Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We can use a simple four-step guide to help us find complex roots, or the nth roots of complex numbers. These guidelines simplify for us the process of using DeMoivre's Theorem to find complex roots. This method for finding complex roots uses the trigonometric form and so we should understand how to convert from rectangular to trigonometric form and from trigonometric to rectangular form.
Let's review some basic facts about the roots of a complex number starting by looking at an example. The fourth roots of -8 plus i times 8 root 3 are 3 root plus i minus 1 plus i root 3 minus root 3 minus i and 1 minus i root 3. Now to get these roots, I first convert my original complex number into trig form and then you'll notice that in order to get the roots I have to take the fourth root of the modulus and that's going to be 2. To get the first root, I divide this number 2pi over 3 the argument by 4 and I get pi over 6 and each of these successive roots is going to be how far apart? Well 2pi one revolution divided by 4, so pi over 2 in each case I"m adding pi over 2 now it looks like I'm adding something else but it's actually 3pi over 6 that I'm adding which is the same as pi over 2 so each of these is pi over 2 apart. And remember, when you're looking for fourth roots there are only going to be 4 of them. If you're looking for sixth roots there will be 6 distinct roots so here is my recap, make sure you know these things when you're looking for the roots of the complex number it'll actually make the process go quicker.
All of the roots are going to have the same modulus right? And you get that modulus by taking the root of the original modulus to get the new one.
The primary root of the first root is going to have argument whatever the original argument was divided by the degree of the root so if you want the fourth root, you take one fourth the original argument in this case we get pi over 6.
Successive roots are found by adding 2pi divided by whatever the index of the root is in this case the fourth root so we divide 2pi by 4, if we were taking fifth roots we would add 2pi over five.
And then the number they're exactly 4 distinct fourth roots, 5 distinct fifth roots, 6 distinct sixth roots and so on.
These are things to remember when you're looking for the roots of the complex number.