Let $S = \{z ∈ C \mid z^8 = 1\}$. Write down $S$ explicitly by listing all its elements as complex numbers in polar form and Cartesian coordinates

Question

Let $S = \{z ∈ C \mid z^8 = 1\}$. Write down $S$ explicitly by listing all its elements as complex numbers in: (a) polar coordinates (b) Cartesian coordinates.

What I did so far is write $z^8 = r^8 (\cos 8\theta +i\sin 8\theta) = 1$ which also equals $r^8e^{i8\theta}$.

I'm stuck, what am I suppose to do? Please help!

Edit: not sure if this helps but I also know that $z\cdot z^{-1} = 1$, so $z^8 = z\cdot z^{-1}$.

Answer 1

Here's a geometric hint. When you square a complex number $z$ you square its length and double its argument (angle). When you cube it you cube its length and triple its angle.

Now think about where you could start in the plane so that taking the $8$th power of the length and multiplying the angle by $8$ will end up at $1 = 1 + 0i$.

I think this will be a useful way to think about the problem even if you solve it algebraically.

Answer 2

$\displaystyle z^8=1 \to (re^{i\theta})^8=1=cos(2k\pi)+isin(2k\pi)=e^{2k\pi i}$

$\to r^8 e^{8\theta i}=1e^{2k\pi i}$

$\to r^8=1$ and $8\theta=2k\pi$

$\to r=1, \theta=k\pi/4$ for $(\theta <2\pi)$

$\to z_1,z_2,...,z_8=e^{k\pi/4}=\cos{(k\pi/4)}+isin{(k\pi/4)}$ for $k=0,1,...,7$

In other words, as you have $z^8-1=0$, so you have 8 roots.