How do you represent the twelfth root of unity graphically?

They are the solutions to the 12th order polynomial x¹² = 1

A 12th order polynomial has exactly 12 roots.

They can be pictured as equally spaced points on the unit circle in the complex plane.

Each root is separated from the next by an angle of 360/12 = 30 degrees.

Starting with the obvious root x = 1 + 0i  on the x-axis, the next one (in a clockwise direction) has coordinates x = cos(30) and y = sin(30) (because the radius of the unit circle is 1, by definition!).  cos(30) = (√3)/2 and sin(30) = 1/2 so this point can be represented by the complex value (√3)/2 + i(1/2).

Just work your way around the unit circle in a similar manner for the others.

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Plot the real parts against the imaginary parts:

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How do you go about finding all of these Alan

and are there no other numbers that would work?

Thanks Alan :)

So if you are finding them all by hand it its going to be quite time consuming?

I knew there were supposed to be 12 answers but how can you prove that there aren't more?

Time consuming:  Not really.  Think about the symmetry.  The off-axis ones are simply all the positive and negative combinations of 1/2 and √3/2.  The on-axis ones are simply, 1, -1, i and -i.

As for proving there are exactly n roots to an n'th order polynomial, you'll have to go to Carl Friedrich Gauss (I think he was the first) for this!

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Thanks Alan 

I am glad you explained all  that :)