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# c06 - 18.03 Class 6 Roots of Unity Euler's formula...

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18.03 Class 6, Feb 21, 2006 Roots of Unity, Euler's formula, Sinusoidal functions [1] Roots of unity Let a > 0 . Since i^2 = -1 , (+- i sqrt(a))^2 = - a : Negative real numbers have square roots in C. Any quadratic polynomial with real coefficients has a root in C , by the quadratic formula x^2 + bx + c = 0 has roots (-b +- sqrt(b^2 - 4c))/2 In fact: "Fundamental Theorem of Algebra": Any polynomial has a root in C (unless it is a constant function). Special case: z^n = 1 : "n-th roots of unity" n = 2 : z = +- 1 In general, if z^n = 1 , then |z^n| = 1 , but Magnitudes Multiply, so |z| = 1 : roots of unity lie on the unit circle. n = 3 : Angles Add, so if z^3 = 1 then the argument of z is 0 ..... no, not quite: it could be 2 pi / 3 , since three times that is 2 pi. It's better to think of the argument of 1 as a choice: 0, or 2 pi, or -2 pi, or 4 pi, or .... This gives ( -1 + sqrt 3 i ) / 2 . Or it could be 4 pi / 3 , which gives ( -1 - sqrt 3 i ) / 2 That's it, there's no other way for it to happen. The cube roots of unity start with 1 and divide the unit circle evenly into 3 parts.

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