Why roots of unity need to be rethought | Abstract Algebra Math Foundations 218 | NJ Wildberger

Roots of unity as regularly spaced points on the unit circle in the complex plane are common-place objects in algebra, but they implicitly rely on the Fundamental theorem of algebra for their existence. But their existence, in fact their definition, is fraught with logical difficulty, at least when we view them in the complex plane.

In this video we summarize some basic algebraic and geometrical properties of complex numbers, centered on the rational parametrization of the unit circle. There is a huge difference between specifying a unit quadrance complex number by an expression like 3/5+4i/5 and cos 4+i sin 4. Once we understand that, we can start to appreciate the logical difficulty in supposing that something like the seventh roots of unity really do exist, perfectly spaced around a regular septagon.

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