1. ## Complex numbers polar form - 2 other answers?

The question asks me to find the solution to $\displaystyle z^{3}=3^{\frac{1}{2}}+i$ in polar form.

I got
$\displaystyle 1^{\frac{1}{3}}cis(\frac{\pi}{18})$
Which is correct, but there 2 other answers but I can't seem to get it which are
$\displaystyle 1^{\frac{1}{3}}cis(\frac{13\pi}{18})$
And
$\displaystyle 1^{\frac{1}{3}}cis(\frac{-11\pi}{18})$

There's also another question I don't understand:
"How many degrees apart are two consecutive roots of $\displaystyle z^{8}=1$"

2. Use

$\displaystyle \displaystyle w_k=r^{1/n}\left(\cos{\left(\frac{\theta+2\pi k}{n}\right)}+i\sin{\left(\frac{\theta+2\pi k}{n}\right)}\right)$

3. There are three cube roots, they are all of the same magnitude and are evenly spaced around a circle.

So they differ by an angle of $\displaystyle \displaystyle \frac{2\pi}{3}$.

4. The point is that adding $\displaystyle 2\pi$ to the argument of a complex number just goes around the "circle" exactly once so that you are back to exactly the same place and same complex number:
$\displaystyle re^{i\theta}= re^{i(\theta+ 2\pi)}= re^{i(\theta+ 4\pi)}= re^{i(\theta+ 6\pi)}= re^{i(\theta+ 8\pi)}$

But when you divide the argument by 3 to find cube root, that changes:
$\displaystyle r^{1/3}e^{i\theta/3}\ne r^{1/3}e^{i(\theta/3+ (2/3)\pi)}\ne r^{1/3}e^{i(\theta/3+ (4/3)\pi)$

Notice, however, that continuing to $\displaystyle 6\pi$, $\displaystyle 8\pi$, etc. does NOT give us anything new:
$\displaystyle r^{1/3}e^{i(\theta/3+ (6/3)\pi)}= r^{1/3}e^{i\theta/3+ 2\pi i}= r^{1/3}e^{i\theta}$
$\displaystyle r^{1/3}e^{i(\theta/3+ (8/3)\pi)}= r^{1/3}e^{i(\theta+ (2/3)\pi)+ 2\pi i}= r^{1/3}e^{i(\theta+ (2/3)\pi)}$
$\displaystyle r^{1/3}e^{i(\theta/3+ (10/3)\pi}= r^{1/3}e^{i(\theta+ (4/3)\pi)+ 2\pi i}= r^{1/3}e^{i(\theta+ (2/3)\pi)}$
so we get exactly 3 distinct third roots.

Show should be able to see that adding any number of multiples of $\displaystyle 2\pi$ to the argument will always give exactly n distinct nth roots of a complex number.

5. Originally Posted by Cthul

There's also another question I don't understand:
"How many degrees apart are two consecutive roots of $\displaystyle z^{8}=1$"
$\displaystyle \displaystyle\ z^8=1\Rightarrow\ z=\sqrt[8]{1+0i}$

There are 8 "8th roots" of 1, all evenly spaced around the circle.

Hence, the roots are all seperated by $\displaystyle \displaystyle\frac{2{\pi}}{8}\;radians=\frac{360^o }{8}$