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.
Let's find the sixth roots of the number -4 root 3 minus 4i. I'm going to call this number z. When you find the roots, you first have to covert to trig form. Let me do that quickly this time. Here's my number graphed and won't surprise you to see that the modulus of this number is a. That's the distance of the number from the origin.
The argument is all this is in degrees this time, 210 degrees. In polar form this number would be 8 times cosine of 210 degrees plus i sine 210. It's really easy to find the roots of the complex number when you have it in trig form. Let's find the first, the primary sixth root.
You want to take the modulus of the original number and take the sixth root of that. The sixth root of 8 to the 1/6 is the same thing as 2 to the 3/6. Because 8 is 2 the to the third power and that's the same as 2 to 1 half. Root 2 is going to be my new modulus. I'll call my roots w, 1 equals root 2 times the cosine of and what's the argument? You just divide this number by 6. 210 divide by 6 is 35. So 35 degrees plus i sine 35. degrees.
Then the rest is easy. In order to get successive roots you just add; normally we would add 2 Pi divided by the number of the root, in this case 6. Since we're working in degrees, we have to add 360 divided by 6 and that's going to be 60 degrees. I'll keep adding 60 degrees. The second root then will be root 2 cosine of 35 plus 60, 95 plus i sine 95, w3 root 2 times cosine of 95 plus 60, 155 plus i sine 155.
Remember we'll expect 6 of these. The fourth one will be root 2 times cosine of 215 plus i sine 215. w5 root 2 adding 60 again 275 cosine 275 plus i sine 275. w6 root 2 cosine 335 plus i sine 335.
Now I've told you about the relationship between the roots before. I want to show you how they relate to one another on Geometry sketchpad so let's take a look. Here we are on Geometry Sketchpad. Notice I've got plotted my number z -4 root 3 minus 4 i. These are the sixth roots that I just came up with. Remember the primary root, w1 has an argument which is 1/6 of the original argument of z. That was 210 degrees divided by 6; 35 degrees. Each of these roots has a modulus which is the sixth root of the modulus of z; 6th root of 8. So root 2 about 1.4. Notice also each of the roots is 60 degrees apart. What's interesting about that, is that if you connect these roots together, connecting consecutive roots you got a perfect regular hexagon.
Notice that that relationship stays the same even if I change the number z whose root I'm taking. Now I'm changing the z, it's not the same z anymore. Look I always get that symmetry. That hexagon whenever I'm taking sixth roots.
That's one of the really neat things about the roots of the complex number, you always get this symmetry. You always get exactly sixth 6 roots. The first root is the most important one. When you find first is going to have a modulus which is the sixth root of the original. It's argument will be one sixth of the original.