11 - 3 Powers of Complex Numbers
 

Demo: Snail Shell (Manipula Math)
Important fact to remember
De Moivre's Theorem says
If z = r cis 0 then
zn = r n cis n0
This formula allows us to make some very cool graphs using argand diagrams. A lot of these graphs emulate many of the spirals and figures that we have in nature. Check out the book on page 409 as an example. Pretty cool. Here's how to calculate them.
 
Example problem:
Find the powers for n = 1, 2, 3, 4, 5, 6 and make an argand diagram for each.
z = 1 + i
Solution:
Find each power first.                                         _______
z = 1.4 cis 45 ( where did I get this? Remember, r = \/a2 + b2 and we can find the angle from tan 0 = (y/x)
r2 = (1.4)2 cis [(2)(45)] = 2 cis 90     (1.4 is the square root of two silly)
z3 = (1.4)3 cis [(3)(45)] = 2.8 cis 135
z4 = (1.4)4 cis [(4)(45)] = 4 cis 180
z5 = (1.4)5 cis [(5)(45)] = 5.6 cis 225
z6 = (1.4)6 cis [(6)(45)] = 8 cis 270
    If you make an argand diagram by sweeping out each angle and going out the appropriate distance, you should end up with a spiral similar to a snail shell. Notice that each angle increases by 45 degrees and each radius gets increasingly larger thus forming a spiral. If you play connect the dots with the tips of each answer, it forms a simple snail shell. The pattern is endless as the higher powers keep increasing in size.  Look at the argand diagram for the above:
     That's about all for this section. Not to bad, but the numbers can make the arithmetic a little sticky.   Have your calculator handy!!
 
    To continue to the last section 11.4 hit 
     To back up and regroup in section 11.2 hit