# nth roots of unity

• Dec 2nd 2007, 02:06 PM
Instigator
nth roots of unity
Hello everybody,

I'm having some difficulties with some problems about complex numbers and I would appreciate it if you could help me. It's mainly about the nth roots of unity.

1. I know that the four 4th roots of unity are 1, i, -1 and -i. I need to determine if it is possible to write them as 1, w, w ² and w³ where w is defined as cis pi/2.

And then I need to show that 1+w+w²+w³ = 0.

2. The second problem is also about this topic. I need to find the 5th roots of unity and then I have to display them on an Argand diagram. The second part of this is assuming that w is the root with the smallest positive argument, I need to show that the roots are 1, w, w², w³ and w. This is very similar to the first problem, so hopefully if you help me with one of them I'm able to do the second one alone.

3. Third: w=x+yi and P(x,y) move in the complex plane, I need the cartesian equation for |w-i| = |w+1+i| [note: |w-i| means modulus of w-i]

4. Next: z1= cos(pi/6) + isin(pi/6) and z2 = cos(pi/4) + isin(pi/4). What is the expression for (z1/z2) in the form of z=a+bi.

5. And the last: What is the fifth root of i ?

Sorry if it seems too easy for you, but unfortunately for me it's not. Thank you for your time and effort.
• Dec 2nd 2007, 02:26 PM
Soroban
Hello, Instigator!

Here are the last two . . .

Quote:

4. $z_1\:=\:\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$ and $z_2\:=\:\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$

What is the expression for: . $\frac{z_1}{z_2}$ in the form of $z\:=\:a+bi$

We have: . $\frac{z_1}{z_2} \;=\;\cos\left(\frac{\pi}{6}-\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{6} - \frac{\pi}{4}\right)\;=\;\cos\left(\text{-}\frac{\pi}{12}\right) + i\sin\left(\text{-}\frac{\pi}{12}\right)$

. . Therefore: . $\frac{z_1}{z_1} \;=\;\cos\frac{\pi}{12} - i\sin\frac{\pi}{12}$

Quote:

5. What is the fifth root of $i$ ?
Since $i \:=\:\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}$

. . then one of the fifth roots is: . $i^{\frac{1}{5}} \;=\;\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)^{\frac{1}{5}}\;=\;\cos\f rac{\pi}{10} + \sin\frac{\pi}{10}$

• Dec 2nd 2007, 02:41 PM
ThePerfectHacker
Quote:

Originally Posted by Instigator
1. I know that the four 4th roots of unity are 1, i, -1 and -i. I need to determine if it is possible to write them as 1, w, w ² and w³ where w is defined as cis pi/2.

And then I need to show that 1+w+w²+w³ = 0.

Let $\omega = e^{2\pi i/n}$ now use geometric series.