# Roots of unity

1. May 31, 2013

### h0llow

1. The problem statement, all variables and given/known data
Use De Moivre's Theorem to solve for the roots of unity 1, ω, ω2
Hence show that the sum of these roots is zero

2. Relevant equations
r(cosθ + isinθ)
r(cos(θ + 2n∏)+isin(θ+isin∏))

3. The attempt at a solution
I know the first root,1, is 1(cos 0 + i sin 0)
but have no clue about the next 2, or of how i would prove they are equal to 0.

2. May 31, 2013

### Staff: Mentor

I don't see anywhere in your post that you are looking for the cube roots of 1.
The cube roots are not equal to 0; you are supposed to show (i.e., prove) that the sum of these three roots is zero.

If z represents one of these roots, what equation should you be trying to solve?

3. May 31, 2013

### h0llow

r(cos(θ + 2n∏)+isin(θ+isin∏))?

4. May 31, 2013

### Staff: Mentor

That's not an equation.

The idea that these are cube roots of unity is important.

5. May 31, 2013

### HallsofIvy

Staff Emeritus
Your problem said "use DeMoivre's theorem".

So, what is DeMoivre's theorem?

6. May 31, 2013

### h0llow

(r(cosθ+isinθ))n

7. May 31, 2013

### Staff: Mentor

Equals what?

And in this problem, what's your best guess as to what n is?

8. May 31, 2013

### h0llow

(r(cosθ+isinθ))n=rneinθ

hmm im guessing for 1, n= 0
for ω, n=1
for ω2, n = 2

9. May 31, 2013

### Staff: Mentor

10. May 31, 2013

### h0llow

well ^3, but im not sure why it is significant.

11. May 31, 2013

### Staff: Mentor

Because you're asked to find the three cube roots. That's why I have be emphasizing cube, since there are three dimensions in a cube.

12. May 31, 2013

### Staff: Mentor

The (square, cube, fourth, fifth, etc.) roots of unity are spread out evenly around the unit circle. Does that suggest something about the angle between each pair of contiguous roots?

Aren't there any examples in your book that are similar to this problem? Hasn't some of this been covered in your class?

13. May 31, 2013

### h0llow

how do you know they are cube roots?

Angle is the same for all values of n?(even integers)

EDIT:
no, this is the first time i have seen a question like this. The maths course in my country is currently changing(introduced in phases), so not everything is in books(99% is, but 1% like this aren't properly explained). And my finals start in 6 days =/.
I would normally ask my teacher, but he said he had no idea where to begin.

Last edited: May 31, 2013
14. May 31, 2013

### Staff: Mentor

Because they asked for three roots - 1, ω, and ω2.

No. For different values of n, you get different angles. For a given n, what is the angle between consecutive roots?

Also, what do mean by "even integers"? The ones that are multiples of two? Or are you saying, "even for integers, the angle is the same"? Either way, I don't understand the question.

15. May 31, 2013

### h0llow

okay, i understand the cube part.

Can you just tell me how you would go about solving it?

16. May 31, 2013

### Staff: Mentor

Use De Moivre's Theorem to solve z3 = 1

Each solution is a cube root of unity (1).

17. May 31, 2013

### h0llow

0_o

z3=1+0i = (1+0i)1/3..........r=1 θ=0

z1 = (1(cos0+isin0))1/3
1(1+0)
=1

z2 = (1(cos2∏+isin2∏))1/3
1(cos(2∏/3)+isin(2∏/3))
= -1/2 + √3/2

z3 = (1(cos(4∏)+isin(4∏))1/3
=1(cos(4∏/3)+isin(4∏/3))
=-1/2 - √3/2

:surprised

1+(-1/2+√3/2)+(-1/2 - √3/2) = 0!

:surprised

Thank you so much!

Maths is amazing when u get it !

P.S. as you increase value of z (z..z1..z2) should θ increase by ∏ or 2∏?..here i used 2∏ but not sure.

18. May 31, 2013

### Staff: Mentor

You're missing the imaginary parts on two of them.
z2 = -1/2 + √3/2 * i
z3 = -1/2 - √3/2 * i
2$\pi$

19. May 31, 2013

### h0llow

o ya oops :P

thanks so much!... the word unity kind of confused me =/