# Thread: Roots of unity and the length from one root to the other roots

1. ## Roots of unity and the length from one root to the other roots

Hello,

I had an assignment that required me to solve for the roots of unity for various equations of the form $z^n -1 = 0$. Then , I was asked to represent the roots of unity for each equation on an argand diagram in the form of a regular polygon.

I did all of that , however, i have a question:

is there a relation ship between the power n and the length from one root to the other roots?

When n = 3,
The roots are $-0.5 \pm 0.8660i$ and 1. The length from one root to the other roots are both 1.7321 (sqrt of 3)

When n = 4,

The roots are $\pm 1, 0 \pm 1i$. The length from one root to the other roots are (sqrt 2, sqrt and 2)

I am wondering , for the equation $z^3 - 1 = 0$ , is there a formula relating the power of z to the length from one roots to the other roots?
Thanks.

2. ## Re: Roots of unity and the length from one root to the other roots

The nth roots of unity lie at the vertices of a polygon with n sides, each vertex having distance from the center of 1. Drawing a line from the center to each vertex divides the polygon into n isosceles triangles, each having two sides of length 1, vertex angle of 360/n degrees and you want to find the length of the third side. If you draw a line from the vertex of such a triangle to the center of the base, you get two right triangles with hypotenuse length 1 and one angle of 180/n. The opposite side of that triangle has length 1(sin(180)/n) so the side of the polygon is twice that :
2sin(180/n).

3. ## Re: Roots of unity and the length from one root to the other roots

Hi, thank you very much for answering the question first.

However, i have already got the conjecture that the length of the polygon would be 2 sin (180/n), but this conjecture cannot be proven algebraically or MI that this statement would be true for any value of n, can it?

I am just wondering is there any other relationships between the length from one root to the other roots? (Lets say, if you draw a line from one root (any root) to the other roots (not only the adjacent root)

n = 3, length is sqrt 3 and sqrt 3
n = 4, length is sqrt 2 , 2 and sqrt 2
n = 5, length is not an exact value.. but what i have got is 1.1756, 1.9021, 1.9021 and 1.1756
n = 6, length is 1, sqrt 3, 2, sqrt 3, 1
n = 7, length is 0.86, 1.56, 1.94, 1.94, 1.56, 0.86

seems like theres a pattern here, but i can't seem to figure it out... would help a lot if someone could tell me if theres a conjecture for this or not, thank you very much.