# Determining if a complex number is a root of unity

How would you determine if $a+ib$ is an nth root of unity?

Obviously, the modulus of $a+ib$ must be $1$. But you would also need to determine if $a+ib$ is located at a vertex of a regular polygon inscribed in the unit circle that has one of its vertices at the point $z=1$. Does that mean checking if the polar angle can be expressed as $\pi r$, where $r \in \mathbb{Q}$?

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Yes, a complex number is a root of unity if and only if its modulus is 1 and its argument in polar form is a rational multiple of $\pi$.