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Roots of Unity

A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

Geometry

         

A "hop" is a movement of 4 sides at a time counterclockwise around the regular nonagon above. From the starting point, What is the minimum numbers of "hops" it will take to end up back at the starting point?

A regular hexagon has a vertex at \((1,0)\) and has its centroid at the origin. Which of these is another coordinate of the hexagon?

Regular polygons are placed on the coordinate plane such that they each have a vertex at \((1,0)\), and centroid at the origin. There is one regular \(n\)-gon for each value of \(n\) between \(3\) and \(50\), inclusive.

How many values of \(n\) are there such that the \(n\)-gon does not share any vertex other than \((1,0)\) with any of the other \(n\)-gons?

The point \((5,-3)\) is rotated \(\dfrac{\pi}{4}\) radians counterclockwise about the point \((2,4)\). What is the resulting image?

Each of the partial sums \(\sum\limits_{k=1}^{n}{e^{k\pi i/3}}\) is graphed on the complex plane for \(n\in\{1,2,3,4,5,6\}\).

Which of the \(6^\text{th}\) roots of unity coincides with one of these partial sums?

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