Friday, January 30, 2009

Interesting occurrence with 7th root of unity

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I've come across a property of that is rather interesting. Letting , we let , we will see that .

Instead of working directly with this difference, we square the expression for something more easily manipulated. . Using the equalities , etc., we can simplify this down to . Since the sum of the 7th roots of unity equals 0 and the sum only neglects the root , we see that . Thus . Amazing stuff, right?

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