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

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Alan N. Share
May 21, 2018

Answer:

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

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Dean R. Share
May 21, 2018

Answer:

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 #k#s. (After that they repeat.)

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