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 