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Question: the n solutions to z re are zre k01 2...

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the n solutions to z- re are: zre, k-0,1, 2,, n-1 Find the 3 third roots of unity, i.e.the 3 solutions to z 1 The angle of the number on the right side is zero, so the three roots of unity are:+ You can (easily) check that z- 1, forj-1, 2,3 Questions: Find all solutions for the equations Find the decimal approximation to all solutions of (1) above, using ecos(t) i sin(t). Then add up all 4 solutions to zt1. Do the same for the solutions to (2) above. Now come up with a conjecture about the n roots of unity. Conjecture: ifZg-e-ה n, k-0, 1, 2, ,n-1, are the n roots of unity, i.e. the n solution to the equation zn-1, then Check your conjecture with the 6 solutions to z6-1 (using Mathematica to simplify your work). Next, draw the solutions to (1) in the complex plane, together with a unit circle. Then draw the solutions to (2) in another complex plane with a unit circle 4 Then draw the solutions to z61 in a coordinate system. Try to connect the pictures with the above conjecture. prove your conjecture. Hint: let ω-e . Then the n roots of unity can be written as 1, a, ω, ω ωη-1. You want to prove that Call that sum S. Multiply S by ω and fiel s-ω S-S(1-ω). |▼mplify the left side, then solve for S. Use the fact that ωη-1 to finish the proof. +

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