) of the complex numbers involved. For instance, for , the magnitude is and the argument .

Now let’s dive into the cube roots of . This number can be expressed in polar form as (magnitude) and an angle of (since it’s purely imaginary). The cube roots can be found by using the formula:

for . Thus, each cube root can be calculated from there!

Now, for a bit more context: The history of polar form stems from the desire to simplify complex number operations, particularly multiplication and division, which can be done through magnitudes and angles instead of wrestling with real and imaginary parts.

And if you’re looking for practical advice, always remember: when adding or subtracting complex numbers, it’s best to combine like terms first. Many mistakes arise when people forget to handle the real and imaginary parts separately!