I'm having some difficulties with some problems about complex numbers and I would appreciate it if you could help me. It's mainly about the nth roots of unity.
1. I know that the four 4th roots of unity are 1, i, -1 and -i. I need to determine if it is possible to write them as 1, w, w ² and w³ where w is defined as cis pi/2.
And then I need to show that 1+w+w²+w³ = 0.
2. The second problem is also about this topic. I need to find the 5th roots of unity and then I have to display them on an Argand diagram. The second part of this is assuming that w is the root with the smallest positive argument, I need to show that the roots are 1, w, w², w³ and w. This is very similar to the first problem, so hopefully if you help me with one of them I'm able to do the second one alone.
3. Third: w=x+yi and P(x,y) move in the complex plane, I need the cartesian equation for |w-i| = |w+1+i| [note: |w-i| means modulus of w-i]
4. Next: z1= cos(pi/6) + isin(pi/6) and z2 = cos(pi/4) + isin(pi/4). What is the expression for (z1/z2) in the form of z=a+bi.
5. And the last: What is the fifth root of i ?
Sorry if it seems too easy for you, but unfortunately for me it's not. Thank you for your time and effort.