# Number of roots of unity on the unit circle

I have to find the number of roots of unity on the unit circle |z|=1 in the argand plane.

I know that there are n roots of the the equation $$z^n=1$$ and all of them lie on the given circle. Does that mean that there are infinite roots of unity on this circle (as the sum 1+2+3..... diverges to infinity) or am I missing something?

• I think you're right. This is the torsion subgroup of the circle group. – Chris Custer Nov 5 '18 at 18:35
• @ChrisCuster Sorry, I am a high school student and have no idea of groups. Can you dumb it down a bit for me? Thanks – Anubhab Das Nov 5 '18 at 18:44
• A group is one of the objects, together with rings and fields, studied in Abstract Algebra. You needn't worry too much about them at this point. – Chris Custer Nov 5 '18 at 18:59

If $$m$$ is a multiple of $$n$$, then all the roots of $$z^n = 1$$ are also roots of $$z^m = 1$$. So counting the roots is not a simple matter of adding up $$1 + 2 + 3 + ...$$, as you are counting the repeats.
However, for any $$m$$, the first root of $$z^m = 1$$ encountered while moving counter-clockwise around the circle from $$1$$ is a "primitive" root. That is, it is not a root of $$z^k = 1$$ for any $$k < m$$. Thus the mapping $$m \mapsto \cos \frac {2\pi}m + i\sin \frac{2\pi}m$$ assigns a unique root of unity to each positive integer $$m$$.
$$^*$$ By applying the Schröder-Bernstein theorem.