Constructing a sequence with specific complex elements

Question

(bad english) Hello everyone,

I am struggling with a (probably) very basic task. In the field ${ \mathbb{C} = \mathbb{R} ^2}$ let ${ \zeta_6:= \frac{1}{2}+ \frac{ \sqrt{3} }{2}i}$. Construct a sequence for the nth power of ${ \zeta_6}$ (${ n \in \mathbb{Z} }$). I calculated the first 7 elements of that sequence, which are:

1. ${ \zeta_6:= \frac{1}{2}+ \frac{ \sqrt{3} }{2}i}$
2. ${ \zeta_6^2:= -\frac{1}{2}+ \frac{ \sqrt{3} }{2}i}$
3. ${ \zeta_6^3:= -1+0i}$
4. ${ \zeta_6^4:= -\frac{1}{2}- \frac{ \sqrt{3} }{2}i}$
5. ${ \zeta_6^5:= \frac{1}{2}- \frac{ \sqrt{3} }{2}i}$
6. ${ \zeta_6^6:= 1+0i}$
7. ${ \zeta_6^7:= \frac{1}{2}+ \frac{ \sqrt{3} }{2}i}$

Define ${ \zeta_6^n=a_n+b_ni}$ . I realize that ${a_n}$ would only jump between ${\frac{1}{2},-\frac{1}{2},-1}$ and ${1}$ for example but i don't see I can construct a sequence that only jumps between those 4 elements for every ${ n \in \mathbb{Z} }$. How should I proceed?

Answer 1

$a_n=\cos(\pi n/3),$ and $b_n=\sin(\pi n/3)$.

Note these have period $6$.

Answer 2

Let n=6k+r where r $\in$ {0,1,2,3,4,5}

Now, $E_6^{6k+r}= e^{i(6k+r)\frac {\pi}{3}}=e^{i(2k\pi+\frac{r\pi}{3})}=e^{i\frac{r\pi}{3}}$

Thus, for any n $E_6^n= e^{i\frac{r\pi}{3}}$ where r=6{$\frac {n}{6}$} Here { } represents fractional part fn