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

P.s. I posted the same thread in algebra forum too because I thought this question concerns both algebra and trigonometry.

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

You are doing just about everything wrong! If there is more than one sub-forum you feel a problem could be in, **choose one**. Do not double post. And you have shown no work of your own. I presume you would prefer hints to help YOU answer the question rather than just being given the answer- and we need to see where you have trouble to offer hints and help.

What, exactly, does DeMoivres' theorem say and how is it relevant to this problem?

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

Quote:

Originally Posted by

**alireza1992** 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

P.s. I posted the same thread in algebra forum too because I thought this question concerns both algebra and trigonometry.

Let $\displaystyle {\zeta _k} = \cos \left( {\frac{2k\pi }{n}} \right) + i\sin \left( {\frac{2k\pi }{n}} \right),\;k=0,1,\cdots,n-1~.$

Now each $\displaystyle \zeta_k$ is a root of $\displaystyle z^n-1=0~.$