Patterns from complex numbers (Part A & B)

Hi, I got my new internal assessment in math today and I found it quite complicated at some points. So, I'd appreciate if you could help me out.

It goes like this:

- Use de Moivre's theorem to obtain solutions to the equation z
^{3}-1=0 - Use graphing software to plot these roots on an Argand diagram as well as a
__unit circle__ with centre origin. - Choose a root and draw line segments from this root to the other two roots.
- Repeat those above for z
^{4}-1=0 and z^{5}-1=0, comment your result and try to formulate a conjecture. - Prove your conjecture.

Part B

- Use de Moivre's theorem to obtain solutions to Z
^{n}=i - Represent each of these solutions on an Arganda diagram
- Generalize and prove your result for Z
^{n}=a+bi, where /a+bi/ = 1 - What happens when /a+bi/ is not equal to 1

Re: Patterns from complex numbers (Part A & B)

This is quite a large assignment you are expecting help with. What have you done so far?

Re: Patterns from complex numbers (Part A & B)

$\displaystyle z^3 =1 $

$\displaystyle Let z = rcis(\theta) $

$\displaystyle r^3cis(3\theta) = cis(0) $

Equate the angles and the r values

Re: Patterns from complex numbers (Part A & B)

Hi, it would be enough if you help me with the first two points of part A and the first point of part B.

Thank you.

Re: Patterns from complex numbers (Part A & B)