FearFactor47 said:
Also, what does 'cis' mean?
Cosine * i Sine. Its a way to express complex numbers as a vector.
Take the standard X Y axis. Place the real numbers on the X axis and the imaginary numbers on the Y axis.
There are 2 ways to express a complex number.
One way is 4 +3 i. The other way is with a radius and an angle.
4+3i is 5 * cos(37) + 5i*sin (37) or 5 cis 37
As for formulae, I was never good at them and usually just re-derived every time.
Lets take a look at the simple case of the square root of 4. Algebra tells us that the answers is +/- 2.
In cis form it is 2 cis pi and 2 cis 0.
4 is equal to 4 cis 0 , 4 cis 2 pi. 4 cis 4 pi, 4 cis 6 pi
The square root absolute number 4 is 2.
0/2 = 0
2pi /2 = pi
4pi /2 = 2 pi
6pi /2 = 3 pi
so the answers for square root of 4 is
2 cis 0 = 2*cos 0 + 2i* sin 0 = 2+0i = 2
2 cis pi = 2*cos pi + 2i* sin pi = -2+0i = -2
2 cis 2 pi = 2*cos 2 pi + 2i* sin 2 pi = 2+0i = 2
2 cis 3 pi = 2*cos 3 pi + 2i* sin 3 pi = -2+0i = -2
The last 2 end up the same as the first 2 so there are only 2 interesting answers.
In this case
-32 = 32 cis pi (180 deg. the cosine of 180 is -1), 32 cis 3pi(540 deg or 180 deg + a full circle), 32 cis 5 pi , etc.
The reason there are multiple angles is that when you can go around the circle any number of times and still end up at the same place.
When you multiply two numbers in vector format you add the angles. So when you raise something to the 5th power you multiply the angle by 5, ergo when you take the 5th root you divide by 5.
the 5th root of 32 cis pi would be 2 cis pi/5.
There are other answers as well since -32 could also be 32 cis 3 pi that would have a different answer It would be 32 cis 3 pi/5.
There are 5 answers before you reach 2 cis 13 pi/ 5 which is the same as 2 cis pi.
