What does the graph of the seventh root of unity look like on a unit circle like the one in the pictured attached?

enter image source here

2 Answers
May 21, 2018

There are seven seventh roots of unity, e^{ {2pi k i }/7}, all on the unit circle, r=1 above. The first one is at theta={2pi}/7 = 360^circ/7 = 51 3/7 \ ^circ , and there are others at {4pi}/7, {6pi}/7, {8pi}/7, {10 pi}/7, {12 pi}/7 and of course at 0 radians, i.e. unity itself.

Explanation:

Euler's Identity to an even integer power of 2k tell us

(e^{i pi})^{2 k} = (-1)^{2k}

e^{2pi k i} = 1

Now we see

1^(1/7) = (e^{2pi k i})^{1/7} = e^{ {2pi k i }/7}

That's seven distinct seventh roots, given by any seven consecutive ks. (After that they repeat.)

May 21, 2018

Another view
See below

Explanation:

It is interesting to note that the n^(th) roots of unity lie on the unit circle on the complex plane and form a regular polygon of with n sides.

The 7 roots of z^7 =1: z in CC are shown below.

enter image source here