Question

question about finding roots of a complex number

If we have: find the cube roots of 1

then we can write z= 1(cos 0 + i sin 0)

and our values of k can range from 0, 1, 2

But our k values could also be -1, 0 and 1 yes?

Answer 1

its been a while since ive done this but lets have a go

Answer 2

cube roots of 1 means this right:

\[z ^{3}=1\]
right?

Answer 3

yes and i did all the values and it looks like this

Answer 4

-240 deg (-0.5 + 0.866i)
-120 deg (-0.5-0.866i)
0 is 1
120 deg (-0.5 + 0.866i)
240 deg (-0.5 -0.866i)

Answer 5

when i did it on the graphics calculator the values i got are:
1
-0.5 + 0.866i
-0.5 - 0.866i

Answer 6

yep, for a cubic polynomial we should always expect one positive real root and a pair of complex conjugates

Answer 7

have you learnt demoviers theorem or what ever?

Answer 8

oh yea i used his formula

Answer 9

simply apply it for k=0,1,2 and you should find those answers that your calc gave you

Answer 10

looking at k values- some youtube clips focus on all positive k values such as 0, 1, 2,

some other youtube videos tend to go to negative k values because the angle exceeds 18- deg because arg z has to lie between -pi and +pi for it to be positive.

but it doesnt really matter ay because of the cyclic nature of this question

Answer 11

correct

Answer 12

besides the calculator gives you the correct answer

Answer 13

well it depends whether or not the question states you have to use a particular method. If it doesn't, simply use your calcualtor

Answer 14

but i would get familiar with the theorems out there so you can grasp the concepts

Answer 15

i have to leave now, but focus on this youtube video, id like to explain it myself but unfortunately i have to run a few errands.
https://www.youtube.com/watch?v=p39QRoe9QwA

Answer 16

good luck

Answer 17

my tip is to use the UNIT CIRCLE all the time. it is your friend! Very very very important in this field of math

Answer 18

thanks heaps!

Answer 19

: )